Table of Integrals, Series, and Products Seventh Edition
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Table of Integrals, Series, and Products Seventh Edition
I.S. Gradshteyn and I.M. Ryzhik Alan Jeffrey, Editor University of Newcastle upon Tyne, England
Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA
Translated from Russian by Scripta Technica, Inc.
AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier
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Contents Preface to the Seventh Edition Acknowledgments The Order of Presentation of the Formulas Use of the Tables Index of Special Functions Notation Note on the Bibliographic References 0
Introduction 0.1 0.11 0.12 0.13 0.14 0.15 0.2 0.21 0.22 0.23–0.24 0.25 0.26 0.3 0.30 0.31 0.32 0.33 0.4 0.41 0.42 0.43 0.44
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Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sums of powers of natural numbers . . . . . . . . . . . . . . . . Sums of reciprocals of natural numbers . . . . . . . . . . . . . . Sums of products of reciprocals of natural numbers . . . . . . . Sums of the binomial coefficients . . . . . . . . . . . . . . . . . Numerical Series and Infinite Products . . . . . . . . . . . . . . The convergence of numerical series . . . . . . . . . . . . . . . Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . Examples of numerical series . . . . . . . . . . . . . . . . . . . Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of infinite products . . . . . . . . . . . . . . . . . . . Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions and theorems . . . . . . . . . . . . . . . . . . . . . . Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . Certain Formulas from Differential Calculus . . . . . . . . . . . . Differentiation of a definite integral with respect to a parameter . The nth derivative of a product (Leibniz’s rule) . . . . . . . . . . The nth derivative of a composite function . . . . . . . . . . . . Integration by substitution . . . . . . . . . . . . . . . . . . . . .
Elementary Functions Power of Binomials . . . . 1.11 Power series . . . . . . . 1.12 Series of rational fractions 1.2 The Exponential Function
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1.21 1.22 1.23 1.3–1.4 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.41 1.42 1.43 1.44–1.45 1.46 1.47 1.48 1.49 1.5 1.51 1.52 1.6 1.61 1.62–1.63 1.64 2
Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The basic functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) . . . . . . . . . . . . . . . . . The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions . . . . . . . . . . . Certain sums of trigonometric and hyperbolic functions . . . . . . . . . . . . . Sums of powers of trigonometric functions of multiple angles . . . . . . . . . . Sums of products of trigonometric functions of multiple angles . . . . . . . . . Sums of tangents of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . Sums leading to hyperbolic tangents and cotangents . . . . . . . . . . . . . . . The representation of cosines and sines of multiples of the angle as finite products The expansion of trigonometric and hyperbolic functions in power series . . . . Expansion in series of simple fractions . . . . . . . . . . . . . . . . . . . . . . Representation in the form of an infinite product . . . . . . . . . . . . . . . . . Trigonometric (Fourier) series . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of products of exponential and trigonometric functions . . . . . . . . . . Series of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Lobachevskiy’s “Angle of Parallelism” Π(x) . . . . . . . . . . . . . . . . . . . The hyperbolic amplitude (the Gudermannian) gd x . . . . . . . . . . . . . . . The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of logarithms (cf. 1.431) . . . . . . . . . . . . . . . . . . . . . . . . . . The Inverse Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . The domain of definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indefinite Integrals of Elementary Functions Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.00 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.01 The basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.02 General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 General integration rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11–2.13 Forms containing the binomial a + bxk . . . . . . . . . . . . . . . . . . . . . . 2.14 Forms containing the binomial 1 ± xn . . . . . . . . . . . . . . . . . . . . . . 2.15 Forms containing pairs of binomials: a + bx and α + βx . . . . . . . . . . . . . 2.16 Forms containing the trinomial a + bxk + cx2k . . . . . . . . . . . . . . . . . . 2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x . . . . 2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx 2.2 Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20 Introduction . . . . . . . . . . . . . . . . .√. . . . . . . . . . . . . . . . . . . 2.21 Forms containing the binomial a + bxk and x . . . . . . . . . . . . . . . . .
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2.22–2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.3 2.31 2.32 2.4 2.41–2.43 2.44–2.45 2.46 2.47 2.48 2.5–2.6 2.50 2.51–2.52 2.53–2.54 2.55–2.56 2.57 2.58–2.62 2.63–2.65 2.66 2.67 2.7 2.71 2.72–2.73 2.74 2.8 2.81 2.82 2.83 2.84 2.85
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n Forms containing √ (a + bx)k . . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing √a + bx and the binomial α + βx . . . . . . . . . . . . . . Forms containing √a + bx + cx2 . . . . . . . . . . . . . . . . . . . . . . . . Forms containing √a + bx + cx2 and integral powers of x . . . . . . . . . . . Forms containing √a + cx2 and integral powers of x . . . . . . . . . . . . . . Forms containing a + bx + cx2 and first- and second-degree polynomials . . Integrals that can be reduced to elliptic or pseudo-elliptic integrals . . . . . . The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing eax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The exponential combined with rational functions of x . . . . . . . . . . . . . Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of sinh x, cosh x, tanh x, and coth x . . . . . . . . . . . . . . . . . . Rational functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . Algebraic functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . Combinations of hyperbolic functions and powers . . . . . . . . . . . . . . . Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . Sines and cosines of multiple angles and of linear and more complicated functions of the argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational functions of√the sine and cosine . . . . . . . . . . . . . . . . . . . . √ Integrals containing a ± b sin x or a ± b cos x . . . . . . . . . . . . . . . . Integrals reducible to elliptic and pseudo-elliptic integrals . . . . . . . . . . . Products of trigonometric functions and powers . . . . . . . . . . . . . . . . Combinations of trigonometric functions and exponentials . . . . . . . . . . . Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . Logarithms and Inverse-Hyperbolic Functions . . . . . . . . . . . . . . . . . . The logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . Arcsines and arccosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The arcsecant, the arccosecant, the arctangent, and the arccotangent . . . . Combinations of arcsine or arccosine and algebraic functions . . . . . . . . . . Combinations of the arcsecant and arccosecant with powers of x . . . . . . . Combinations of the arctangent and arccotangent with algebraic functions . .
3–4 Definite Integrals of Elementary Functions 3.0 Introduction . . . . . . . . . . . . . . . 3.01 Theorems of a general nature . . . . . . 3.02 Change of variable in a definite integral . 3.03 General formulas . . . . . . . . . . . . . 3.04 Improper integrals . . . . . . . . . . . . 3.05 The principal values of improper integrals 3.1–3.2 Power and Algebraic Functions . . . . . 3.11 Rational functions . . . . . . . . . . . .
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3.12 3.13–3.17 3.18 3.19–3.23 3.24–3.27 3.3–3.4 3.31 3.32–3.34 3.35 3.36–3.37 3.38–3.39 3.41–3.44 3.45 3.46–3.48 3.5 3.51 3.52–3.53 3.54 3.55–3.56 3.6–4.1 3.61 3.62 3.63 3.64–3.65 3.66 3.67 3.68 3.69–3.71 3.72–3.74 3.75 3.76–3.77 3.78–3.81 3.82–3.83 3.84 3.85–3.88 3.89–3.91 3.92 3.93 3.94–3.97 3.98–3.99 4.11–4.12 4.13
Products of rational functions and expressions that can be reduced to square roots of first- and second-degree polynomials . . . . . . . . . . . . . . . . . . . Expressions that can be reduced to square roots of third- and fourth-degree polynomials and their products with rational functions . . . . . . . . . . . . . . Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions . . . . . . . . . . . . . . . . . . . . . Combinations of powers of x and powers of binomials of the form (α + βx) . . Powers of x, of binomials of the form α + βxp and of polynomials in x . . . . . Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponentials of more complicated arguments . . . . . . . . . . . . . . . . . . . Combinations of exponentials and rational functions . . . . . . . . . . . . . . . Combinations of exponentials and algebraic functions . . . . . . . . . . . . . . Combinations of exponentials and arbitrary powers . . . . . . . . . . . . . . . . Combinations of rational functions of powers and exponentials . . . . . . . . . Combinations of powers and algebraic functions of exponentials . . . . . . . . . Combinations of exponentials of more complicated arguments and powers . . . Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of hyperbolic functions and algebraic functions . . . . . . . . . . Combinations of hyperbolic functions and exponentials . . . . . . . . . . . . . Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . . Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational functions of sines and cosines and trigonometric functions of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . Powers of trigonometric functions and trigonometric functions of linear functions Powers and rational functions of trigonometric functions . . . . . . . . . . . . . Forms containing powers of linear functions of trigonometric functions . . . . . Square roots of expressions containing trigonometric functions . . . . . . . . . Various forms of powers of trigonometric functions . . . . . . . . . . . . . . . . Trigonometric functions of more complicated arguments . . . . . . . . . . . . . Combinations of trigonometric and rational functions . . . . . . . . . . . . . . Combinations of trigonometric and algebraic functions . . . . . . . . . . . . . . Combinations of trigonometric functions and powers . . . . . . . . . . . . . . . Rational functions of x and of trigonometric functions . . . . . . . . . . . . . . Powers of trigonometric with other powers . . . . . . . . . functions combined √ 2 2 2 Integrals containing 1 − k sin x, 1 − k cos2 x, and similar expressions . . Trigonometric functions of more complicated arguments combined with powers Trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . . Trigonometric functions of more complicated arguments combined with exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric and exponential functions of trigonometric functions . . . . . . . Combinations involving trigonometric functions, exponentials, and powers . . . Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . . Combinations involving trigonometric and hyperbolic functions and powers . . . Combinations of trigonometric and hyperbolic functions and exponentials . . . .
254 254 313 315 322 334 334 336 340 344 346 353 363 364 371 371 375 382 386 390 390 395 397 401 405 408 411 415 423 434 436 447 459 472 475 485 493 495 497 509 516 522
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4.14 4.2–4.4 4.21 4.22 4.23 4.24 4.25 4.26–4.27 4.28 4.29–4.32 4.33–4.34 4.35–4.36 4.37 4.38–4.41 4.42–4.43 4.44 4.5 4.51 4.52 4.53–4.54 4.55 4.56 4.57 4.58 4.59 4.6 4.60 4.61 4.62 4.63–4.64 5
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Combinations of trigonometric and hyperbolic functions, exponentials, and powers Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithms of more complicated arguments . . . . . . . . . . . . . . . . . . . . Combinations of logarithms and rational functions . . . . . . . . . . . . . . . . Combinations of logarithms and algebraic functions . . . . . . . . . . . . . . . Combinations of logarithms and powers . . . . . . . . . . . . . . . . . . . . . . Combinations involving powers of the logarithm and other powers . . . . . . . . Combinations of rational functions of ln x and powers . . . . . . . . . . . . . . Combinations of logarithmic functions of more complicated arguments and powers Combinations of logarithms and exponentials . . . . . . . . . . . . . . . . . . . Combinations of logarithms, exponentials, and powers . . . . . . . . . . . . . . Combinations of logarithms and hyperbolic functions . . . . . . . . . . . . . . . Logarithms and trigonometric functions . . . . . . . . . . . . . . . . . . . . . . Combinations of logarithms, trigonometric functions, and powers . . . . . . . . Combinations of logarithms, trigonometric functions, and exponentials . . . . . Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of arcsines, arccosines, and powers . . . . . . . . . . . . . . . . . Combinations of arctangents, arccotangents, and powers . . . . . . . . . . . . . Combinations of inverse trigonometric functions and exponentials . . . . . . . . A combination of the arctangent and a hyperbolic function . . . . . . . . . . . Combinations of inverse and direct trigonometric functions . . . . . . . . . . . A combination involving an inverse and a direct trigonometric function and a power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of inverse trigonometric functions and logarithms . . . . . . . . . Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of variables in multiple integrals . . . . . . . . . . . . . . . . . . . . . Change of the order of integration and change of variables . . . . . . . . . . . Double and triple integrals with constant limits . . . . . . . . . . . . . . . . . . Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indefinite Integrals of Special Functions Elliptic Integrals and Functions . . . . . . . . . . . . . . . . Complete elliptic integrals . . . . . . . . . . . . . . . . . . . 5.11 5.12 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . 5.14 Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . 5.2 The Exponential Integral Function . . . . . . . . . . . . . . 5.21 The exponential integral function . . . . . . . . . . . . . . . 5.22 Combinations of the exponential integral function and powers 5.23 Combinations of the exponential integral and the exponential 5.3 The Sine Integral and the Cosine Integral . . . . . . . . . . . 5.4 The Probability Integral and Fresnel Integrals . . . . . . . . . 5.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . 5.1
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CONTENTS
6–7 Definite Integrals of Special Functions 6.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Forms containing F (x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Forms containing E (x, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Integration of elliptic integrals with respect to the modulus . . . . . . . . . . 6.14–6.15 Complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 The theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2–6.3 The Exponential Integral Function and Functions Generated by It . . . . . . . 6.21 The logarithm integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.22–6.23 The exponential integral function . . . . . . . . . . . . . . . . . . . . . . . . 6.24–6.26 The sine integral and cosine integral functions . . . . . . . . . . . . . . . . . 6.27 The hyperbolic sine integral and hyperbolic cosine integral functions . . . . . 6.28–6.31 The probability integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.32 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Gamma Function and Functions Generated by It . . . . . . . . . . . . . 6.41 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.42 Combinations of the gamma function, the exponential, and powers . . . . . . 6.43 Combinations of the gamma function and trigonometric functions . . . . . . . 6.44 The logarithm of the gamma function∗ . . . . . . . . . . . . . . . . . . . . . 6.45 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . 6.46–6.47 The function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5–6.7 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.51 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.52 Bessel functions combined with x and x2 . . . . . . . . . . . . . . . . . . . . 6.53–6.54 Combinations of Bessel functions and rational functions . . . . . . . . . . . . 6.55 Combinations of Bessel functions and algebraic functions . . . . . . . . . . . 6.56–6.58 Combinations of Bessel functions and powers . . . . . . . . . . . . . . . . . . 6.59 Combinations of powers and Bessel functions of more complicated arguments 6.61 Combinations of Bessel functions and exponentials . . . . . . . . . . . . . . . 6.62–6.63 Combinations of Bessel functions, exponentials, and powers . . . . . . . . . . 6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.66 Combinations of Bessel, hyperbolic, and exponential functions . . . . . . . . . 6.67–6.68 Combinations of Bessel and trigonometric functions . . . . . . . . . . . . . . 6.69–6.74 Combinations of Bessel and trigonometric functions and powers . . . . . . . . 6.75 Combinations of Bessel, trigonometric, and exponential functions and powers 6.76 Combinations of Bessel, trigonometric, and hyperbolic functions . . . . . . . 6.77 Combinations of Bessel functions and the logarithm, or arctangent . . . . . . 6.78 Combinations of Bessel and other special functions . . . . . . . . . . . . . . . 6.79 Integration of Bessel functions with respect to the order . . . . . . . . . . . . 6.8 Functions Generated by Bessel Functions . . . . . . . . . . . . . . . . . . . . 6.81 Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.82 Combinations of Struve functions, exponentials, and powers . . . . . . . . . . 6.83 Combinations of Struve and trigonometric functions . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
631 631 631 632 632 632 633 635 636 636 638 639 644 645 649 650 650 652 655 656 657 658 659 659 664 670 674 675 689 694 699
.
708
. . . . . . . . . . . . .
711 713 717 727 742 747 747 748 749 753 753 754 755
CONTENTS
6.84–6.85 6.86 6.87 6.9 6.91 6.92 6.93 6.94 7.1–7.2 7.11 7.12–7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.21 7.22 7.23 7.24 7.25 7.3–7.4 7.31 7.32 7.325∗ 7.33 7.34 7.35 7.36 7.37–7.38 7.39 7.41–7.42 7.5 7.51 7.52 7.53 7.54 7.6 7.61 7.62–7.63 7.64 7.65
xi
Combinations of Struve and Bessel functions . . . . . . . . . . . . . . . . . . . Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of Mathieu, hyperbolic, and trigonometric functions . . . . . . . Combinations of Mathieu and Bessel functions . . . . . . . . . . . . . . . . . . Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . Associated Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of associated Legendre functions and powers . . . . . . . . . . . Combinations of associated Legendre functions, exponentials, and powers . . . Combinations of associated Legendre and hyperbolic functions . . . . . . . . . Combinations of associated Legendre functions, powers, and trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A combination of an associated Legendre function and the probability integral . Combinations of associated Legendre and Bessel functions . . . . . . . . . . . . Combinations of associated Legendre functions and functions generated by Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of associated Legendre functions with respect to the order . . . . . Combinations of Legendre polynomials, rational functions, and algebraic functions Combinations of Legendre polynomials and powers . . . . . . . . . . . . . . . . Combinations of Legendre polynomials and other elementary functions . . . . . Combinations of Legendre polynomials and Bessel functions . . . . . . . . . . . Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of Gegenbauer polynomials Cnν (x) and powers . . . . . . . . . . Combinations of Gegenbauer polynomials Cnν (x) and elementary functions . . . Complete System of Orthogonal Step Functions . . . . . . . . . . . . . . . . . Combinations of the polynomials C νn (x) and Bessel functions; Integration of Gegenbauer functions with respect to the index . . . . . . . . . . . . . . . . . Combinations of Chebyshev polynomials and powers . . . . . . . . . . . . . . . Combinations of Chebyshev polynomials and elementary functions . . . . . . . Combinations of Chebyshev polynomials and Bessel functions . . . . . . . . . . Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of hypergeometric functions and powers . . . . . . . . . . . . . . Combinations of hypergeometric functions and exponentials . . . . . . . . . . . Hypergeometric and trigonometric functions . . . . . . . . . . . . . . . . . . . Combinations of hypergeometric and Bessel functions . . . . . . . . . . . . . . Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . Combinations of confluent hypergeometric functions and powers . . . . . . . . Combinations of confluent hypergeometric functions and exponentials . . . . . Combinations of confluent hypergeometric and trigonometric functions . . . . . Combinations of confluent hypergeometric functions and Bessel functions . . .
756 760 761 763 763 763 767 767 769 769 770 776 778 779 781 782 787 788 789 791 792 794 795 795 797 798 798 800 802 803 803 806 808 812 812 814 817 817 820 820 822 829 830
xii
CONTENTS
7.66 7.67 7.68 7.69 7.7 7.71 7.72 7.73 7.74 7.75 7.76 7.77 7.8 7.81 7.82 7.83
Combinations of confluent hypergeometric functions, Bessel functions, and powers Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of confluent hypergeometric functions and other special functions Integration of confluent hypergeometric functions with respect to the index . . Parabolic Cylinder Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of parabolic cylinder functions, powers, and exponentials . . . . . Combinations of parabolic cylinder and hyperbolic functions . . . . . . . . . . . Combinations of parabolic cylinder and trigonometric functions . . . . . . . . . Combinations of parabolic cylinder and Bessel functions . . . . . . . . . . . . . Combinations of parabolic cylinder functions and confluent hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of a parabolic cylinder function with respect to the index . . . . . . Meijer’s and MacRobert’s Functions (G and E) . . . . . . . . . . . . . . . . . Combinations of the functions G and E and the elementary functions . . . . . Combinations of the functions G and E and Bessel functions . . . . . . . . . . Combinations of the functions G and E and other special functions . . . . . . .
8–9 Special Functions 8.1 Elliptic Integrals and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12 Functional relations between elliptic integrals . . . . . . . . . . . . . . . . . . . 8.13 Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.14 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.15 Properties of Jacobian elliptic functions and functional relationships between them 8.16 The Weierstrass function ℘(u) . . . . . . . . . . . . . . . . . . . . . . . . . . 8.17 The functions ζ(u) and σ(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . Theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.18–8.19 8.2 The Exponential Integral Function and Functions Generated by It . . . . . . . . 8.21 The exponential integral function Ei(x) . . . . . . . . . . . . . . . . . . . . . . 8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x . . . 8.23 The sine integral and the cosine integral: si x and ci x . . . . . . . . . . . . . . 8.24 The logarithm integral li(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.25 The probability integral Φ(x), the Fresnel integrals S (x) and C (x), the error function erf(x), and the complementary error function erfc(x) . . . . . . . . . 8.26 Lobachevskiy’s function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Euler’s Integrals of the First and Second Kinds . . . . . . . . . . . . . . . . . . 8.31 The gamma function (Euler’s integral of the second kind): Γ(z) . . . . . . . . 8.32 Representation of the gamma function as series and products . . . . . . . . . . 8.33 Functional relations involving the gamma function . . . . . . . . . . . . . . . . 8.34 The logarithm of the gamma function . . . . . . . . . . . . . . . . . . . . . . . 8.35 The incomplete gamma function . . . . . . . . . . . . . . . . . . . . . . . . . 8.36 The psi function ψ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.37 The function β(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.38 The beta function (Euler’s integral of the first kind): B(x, y) . . . . . . . . . . 8.39 The incomplete beta function Bx (p, q) . . . . . . . . . . . . . . . . . . . . . . 8.4–8.5 Bessel Functions and Functions Associated with Them . . . . . . . . . . . . . .
831 834 839 841 841 841 842 843 844 845 849 849 850 850 854 856 859 859 859 863 865 866 870 873 876 877 883 883 886 886 887 887 891 892 892 894 895 898 899 902 906 908 910 910
CONTENTS
8.40 8.41 8.42 8.43 8.44 8.45 8.46 8.47–8.48 8.49 8.51–8.52 8.53 8.54 8.55 8.56 8.57 8.58 8.59 8.6 8.60 8.61 8.62 8.63 8.64 8.65 8.66 8.67 8.7–8.8 8.70 8.71 8.72 8.73–8.74 8.75 8.76 8.77 8.78 8.79 8.81 8.82–8.83 8.84 8.85 8.9 8.90 8.91 8.919 8.92 8.93 8.94
xiii
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral representations of the functions Jν (z) and Nν (z) . . . . . . . . (1) (2) Integral representations of the functions Hν (z) and Hν (z) . . . . . . Integral representations of the functions Iν (z) and Kν (z) . . . . . . . . Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic expansions of Bessel functions . . . . . . . . . . . . . . . . Bessel functions of order equal to an integer plus one-half . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential equations leading to Bessel functions . . . . . . . . . . . . . Series of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . Expansion in products of Bessel functions . . . . . . . . . . . . . . . . . The zeros of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomson functions and their generalizations . . . . . . . . . . . . . . . Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anger and Weber functions Jν (z) and Eν (z) . . . . . . . . . . . . . . . Neumann’s and Schl¨afli’s polynomials: O n (z) and S n (z) . . . . . . . . Mathieu Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathieu’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periodic Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . (2n) (2n+1) (2n+1) (2n+2) Recursion relations for the coefficients A2r , A2r+1 , B2r+1 , B2r+2 Mathieu functions with a purely imaginary argument . . . . . . . . . . . Non-periodic solutions of Mathieu’s equation . . . . . . . . . . . . . . . Mathieu functions for negative q . . . . . . . . . . . . . . . . . . . . . Representation of Mathieu functions as series of Bessel functions . . . . The general theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic series for large values of |ν| . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special cases and particular values . . . . . . . . . . . . . . . . . . . . Derivatives with respect to the order . . . . . . . . . . . . . . . . . . . Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . The zeros of associated Legendre functions . . . . . . . . . . . . . . . . Series of associated Legendre functions . . . . . . . . . . . . . . . . . . Associated Legendre functions with integer indices . . . . . . . . . . . . Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of products of Legendre and Chebyshev polynomials . . . . . . . Series of Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . Gegenbauer polynomials Cnλ (t) . . . . . . . . . . . . . . . . . . . . . . The Chebyshev polynomials Tn (x) and Un (x) . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
910 912 914 916 918 920 924 926 931 933 940 941 942 944 945 948 949 950 950 951 951 952 953 953 954 957 958 958 960 962 964 968 969 970 972 972 974 975 980 981 982 982 983 988 988 990 993
xiv
CONTENTS
8.95 8.96 8.97 9.1 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.2 9.20 9.21 9.22–9.23 9.24–9.25 9.26 9.3 9.30 9.31 9.32 9.33 9.34 9.4 9.41 9.42 9.5 9.51 9.52 9.53 9.54 9.55 9.56 9.6 9.61 9.62 9.63 9.64 9.65 9.7 9.71 9.72
The Hermite polynomials H n (x) . . . . . . . . . . . . . . . . . . . . . . . . . Jacobi’s polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of elementary functions in terms of a hypergeometric functions . Transformation formulas and the analytic continuation of functions defined by hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A generalized hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . The hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . Riemann’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . Representing the solutions to certain second-order differential equations using a Riemann scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypergeometric functions of two variables . . . . . . . . . . . . . . . . . . . . A hypergeometric function of several variables . . . . . . . . . . . . . . . . . . Confluent Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The functions Φ(α, γ; z) and Ψ(α, γ; z) . . . . . . . . . . . . . . . . . . . . . . The Whittaker functions Mλ,μ (z) and Wλ,μ (z) . . . . . . . . . . . . . . . . . . Parabolic cylinder functions Dp (z) . . . . . . . . . . . . . . . . . . . . . . . . Confluent hypergeometric series of two variables . . . . . . . . . . . . . . . . . Meijer’s G-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A differential equation for the G-function . . . . . . . . . . . . . . . . . . . . . Series of G-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connections with other special functions . . . . . . . . . . . . . . . . . . . . . MacRobert’s E-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation by means of multiple integrals . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s) Definition and integral representations . . . . . . . . . . . . . . . . . . . . . . Representation as a series or as an infinite product . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singular points and zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Lerch function Φ(z, s, v) . . . . . . . . . . . . . . . . . . . . . . . . . . . The function ξ (s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernoulli Numbers and Polynomials, Euler Numbers . . . . . . . . . . . . . . . Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y) . . . . . . . . . . Euler polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
996 998 1000 1005 1005 1005 1006 1008 1010 1010 1014 1017 1018 1022 1022 1022 1023 1024 1028 1031 1032 1032 1033 1034 1034 1034 1035 1035 1035 1036 1036 1037 1037 1038 1039 1040 1040 1040 1041 1043 1043 1044 1045 1045 1045
CONTENTS
9.73 9.74
xv
Euler’s and Catalan’s constants . . . . . . . . . . . . . . . . . . . . . . . . . . Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Vector Field Theory 10.1–10.8 Vectors, Vector Operators, and Integral 10.11 Products of vectors . . . . . . . . . . 10.12 Properties of scalar product . . . . . . 10.13 Properties of vector product . . . . . . 10.14 Differentiation of vectors . . . . . . . . 10.21 Operators grad, div, and curl . . . . . 10.31 Properties of the operator ∇ . . . . . 10.41 Solenoidal fields . . . . . . . . . . . . 10.51–10.61 Orthogonal curvilinear coordinates . . 10.71–10.72 Vector integral theorems . . . . . . . . 10.81 Integral rate of change theorems . . .
1046 1046
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1049 1049 1049 1049 1049 1050 1050 1051 1052 1052 1055 1057
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1059 1059 1059 1060 1061
12 Integral Inequalities 12.11 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . 12.111 First mean value theorem . . . . . . . . . . . . . . . . . . 12.112 Second mean value theorem . . . . . . . . . . . . . . . . . 12.113 First mean value theorem for infinite integrals . . . . . . . 12.114 Second mean value theorem for infinite integrals . . . . . . 12.21 Differentiation of Definite Integral Containing a Parameter 12.211 Differentiation when limits are finite . . . . . . . . . . . . . 12.212 Differentiation when a limit is infinite . . . . . . . . . . . . 12.31 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . 12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals . . . 12.312 H¨older’s inequality for integrals . . . . . . . . . . . . . . . 12.313 Minkowski’s inequality for integrals . . . . . . . . . . . . . 12.314 Chebyshev’s inequality for integrals . . . . . . . . . . . . . 12.315 Young’s inequality for integrals . . . . . . . . . . . . . . . 12.316 Steffensen’s inequality for integrals . . . . . . . . . . . . . 12.317 Gram’s inequality for integrals . . . . . . . . . . . . . . . . 12.318 Ostrowski’s inequality for integrals . . . . . . . . . . . . . 12.41 Convexity and Jensen’s Inequality . . . . . . . . . . . . . . 12.411 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . 12.412 Carleman’s inequality for integrals . . . . . . . . . . . . . . 12.51 Fourier Series and Related Inequalities . . . . . . . . . . . 12.511 Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . . 12.512 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . 12.513 Parseval’s theorem for trigonometric Fourier series . . . . . 12.514 Integral representation of the nth partial sum . . . . . . . .
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1063 1063 1063 1063 1063 1064 1064 1064 1064 1064 1064 1064 1065 1065 1065 1065 1065 1066 1066 1066 1066 1066 1067 1067 1067 1067
Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Algebraic Inequalities 11.1–11.3 General Algebraic Inequalities . . . . . . . . . . 11.11 Algebraic inequalities involving real numbers . . 11.21 Algebraic inequalities involving complex numbers 11.31 Inequalities for sets of complex numbers . . . .
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xvi
CONTENTS
12.515 12.516 12.517
Generalized Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bessel’s inequality for generalized Fourier series . . . . . . . . . . . . . . . . . Parseval’s theorem for generalized Fourier series . . . . . . . . . . . . . . . . .
1067 1068 1068
13 Matrices and Related Results 13.11–13.12 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.111 Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.112 Identity matrix and null matrix . . . . . . . . . . . . . . . . . . . . . . . 13.113 Reducible and irreducible matrices . . . . . . . . . . . . . . . . . . . . . . 13.114 Equivalent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.115 Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.116 Adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.117 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.118 Trace of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.119 Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.120 Skew-symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.121 Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.122 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.123 Hermitian transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . 13.124 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.125 Unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.126 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . 13.127 Nilpotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.128 Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.129 Positive definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.130 Non-negative definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.131 Diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.21 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.211 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.212 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.213 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.214 Positive definite and semidefinite quadratic form . . . . . . . . . . . . . . 13.215 Basic theorems on quadratic forms . . . . . . . . . . . . . . . . . . . . . 13.31 Differentiation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 13.41 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.411 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1069 1069 1069 1069 1069 1069 1069 1070 1070 1070 1070 1070 1070 1070 1070 1070 1071 1071 1071 1071 1071 1071 1071 1071 1072 1072 1072 1072 1072 1073 1074 1074
14 Determinants 14.11 14.12 14.13 14.14 14.15* 14.16 14.17 14.18 14.21 14.31
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1075 1075 1075 1075 1076 1076 1076 1077 1077 1077 1078
Expansion of Second- and Third-Order Determinants Basic Properties . . . . . . . . . . . . . . . . . . . . Minors and Cofactors of a Determinant . . . . . . . . Principal Minors . . . . . . . . . . . . . . . . . . . . Laplace Expansion of a Determinant . . . . . . . . . Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . Hadamard’s Theorem . . . . . . . . . . . . . . . . . Hadamard’s Inequality . . . . . . . . . . . . . . . . . Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . Some Special Determinants . . . . . . . . . . . . . .
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CONTENTS
14.311 14.312 14.313 14.314 14.315 14.316 14.317 15 Norms 15.1–15.9 15.11 15.21 15.211 15.212 15.213 15.31 15.311 15.312 15.313 15.41 15.411 15.412 15.413 15.51 15.511 15.512 15.61 15.611 15.612 15.71 15.711 15.712 15.713 15.714 15.715 15.81–15.82 15.811 15.812 15.813 15.814 15.815 15.816 15.817 15.818 15.819 15.820 15.821 15.822
xvii
Vandermonde’s determinant (alternant) . . . . . Circulants . . . . . . . . . . . . . . . . . . . . . Jacobian determinant . . . . . . . . . . . . . . Hessian determinants . . . . . . . . . . . . . . Wronskian determinants . . . . . . . . . . . . . Properties . . . . . . . . . . . . . . . . . . . . Gram-Kowalewski theorem on linear dependence
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1078 1078 1078 1079 1079 1079 1080
Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . General Properties . . . . . . . . . . . . . . . . . . . . . . . Principal Vector Norms . . . . . . . . . . . . . . . . . . . . The norm ||x||1 . . . . . . . . . . . . . . . . . . . . . . . . The norm ||x||2 (Euclidean or L2 norm) . . . . . . . . . . . The norm ||x||∞ . . . . . . . . . . . . . . . . . . . . . . . . Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . General properties . . . . . . . . . . . . . . . . . . . . . . . Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . Natural norm of unit matrix . . . . . . . . . . . . . . . . . . Principal Natural Norms . . . . . . . . . . . . . . . . . . . . Maximum absolute column sum norm . . . . . . . . . . . . . Spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . Maximum absolute row sum norm . . . . . . . . . . . . . . . Spectral Radius of a Square Matrix . . . . . . . . . . . . . . Inequalities concerning matrix norms and the spectral radius Deductions from Gerschgorin’s theorem (see 15.814) . . . . Inequalities Involving Eigenvalues of Matrices . . . . . . . . . Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . Inequalities for the Characteristic Polynomial . . . . . . . . . Named and unnamed inequalities . . . . . . . . . . . . . . . Parodi’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Corollary of Brauer’s theorem . . . . . . . . . . . . . . . . . Ballieu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Routh-Hurwitz theorem . . . . . . . . . . . . . . . . . . . . Named Theorems on Eigenvalues . . . . . . . . . . . . . . . Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . Sturmian separation theorem . . . . . . . . . . . . . . . . . Poincare’s separation theorem . . . . . . . . . . . . . . . . . Gerschgorin’s theorem . . . . . . . . . . . . . . . . . . . . . Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . Wielandt’s theorem . . . . . . . . . . . . . . . . . . . . . . Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . . . . First theorem due to Lyapunov . . . . . . . . . . . . . . . . Second theorem due to Lyapunov . . . . . . . . . . . . . . .
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1081 1081 1081 1081 1081 1081 1081 1082 1082 1082 1082 1082 1082 1082 1083 1083 1083 1083 1084 1084 1084 1084 1085 1086 1086 1086 1086 1087 1087 1087 1087 1088 1088 1088 1088 1088 1088 1089 1089 1089
xviii
CONTENTS
15.823 15.91 15.911 15.912
Hermitian matrices and diophantine relations rational angles due to Calogero and Perelomov Variational Principles . . . . . . . . . . . . . . Rayleigh quotient . . . . . . . . . . . . . . . . Basic theorems . . . . . . . . . . . . . . . . .
involving . . . . . . . . . . . . . . . . . . . .
circular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1089 1091 1091 1091
16 Ordinary Differential Equations 1093 16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations . . . . . . . 1093 16.11 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 16.111 Solution of a first-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093 16.112 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 16.113 Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093 16.114 Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.21 Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094 16.211 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.212 Comparison of approximate solutions of a differential equation . . . . . . . . . 1094 16.31 First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.311 Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094 16.312 Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 16.313 Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095 16.314 Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095 16.315 Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096 16.316 First-order linear differential equation . . . . . . . . . . . . . . . . . . . . . . . 1096 16.317 Linear systems of differential equations . . . . . . . . . . . . . . . . . . . . . . 1096 16.41 Some Special Types of Elementary Differential Equations . . . . . . . . . . . . 1097 16.411 Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.412 Exact differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.413 Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.414 Homogeneous differential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097 16.51 Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 16.511 Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098 16.512 Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 16.513 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 16.514 The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 16.515 Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Solution of a second-order linear differential equation . . . . . . . . . . . . . . 1100 16.516 16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100 16.611 First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100 16.622 Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.623 Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.624 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.625 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.626 Szeg¨ o’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 16.627 Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 16.628 Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 16.629 Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 16.71 Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103 16.711 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103
CONTENTS
16.712 16.81–16.82 16.811 16.822 16.823 16.91 16.911 16.912 16.913 16.914 16.92 16.921 16.922 16.923 16.924 16.93
xix
Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . . Kneser’s non-oscillation theorem . . . . . . . . . . . . . . . . . Comparison theorem for non-oscillation . . . . . . . . . . . . . . Necessary and sufficient conditions for non-oscillation . . . . . . Some Growth Estimates for Solutions of Second-Order Equations Strictly increasing and decreasing solutions . . . . . . . . . . . . General result on dominant and subdominant solutions . . . . . Estimate of dominant solution . . . . . . . . . . . . . . . . . . . A theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . Boundedness Theorems . . . . . . . . . . . . . . . . . . . . . . All solutions of the equation . . . . . . . . . . . . . . . . . . . . If all solutions of the equation . . . . . . . . . . . . . . . . . . . If a(x) → ∞ monotonically as x → ∞, then all solutions of . . . Consider the equation . . . . . . . . . . . . . . . . . . . . . . . Growth of maxima of |y| . . . . . . . . . . . . . . . . . . . . . .
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1103 1103 1103 1104 1104 1104 1104 1104 1105 1105 1106 1106 1106 1106 1106 1106
17 Fourier, Laplace, and Mellin Transforms 17.1–17.4 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . 17.11 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . 17.12 Basic properties of the Laplace transform . . . . . . . . . . . . . . 17.13 Table of Laplace transform pairs . . . . . . . . . . . . . . . . . . 17.21 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.22 Basic properties of the Fourier transform . . . . . . . . . . . . . . 17.23 Table of Fourier transform pairs . . . . . . . . . . . . . . . . . . . 17.24 Table of Fourier transform pairs for spherically symmetric functions 17.31 Fourier sine and cosine transforms . . . . . . . . . . . . . . . . . . 17.32 Basic properties of the Fourier sine and cosine transforms . . . . . 17.33 Table of Fourier sine transforms . . . . . . . . . . . . . . . . . . . 17.34 Table of Fourier cosine transforms . . . . . . . . . . . . . . . . . . 17.35 Relationships between transforms . . . . . . . . . . . . . . . . . . 17.41 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.42 Basic properties of the Mellin transform . . . . . . . . . . . . . . 17.43 Table of Mellin transforms . . . . . . . . . . . . . . . . . . . . . .
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1107 1107 1107 1107 1108 1117 1118 1118 1120 1121 1121 1122 1126 1129 1129 1130 1131
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18 The z-Transform 18.1–18.3 Definition, Bilateral, and 18.1 Definitions . . . . . . . 18.2 Bilateral z-transform . . 18.3 Unilateral z-transform . References Supplemental references Index of Functions and Constants General Index of Concepts
Unilateral z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1135 1135 1135 1136 1138 1141 1145 1151 1161
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Preface to the Seventh Edition Since the publication in 2000 of the completely reset sixth edition of Gradshteyn and Ryzhik, users of the reference work have continued to submit corrections, new results that extend the work, and suggestions for changes that improve the presentation of existing entries. It is a matter of regret to us that the structure of the book makes it impossible to acknowledge these individual contributions, so, as usual, the names of the many new contributors have been added to the acknowledgment list at the front of the book. This seventh edition contains the corrections received since the publication of the sixth edition in 2000, together with a considerable amount of new material acquired from isolated sources. Following our previous conventions, an amended entry has a superscript “11” added to its entry reference number, where the equivalent superscript number for the sixth edition was “10.” Similarly, an asterisk on an entry’s reference number indicates a new result. When, for technical reasons, an entry in a previous edition has been removed, to preserve the continuity of numbering between the new and older editions the subsequent entries have not been renumbered, so the numbering will jump. We wish to express our gratitude to all who have been in contact with us with the object of improving and extending the book, and we want to give special thanks to Dr. Victor H. Moll for his interest in the book and for the many contributions he has made over an extended period of time. We also wish to acknowledge the contributions made by Dr. Francis J. O’Brien Jr. of the Naval Station in Newport, in particular for results involving integrands where exponentials are combined with algebraic functions. Experience over many years has shown that each new edition of Gradshteyn and Ryzhik generates a fresh supply of suggestions for new entries, and for the improvement of the presentation of existing entries and errata. In view of this, we do not expect this new edition to be free from errors, so all users of this reference work who identify errors, or who wish to propose new entries, are invited to contact the authors, whose email addresses are listed below. Corrections will be posted on the web site www.az-tec.com/gr/errata. Alan Jeffrey
[email protected] Daniel Zwillinger
[email protected] xxi
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Acknowledgments The publisher and editors would like to take this opportunity to express their gratitude to the following users of the Table of Integrals, Series, and Products who, either directly or through errata published in Mathematics of Computation, have generously contributed corrections and addenda to the original printing. Dr. A. Abbas Dr. P. B. Abraham Dr. Ari Abramson Dr. Jose Adachi Dr. R. J. Adler Dr. N. Agmon Dr. M. Ahmad Dr. S. A. Ahmad Dr. Luis Alvarez-Ruso Dr. Maarten H P Ambaum Dr. R. K. Amiet Dr. L. U. Ancarani Dr. M. Antoine Dr. C. R. Appledorn Dr. D. R. Appleton Dr. Mitsuhiro Arikawa Dr. P. Ashoshauvati Dr. C. L. Axness Dr. E. Badralexe Dr. S. B. Bagchi Dr. L. J. Baker Dr. R. Ball Dr. M. P. Barnett Dr. Florian Baumann Dr. Norman C. Beaulieu Dr. Jerome Benoit Mr. V. Bentley Dr. Laurent Berger Dr. M. van den Berg Dr. N. F. Berk Dr. C. A. Bertulani
Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.
J. Betancort-Rijo P. Bickerstaff Iwo Bialynicki-Birula Chris Bidinosti G. R. Bigg Ian Bindloss L. Blanchet Mike Blaskiewicz R. D. Blevins Anders Blom L. M. Blumberg R. Blumel S. E. Bodner M. Bonsager George Boros S. Bosanac B. Van den Bossche A. Bostr¨ om J. E. Bowcock T. H. Boyer K. M. Briggs D. J. Broadhurst Chris Van Den Broeck W. B. Brower H. N. Browne Christoph Bruegger William J. Bruno Vladimir Bubanja D. J. Buch D. J. Bukman F. M. Burrows
xxiii
Dr. R. Caboz Dr. T. Calloway Dr. F. Calogero Dr. D. Dal Cappello Dr. David Cardon Dr. J. A. Carlson Gallos Dr. B. Carrascal Dr. A. R. Carr Dr. S. Carter Dr. G. Cavalleri Mr. W. H. L. Cawthorne Dr. A. Cecchini Dr. B. Chan Dr. M. A. Chaudhry Dr. Sabino Chavez-Cerda Dr. Julian Cheng Dr. H. W. Chew Dr. D. Chin Dr. Young-seek Chung Dr. S. Ciccariello Dr. N. S. Clarke Dr. R. W. Cleary Dr. A. Clement Dr. P. Cochrane Dr. D. K. Cohoon Dr. L. Cole Dr. Filippo Colomo Dr. J. R. D. Copley Dr. D. Cox Dr. J. Cox Dr. J. W. Criss
xxiv
Dr. A. E. Curzon Dr. D. Dadyburjor Dr. D. Dajaputra Dr. C. Dal Cappello Dr. P. Daly Dr. S. Dasgupta Dr. John Davies Dr. C. L. Davis Dr. A. Degasperis Dr. B. C. Denardo Dr. R. W. Dent Dr. E. Deutsch Dr. D. deVries Dr. P. Dita Dr. P. J. de Doelder Dr. Mischa Dohler Dr. G. Dˆ ome Dr. Shi-Hai Dong Dr. Balazs Dora Dr. M. R. D’Orsogna Dr. Adrian A. Dragulescu Dr. Eduardo Duenez Mr. Tommi J. Dufva Dr. E. B. Dussan, V Dr. C. A. Ebner Dr. M. van der Ende Dr. Jonathan Engle Dr. G. Eng Dr. E. S. Erck Dr. Jan Erkelens Dr. Olivier Espinosa Dr. G. A. Est´evez Dr. K. Evans Dr. G. Evendon Dr. V. I. Fabrikant Dr. L. A. Falkovsky Dr. K. Farahmand Dr. Richard J. Fateman Dr. G. Fedele Dr. A. R. Ferchmin Dr. P. Ferrant Dr. H. E. Fettis Dr. W. B. Fichter Dr. George Fikioris Mr. J. C. S. S. Filho Dr. L. Ford Dr. Nicolao Fornengo
Acknowledgments
Dr. J. France Dr. B. Frank Dr. S. Frasier Dr. Stefan Fredenhagen Dr. A. J. Freeman Dr. A. Frink Dr. Jason M. Gallaspy Dr. J. A. C. Gallas Dr. J. A. Carlson Gallas Dr. G. R. Gamertsfelder Dr. T. Garavaglia Dr. Jaime Zaratiegui Garcia Dr. C. G. Gardner Dr. D. Garfinkle Dr. P. N. Garner Dr. F. Gasser Dr. E. Gath Dr. P. Gatt Dr. D. Gay Dr. M. P. Gelfand Dr. M. R. Geller Dr. Ali I. Genc Dr. Vincent Genot Dr. M. F. George Dr. P. Germain Dr. Ing. Christoph Gierull Dr. S. P. Gill Dr. Federico Girosi Dr. E. A. Gislason Dr. M. I. Glasser Dr. P. A. Glendinning Dr. L. I. Goldfischer Dr. Denis Golosov Dr. I. J. Good Dr. J. Good Mr. L. Gorin Dr. Martin G¨ otz Dr. R. Govindaraj Dr. M. De Grauf Dr. L. Green Mr. Leslie O. Green Dr. R. Greenwell Dr. K. D. Grimsley Dr. Albert Groenenboom Dr. V. Gudmundsson Dr. J. Guillera Dr. K. Gunn
Dr. D. L. Gunter Dr. Julio C. Guti´errez-Vega Dr. Roger Haagmans Dr. H. van Haeringen Dr. B. Hafizi Dr. Bahman Hafizi Dr. T. Hagfors Dr. M. J. Haggerty Dr. Timo Hakulinen Dr. Einar Halvorsen Dr. S. E. Hammel Dr. E. Hansen Dr. Wes Harker Dr. T. Harrett Dr. D. O. Harris Dr. Frank Harris Mr. Mazen D. Hasna Dr. Joel G. Heinrich Dr. Sten Herlitz Dr. Chris Herzog Dr. A. Higuchi Dr. R. E. Hise Dr. Henrik Holm Dr. Helmut H¨ olzler Dr. N. Holte Dr. R. W. Hopper Dr. P. N. Houle Dr. C. J. Howard Dr. J. H. Hubbell Dr. J. R. Hull Dr. W. Humphries Dr. Jean-Marc Hur´e Dr. Ben Yu-Kuang Hu Dr. Y. Iksbe Dr. Philip Ingenhoven Mr. L. Iossif Dr. Sean A. Irvine ´ ´Isberg Dr. Ottar Dr. Cyril-Daniel Iskander Dr. S. A. Jackson Dr. John David Jackson Dr. Francois Jaclot Dr. B. Jacobs Dr. E. C. James Dr. B. Jancovici Dr. D. J. Jeffrey Dr. H. J. Jensen
Acknowledgments
Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.
Edwin F. Johnson I. R. Johnson Steven Johnson I. Johnstone Y. P. Joshi Jae-Hun Jung Damir Juric Florian Kaempfer S. Kanmani Z. Kapal Dave Kasper M. Kaufman B. Kay Avinash Khare Ilki Kim Youngsun Kim S. Klama L. Klingen C. Knessl M. J. Knight Mel Knight Yannis Kohninos D. Koks L. P. Kok K. S. K¨ olbig Y. Komninos D. D. Konowalow Z. Kopal I. Kostyukov R. A. Krajcik Vincent Krakoviack Stefan Kramer Tobias Kramer Hermann Krebs J. W. Krozel E. D. Krupnikov Kun-Lin Kuo E. A. Kuraev Konstantinos Kyritsis Velimir Labinac A. D. J. Lambert A. Lambert A. Larraza K. D. Lee M. Howard Lee M. K. Lee P. A. Lee
xxv
Dr. Todd Lee Dr. J. Legg Dr. Armando Lemus Dr. S. L. Levie Dr. D. Levi Dr. Michael Lexa Dr. Kuo Kan Liang Dr. B. Linet Dr. M. A. Lisa Dr. Donald Livesay Dr. H. Li Dr. Georg Lohoefer Dr. I. M. Longman Dr. D. Long Dr. Sylvie Lorthois Dr. Y. L. Luke Dr. W. Lukosz Dr. T. Lundgren Dr. E. A. Luraev Dr. R. Lynch Dr. R. Mahurin Dr. R. Mallier Dr. G. A. Mamon Dr. A. Mangiarotti Dr. I. Manning Dr. J. Marmur Dr. A. Martin Sr. Yuzo Maruyama Dr. David J. Masiello Dr. Richard Marthar Dr. H. A. Mavromatis Dr. M. Mazzoni Dr. K. B. Ma Dr. P. McCullagh Dr. J. H. McDonnell Dr. J. R. McGregor Dr. Kim McInturff Dr. N. McKinney Dr. David McA McKirdy Dr. Rami Mehrem Dr. W. N. Mei Dr. Angelo Melino Mr. Jos´e Ricardo Mendes Dr. Andy Mennim Dr. J. P. Meunier Dr. Gerard P. Michon Dr. D. F. R. Mildner
Dr. D. L. Miller Dr. Steve Miller Dr. P. C. D. Milly Dr. S. P. Mitra Dr. K. Miura Dr. N. Mohankumar Dr. M. Moll Dr. Victor H. Moll Dr. D. Monowalow Mr. Tony Montagnese Dr. Jim Morehead Dr. J. Morice Dr. W. Mueck Dr. C. Muhlhausen Dr. S. Mukherjee Dr. R. R. M¨ uller Dr. Pablo Parmezani Munhoz Dr. Paul Nanninga Dr. A. Natarajan Dr. Stefan Neumeier Dr. C. T. Nguyen Dr. A. C. Nicol Dr. M. M. Nieto Dr. P. Noerdlinger Dr. A. N. Norris Dr. K. H. Norwich Dr. A. H. Nuttall Dr. Frank O’Brien Dr. R. P. O’Keeffe Dr. A. Ojo Dr. P. Olsson Dr. M. Ortner Dr. S. Ostlund Dr. J. Overduin Dr. J. Pachner Dr. John D. Paden Mr. Robert A. Padgug Dr. D. Papadopoulos Dr. F. J. Papp Mr. Man Sik Park Dr. Jong-Do Park Dr. B. Patterson Dr. R. F. Pawula Dr. D. W. Peaceman Dr. D. Pelat Dr. L. Peliti Dr. Y. P. Pellegrini
xxvi
Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.
Acknowledgments
G. J. Pert Nicola Pessina J. B. Peterson Rickard Petersson Andrew Plumb Dror Porat E. A. Power E. Predazzi William S. Price Paul Radmore F. Raynal X. R. Resende J. M. Riedler Thomas Richard E. Ringel T. M. Roberts N. I. Robinson P. A. Robinson D. M. Rosenblum R. A. Rosthal J. R. Roth Klaus Rottbrand D. Roy E. Royer D. Rudermann Sanjib Sabhapandit C. T. Sachradja J. Sadiku A. Sadiq Motohiko Saitoh Naoki Saito A. Salim J. H. Samson Miguel A. Sanchis-Lozano J. A. Sanders M. A. F. Sanjun P. Sarquiz Avadh Saxena Vito Scarola O. Sch¨ arpf A. Scherzinger B. Schizer Martin Schmid J. Scholes Mel Schopper H. J. Schulz G. J. Sears
Dr. Kazuhiko Seki Dr. B. Seshadri Dr. A. Shapiro Dr. Masaki Shigemori Dr. J. S. Sheng Dr. Kenneth Ing Shing Dr. Tomohiro Shirai Dr. S. Shlomo Dr. D. Siegel Dr. Matthew Stapleton Dr. Steven H. Simon Dr. Ashok Kumar Singal Dr. C. Smith Dr. G. C. C. Smith Dr. Stefan Llewellyn Smith Dr. S. Smith Dr. G. Solt Dr. J. Sondow Dr. A. Sørenssen Dr. Marcus Spradlin Dr. Andrzej Staruszkiewicz Dr. Philip C. L. Stephenson Dr. Edgardo Stockmeyer Dr. J. C. Straton Mr. H. Suraweera Dr. N. F. Svaiter Dr. V. Svaiter Dr. R. Szmytkowski Dr. S. Tabachnik Dr. Erik Talvila Dr. G. Tanaka Dr. C. Tanguy Dr. G. K. Tannahill Dr. B. T. Tan Dr. C. Tavard Dr. Gon¸calo Tavares Dr. Aba Teleki Dr. Arash Dahi Taleghani Dr. D. Temperley Dr. A. J. Tervoort Dr. Theodoros Theodoulidis Dr. D. J. Thomas Dr. Michael Thorwart Dr. S. T. Thynell Dr. D. C. Torney Dr. R. Tough Dr. B. F. Treadway
Dr. Ming Tsai Dr. N. Turkkan Dr. Sandeep Tyagi Dr. J. J. Tyson Dr. S. Uehara Dr. M. Vadacchino Dr. O. T. Valls Dr. D. Vandeth Mr. Andras Vanyolos Dr. D. Veitch Mr. Jose Lopez Vicario Dr. K. Vogel Dr. J. M. M. J. Vogels Dr. Alexis De Vos Dr. Stuart Walsh Dr. Reinhold Wannemacher Dr. S. Wanzura Dr. J. Ward Dr. S. I. Warshaw Dr. R. Weber Dr. Wei Qian Dr. D. H. Werner Dr. E. Wetzel Dr. Robert Whittaker Dr. D. T. Wilton Dr. C. Wiuf Dr. K. T. Wong Mr. J. N. Wright Dr. J. D. Wright Dr. D. Wright Dr. D. Wu Dr. Michel Daoud Yacoub Dr. Yu S. Yakovlev Dr. H.-C. Yang Dr. J. J. Yang Dr. Z. J. Yang Dr. J. J. Wang Dr. Peter Widerin Mr. Chun Kin Au Yeung Dr. Kazuya Yuasa Dr. S. P. Yukon Dr. B. Zhang Dr. Y. C. Zhang Dr. Y. Zhao Dr. Ralf Zimmer
The Order of Presentation of the Formulas The question of the most expedient order in which to give the formulas, in particular, in what division to include particular formulas such as the definite integrals, turned out to be quite complicated. The thought naturally occurs to set up an order analogous to that of a dictionary. However, it is almost impossible to create such a system for the formulas of integral calculus. Indeed, in an arbitrary formula of the form b f (x) dx = A a
one may make a large number of substitutions of the form x = ϕ(t) and thus obtain a number of “synonyms” of the given formula. We must point out that the table of definite integrals by Bierens de Haan and the earlier editions of the present reference both sin in the plethora of such “synonyms” and formulas of complicated form. In the present edition, we have tried to keep only the simplest of the “synonym” formulas. Basically, we judged the simplicity of a formula from the standpoint of the simplicity of the arguments of the “outer” functions that appear in the integrand. Where possible, we have replaced a complicated formula with a simpler one. Sometimes, several complicated formulas were thereby reduced to a single, simpler one. We then kept only the simplest formula. As a result of such substitutions, we sometimes obtained an integral that could be evaluated by use of the formulas of Chapter Two and the Newton–Leibniz formula, or to an integral of the form a f (x) dx, −a
where f (x) is an odd function. In such cases, the complicated integrals have been omitted. Let us give an example using the expression
π/4 0
p−1
(cot x − 1) sin2 x
ln tan x dx = −
π cosec pπ. p
By making the natural substitution u = cot x − 1, we obtain ∞ π up−1 ln(1 + u) du = cosec pπ. p 0 Integrals similar to formula (0.1) are omitted in this new edition. Instead, we have formula (0.2).
xxvii
(0.1)
(0.2)
xxviii
The Order of Presentation of the Formulas
As a second example, let us take
π/2
ln (tanp x + cotp x) ln tan x dx = 0.
I= 0
The substitution u = tan x yields
∞
I= 0
ln (up + u−p ) ln u du. 1 + u2
If we now set υ = ln u, we obtain ∞ ∞ pυ υeυ ln (2 cosh pυ) −pυ dυ. dυ = ln e + e υ I= 2υ 2 cosh υ −∞ 1 + e −∞ The integrand is odd, and, consequently, the integral is equal to 0. Thus, before looking for an integral in the tables, the user should simplify as much as possible the arguments (the “inner” functions) of the functions in the integrand. The functions are ordered as follows: First we have the elementary functions: 1. 2. 3. 4. 5. 6. 7.
The function f (x) = x. The exponential function. The hyperbolic functions. The trigonometric functions. The logarithmic function. The inverse hyperbolic functions. (These are replaced with the corresponding logarithms in the formulas containing definite integrals.) The inverse trigonometric functions.
Then follow the special functions: 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
Elliptic integrals. Elliptic functions. The logarithm integral, the exponential integral, the sine integral, and the cosine integral functions. Probability integrals and Fresnel’s integrals. The gamma function and related functions. Bessel functions. Mathieu functions. Legendre functions. Orthogonal polynomials. Hypergeometric functions. Degenerate hypergeometric functions. Parabolic cylinder functions. Meijer’s and MacRobert’s functions. Riemann’s zeta function.
The integrals are arranged in order of outer function according to the above scheme: the farther down in the list a function occurs, (i.e., the more complex it is) the later will the corresponding formula appear
The Order of Presentation of the Formulas
xxix
in the tables. Suppose that several expressions have the same outer function. For example, consider sin ex , sin x, sin ln x. Here, the outer function is the sine function in all three cases. Such expressions are then arranged in order of the inner function. In the present work, these functions are therefore arranged in the following order: sin x, sin ex , sin ln x. Our list does not include polynomials, rational functions, powers, or other algebraic functions. An algebraic function that is included in tables of definite integrals can usually be reduced to a finite combination of roots of rational power. Therefore, for classifying our formulas, we can conditionally treat a power function as a generalization of an algebraic and, consequently, of a rational function.∗ We shall distinguish between all these functions and those listed above, and we shall treat them as operators. Thus, in the expression sin2 ex , we shall think of the squaring operator as applied to the outer function, sin x+cos x namely, the sine. In the expression sin x−cos x , we shall think of the rational operator as applied to the trigonometric functions sine and cosine. We shall arrange the operators according to the following order: 1. 2. 3. 4.
Polynomials (listed in order of their degree). Rational operators. Algebraic operators (expressions of the form Ap/q , where q and p are rational, and q > 0; these are listed according to the size of q). Power operators.
Expressions with the same outer and inner functions are arranged in the order of complexity of the operators. For example, the following functions [whose outer functions are all trigonometric, and whose inner functions are all f (x) = x] are arranged in the order shown: sin x,
sin x cos x,
1 = cosec x, sin x
sin x = tan x, cos x
sin x + cos x , sin x − cos x
sinm x,
sinm x cos x.
Furthermore, if two outer functions ϕ1 (x) and ϕ2 (x), where ϕ1 (x) is more complex than ϕ2 (x), appear in an integrand and if any of the operations mentioned are performed on them, the corresponding integral will appear [in the order determined by the position of ϕ2 (x) in the list] after all integrals containing only the function ϕ1 (x). Thus, following the trigonometric functions are the trigonometric and power functions [that is, ϕ2 (x) = x]. Then come • combinations of trigonometric and exponential functions, • combinations of trigonometric functions, exponential functions, and powers, etc., • combinations of trigonometric and hyperbolic functions, etc. Integrals containing two functions ϕ1 (x) and ϕ2 (x) are located in the division and order corresponding to the more complicated function of the two. However, if the positions of several integrals coincide because they contain the same complicated function, these integrals are put in the position defined by the complexity of the second function. To these rules of a general nature, we need to add certain particular considerations that will be easily 1 understood from the tables. For example, according to the above remarks, the function e x comes after 1 1 ex as regards complexity, but ln x and ln are equally complex since ln = − ln x. In the section on x x “powers and algebraic functions,” polynomials, rational functions, and powers of powers are formed from power functions of the form (a + bx)n and (α + βx)ν . ∗ For any natural number n, the involution (a + bx)n of the binomial a + bx is a polynomial. If n is a negative integer, (a + bx)n is a rational function. If n is irrational, the function (a + bx)n is not even an algebraic function.
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Use of the Tables∗ For the effective use of the tables contained in this book, it is necessary that the user should first become familiar with the classification system for integrals devised by the authors Ryzhik and Gradshteyn. This classification is described in detail in the section entitled The Order of Presentation of the Formulas (see page xxvii) and essentially involves the separation of the integrand into inner and outer functions. The principal function involved in the integrand is called the outer function, and its argument, which is itself usually another function, is called the inner function. Thus, if the integrand comprised the expression ln sin x, the outer function would be the logarithmic function while its argument, the inner function, would be the trigonometric function sin x. The desired integral would then be found in the section dealing with logarithmic functions, its position within that section being determined by the position of the inner function (here a trigonometric function) in Gradshteyn and Ryzhik’s list of functional forms. It is inevitable that some duplication of symbols will occur within such a large collection of integrals, and this happens most frequently in the first part of the book dealing with algebraic and trigonometric integrands. The symbols most frequently involved are α, β, γ, δ, t, u, z, zk , and Δ. The expressions associated with these symbols are used consistently within each section and are defined at the start of each new section in which they occur. Consequently, reference should be made to the beginning of the section being used in order to verify the meaning of the substitutions involved. Integrals of algebraic functions are expressed as combinations of roots with rational power indices, and definite integrals of such functions are frequently expressed in terms of the Legendre elliptic integrals F (φ, k), E (φ, k) and Π(φ, n, k), respectively, of the first, second, and third kinds. The four inverse hyperbolic functions arcsinh z, arccosh z, arctanh z, and arccoth z are introduced through the definitions 1 arcsinh(iz) i 1 arccos z = arccosh(z) i 1 arctan z = arctanh(iz) i arccot z = i arccoth(iz) arcsin z =
∗ Prepared
by Alan Jeffrey for the English language edition.
xxxi
xxxii
Use of the Tables
or 1 arcsin(iz) i arccosh z = i arccos z 1 arctanh z = arctan(iz) i 1 arccoth z = arccot(−iz) i arcsinh z =
The numerical constants C and G which often appear in the definite integrals denote Euler’s constant and Catalan’s constant, respectively. Euler’s constant C is defined by the limit s 1 − ln s = 0.577215 . . . . C = lim s→∞ m m=1 On occasion, other writers denote Euler’s constant by the symbol γ, but this is also often used instead to denote the constant γ = eC = 1.781072 . . . . Catalan’s constant G is related to the complete elliptic integral π/2 dx K ≡ K(k) ≡ 0 1 − k 2 sin2 x by the expression ∞ (−1)m 1 1 K dk = = 0.915965 . . . . G= 2 0 (2m + 1)2 m=0 Since the notations and definitions for higher transcendental functions that are used by different authors are by no means uniform, it is advisable to check the definitions of the functions that occur in these tables. This can be done by identifying the required function by symbol and name in the Index of Special Functions and Notation on page xxxix, and by then referring to the defining formula or section number listed there. We now present a brief discussion of some of the most commonly used alternative notations and definitions for higher transcendental functions. Bernoulli and Euler Polynomials and Numbers Extensive use is made throughout the book of the Bernoulli and Euler numbers Bn and En that are defined in terms of the Bernoulli and Euler polynomials of order n, B n (x) and E n (x), respectively. These polynomials are defined by the generating functions ∞ tn text = for |t| < 2π B n (x) t e − 1 n=0 n! and ∞ 2ext tn = for |t| < π. E n (x) t e + 1 n=0 n! The Bernoulli numbers are always denoted by Bn and are defined by the relation Bn = Bn (0)
for n = 0, 1, . . . ,
when B0 = 1,
1 B1 = − , 2
B2 =
1 , 6
B4 = −
1 ,.... 30
Use of the Tables
xxxiii
The Euler numbers En are defined by setting
1 En = 2 E n for n = 0, 1, . . . 2 The En are all integral, and E0 = 1, E2 = −1, E4 = 5, E6 = −61, . . . . An alternative definition of Bernoulli numbers, which we shall denote by the symbol Bn∗ , uses the same generating function but identifies the Bn∗ differently in the following manner: 1 t2 t4 t = 1 − t + B1∗ − B2∗ + . . . . t e −1 2 2! 4! This definition then gives rise to the alternative set of Bernoulli numbers n
B1∗ = 1/6, B2∗ = 1/30, B3∗ = 1/42, B4∗ = 1/30, B5∗ = 5/66, B6∗ = 691/2730, B7∗ = 7/6, B8∗ = 3617/510, . . . . These differences in notation must also be taken into account when using the following relationships that exist between the Bernoulli and Euler polynomials: n 1 n n = 0, 1, . . . Bn−k E k (2x) k 2n k=0
x x+1 2n E n−1 (x) = Bn − Bn n 2 2
B n (x) =
or E n−1 (x) = and
x 2 B n (x) − 2n B n n 2
n = 1, 2, . . .
n n−k 2 − 1 Bn−k B n (x) n = 2, 3, . . . k 2 k=0 There are also alternative definitions of the Euler polynomial of order n, and it should be noted that some authors, using a modification of the third expression above, call
x 2 B n (x) − 2n B n n+1 2 the Euler polynomial of order n. E n−2 (x) = 2
n
−1
n−2
Elliptic Functions and Elliptic Integrals The following notations are often used in connection with the inverse elliptic functions sn u, cn u, and dn u: 1 sn u sn u sc u = cn u sn u sd u = dn u
ns u =
1 cn u cn u cs u = sn u cn u cd u = dn u nc u =
1 dn u dn u ds u = sn u dn u dc u = cn u
nd u =
xxxiv
Use of the Tables
The elliptic integral of the third kind is defined by Gradshteyn and Ryzhik to be ϕ da 2 Π ϕ, n , k = 2 2 0 1 − n sin a 1 − k 2 sin2 a −∞ < n2 < ∞ sin ϕ dx = 2 2 (1 − n x ) (1 − x2 ) (1 − k 2 x2 ) 0 The Jacobi Zeta Function and Theta Functions The Jacobi zeta function zn(u, k), frequently written Z(u), is defined by the relation u E E 2 zn(u, k) = Z(u) = dn υ − dυ = E(u) − u. K K 0 This is related to the theta functions by the relationship ∂ zn(u, k) = ln Θ(u) ∂u giving πu π ϑ1 2K cn u dn u πu − (i). zn(u, k) = 2K ϑ sn u 1 2K πu ϑ 2 π dn u sn u 2K − (ii). zn(u, k) = 2K ϑ πu cn u 2 2K πu ϑ π 3 2K sn u cn u − k2 (iii). zn(u, k) = 2K ϑ πu dn u 3 2K πu π ϑ4 2K (iv). zn(u, k) = 2K ϑ πu 4 2K Many different notations for the theta function are in current use. The most common variants are the replacement of the argument u by the argument u/π and, occasionally, a permutation of the identification of the functions ϑ1 to ϑ4 with the function ϑ4 replaced by ϑ. The Factorial (Gamma) Function In older reference texts, the gamma function Γ(z), defined by the Euler integral ∞ Γ(z) = tz−1 e−t dt, 0
is sometimes expressed in the alternative notation Γ(1 + z) = z! = Π(z). On occasions, the related derivative of the logarithmic factorial function Ψ(z) is used where
(z!) d (ln z!) = = Ψ(z). dz z!
Use of the Tables
xxxv
This function satisfies the recurrence relation Ψ(z) = Ψ(z − 1) + and is defined by the series Ψ(z) = −C +
∞ n=0
1 z−1
1 1 − n+1 z+n
.
The derivative Ψ (z) satisfies the recurrence relation Ψ (z + 1) = Ψ (z) − and is defined by the series Ψ (z) =
1 z2
∞
1 . (z + n)2 n=0
Exponential and Related Integrals The exponential integrals E n (z) have been defined by Schloemilch using the integral ∞ E n (z) = e−zt t−n dt (n = 0, 1, . . . , Re z > 0) . 1
They should not be confused with the Euler polynomials already mentioned. The function E 1 (z) is related to the exponential integral Ei(z) through the expressions ∞ E 1 (z) = − Ei(−z) = e−t t−1 dt z
and
z
dt = Ei (ln z) [z > 1] . ln t 0 The functions E n (z) satisfy the recurrence relations 1 −z e − z E n−1 (z) E n (z) = [n > 1] n−1 and E n (z) = − E n−1 (z) with li(z) =
E 0 (z) = e−z /z. The function E n (z) has the asymptotic expansion
e−z 3π n n(n + 1) n(n + 1)(n + 2) − + · · · |arg z| < E n (z) ∼ 1− + z z z2 z3 2 while for large n, n e−x n(n − 2x) n 6x2 − 8nx + n2 1+ + + + R(n, x) , E n (x) = x+n (x + n)2 (x + n)4 (x + n)6 where
1 −4 −0.36n ≤ R(n, x) ≤ 1 + [x > 0] . n−4 x+n−1 The sine and cosine integrals si(x) and ci(x) are related to the functions Si(x) and Ci(x) by the integrals x π sin t Si(x) = dt = si(x) + t 2 0 and
xxxvi
Use of the Tables
(cos t − 1) dt. t 0 The hyperbolic sine and cosine integrals shi(x) and chi(x) are defined by the relations x sinh t shi(x) = dt t 0 and x (cosh t − 1) chi(x) = C + ln x + dt. t 0 Some authors write x (1 − cos t) Cin(x) = dt t 0 so that Cin(x) = − Ci(x) + ln x + C. The error function erf(x) is defined by the relation x 2 2 erf(x) = Φ(x) = √ e−t dt, π 0 and the complementary error function erfc(x) is related to the error function erfc(x) and to Φ(x) by the expression erfc(x) = 1 − erf(x). The Fresnel integrals S (x) and C (x) are defined by Gradshteyn and Ryzhik as x 2 S (x) = √ sin t2 dt 2π 0 and x 2 C (x) = √ cos t2 dt. 2π 0 Other definitions that are in use are x x πt2 πt2 dt, C 1 (x) = dt, S 1 (x) = sin cos 2 2 0 0 and x x sin t cos t 1 1 √ dt, √ dt. S 2 (x) = √ C 2 (x) = √ t t 2π 0 2π 0 These are related by the expressions 2 = S 2 x2 S (x) = S 1 x π and 2 = C 2 x2 C (x) = C 1 x π x
Ci(x) = C + ln x +
Hermite and Chebyshev Orthogonal Polynomials The Hermite polynomials H n (x) are related to the Hermite polynomials He n (x) by the relations
x −n/2 He n (x) = 2 Hn √ 2 and √ H n (x) = 2n/2 He n x 2 .
Use of the Tables
xxxvii
These functions satisfy the differential equations dHn d2 H n + 2n H n = 0 − 2x 2 dx dx and d2 He n d He n + n He n = 0. −x dx2 dx They obey the recurrence relations H n+1 = 2x H n −2n H n−1 and He n+1 = x He n −n He n−1 . The first six orthogonal polynomials He n are He 0 = 1, He 1 = x, He 2 = x2 − 1, He 3 = x3 − 3x, He 4 = x4 − 6x2 + 3, He 5 = x5 − 10x3 + 15x. Sometimes the Chebyshev polynomial U n (x) of the second kind is defined as a solution of the equation d2 y dy + n(n + 2)y = 0. 1 − x2 − 3x dx2 dx Bessel Functions A variety of different notations for Bessel functions are in use. Some common ones involve the replacement of Y n (z) by N n (z) and the introduction of the symbol −n 1 Λn (z) = z Γ(n + 1) J n (z). 2 1 In the book by Gray, Mathews, and MacRobert, the symbol Y n (z) is used to denote π Y n (z) + 2 (ln 2 − C) J n (z) while Neumann uses the symbol Y (n) (z) for the identical quantity. (1) (2) The Hankel functions H ν (z) and H ν (z) are sometimes denoted by Hsν (z) and Hiν (z), and some 1 authors write Gν (z) = πi H (1) ν (z). 2 The Neumann polynomial O n (t) is a polynomial of degree n + 1 in 1/t, with O 0 (t) = 1/t. The polynomials O n (t) are defined by the generating function ∞ 1 = J 0 (z) O 0 (t) + 2 J k (z) O k (t), t−z k=1 giving n−2k+1 [n/2] 1 n(n − k − 1)! 2 O n (t) = for n = 1, 2, . . . , 4 k! t k=0 where 12 n signifies the integral part of 12 n. The following relationship holds between three successive polynomials: 2 n2 − 1 nπ 2n O n (t) = sin2 . (n − 1) O n+1 (t) + (n + 1) O n−1 (t) − t t 2
xxxviii
Use of the Tables
The Airy functions Ai(z) and Bi(z) are independent solutions of the equation d2 u − zu = 0. dz 2 The solutions can be represented in terms of Bessel functions by the expressions
2 3/2 2 3/2 2 3/2 1√ 1 z Ai(z) = z z K 1/3 z z I −1/3 − I 1/3 = 3 3 3 π 3 3
2 3/2 2 3/2 1√ Ai(−z) = z z z J 1/3 + J −1/3 3 3 3 and by
2 3/2 2 3/2 z z z + I 1/3 , I −1/3 3 3 3
2 3/2 2 3/2 z z z Bi(−z) = − J 1/3 . J −1/3 3 3 3 Bi(z) =
Parabolic Cylinder Functions and Whittaker Functions The differential equation d2 y 2 + az + bz + c y = 0 dz 2 has associated with it the two equations
1 2 1 2 d2 y d2 y z z + + a y = 0 and − + a y = 0, dz 2 4 dz 2 4 the solutions of which are parabolic cylinder functions. The first equation can be derived from the second by replacing z by zeiπ/4 and a by −ia. The solutions of the equation
1 2 d2 y z +a y =0 − dz 2 4 are sometimes written U (a, z) and V (a, z). These solutions are related to Whittaker’s function D p (z) by the expressions U (a, z) = D −a− 12 (z) and
1 1 D −a− 12 (−z) + (sin πa) D −a− 12 (z) . V (a, z) = Γ +a π 2 Mathieu Functions There are several accepted notations for Mathieu functions and for their associated parameters. The defining equation used by Gradshteyn and Ryzhik is d2 y + a − 2k 2 cos 2z y = 0 with k 2 = q. dz 2 Different notations involve the replacement of a and q in this equation by h and θ, λ and h2 , and √ b and c = 2 q, respectively. The periodic solutions sen (z, q) and cen (z, q) and the modified periodic solutions Sen (z, q) and Cen (z, q) are suitably altered and, sometimes, re-normalized. A description of these relationships together with the normalizing factors is contained in: Tables Relating to Mathieu Functions. National Bureau of Standards, Columbia University Press, New York, 1951.
Index of Special Functions Notation β(x) Γ(z) γ(a, x), Γ(a, x) Δ(n − k) ξ(s) λ(x, y) μ(x, β), μ(x, β, α) ν(x), ν(x, α) Π(x) Π(ϕ, n, k) ζ(u) ζ(z, q), ζ(z) πu πu , Θ1 (u) = ϑ⎫ Θ(u) ⎧ = ϑ4 2K 3 2K ⎪ ⎨ ϑ0 (υ | τ ) = ϑ4 (υ | τ ), ⎪ ⎬ ϑ1 (υ | τ ), ϑ2 (υ | τ ), ⎪ ⎪ ⎩ ⎭ ϑ3 (υ | τ ) σ(u) Φ(x) Φ(z, s, υ) Φ(a, c; x) = 1 F 1 (α; γ; ⎧ ⎫ x) Φ (α, β, γ, x, y) ⎪ ⎪ ⎨ 1 ⎬ Φ2 (β, β , γ, x, y) ⎪ ⎪ ⎩ ⎭ Φ3 (β, γ, x, y) ψ(x) ℘(u) am(u, k) Bn B n (x) B(x, y) Bx (p, q) bei(z), ber(z)
Name of the function and the number of the formula containing its definition
Lobachevskiy’s angle of parallelism Elliptic integral of the third kind Weierstrass zeta function Riemann’s zeta functions Jacobian theta function
8.37 8.31–8.33 8.35 18.1 9.56 9.640 9.640 9.640 1.48 8.11 8.17 9.51–9.54 8.191–8.196
Elliptic theta functions
8.18, 8.19
Weierstrass sigma function See probability integral Lerch function Confluent hypergeometric function
8.17 8.25 9.55 9.21
Degenerate hypergeometric series in two variables
9.26
Euler psi function Weierstrass elliptic function Amplitude (of an elliptic function) Bernoulli numbers Bernoulli polynomials Beta functions Incomplete beta functions Thomson functions
8.36 8.16 8.141 9.61, 9.71 9.620 8.38 8.39 8.56
Gamma function Incomplete gamma functions Unit integer pulse function
continued on next page
xxxix
xl
Index of Special Functions
continued from previous page
Notation C C (x) C ν (a) C λn (t) C λn (x) ce2n (z, q),
ce2n+1 (z, q)
Ce2n (z, q),
Ce2n+1 (z, q)
chi(x) ci(x) cn(u) D(k) ≡ D D(ϕ, k) D n (z), D p (z) dn u e1 , e2 , e3 En E (ϕ, k) E(k) = E E(k ) = E E(p; ar : q; s : x) Eν (z) Ei(z) erf(x) erfc(x) = 1 − erf(x) F (ϕ, k) (α , . . . , αp ; β1 , . . . , βq ; z) F p q 1 (α, β; γ; z) = F (α, β; γ; z) F 2 1 (α; γ; z) = Φ(α, γ; z) F 1 1 FΛ (α :β1 , . . . , βn ; γ1 , . . . . . . , γn : z1 , . . . , zn ) F1 , F2 , F3 , F4 fen (z, q), Fen (z, q) . . . Feyn (z, q), Fekn (z, q) . . . G g2 , g3 gd x gen (z, q), Gen (z, q) Geyn (z, q), Gekn (z, q) ! ! ,... ,ap x ! ab11,... G m,n p,q ,bq
Name of the function and the number of the formula containing its definition Euler constant 9.73, 8.367 Fresnel cosine integral 8.25 Young functions 3.76 Gegenbauer polynomials 8.93 Gegenbauer functions 8.932 1 Periodic Mathieu functions (Mathieu 8.61 functions of the first kind) Associated (modified) Mathieu functions of 8.63 the first kind Hyperbolic cosine integral function 8.22 Cosine integral 8.23 Cosine amplitude 8.14 Elliptic integral 8.112 Elliptic integral 8.111 Parabolic cylinder functions 9.24–9.25 Delta amplitude 8.14 (used with the Weierstrass function) 8.162 Euler numbers 9.63, 9.72 Elliptic integral of the second kind 8.11–8.12 Complete elliptic integral of the second 8.11-8.12 kind MacRobert’s function 9.4 Weber function 8.58 Exponential integral function 8.21 Error function 8.25 Complementary error function 8.25 Elliptic integral of the first kind 8.11–8.12 Generalized hypergeometric series 9.14 Gauss hypergeometric function 9.10–9.13 Degenerate hypergeometric function 9.21 Hypergeometric function of several 9.19 variables Hypergeometric functions of two variables 9.18 Other nonperiodic solutions of Mathieu’s 8.64, 8.663 equation Catalan constant 9.73 Invariants of the ℘(u)-function 8.161 Gudermannian 1.49 Other nonperiodic solutions of Mathieu’s equation
8.64, 8.663
Meijer’s functions
9.3 continued on next page
Index of Special Functions
xli
continued from previous page
Notation h(n) heiν (z), herν (z) H (1) ν (z),
H (2) ν (z) πu H (u) = ϑ1 2K πu H 1 (u) = ϑ2 2K H n (z) Hν (z) I ν (z) I x (p, q) J ν (z) Jν (z) kν (x) K(k) = K, K(k ) = K K ν (z) kei(z), ker(z) L(x) Lν (z) Lα n (z) li(x) M λ,μ (z) O n (x) P μν (z),
P μν (x)
P ν (z), P ν (x) ⎫ ⎧ ⎨a b c ⎬ P α β γ δ ⎩ ⎭ α β γ P (α,β) (x) n Q μν (z),
Q μν (x)
Q ν (z), Q ν (x) S (x) S n (x) s μ,ν (z), S μ,ν (z) se2n+1 (z, q), se2n+2 (z, q) Se2n+1 (z, q),
Se2n+2 (z, q)
shi(x) si(x) sn u T n (x) U n (x)
Name of the function and the number of the formula containing its definition Unit integer function 18.1 Thomson functions 8.56 Hankel functions of the first and second 8.405, 8.42 kinds Theta function 8.192 Theta function 8.192 Hermite polynomials 8.95 Struve functions 8.55 Bessel functions of an imaginary argument 8.406, 8.43 Normalized incomplete beta function 8.39 Bessel function 8.402, 8.41 Anger function 8.58 Bateman’s function 9.210 3 Complete elliptic integral of the first kind 8.11–8.12 Bessel functions of imaginary argument 8.407, 8.43 Thomson functions 8.56 Lobachevskiy’s function 8.26 Modified Struve function 8.55 Laguerre polynomials 8.97 Logarithm integral 8.24 Whittaker functions 9.22, 9.23 Neumann’s polynomials 8.59 Associated Legendre functions of the first 8.7, 8.8 kind Legendre functions and polynomials 8.82, 8.83, 8.91 Riemann’s differential equation
9.160
Jacobi’s polynomials Associated Legendre functions of the second kind Legendre functions of the second kind Fresnel sine integral Schl¨ afli’s polynomials Lommel functions Periodic Mathieu functions Mathieu functions of an imaginary argument Hyperbolic sine integral Sine integral Sine amplitude Chebyshev polynomial of the 1st kind Chebyshev polynomials of the 2nd kind
8.96 8.7, 8.8 8.82, 8.83 8.25 8.59 8.57 8.61 8.63 8.22 8.23 8.14 8.94 8.94 continued on next page
xlii
Index of Special Functions
continued from previous page
Notation U ν (w, z), V ν (w, z) W λ,μ (z) Y ν (z) Z ν (z) Zν (z)
Name of the function and the number of the formula containing its definition Lommel functions of two variables 8.578 Whittaker functions 9.22, 9.23 Neumann functions 8.403, 8.41 Bessel functions 8.401 Bessel functions
Notation Symbol
Meaning
x
The integral part of the real number x (also denoted by [x])
(b+) a
(b−) a
" C
PV
Contour integrals; the path of integration starting at the point a extends to the point b (along a straight line unless there is an indication to the contrary), encircles the point b along a small circle in the positive (negative) direction, and returns to the point a, proceeding along the original path in the opposite direction. Line integral along the curve C
"
Principal value integral
z = x − iy
The complex conjugate of z = x + iy
n!
= 1 · 2 · 3 . . . n,
(2n + 1)!!
= 1 · 3 . . . (2n + 1). (double factorial notation)
(2n)!!
= 2 · 4 . . . (2n). (double factorial notation)
0!! = 1 and (−1)!! = 1
(cf. 3.372 for n = 0)
00 = 1
(cf. 0.112 and 0.113 for q = 0)
p
p! p(p − 1) . . . (p − n + 1) = , 1 · 2...n n!(p − n)! [n = 1, 2, . . . , p ≥ n] =
n x
k=m n
# ,
= a(a + 1) . . . (a + n − 1) =
0
= 1,
p n
=
p! n!(p − n)!
Γ(a+n) Γ(a)
(Pochhammer symbol) n = um + um+1 + . . . + un . If n < m, we define uk = 0
uk ,
p
= x(x − 1) . . . (x − n + 1)/n! [n = 0, 1, . . . ]
n (a)n n
0! = 1
m,n
$
k=m
Summation over all integral values of n excluding n = 0, and summation over all integral values of n and m excluding m = n = 0, respectively. # $ An empty has value 0, and an empty has value 1 continued on next page
xliii
xliv
Notation
continued from previous page
Symbol 1 i=j δij = 0 i=
j
Meaning Kronecker delta
τ
Theta function parameter (cf. 8.18)
× and ∧
Vector product (cf. 10.11)
·
Scalar product (cf. 10.11)
∇ or “del”
Vector operator (cf. 10.21)
∇2
Laplacian (cf. 10.31)
∼
Asymptotically equal to
arg z
The argument of the complex number z = x + iy
curl or rot
Vector operator (cf. 10.21)
div
Vector operator (divergence) (cf. 10.21)
F
Fourier transform (cf. 17.21)
Fc
Fourier cosine transform (cf. 17.31)
Fs
Fourier sine transform (cf. 17.31)
grad
Vector operator (gradient) (cf. 10.21)
hi and gij
Metric coefficients (cf. 10.51)
H
Hermitian transpose of a vector or matrix (cf. 13.123)
0 x 0 such that |g(z)| ≤ M |f (z)| in some sufficiently small neighborhood of the point z0 , we write g(z) = O(f (z)). continued on next page
Notation
xlv
continued from previous page
Symbol
Meaning
q
The nome, a theta function parameter (cf. 8.18)
R
The real numbers
R(x)
A rational function
Re z ≡ x
The real part of the complex number z = x + iy
Snm
Stirling number of the first kind (cd. 9.74)
Sm n
Stirling number of the second kind (cd. 9.74)
⎧ ⎪ ⎨+1 sign x = 0 ⎪ ⎩ −1
x>0 x=0 x 1] 2 n+1 + 2n − 2n − 1 x − n2 xn + x2 + x (1 − x)3
JO (5)
0.12 Sums of powers of natural numbers 0.121
n k=1
1.
n k=1
2.
3.
n
nq 1 q nq+1 1 q 1 q + + B2 nq−1 + B4 nq−3 + B6 nq−5 + · · · q+1 2 2 1 4 3 6 5 nq+1 nq qnq−1 q(q − 1)(q − 2) q−3 q(q − 1)(q − 2)(q − 3)(q − 4) q−5 = + + − n n + − ··· q+1 2 12 720 30, 240 [last term contains either n or n2 ] CE 332
kq =
k=
n(n + 1) 2
CE 333
n(n + 1)(2n + 1) 6 k=1 2 n n(n + 1) 3 k = 2 k2 =
CE 333
CE 333
k=1
1
2
Finite Sums
4.
5.
6.
7.
n k=1 n k=1 n k=1 n
k4 =
1 n(n + 1)(2n + 1)(3n2 + 3n − 1) 30
CE 333
k5 =
1 2 n (n + 1)2 (2n2 + 2n − 1) 12
CE 333
k6 =
1 n(n + 1)(2n + 1)(3n4 + 6n3 − 3n + 1) 42
CE 333
k7 =
1 2 n (n + 1)2 (3n4 + 6n3 − n2 − 4n + 2) 24
CE 333
k=1
0.122
0.122
n
2q q+1 1 q q−1 1 q q−3 3 n 2 B2 nq−1 − 2 2 − 1 B4 nq−3 − · · · − q+1 2 1 4 3
(2k − 1)q =
k=1
[last term contains either n or n2 .] 1.
2.
3. 4.11 5.10 6.10
n k=1 n k=1 n k=1 n k=1 n k=1 n
(2k − 1) = n2 1 n(4n2 − 1) 3
(2k − 1)2 =
(2k − 1)3 = n2 (2n2 − 1) (mk − 1) =
n
(mk − 1)2 =
1 n[m2 (n + 1)(2n + 1) − 6m(n + 1) + 6] 6
(mk − 1)3 =
1 n[m3 n(n + 1)2 − 2m2 (n + 1)(2n + 1) + 6m(n + 1) − 4] 4
k(k + 1)2 =
k=1
0.124 1. 2.10
1 n(n + 1)(n + 2)(3n + 5) 12
q 1 k n2 − k 2 = q(q + 1) 2n2 − q 2 − q 4
k=1 n
k(k + 1)3 =
k=1
0.125 0.126
JO (32b)
n [m(n + 1) − 2] 2
k=1
0.123
n k=1 n k=0
JO (32a)
1 n(n + 1) 12n3 + 63n2 + 107n + 58 60
k! · k = (n + 1)! − 1 (n + k)! = k!(n − k)!
[q = 1, 2, . . .]
e K 1 π n+ 2
AD (188.1)
1 2
WA 94
0.142
Sums of the binomial coefficients
3
0.13 Sums of reciprocals of natural numbers ∞ n 1 1 Ak = C + ln n + − , k 2n n(n + 1) . . . (n + k − 1)
0.13111
k=1
JO (59), AD (1876)
k=2
where 1 1 x(1 − x)(2 − x)(3 − x) · · · (k − 1 − x) dx k 0 1 1 A2 = , A3 = 12 12 19 9 A4 = , A5 = , 120 20 3 n 2 − 1 B4 1 B2 1 7 0.132 = (C + ln n) + ln 2 + 2 + + ... 2k − 1 2 8n 64n4 k=1 n 3 2n + 1 1 = − 0.133 k2 − 1 4 2n(n + 1) Ak =
JO (71a)a JO (184f)
k=2
0.14 Sums of products of reciprocals of natural numbers 1.
n k=1
2.
n k=1
3.
4.
n 1 = [p + (k − 1)q](p + kq) p(p + nq)
GI III (64)a
n(2p + nq + q) 1 = [p + (k − 1)q](p + kq)[p + (k + 1)q] 2p(p + q)(p + nq)[p + (n + 1)q]
GI III (65)a
n
1 [p + (k − 1)q](p + kq) . . . [p + (k + l)q] k=1
1 1 1 − = (l + 1)q p(p + q) . . . (p + lq) (p + nq)[p + (n + 1)q] . . . [p + (n + l)q] n k=1
1 1 = [1 + (k − 1)q][1 + (k − l)q + p] p
%
n k=1
1 − 1 + (k − 1)q
n k=1
1 1 + (k − 1)q + p
&
AD (1856)a
GI III (66)a
0.142
n k2 + k − 1 k=1
(k + 2)!
=
1 n+1 − 2 (n + 2)!
JO (157)
0.15 Sums of the binomial coefficients Notation: n is a natural number.
m n+k n+m+1 1. = n n+1 k=0 n n 2. 1+ + + . . . = 2n−1 2 4
KR 64 (70.1) KR 62 (58.1)
4
Finite Sums
3. 4.
n 1 m
+
n
(−1)k
3 n k
k=0
0.152 1. 2. 3. 0.153 1. 2. 3. 4.
+
n 5
0.152
+ . . . = 2n−1
= (−1)m
n−1 m
KR 62 (58.1)
[n ≥ 1]
KR 64 (70.2)
nπ 1 n 2 + 2 cos 3 3 0 3 6
n n n 1 (n − 2)π + + + ... = 2n + 2 cos 3 3 1 4 7
n n n 1 (n − 4)π n + + + ... = 2 + 2 cos 3 3 2 5 8 n
+
n
+
n
+ ... =
KR 62 (59.1) KR 62 (59.2) KR 62 (59.3)
n 1 n−1 nπ 2 + 2 2 cos 2 4 0 4 8 n n n n 1 n−1 nπ 2 + + + ... = + 2 2 sin 2 4 1 5 9 n n n n 1 n−1 nπ 2 + + + ... = − 2 2 cos 2 4 2 6 10 n n n n 1 n−1 nπ 2 + + + ... = − 2 2 sin 2 4 3 7 11
n
+
n
+
n
+ ... =
KR 63 (60.1) KR 63 (60.2) KR 63 (60.3) KR 63 (60.4)
0.154 1.
n
(k + 1)
n
k=0
2.
n
k
(−1)k+1 k
n k
k=1
3.
N
(−1)k
k=0
4.
n
(−1)k
5.
n
(−1)k
N k=0
0.155 1.
(−1)k
k n−1 = 0
n k N k
=0
KR 63 (66.1)
[n ≥ 2]
KR 63 (66.2)
[N ≥ n ≥ 1;
00 ≡ 1]
kn = (−1)n n!
[n ≥ 0;
00 ≡ 1]
(α + k)n = (−1)n n!
[n ≥ 0;
00 ≡ 1]
(α + k)n−1 = 0
n (−1)k+1 n n = k+1 k n+1
k=1
[n ≥ 0]
k
k=0
6.
N k
n
k=0
= 2n−1 (n + 2)
[N ≥ n ≥ 1,
00 ≡ 1 N, n ∈ N + ]
KR 63 (67)
0.159
2.
Sums of the binomial coefficients
1 n 2n+1 − 1 = k+1 k n+1
n k=0
3.
5
KR 63 (68.1)
n αk+1 n (α + 1)n+1 − 1 = k+1 k n+1
KR 63 (68.2)
k=0
4.
n n (−1)k+1 n 1 = k k m m=1
KR 64 (69)
k=1
0.156 1.
p n m n+m = k p−k p
[m is a natural number]
KR 64 (71.1)
k=0
2.
n−p k=0
n k
n p+k
0.157 1.
n n
k
k=0
2.
2n
2
(−1)k
k=0
3.
2n+1
(−1)k
k=0
4. 0.15810 1.
2.
3.
4.
= 2n k
=
2n n
(2n)! (n − p)!(n + p)!
KR 64 (72.1)
2
= (−1)n
2n + 1 k
2n n
1.
2
=0
n n (2n − 1)! 2 k = 2 k [(n − 1)!] k=1
n 2n 2n − k 2n − k − 1 k k+1 n 2 −2 k=4 − n n−k n−k−1 k=1
n 2n 2n − k 2n − k − 1 3 · 4n 2k − 2k+1 k 2 = 4n − n n−k n−k−1 k=1
n 2n 2n − k 2n − k − 1 2k − 2k+1 k 3 = (6n + 13)4n − 18n n n−k n−k−1 k=1
n 2n 2n − k 2n − k − 1 − (60n + 75)4n 2k − 2k+1 k 4 = (32n2 + 104n) n n−k n−k−1
n 2n 2n 2n 1 n − k= 4 − n n−k n−k−1 2
k=0
KR 64 (72.2)
k=1
0.15910
KR 64 (71.2)
KR 64 (72.3)
KR 64 (72.4)
6
Numerical Series and Infinite Products
2.
3.
0.160
n 2n 2n 2n 1 − k2 = (2n + 1) − 4n n−k n−k−1 2 n k=0
n 2n 2n (3n + 2) n 1 2n ·4 − − k3 = (3n + 1) n−k n−k−1 4 2 n k=0
0.16010 1.
k
2n n−1 2n α 1 2n (1 + α)2n−1 (1 − α) 2k 1 = (1 + α)2n αk + αn + 2 k k 2 n 2 (1 + α) 2
k=n+1
2.
k=0
n Γ (r + b) B (n + a − b, b) = (−1)r r Γ (r + a) Γ (a − b) r=0
n
0.2 Numerical Series and Infinite Products 0.21 The convergence of numerical series The series ∞ 0.211 uk = u1 + u2 + u3 + . . . k=1
is said to converge absolutely if the series ∞ 0.212 |uk | = |u1 | + |u2 | + |u3 | + · · · , k=1
composed of the absolute values of its terms converges. If the series 0.211 converges and the series 0.212 diverges, the series 0.211 is said to converge conditionally. Every absolutely convergent series converges.
0.22 Convergence tests Suppose that lim |uk |
k→∞
1/k
=q
If q < 1, the series 0.211 converges absolutely. On the other hand, if q > 1, the series 0.211 diverges. (Cauchy) 0.222 Suppose that ! ! ! uk+1 ! !=q lim !! k→∞ uk ! ! ! ! uk+1 ! ! Here, if q < 1, the series 0.211 converges absolutely. If q > 1, the series 0.211 diverges. If !! uk ! approaches 1 but remains greater than unity, then the series 0.211 diverges. (d’Alembert) 0.223
Suppose that
!
! ! uk ! ! ! lim k ! −1 =q k→∞ uk+1 ! Here, if q > 1, the series 0.211 converges absolutely. If q < 1, the series 0.211 diverges. (Raabe)
0.229
Convergence tests
7
0.224
Suppose that f (x) is a positive decreasing function and that ek f ek lim =q k→∞ f (k) #∞ for natural k. If q < 1, the series k=1 f (k) converges. If q > 1, this series diverges. (Ermakov) 0.225 Suppose that ! ! ! uk ! q |vk | ! ! ! uk+1 ! = 1 + k + k p , where p > 1 and the |vk | are bounded, that is, the |vk | are all less than some M , which is independent of k. Here, if q > 1, the series 0.211 converges absolutely. If q ≤ 1, this series diverges. (Gauss) 0.226 Suppose that a function f (x) defined for x ≥ q ≥ 1 is continuous, positive, and decreasing. Under these conditions, the series ∞ f (k) k=1
converges or diverges accordingly as the integral ∞ f (x) dx q
converges or diverges (the Cauchy integral test). 0.227 Suppose that all terms of a sequence u1 , u2 , . . . , un are positive. In such a case, the series 1.
∞
(−1)k+1 uk = u1 − u2 + u3 − . . .
k=1
is called an alternating series. If the terms of an alternating series decrease monotonically in absolute value and approach zero, that is, if 2.
3.
uk+1 < uk and lim uk = 0, k→∞
the series 0.227 1 converges. Here, the remainder of the series is ! !∞ ∞ n ! ! ! ! (−1)k−n+1 uk = ! (−1)k+1 uk − (−1)k+1 uk < un+1 ! ! ! k=n+1
0.228 1.
k=1
(Leibniz)
k=1
If the series ∞
vk = v1 + v2 + . . . + vk + . . .
k=1
2.
converges and the numbers uk form a monotonic bounded sequence, that is, if |uk | < M for some number M and for all k, the series ∞ uk vk = u1 v1 + u2 v2 + . . . + uk vk + . . . FI II 354 k=1
converges. (Abel) 0.229 If the partial sums of the series 0.228 1 are bounded and if the numbers uk constitute a monotonic sequence that approaches zero, that is, if
8
Numerical Series and Infinite Products
! ! n ! ! ! ! vk ! < M ! ! !
[n = 1, 2, . . .]
0.231
and lim uk = 0,
k=1
FI II 355
k→∞
then the series 0.228 2 converges (Dirichlet).
0.23–0.24 Examples of numerical series 0.231
Progressions ∞
1.
aq k =
k=0 ∞
2.
a 1−q
(a + kr)q k =
k=0
0.232 1.
∞
∞
1 = ln 2 k
(−1)k+1
π 1 1 =1−2 = 2k − 1 (4k − 1)(4k + 1) 4
3.
(cf. 0.113)
(cf. 1.511)
k=1
∞ ka k=1
[|q| < 1]
∞
k=1
∗
rq a + 1 − q (1 − q)2
(−1)k+1
k=1
2.
[|q| < 1]
(cf. 1.643) ⎤
⎡
a i 1 (−1)j (a + 1)!(i − j)a ⎦ 1 ⎣ = bk (b − 1)a+1 i=1 ba−i j=0 j!(a + 1 − j)!
[a = 1, 2, 3, . . . , 0.233 1.
∞ 1 1 1 = 1 + p + p + . . . = ζ(p) kp 2 3
b = 1]
[Re p > 1]
WH
[Re p > 0]
WH
k=1
2.
∞
(−1)k+1
k=1
3.10
∞ 1 22n−1 π 2n |B2n |, = k 2n (2n)!
k=1
4.
∞
(−1)k+1
k=1
5.
∞ k=1
6.
1 = (1 − 21−p ) ζ(p) kp
∞ k=1
∞ 1 π2 = k2 6
1 (22n−1 − 1)π 2n |B2n | = k 2n (2n)!
1 (22n − 1)π 2n |B2n | = 2n (2k − 1) 2 · (2n)! (−1)k+1
FI II 721
k=1
1 π 2n+1 |E2n | = 2n+2 2n+1 (2k − 1) 2 (2n)!
JO (165)
JO (184b)
JO (184d)
0.236
0.234 1.
2.
3.
4.
5.
6.
7. 8.6 9.∗
Examples of numerical series
∞ k=1 ∞ k=1 ∞ k=0 ∞ k=1 ∞ k=1 ∞ k=1 ∞
(−1)k+1
1 π2 = 2 k 12
9
EU
1 π2 = 2 (2k − 1) 8
EU
(−1)k =G (2k + 1)2
FI II 482
(−1)k+1 π3 = (2k − 1)3 32
EU
1 π4 = 4 (2k − 1) 96
EU
(−1)k+1 5π 5 = (2k − 1)5 1536
EU
(−1)k+1
k=1 ∞
k π2 − ln 2 = (k + 1)2 12
1 = 2 − 2 ln 2 k(2k + 1) k=1 ∞ √ Γ n + 12 = π ln 4 2 n Γ (n) n=1
0.235 Sn =
∞ k=1
S1 =
(4k 2 1 , 2
1 n − 1)
S2 =
π2 − 8 , 16
S3 =
32 − 3π 2 , 64
S4 =
π 4 + 30π 2 − 384 768 JO (186)
0.236 1.
2.
3.
4.
5.
∞ k=1 ∞ k=1 ∞ k=1 ∞
1 = 2 ln 2 − 1 k (4k 2 − 1)
BR 51a
3 1 = (ln 3 − 1) 2 k (9k − 1) 2
BR 51a
3 1 = −3 + ln 3 + 2 ln 2 k (36k 2 − 1) 2 k
=
2
(4k 2 − 1) k=1 ∞ k=1
1 k (4k 2
2
− 1)
1 8
=
BR 52, AD (6913.3)
BR 52
3 − 2 ln 2 2
BR 52
10
6.
Numerical Series and Infinite Products ∞
2
k=1
7.6
12k 2 − 1
∞ k=1
k (4k 2 − 1)
0.237
= 2 ln 2
AD (6917.3), BR 52
1 π2 − 2 ln 2 =4− 2 k(2k + 1) 4
0.237 1.
∞ k=1
2.
∞ k=1
3.
∞ k=2
4.
1 1 = (2k − 1)(2k + 1) 2 1 π 1 = − (4k − 1)(4k + 1) 2 8 3 1 = (k − 1)(k + 1) 4
∞ k=1,k=m
5.
AD (6917.2), BR 52
∞ k=1,k=m
[cf. 0.133],
3 1 =− 2 (m + k)(m − k) 4m
[m is an integer]
AD (6916.1)
3 (−1)k−1 = (m − k)(m + k) 4m2
[m is an even number]
AD (6916.2)
0.238 1.
∞ k=1
2.
∞ k=1
3. 0.239 1.11 2.7
3.
1 (−1)k+1 = (1 − ln 2) (2k − 1)2k(2k + 1) 2
∞
1 1 π 1 = − ln 3 + √ (3k + 1)(3k + 2)(3k + 3)(3k + 4) 6 4 12 3 k=0
π √ + ln 2 3 k=1
∞ 1 π 1 √ − ln 2 = (−1)k+1 3k − 1 3 3 k=1 ∞
∞ k=1
4.
1 1 = ln 2 − (2k − 1)2k(2k + 1) 2
∞ k=1
(−1)k+1
(−1)k+1
1 1 = 3k − 2 3
GI III (93)
GI III (94)a
GI III (95)
√ , 1 + 1 = √ π + 2 ln 2+1 4k − 3 4 2
k+3 1 π 1 (−1)[ 2 ] = + ln 2 k 4 2
GI III (85), BR∗ 161 (1)
BR∗ 161 (1)
BR∗ 161 (1)
GI III (87)
0.241
Examples of numerical series ∞
5.
k+3 (−1)[ 2 ]
k=1 ∞
6.
k+5 (−1)[ 3 ]
k=1 ∞
7.
k=1
11
1 π = √ 2k − 1 2 2 1 5π = 2k − 1 12
GI III (88)
1 π √ 1 = − 2+1 (8k − 1)(8k + 1) 2 16
0.241 ∞ 1 = ln 2 2k k
1.
JO (172g)
k=1 ∞
2.
k=1
3.11
1 2k k 2
=
∞ 2n n
n=0
1 π2 2 − (ln 2) 12 2
pn = √
JO (174)
1 1 − 4p
0≤p
0,
q 1 1 1 = (k + p)(k + q) q − p m=p+1 m
Summations of reciprocals of factorials 0.245 1.
∞ 1 = e = 2.71828 . . . k!
k=0
2.11
∞ (−1)k k=0
k!
=
1 ≈ 0.1839397 . . . 2e
q > −1, q > 0]
[q > p > −1,
p = q]
GI III (90)
BR∗ 161 (1)
p and q integers]
0.249
3.
4.
5.
6.
7.
8.
Examples of numerical series ∞ k=1 ∞
1.
2.
3.
4.
5.
6.
k=0 ∞ k=0 ∞
∞ k=0 ∞ k=0 ∞ k=0 ∞
0.248
(−1)k = cos 1 = 0.54030 . . . (2k)! (−1)k−1 = sin 1 = 0.84147 . . . (2k − 1)! 1 (k!)
2
k=0 ∞
∞ k=1 ∞ k=1
= I 0 (2) = 2.27958530 . . .
1 = I 1 (2) = 1.590636855 . . . k!(k + 1)! 1 = I n (2) k!(k + n)! (−1)k 2
k=0 ∞
k=0
0.247
1 k = = 0.36787 . . . (2k + 1)! e
k =1 (k + 1)! k=1
∞ 1 1 1 = e+ = 1.54308 . . . (2k)! 2 e k=0
∞ 1 1 1 = e− = 1.17520 . . . (2k + 1)! 2 e
k=0
0.246
(k!)
= J 0 (2) = 0.22389078 . . .
(−1)k = J 1 (2) = 0.57672481 . . . k!(k + 1)! (−1)k = J n (2) k!(k + n)! 1 k! = (n + k − 1)! (n − 2) · (n − 1)! kn = Sn , k! S1 = e, S5 = 52e,
0.2497
13
∞ (k + 1)3 k=0
k!
= 15e
S2 = 2e, S6 = 203e,
S3 = 5e, S7 = 877e,
S4 = 15e S8 = 4140e
14
Numerical Series and Infinite Products
0.250
0.25 Infinite products 0.250
Suppose that a sequence of numbers a1 , a2 , . . . , ak , . . . is given. If the limit lim
n→∞
n -
(1 + ak )
k=1
exists, whether finite or infinite (but of definite sign), this limit is called the value of the infinite product ∞ (1 + ak ), and we write k=1
1.
lim
n→∞
n -
(1 + ak ) =
k=1
∞ -
(1 + ak )
k=1
If an infinite product has a finite nonzero value, it is said to converge. Otherwise, the infinite product is said to diverge. We assume that no ak is equal to −1. FI II 400 0.251 For the infinite product 0.250 1. to converge, it is necessary that lim ak = 0. FI II 403 k→∞
0.252 If ak > 0 or ak < 0 for all values of the index k starting with some# particular value, then, for the ∞ product 0.250 1 to converge, it is necessary and sufficient that the series k=1 ak converge. ∞ ∞ 0.253 The product (1 + ak ) is said to converge absolutely if the product (1 + |ak |) converges. k=1
k=1
FI II 403
0.254 0.255
Absolute convergence of an infinite product implies its convergence. ∞ ∞ (1 + ak ) converges absolutely if, and only if, the series ak converges abThe product k=1
k=1
solutely.
FI II 406
0.26 Examples of infinite products 0.261
∞ k=1
0.262 1.
∞ -
(−1)k+1 1+ 2k − 1
1−
k=2
2.
3.
∞ -
1 k2
=
=
√ 2
1 2 =
k=1
0.263 1. 2.∗
EU
FI II 401
2 π k=1
∞ 1 π 1− = 2 (2k + 1) 4 1−
1 (2k)2
1/2
1/4
1/8 4 6·8 10 · 12 · 14 · 16 2 e= · ... 1 3 5·7 9 · 11 · 13 · 15 1/2 2 1/3 3 1/4 4 4 1/5 2 2 2 ·4 2 ·4 e= ··· 3 1 1·3 1·3 1 · 36 · 5
FI II 401
FI II 401
0.303
3.
∗
Definitions and theorems
π = 2
1/2 2 1/4 3 1/8 4 4 1/16 1 2 2 ·4 2 ·4 ··· 2 1·3 1 · 33 1 · 36 · 5
where the nth factor is the (n + 1)th root of the product 0.264 1.
2.∗
15
$n
k=0 (k
+ 1)(−1)
k+1
( nk ) .
√ k e 1 k=1 1 + k 1/2 2 1/3 3 1/4 4 4 1/5 2 2 2 ·4 2 ·4 eC = ··· 1 1·3 1 · 33 1 · 36 · 5 eC =
∞ -
FI II 402
$n k+1 n where the nth factor is the (n + 1)th root of the product k=0 (k + 1)(−1) ( k ) . Here C is the Euler constant, denoted in other works by γ. / . . 0 0 2 1 1 1 1 11 1 1 1 1 0.265 = · + · + + ... FI II 402 π 2 2 2 2 2 2 2 2 2 ∞ k 1 1 + x2 = [0 < x < 1] 0.2668 FI II 401 1−x k=0
0.3 Functional Series 0.30 Definitions and theorems 0.301 1.
The series ∞
fk (x),
k=1
the terms of which are functions, is called a functional series. The set of values of the independent variable x for which the series 0.301 1 converges constitutes what is called the region of convergence of that series. 0.302 A series that converges for all values of x in a region M is said to converge uniformly in that region if, for every ε ≥ 0, there exists a number N such that, for n > N , the inequality ! ! ∞ ! ! ! ! fk (x)! < ε ! ! ! k=n+1
holds for all x in M . 0.303
If the terms of the functional series 0.301 1 satisfy the inequalities: |fk (x)| < uk (k = 1, 2, 3, . . .) , throughout the region M , where the uk are the terms of some convergent numerical series ∞ uk = u1 + u2 + . . . + uk + . . . , k=1
the series 0.301 1 converges uniformly in M . (Weierstrass)
16
Functional Series
0.304
0.304 Suppose that the series 0.301 1 converges uniformly in a region M and that a set of functions gk (x) constitutes (for each x) a monotonic sequence, and that these functions are uniformly bounded; that is, suppose that a number L exists such that the inequalities 1.
|gn (x)| ≤ L
2.
hold for all n and x. Then, the series ∞ fk (x)gk (x) k=1
converges uniformly in the region M . (Abel)
FI II 451
0.305 Suppose that the partial sums of the series 0.301 1 are uniformly bounded; that is, suppose that, for some L and for all n and x in M , the inequalities ! n ! ! ! ! ! fk (x)! < L ! ! ! k=1
hold. Suppose also that for each x the functions gn (x) constitute a monotonic sequence that approaches zero uniformly in the region M . Then, the series 0.304 2 converges uniformly in the region M . (Dirichlet) FI II 451 6
0.306 If the functions fk (x) (for k = 1, 2, 3, . . . ) are integrable on the interval [a, b] and if the series 0.301 1 made up of these functions converges uniformly on that interval, this series may be integrated termwise; that is, b ∞ ∞ b fk (x) dx = fk (x) dx [a ≤ x ≤ b] FI II 459 a
k=1
k=1
a
0.307 Suppose that the functions fk (x) (for k = 1, 2, 3, . . . ) have continuous derivatives #∞ fk (x) on the interval [a, b]. If the series 0.301 1 converges on this interval and if the series k=1 fk (x) of these derivatives converges uniformly, the series 0.301 1 may be differentiated termwise; that is, ∞ ∞ fk (x) = fk (x) FI II 460 k=1
k=1
0.31 Power series 0.311 1.
A functional series of the form ∞
ak (x − ξ)k = a0 + a1 (x − ξ) + a2 (x − ξ)2 + . . .
k=0
is called a power series. The following is true of any power series: if it is not everywhere convergent, the region of convergence is a circle with its center at the point ξ and a radius equal to R; at every interior point of this circle, the power series 0.311 1 converges absolutely, and outside this circle, it diverges. This circle is called the circle of convergence, and its radius is called the radius of convergence. If the series converges at all points of the complex plane, we say that the radius of convergence is infinite (R = +∞).
0.315
0.312 is,
Power series
17
Power series may be integrated and differentiated termwise inside the circle of convergence; that
x
ξ
∞
ak (x − ξ)
k
k=0
d dx
∞
∞ ak (x − ξ)k+1 , k+1
dx =
k=0 ∞
ak (x − ξ)k
=
k=0
kak (x − ξ)k−1 .
k=1
The radius of convergence of a series that is obtained from termwise integration or differentiation of another power series coincides with the radius of convergence of the original series. Operations on power series 0.313
Division of power series. ∞ k=0 ∞
bk xk = ak xk
∞ 1 ck xk , a0 k=0
k=0
where
1 cn−k ak − bn = 0, a0 n
cn +
k=1
or
⎡
k=0
where c0 =
a0 a1 a2 .. .
0 a0 a1 .. .
···
⎢ ⎢ (−1) ⎢ ⎢ cn = ⎢ .. an0 ⎢ . ⎢ ⎣an−1 b0 − a0 bn−1 an−2 an−3 · · · an b 0 − a0 b n an−1 an−2 · · · Power series raised to powers. n ∞ ∞ k ak x = ck xk , n
0.314
a1 b 0 − a0 b 1 a2 b 0 − a0 b 2 a3 b 0 − a0 b 3 .. .
an0 ,
cm
m 1 = (kn − m + k)ak cm−k ma0
⎤ 0 0⎥ ⎥ 0⎥ ⎥ ⎥ ⎥ ⎥ a0 ⎦ a1
AD (6360)
k=0
for m ≥ 1
[n is a natural number]
AD (6361)
k=1
0.315
The substitution of one series into another. ∞ ∞ ∞ bk y k = ck xk y= ak xk ; k=1
c 1 = a1 b 1 ,
k=1
c2 = a2 b1 + a21 b2 ,
k=1
c3 = a3 b1 + 2a1 a2 b2 + a31 b3 ,
c4 = a4 b1 + a22 b2 + 2a1 a3 b2 + 3a21 a2 b3 + a41 b4 ,
...
AD (6362)
18
0.316
Functional Series
Multiplication of power series ∞ ∞ ∞ ak xk bk xk = ck xk k=0
k=0
cn =
k=0
0.316
n
ak bn−k
FI II 372
k=0
Taylor series 0.317 If a function f (x) has derivatives of all orders throughout a neighborhood of a point ξ, then we may write the series 1.
(x − ξ) (x − ξ)2 (x − ξ)3 f (ξ) + f (ξ) + f (ξ) + . . . , 1! 2! 3! which is known as the Taylor series of the function f (x).
f (ξ) +
The Taylor series converges to the function f (x) if the remainder 2.
Rn (x) = f (x) − f (ξ) −
n (x − ξ)k k=1
k!
f (k) (ξ)
approaches zero as n → ∞. The following are different forms for the remainder of a Taylor series: 3.
Rn (x) =
4.
Rn (x) =
5.
(x − ξ)n+1 (n+1) f (ξ + θ(x − ξ)) (n + 1)!
[0 < θ < 1]
(Lagrange)
(x − ξ)n+1 (1 − θ)n f (n+1) (ξ + θ(x − ξ)) [0 < θ < 1] n! ψ(x − ξ) − ψ(0) (x − ξ)n (1 − θ)n (n+1) f Rn (x) = (ξ + θ(x − ξ)) ψ [(x − ξ)(1 − θ)] n! [0 < θ < 1] ,
(Cauchy)
(Schl¨ omilch)
where ψ(x) is an arbitrary function satisfying the following two conditions: (1) It and its derivative ψ (x) are continuous in the interval (0, x − ξ); and (2) the derivative ψ (x) does not change sign in that interval. If we set ψ(x) = xp+1 , we obtain the following form for the remainder: (x − ξ)n+1 (1 − θ)n−p−1 (n+1) f (ξ + θ(x − ξ)) (p + 1)n! 1 x (n+1) Rn (x) = f (t)(x − t)n dt n! ξ Rn (x) =
6. 0.318 1.11
0 < θ < 1]
(Rouch´e)
Other forms in which a Taylor series may be written: f (a + x) =
∞ xk k=0
2.
[0 < p ≤ n;
∞ xk k=0
k!
k!
f (k) (a) = f (a) +
f (k) (0) = f (0) +
x x2 f (a) + f (a) + . . . 1! 2!
x x2 f (0) + f (0) + . . . 1! 2!
(Maclaurin series)
0.323
0.319
Fourier series
19
The Taylor series of functions of several variables:
∂f (ξ, η) ∂f (ξ, η) + (y − η) f (x, y) = f (ξ, η) + (x − ξ) ∂x ∂y
2 2 ∂ f (ξ, η) ∂ 2 f (ξ, η) 1 2 2 ∂ f (ξ, η) + (y − η) + 2(x − ξ)(y − η) + + ... (x − ξ) 2! ∂x2 ∂x ∂y ∂y 2
0.32 Fourier series 0.320 Suppose that f (x) is a periodic function of period 2l and that it is absolutely integrable (possibly improperly) over the interval (−l, l). The following trigonometric series is called the Fourier series of f (x):
∞ a0 kπx kπx + + bk sin , 1. ak cos 2 l l k=1
2. 3.11
the coefficients of which (the Fourier coefficients) are given by the formulas 1 α+2l 1 l kπt kπt ak = dt = dt (k = 0, 1, 2, . . .) f (t) cos f (t) cos l −l l l α l 1 α+2l 1 l kπt kπt dt = dt (k = 1, 2, . . .) bk = f (t) sin f (t) sin l −l l l α l
Convergence tests 0.321
The Fourier series of a function f (x) at a point x0 converges to the number f (x0 + 0) + f (x0 − 0) , 2 if, for some h > 0, the integral h |f (x0 + t) + f (x0 − t) − f (x0 + 0) − f (x0 − 0)| dt t 0 exists. Here, it is assumed that the function f (x) either is continuous at the point x0 or has a discontinuity of the first kind (a saltus) at that point and that both one-sided limits f (x0 + 0) and f (x0 − 0) exist. FI III 524 (Dini) 0.322 The Fourier series of a periodic function f (x) that satisfies the Dirichlet conditions on the interval [a, b] converges at every point x0 to the value 12 [f (x0 + 0) + f (x0 − 0)]. (Dirichlet) We say that a function f (x) satisfies the Dirichlet conditions on the interval [a, b] if it is bounded on that interval and if the interval [a, b] can be partitioned into a finite number of subintervals inside each of which the function f (x) is continuous and monotonic. 0.323 The Fourier series of a function f (x) at a point x0 converges to 12 [f (x0 + 0) + f (x0 − 0)] if f (x) is FI III 528 of bounded variation in some interval (x0 − h, x0 + h) with center at x0 . (Jordan–Dirichlet) The definition of a function of bounded variation. Suppose that a function f (x) is defined on some interval [a, b], where z < b. Let us partition this interval in an arbitrary manner into subintervals with the dividing points a = x0 < x1 < x2 < . . . < xn−1 < xn = b and let us form the sum
20
Functional Series n
0.324
|f (xk ) − f (xk−1 )|
k=1
Different partitions of the interval [a, b] (that is, different choices of points of division xi ) yield, generally speaking, different sums. If the set of these sums is bounded above, we say that the function f (x) is of bounded variation on the interval [a, b]. The least upper bound of these sums is called the total variation of the function f (x) on the interval [a, b]. 0.324 Suppose that a function f (x) is piecewise-continuous on the interval [a, b] and that in each interval of continuity it has a piecewise-continuous derivative. Then, at every point x0 of the interval [a, b], the Fourier series of the function f (x) converges to 12 [f (x0 + 0) + f (x0 − 0)]. 0.325 A function f (x) defined in the interval (0, l) can be expanded in a cosine series of the form ∞
1.
a0 kπx + , ak cos 2 l k=1
where 2. 0.326 1.
ak =
2 l
l
f (t) cos 0
A function f (x) defined in the interval (0, l) can be expanded in a sine series of the form ∞
bk sin
k=1
where 2.
kπt dt l
bk =
2 l
kπx , l l
f (t) sin 0
kπt dt l
The convergence tests for the series 0.325 1 and 0.326 1 are analogous to the convergence tests for the series 0.320 1 (see 0.321–0.324). 0.327 The Fourier coefficients ak and bk (given by formulas 0.320 2 and 0.320 3) of an absolutely integrable function approach zero as k → ∞. If a function f (x) is square-integrable on the interval (−l, l), the equation of closure is satisfied: ∞ 1 l 2 a20 2 + ak + b2k = f (x) dx (A. M. Lyapunov) FI III 705 2 l −l k=1
0.328 Suppose that f (x) and ϕ(x) are two functions that are square-integrable on the interval (−l, l) and that ak , bk and αk , βk are their Fourier coefficients. For such functions, the generalized equation of closure (Parseval’s equation) holds: ∞ 1 l a0 α0 + (ak αk + bk βk ) = f (x)ϕ(x) dx FI III 709 2 l −l k=1 For examples of Fourier series, see 1.44 and 1.45.
0.411
Differentiation of a definite integral with respect to a parameter
21
0.33 Asymptotic series 0.330 Included in the collection of all divergent series is the broad class of series known as asymptotic or semiconvergent series. Despite the fact that these series diverge, the values of the functions that they represent can be calculated with a high degree of accuracy if we take the sum of a suitable number of terms of such series. In the case of alternating asymptotic series, we obtain greatest accuracy if we break off the series in question at whatever term is of lowest absolute value. In this case, the error (in absolute value) does not exceed the absolute value of the first of the discarded terms (cf. 0.227 3). Asymptotic series have many properties that are analogous to the properties of convergent series, and, for that reason, they play a significant role in analysis. The asymptotic expansion of a function is denoted as follows: ∞ An z −n f (z) ∼ n=0
∞ An This is the definition of an asymptotic expansion. The divergent series is called the asymptotic zn n=0 expansion of a function f (z) in a given region of values of arg z if the expression Rn (z) = z n [f (z) − Sn (z)], n Ak , satisfies the condition lim Rn (z) = 0 for fixed n. FI II 820 where Sn (z) = zk |z|→∞ k=0 A divergent series that represents the asymptotic expansion of some function is called an asymptotic series. 0.331 Properties of asymptotic series
1.
The operations of addition, subtraction, multiplication, and raising to a power can be performed on asymptotic series just as on absolutely convergent series. The series obtained as a result of these operations will also be asymptotic.
2.
One asymptotic series can be divided by another, provided that the first term A0 of the divisor is not equal to zero. The series obtained as a result of division will also be asymptotic. FI II 823-825
3.
An asymptotic series can be integrated termwise, and the resultant series will also be asymptotic. FI II 824 In contrast, differentiation of an asymptotic series is, in general, not permissible.
4.
A single asymptotic expansion can represent different functions. On the other hand, a given function can be expanded in an asymptotic series in only one manner.
0.4 Certain Formulas from Differential Calculus 0.41 Differentiation of a definite integral with respect to a parameter 0.410 0.411 1. 2.
ϕ(a) dψ(a) d d ϕ(a) dϕ(a) − f (ψ(a), a) + f (x, a) dx f (x, a) dx = f (ϕ(a), a) da ψ(a) da da da ψ(a) In particular, d a f (x) dx = f (a) da b d a f (x) dx = −f (b) db b
FI II 680
22
Certain Formulas from Differential Calculus
0.430
0.42 The nth derivative of a product (Leibniz’s rule) Suppose that u and v are n-times-differentiable functions of x. Then, dn v n du dn−1 v n d2 u dn−2 v n d3 u dn−3 v dn u dn (uv) =u n + + + + ··· + v n n n−1 2 n−2 3 n−3 dx dx dx 1 dx dx 2 dx dx 3 dx dx or, symbolically, dn (uv) = (u + v)(n) dxn
FI I 272
0.43 The nth derivative of a composite function 0.430 1.
If f (x) = F (y)and y = ϕ(x), then dn U1 U2 U3 Un (n) F (y) + F (y) + F (y) + . . . + F (y), f (x) = dxn 1! 2! 3! n! where n dn k k dn k−1 k(k − 1) 2 dn k−2 k−1 k−1 d y y y y − y + y − . . . + (−1) ky AD (7361) GO dxn 1! dxn 2! dxn dxn (l) k i j h y y y dm F y dn n! f (x) = ··· , dxn i!j!h! . . . k! dy m 1! 2! 3! l! # Here, the symbol indicates summation over all solutions in non-negative integers of the equation i + 2j + 3h + . . . + lk = n and m = i + j + h + . . . + k.
Uk =
2.
0.431 1.
dn (−1) F dxn n
1 1 (n) 1 n − 1 n (n−1) 1 = 2n F + 2n−1 F x x x x 1! x (n − 1)(n − 2) n(n − 1) (n−2) 1 F + + ... x2n−2 2! x ⎡
2.
(−1)n
AD (7362.1)
n a n−1 n a n−2 dn a 1 a ⎣ a n x = x e e + (n − 1) + (n − 1)(n − 2) dxn xn x x x 1 2 ⎤ n a n−3 + (n − 1)(n − 2)(n − 3) + . . .⎦ x 3 AD (7362.2)
0.432 1.
n(n − 1) dn 2 (2x)n−2 F (n−1) x2 F x = (2x)n F (n) x2 + n dx 1! n(n − 1)(n − 2)(n − 3) + (2x)n−4 F (n−2) x2 + 2! n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) + (2x)n−6 F (n−3) x2 + . . . 3! AD (7363.1)
0.440
Integration by substitution
23
⎡ n(n − 1) n(n − 1)(n − 2)(n − 3) d ax + e = (2ax)n eax ⎣1 + 2 dxn 1! (4ax2 ) 2! (4ax2 ) ⎤ n(n − 1)(n − 2)(n − 3)(n − 4)(n − 5) + + · · ·⎦ 3 3! (4ax2 ) n
2.
2
2
AD (7363.2)
p p(p − 1)(p − 2) . . . (p − n + 1)(2ax) d 1 + ax2 = n−p dxn (1 + ax2 )
2 n(n − 1) 1 + ax2 n(n − 1)(n − 2)(n − 3) 1 + ax2 × 1+ + + ... , 1!(p − n + 1) 4ax2 2!(p − n + 1)(p − n + 2) 4ax2 n
3.
n
AD (7363.3)
4. 5.
m− 12 (2m − 1)!! dm−1 sin (m arccos x) 1 − x2 = (−1)m−1 m−1 dx m
n+1
2k n a a n+1 b n ∂ k (−1) (−1) = n! n 2 2 2 2 2k ∂a a +b a +b a
AD (7363.4) (3.944.12)
0≤2k≤n+1
6. 0.433 1.
(−1)n
∂n ∂an
b 2 a + b2
= n!
a 2 a + b2
n+1 0≤2k≤n
(−1)k
n+1 2k + 1
2k+1 b a
(3.944.11)
√ √ √ dn √ F (n) ( x) n(n − 1) F (n−1) ( x) (n + 1)n(n − 1)(n − 2) F (n−2) ( x) F x = − + √ √ n+1 √ n+2 − . . . n dxn 1! 2! (2 x) (2 x) (2 x) AD (7364.1)
2.
0.434 0.435
n−1 √ 2n−1 dn (2n − 1)!! a 1 2 √ 1 + a x = − a dxn 2n x x 2
n 1 n 1 n n y n−p dn p p−1 d y p−2 d y y y =p − + − ... n dxn dxn dxn 1 p−1 2 p−2 n 1 dn y n 1 dn y 2 dn y 3 dn n ln y = − + x − ... 1 1 · y dxn dxn dxn 2 2 · y 2 dxn
AD (7364.2)
AD (737.1)
AD (737.2)
0.44 Integration by substitution 0.44011 Let f (g(x)) and g(x) be continuous in [a, b]. Further, let g (x) exist and be continuous there. b g(b) Then f [g(x)]g (x) dx = f (u) du a
g(a)
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1 Elementary Functions 1.1 Power of Binomials 1.11 Power series ∞ q q(q − 1) 2 q(q − 1) . . . (q − k + 1) k xk x + ··· + x + ··· = k 2! k! k=0 If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for |x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1. For x = 1, the series converges absolutely for q > 0. For x = −1, it converges absolutely for q > 0 and diverges for q < 0. If FI II 425 q = n is a natural number, the series 1.110 is reduced to the finite sum 1.111. n n xk an−k 1.111 (a + x)n = k
1.110 (1 + x)q = 1 + qx +
k=0
1.112 1.
(1 + x)−1 = 1 − x + x2 − x3 + · · · =
∞
(−1)k−1 xk−1
k=1
(see also 1.121 2) 2.
(1 + x)−2 = 1 − 2x + 3x2 − 4x3 + · · · =
∞
(−1)k−1 kxk−1
k=1
3.11 4.
1·1 2 1·1·3 3 1·1·3·5 4 1 x + x − x + ... (1 + x)1/2 = 1 + x − 2 2·4 2·4·6 2·4·6·8 1·3 2 1·3·5 3 1 (1 + x)−1/2 = 1 − x + x − x + ... 2 2·4 2·4·6 ∞
x = kxk (1 − x)2
1.113
x2 < 1
k=1
1.114 1.
√ q q x q(q − 3) x 2 q(q − 4)(q − 5) x 3 q 1+ 1+x =2 1+ + + ... + 1! 4 2! 4 4 2 3! x < 1, q is a real number AD (6351.1)
25
26
The Exponential Function
2.
x+
1 + x2
q
1.121
∞ q 2 q 2 − 22 q 2 − 42 . . . q 2 − (2k)2 x2k+2 =1+ (2k + 2)! k=0 2 ∞ 2 2 q −1 q − 32 . . . q 2 − (2k − 1)2 2k+1 +qx + q x (2k + 1)! k=1 2 x < 1, q is a real number
AD(6351.2)
1.12 Series of rational fractions 1.121 ∞
1.
∞
2k−1 x2 x2 x = k−1 = 2 1−x 1+x 1 − x2k k−1
k=1
k−1
x2 < 1
AD (6350.3)
k=1
∞
2.
2k−1 1 = x−1 x2k−1 + 1
x2 > 1
AD (6350.3)
k=1
1.2 The Exponential Function 1.21 Series representation 1.211 1.11
ex =
∞ xk k=0
2.
ax =
k!
∞ k (x ln a)
k!
k=0
3.
e−x = 2
∞ k=0
4.∗
ex = lim
n→∞
x2k k! x n
(−1)k
1+
n
∞
xk (k + 1) k! k=0 ∞ x x B2k x2k 1.213 = 1 − + ex − 1 2 (2k)! k=1
2 5x3 15x4 2x ex + + + ... 1.214 e = e 1 + x + 2! 3! 4! 1.215 3x4 8x5 3x6 56x7 x2 1. esin x = 1 + x + − − − + + ... 2! 4! 5! 6! 7!
4x4 31x6 x2 + − + ... 2. ecos x = e 1 − 2! 4! 6! 1.212 ex (1 + x) =
[x < 2π]
FI II 520 AD (6460.3)
AD (6460.4) AD (6460.5)
1.232
Series of exponentials
etan x = 1 + x +
3.
27
3x3 9x4 37x5 x2 + + + + ... 2! 3! 4! 5!
AD (6460.6)
1.216 2x3 5x4 x2 + + + ... 2! 3! 4! x3 7x4 x2 − − + ... =1+x+ 2! 3! 4!
1.
earcsin x = 1 + x +
AD (6460.7)
2.
earctan x
AD (6460.8)
1.217 1.
π
∞ 1 eπx + e−πx = x πx −πx 2 e −e x + k2
(cf. 1.421 3)
AD (6707.1)
(cf. 1.422 3)
AD (6707.2)
k=−∞
∞ (−1)k 2π = x πx −πx e −e x2 + k 2
2.
k=−∞
1.22 Functional relations 1.221 1. ax = ex ln a 1
aloga x = a logx a = x
2. 1.222 1.
ex = cosh x + sinh x
2.
eix = cos x + i sin x
1.223 eax − ebx = (a − b)x exp
∞ 1 (a − b)2 x2 (a + b)x 1+ 2 2k 2 π 2 k=1
1.23 Series of exponentials 1.231
∞
akx =
k=0
1 1 − ax
[a > 1 and x < 0 or 0 < a < 1 and x > 0]
1.232 1.
∞
(−1)k e−2kx
[x > 0]
(−1)k e−(2k+1)x
[x > 0]
tanh x = 1 + 2
k=1
2.
sech x = 2
∞ k=0
3.
cosech x = 2
∞
e−(2k+1)x
k=0
%
4.
∗
∞ cos2n x sin x = exp − 2n n=1
[x > 0] & [0 ≤ x ≤ π]
MO 216
28
Trigonometric and Hyperbolic Functions
1.311
1.3–1.4 Trigonometric and Hyperbolic Functions 1.30 Introduction The trigonometric and hyperbolic sines are related by the identities sinh x =
1 sin(ix), i
sin x =
1 sinh(ix). i
The trigonometric and hyperbolic cosines are related by the identities cosh x = cos(ix),
cos x = cosh(ix).
Because of this duality, every relation involving trigonometric functions has its formal counterpart involving the corresponding hyperbolic functions, and vice versa. In many (though not all) cases, both pairs of relationships are meaningful. The idea of matching the relationships is carried out in the list of formulas given below. However, not all the meaningful “pairs” are included in the list.
1.31 The basic functional relations 1.311
1 ix e − e−ix 2i = −i sinh(ix)
1.
sin x =
2.
sinh x =
3.
cos x =
4.
cosh x =
5. 6. 7. 8.
1 x e − e−x 2 = −i sin(ix)
1 ix e + e−ix 2 = cosh(ix) 1 x e + e−x 2 = cos(ix)
1 sin x = tanh(ix) cos x i 1 sinh x = tan(ix) tanh x = cosh x i cos x 1 cot x = = = i coth(ix) sin x tan x 1 cosh x = = i cot (ix) coth x = sinh x tanh x tan x =
1.312 1.
cos2 x + sin2 x = 1
1.314
2.
The basic functional relations
cosh2 x − sinh2 x = 1
1.313 1. sin (x ± y) = sin x cos y ± sin y cos x 2.
sinh (x ± y) = sinh x cosh y ± sinh y cosh x
3.
sin (x ± iy) = sin x cosh y ± i sinh y cos x
4.
sinh (x ± iy) = sinh x cos y ± i sin y cosh x
5.
cos (x ± y) = cos x cos y ∓ sin x sin y
6.
cosh (x ± y) = cosh x cosh y ± sinh x sinh y
7.
cos (x ± iy) = cos x cosh y ∓ i sin x sinh y
8.
cosh (x ± iy) = cosh x cos y ± i sinh x sin y tan x ± tan y tan (x ± y) = 1 ∓ tan x tan y tanh x ± tanh y tanh (x ± y) = 1 ± tanh x tanh y tan x ± i tanh y tan (x ± iy) = 1 ∓ i tan x tanh y tanh x ± i tan y tanh (x ± iy) = 1 ± i tanh x tan y
9. 10. 11. 12. 1.314 1. 2. 3. 4. 5. 6. 7. 8. 9.∗
1 1 (x ± y) cos (x ∓ y) 2 2 1 1 sinh x ± sinh y = 2 sinh (x ± y) cosh (x ∓ y) 2 2 1 1 cos x + cos y = 2 cos (x + y) cos (x − y) 2 2 1 1 cosh x + cosh y = 2 cosh (x + y) cosh (x − y) 2 2 1 1 cos x − cos y = 2 sin (x + y) sin (y − x) 2 2 1 1 cosh x − cosh y = 2 sinh (x + y) sinh (x − y) 2 2 sin (x ± y) tan x ± tan y = cos x cos y sinh (x ± y) tanh x ± tanh y = cosh x cosh y 1 π 1 π sin x ± cos y = ±2 sin (x + y) ± sin (x − y) ± 4 4 2 2 1 π 1 π = ±2 cos (x + y) ∓ cos (x − y) ∓ 2 4 2 4 1 π 1 π = 2 sin (x ± y) ± cos (x ∓ y) ∓ 2 4 2 4 sin x ± sin y = 2 sin
29
30
Trigonometric and Hyperbolic Functions
. 10.∗
a sin x ± b cos x = a 1 +
2 b b sin x ± arctan a a
11.∗
±a sin x + b cos x = b
1+
[a = 0] a 2 b
+ a , cos x ∓ arctan b
[b = 0] 2 r r 1 a sin x ± b cos y = q 1 + sin (x ± y) + arctan q 2 q 1 1 [q = 0] q = (a + b) cos (x ∓ y) , r = (a − b) sin (x ∓ y) 2 2 s s 2 1 a cos x + b cos y = t 1 + cos (x ∓ y) + arctan [t = 0] t 2 t . 2 t t 1 cos (x ∓ y) − arctan = −s 1 + [s = 0] s 2 s 1 1 s = (a − b) sin (x ± y) , t = (a + b) cos (x ± y) 2 2 .
12.∗
13.∗
1.315 1.
sin2 x − sin2 y = sin(x + y) sin(x − y) = cos2 y − cos2 x
2.
sinh2 x − sinh2 y = sinh(x + y) sinh(x − y) = cosh2 x − cosh2 y
3.
cos2 x − sin2 y = cos(x + y) cos(x − y) = cos2 y − sin2 x
4.
sinh2 x + cosh2 y = cosh(x + y) cosh(x − y) = cosh2 x + sinh2 y
1.316 1. 2. 1.317 1. 2. 3. 4. 5.
1.315
n
(cos x + i sin x) = cos nx + i sin nx n
(cosh x + sinh x) = sinh nx + cosh nx x 1 sin = ± (1 − cos x) 2 2 x 1 sinh = ± (cosh x − 1) 2 2 x 1 cos = ± (1 + cos x) 2 2 x 1 cosh = (cosh x + 1) 2 2 1 − cos x sin x x = tan = 2 sin x 1 + cos x
[n is an integer] [n is an integer]
1.321
6.
Trigonometric and hyperbolic functions: expansion in multiple angles
tanh
31
cosh x − 1 sinh x x = = 2 sinh x cosh x + 1
The signs in front of the radical in formulas 1.317 1, 1.317 2, and 1.317 3 are taken so as to agree with the signs of the left-hand members. The sign of the left hand members depends in turn on the value of x.
1.32 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) 1.320 1.
2.
n−1
2n 2n n−k sin (−1) 2 cos 2(n − k)x + k n k=0 n−1
2n 2n (−1)n 2n n−k sinh x = 2n (−1) 2 cosh 2(n − k)x + k n 2 2n
1 x = 2n 2
k=0
3.
4.
5.
6.
sin2n−1 x =
n−1
1 22n−2
(−1)n+k−1
k=0
8.
cos2n−1 x =
cosh
2n−1
n−1
1 22n−2
x=
k=0
1 22n−2
2n − 1 k
3. 4.
KR 56 (10, 4)
n−1 k=0
2n − 1 k
cos(2n − 2k − 1)x
cosh(2n − 2k − 1)x
1 (− cos 2x + 1) 2 1 sin3 x = (− sin 3x + 3 sin x) 4 1 sin4 x = (cos 4x − 4 cos 2x + 3) 8 1 sin5 x = (sin 5x − 5 sin 3x + 10 sin x) 16 sin2 x =
KR 56 (10, 1)
1.321
2.
sin(2n − 2k − 1)x
n−1 (−1)n−1 n+k−1 2n − 1 sinh x = 2n−2 (−1) sinh(2n − 2k − 1)x k 2 k=0 n−1
2n 2n 1 2n cos x = 2n 2 cos 2(n − k)x + k n 2 k=0 n−1
2n 2n 1 2n cosh x = 2n 2 cosh 2(n − k)x + k n 2
Special cases
1.
2n−1
k=0
7.
2n − 1 k
KR 56 (10, 2)
KR 56 (10, 3)
32
5. 6.
Trigonometric and Hyperbolic Functions
1 (− cos 6x + 6 cos 4x − 15 cos 2x + 10) 32 1 sin7 x = (− sin 7x + 7 sin 5x − 21 sin 3x + 35 sin x) 64
sin6 x =
1.322 1.
sinh2 x =
2.
sinh3 x =
3.
sinh4 x =
4.
sinh5 x =
5.
sinh6 x =
6.
sinh7 x =
1 (cosh 2x − 1) 2 1 (sinh 3x − 3 sinh x) 4 1 (cosh 4x − 4 cosh 2x + 3) 8 1 (sinh 5x − 5 sinh 3x + 10 sinh x) 16 1 (cosh 6x − 6 cosh 4x + 15 cosh 2x + 10) 32 1 (sinh 7x − 7 sinh 5x + 21 sinh 3x + 35 sinh x) 64
1.323 1.
cos2 x =
2.
cos3 x =
3.
cos4 x =
4.
cos5 x =
5.
cos6 x =
6.
cos7 x =
1 (cos 2x + 1) 2 1 (cos 3x + 3 cos x) 4 1 (cos 4x + 4 cos 2x + 3) 8 1 (cos 5x + 5 cos 3x + 10 cos x) 16 1 (cos 6x + 6 cos 4x + 15 cos 2x + 10) 32 1 (cos 7x + 7 cos 5x + 21 cos 3x + 35 cos x) 64
1.324 1.
cosh2 x =
2.
cosh3 x =
3.
cosh4 x =
4.
cosh5 x =
5.
cosh6 x =
6.
cosh7 x =
1 (cosh 2x + 1) 2 1 (cosh 3x + 3 cosh x) 4 1 (cosh 4x + 4 cosh 2x + 3) 8 1 (cosh 5x + 5 cosh 3x + 10 cosh x) 16 1 (cosh 6x + 6 cosh 4x + 15 cosh 2x + 10) 32 1 (cosh 7x + 7 cosh 5x + 21 cosh 3x + 35 cosh x) 64
1.322
1.332
Trigonometric and hyperbolic functions: expansion in powers
33
1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions 1.331 1.7
n n sin nx = n cosn−1 x sin x − cosn−3 x sin3 x + cosn−5 x sin5 x − . . . ; 3 5 ⎧
⎨ n − 2 n−3 2 cosn−3 x = sin x 2n−1 cosn−1 x − ⎩ 1 ⎫
⎬ n − 3 n−5 n − 4 n−7 + 2 cosn−5 x − 2 cosn−7 x + . . . ⎭ 2 3 AD (3.175) [(n+1)/2]
2.
sinh nx = x
n 2k − 1
k=1 [(n−1)/2]
= sinh x
sinh2k−2 x coshn−2k+1 x
(−1)k
k=0
3.
cos nx = cosn x −
n
n − k − 1 n−2k−1 coshn−2k−1 x 2 k n
cosn−4 x sin4 x − . . . ; 4
n n−3 n n − 3 n−5 n−1 n n−2 =2 cos x − 2 cos x+ cosn−4 x 2 1 2 1
n n − 4 n−7 − cosn−6 x + . . . 2 3 2 2
cosn−2 x sin2 x +
AD (3.175)
4.3
[n/2]
n sinh2k x coshn−2k x 2k k=0
[n/2] n n−1 k1 n−k−1 cosh x + n (−1) =2 2n−2k−1 coshn−2k x k k−1
cosh nx =
k=1
1.332 1.
2 4n − 22 4n2 − 42 4n2 − 22 3 5 sin 2nx = 2n cos x sin x − sin x + sin x − . . . 3! 5!
2n − 2 2n−3 2n−3 2 = (−1)n−1 cos x 22n−1 sin2n−1 x − sin x 1! (2n − 3)(2n − 4) 2n−5 2n−5 2 sin x + 2! (2n − 4)(2n − 5)(2n − 6) 2n−7 2n−7 2 − sin x + ... 3!
AD (3.171)
AD (3.173)
34
2.
3.
Trigonometric and Hyperbolic Functions
1.333
(2n − 1)2 − 12 sin3 x sin(2n − 1)x = (2n − 1) sin x − 3! (2n − 1)2 − 12 (2n − 1)2 − 32 5 sin x − . . . AD (3.172) + 5!
2n − 1 2n−4 2n−3 2 = (−1)n−1 22n−2 sin2n−1 x − sin x 1! (2n − 1)(2n − 4) 2n−6 2n−5 2 sin x + 2! (2n − 1)(2n − 5)(2n − 6) 2n−8 2n−7 2 − sin x + ... AD (3.174) 3! 4n2 4n2 − 22 4n2 4n2 − 2 4n2 − 42 4n2 sin2 x + sin4 x − sin6 x + . . . cos 2nx = 1 − 2! 4! 6!
2n 2n−3 2n−2 2 = (−1)n 22n−1 sin2n x − sin x 1!
AD (3.171)
2n(2n − 3) 2n−5 2n−4 2n(2n − 4)(2n − 5) 2n−7 2n−6 2 2 + sin x− sin x + ... 2! 3! AD (3.173)a
4.
(2n − 1)2 − 12 sin2 x cos(2n − 1)x = cos x 1 − 2! (2n − 1)2 − 12 (2n − 1)2 − 32 4 sin x − . . . + 4!
2n − 3 2n−4 2n−4 2 = (−1)n−1 cos x 22n−2 sin2n−2 x − sin x 1! (2n − 4)(2n − 5) 2n−6 2n−6 2 sin x + 2! (2n − 5)(2n − 6)(2n − 7) 2n−8 2n−8 2 − sin x + ... 3!
AD (3.172)
AD (3.174)
By using the formulas and values of 1.30, we can write formulas for sinh 2nx, sinh(2n − 1)x, cosh 2nx, and cosh(2n − 1)x that are analogous to those of 1.332, just as was done in the formulas in 1.331. Special cases 1.333 1.
sin 2x = 2 sin x cos x
2.
sin 3x = 3 sin x − 4 sin3 x sin 4x = cos x 4 sin x − 8 sin3 x
3. 4. 5.
sin 5x = 5 sin x − 20 sin3 x + 16 sin5 x sin 6x = cos x 6 sin x − 32 sin3 x + 32 sin5 x
1.337
6.
Trigonometric and hyperbolic functions: expansion in powers
sin 7x = 7 sin x − 56 sin3 x + 112 sin5 x − 64 sin7 x
1.334 1.
sinh 2x = 2 sinh x cosh x
2.
sinh 3x = 3 sinh x + 4 sinh3 x sinh 4x = cosh x 4 sinh x + 8 sinh3 x
3.11 5.11
sinh 5x = 5 sinh x + 20 sinh3 x + 16 sinh5 x sinh 6x = cosh x 6 sinh x + 32 sinh3 x + 32 sinh5 x
6.
sinh 7x = 7 sinh x + 56 sinh3 x + 112 sinh5 x + 64 sinh7 x
4.
1.335 1.
cos 2x = 2 cos2 x − 1
2.
cos 3x = 4 cos3 x − 3 cos x
3.
cos 4x = 8 cos4 x − 8 cos2 x + 1
4.
cos 5x = 16 cos5 x − 20 cos3 x + 5 cos x
5.
cos 6x = 32 cos6 x − 48 cos4 x + 18 cos2 x − 1
6.
cos 7x = 64 cos7 x − 112 cos5 x + 56 cos3 x − 7 cos x
1.336 1.
cosh 2x = 2 cosh2 x − 1
2.
cosh 3x = 4 cosh3 x − 3 cosh x
3.
cosh 4x = 8 cosh4 x − 8 cosh2 x + 1
4.
cosh 5x = 16 cosh5 x − 20 cosh3 x + 5 cosh x
5.
cosh 6x = 32 cosh6 x − 48 cosh4 x + 18 cosh2 x − 1
6.
cosh 7x = 64 cosh7 x − 112 cosh5 x + 56 cosh3 x − 7 cosh x
1.337 1.∗ 2.∗ 3.∗ 4.∗ 5.∗ 6.∗
cos 3x cos3 x cos 4x cos4 x cos 5x cos5 x cos 6x cos6 x sin 3x cos3 x sin 4x cos4 x
= 1 − 3 tan2 x = 1 − 6 tan2 x + tan4 x = 1 − 10 tan2 x + 5 tan4 x = 1 − 15 tan2 x + 15 tan4 x − tan6 x = 3 tan x − tan3 x = 4 tan x − 4 tan3 x
35
36
7.∗ 8.∗ 9.∗ 10.∗ 11.∗ 12.∗ 13.∗ 14.∗ 15.∗ 16.∗
Trigonometric and Hyperbolic Functions
1.341
sin 5x = 5 tan x − 10 tan3 x + tan5 x cos5 x sin 6x = 6 tan x − 20 tan3 x + 6 tan5 x cos6 x cos 3x = cot3 x − 3 cot x sin3 x cos 4x = cot4 x − 6 cot2 x + 1 sin4 x cos 5x = cot5 x − 10 cot3 x + 5 cot x sin5 x cos 6x = cot6 x − 15 cot4 x + 15 cot2 x − 1 sin6 x sin 3x = 3 cot2 x − 1 sin3 x sin 4x = 4 cot3 x − 4 cot x sin4 x sin 5x = 5 cot4 x − 10 cot2 x + 1 sin5 x sin 6x = 6 cot5 x − 20 cot3 x + 6 cot x sin6 x
1.34 Certain sums of trigonometric and hyperbolic functions 1.341 1.
n−1 k=0
2.
3.
ny y n−1 y sin cosec sin(x + ky) = sin x + 2 2 2
ny 1 n−1 y sinh sinh(x + ky) = sinh x + 2 2 sinh y k=0 2
n−1 ny y n−1 y sin cosec cos(x + ky) = cos x + 2 2 2
AD (361.8)
n−1
AD (361.9)
k=0
4.
5.
ny 1 n−1 y sinh cosh(x + ky) = cosh x + 2 2 sinh y k=0 2
2n−1 y 2n − 1 y sin ny sec (−1)k cos(x + ky) = sin x + 2 2 n−1
JO (202)
k=0
6.
n(y + π) y n−1 (y + π) sin sec (−1)k sin(x + ky) = sin x + 2 2 2
n−1 k=0
AD (202a)
1.351
Sums of powers of trigonometric functions of multiple angles
37
Special cases 1.342 1. 2.10
n k=1 n
sin kx = sin
nx x n+1 x sin cosec 2 2 2
AD (361.1)
nx x n+1 x sin cosec + 1 2 2 2 sin n + 12 x nx n+1 x 1 = cos sin x cosec = 1+ 2 2 2 2 sin x2
cos kx = cos
k=0
AD (361.2)
3.
4.
n k=1 n
sin(2k − 1)x = sin2 nx cosec x 1 sin 2nx cosec x 2
cos(2k − 1)x =
k=1
1.343 1.
2.
3.
1 (−1)n cos 2n+1 2 x (−1) cos kx = − + x 2 2 cos k=1 2 n sin 2nx (−1)k+1 sin(2k − 1)x = (−1)n+1 2 cos x n
k=1 n
k
cos(4k − 3)x +
k=1
AD (361.7)
n
JO (207)
AD (361.11)
AD (361.10)
sin(4k − 1)x = sin 2nx (cos 2nx + sin 2nx) (cos x + sin x) cosec 2x
k=1
JO (208)
1.344 1.
n−1 k=1
2.
n−1 k=1
3.
n−1 k=0
sin
π πk = cot n 2n
√ nπ nπ n 2πk 2 = 1 + cos − sin sin n 2 2 2 √ nπ nπ n 2πk 2 = 1 + cos + sin cos n 2 2 2
AD (361.19)
AD (361.18)
AD (361.17)
1.35 Sums of powers of trigonometric functions of multiple angles 1.351 1.
n k=1
1 [(2n + 1) sin x − sin(2n + 1)x] cosec x 4 n cos(n + 1)x sin nx = − 2 2 sin x
sin2 kx =
AD (361.3)
38
2.
Trigonometric and Hyperbolic Functions n k=1
1.352
n−1 1 + cos nx sin(n + 1)x cosec x 2 2 n cos(n + 1)x sin nx = + 2 2 sin x
cos2 kx =
AD (361.4)a
3.
n
sin3 kx =
n+1 nx x 1 3(n + 1)x 3nx 3x 3 sin x sin cosec − sin sin cosec 4 2 2 2 4 2 2 2
JO (210)
cos3 kx =
n+1 nx x 1 3(n + 1) 3nx 3x 3 cos x sin cosec + cos x sin cosec 4 2 2 2 4 2 2 2
JO (211)a
sin4 kx =
1 [3n − 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] 8
JO (212)
cos4 kx =
1 [3n + 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] 8
JO (213)
k=1
4.
n k=1
5.
n k=1
6.
n k=1
1.352 1.11
2.11
n cos 2n−1 sin nx 2 x − 2 sin x2 4 sin2 x2 k=1 n−1 n sin 2n−1 1 − cos nx 2 x k cos kx = − 2 sin x2 4 sin2 x2 n−1
k sin kx =
AD (361.5)
AD (361.6)
k=1
1.353 1.
n−1
pk sin kx =
k=1
2.
n−1
p sin x − pn sin nx + pn+1 sin(n − 1)x 1 − 2p cos x + p2
pk sinh kx =
k=1
3.
n−1 k=0
4.
n−1 k=0
pk cos kx =
p sinh x − pn sinh nx + pn+1 sinh(n − 1)x 1 − 2p cosh x + p2
1 − p cos x − pn cos nx + pn+1 cos(n − 1)x 1 − 2p cos x + p2
pk cosh kx =
AD (361.12)a
1 − p cosh x − pn cosh nx + pn+1 cosh(n − 1)x 1 − 2p cosh x + p2
AD (361.13)a¡
JO (396)
1.36 Sums of products of trigonometric functions of multiple angles 1.361 1.
n
sin kx sin(k + 1)x =
1 [(n + 1) sin 2x − sin 2(n + 1)x] cosec x 4
JO (214)
sin kx sin(k + 2)x =
1 n cos 2x − cos(n + 3)x sin nx cosec x 2 2
JO (216)
k=1
2.
n k=1
1.381
3.
Sums leading to hyperbolic tangents and cotangents
2
39
n(x + 2y) x + 2y n+1 x sin cosec sin kx cos(2k − 1)y = sin ny + 2 2 2
k=1 n(2y − x) 2y − x n+1 x sin cosec − sin ny − 2 2 2 n
JO (217)
1.362 1.
2.
n
x 2 n x 2 = 2 sin − sin2 x 2k 2n k=1
2
2 n 1 1 x x 2 sec k = cosec x − cosec n 2k 2 2n 2 2k sin2
AD (361.15)
AD (361.14)
k=1
1.37 Sums of tangents of multiple angles 1.371 1.
n 1 x 1 x tan k = n cot n − 2 cot 2x k 2 2 2 2
AD (361.16)
n 1 x 22n+2 − 1 1 2 x tan = + 4 cot2 2x − 2n cot2 n 22k 2k 3 · 22n−1 2 2
AD (361.20)
k=0
2.
k=0
1.38 Sums leading to hyperbolic tangents and cotangents 1.381
⎞
⎛
⎟ 1
⎟ ⎠ 2k + 1 π n sin2 4n = tanh (2nx) 2 tanh x
1+ 2k + 1 π tan2 4n ⎞ ⎛
⎜ tanh ⎜ ⎝x 1.
n−1 k=0
⎜ tanh ⎜ ⎝x 2.
n−1 k=1
1
kπ 2n 2 tanh x
1+ kπ 2 tan 2n n sin2
JO (402)a
⎟
⎟ ⎠ = coth (2nx) −
1 (tanh x + coth x) 2n
JO (403)
40
Trigonometric and Hyperbolic Functions
⎞
⎛
⎟
⎟ ⎠ 2k + 1 π (2n + 1) sin2 tanh x 2(2n + 1) = tanh (2n + 1) x − 2n + 1 tanh2 x
1+ 2k + 1 π tan2 2(2n + 1) ⎞ ⎛
⎜ tanh ⎜ ⎝x 3.
n−1 k=0
⎜ tanh ⎜ ⎝x 4.
1.382
2
2
kπ 2(2n + 1) tanh2 x
1+ kπ 2 tan (2n + 1)
n k=1
(2n + 1) sin2
JO (404)
⎟
⎟ ⎠ = coth (2n + 1) x −
coth x 2n + 1
JO (405)
1.382 1.
n−1 k=0
2.
n−1 k=1
3.
n−1 k=0
4.
n k=1
⎛
1
⎞ = 2n tanh (nx) x ⎟ 1 ⎟ + tanh 2 2 ⎠
JO (406)
⎞ = 2n coth (nx) − 2 coth x x ⎟ 1 ⎟ + tanh 2 2 ⎠
JO (407)
2k + 1 π ⎜ sin 4n ⎜ ⎝ sinh x 2
⎛
⎛
kπ 2 ⎜ sin 2n ⎜ ⎝ sinh x
2
⎜ sin ⎜ ⎝
⎛
1
1
⎞ = (2n + 1) tanh 2k + 1 π x ⎟ 1 2(2n + 1) ⎟ + tanh sinh x 2 2 ⎠
kπ 2 ⎜ sin 2n + 1 ⎜ ⎝ sinh x
1
⎞ = (2n + 1) coth +
x ⎟ 1 ⎟ tanh 2 2 ⎠
(2n + 1)x 2
(2n + 1)x 2
− tanh
− coth
x 2
x 2
JO (408)
JO (409)
1.395
Representing sines and cosines as finite products
41
1.39 The representation of cosines and sines of multiples of the angle as finite products 1.391 1.
sin nx = n sin x cos x
n−2 2 k=1
2.
3.
4.
1.392 1.
2.
n 2 -
⎛
⎛
⎞ 2
sin x ⎟ ⎜ ⎝1 − kπ ⎠ sin2 n ⎞
sin2 x ⎟ ⎠ (2k − 1)π 2 k=1 sin 2n ⎛ ⎞ n−1 2 2 sin x ⎟ ⎜ sin nx = n sin x ⎝1 − kπ ⎠ k=1 sin2 n ⎛ ⎞ n−1 2 2 - ⎜ sin x ⎟ cos nx = cos x ⎝1 − ⎠ (2k − 1)π 2 k=1 sin 2n cos nx =
⎜ ⎝1 −
[n is even]
JO (568)
[n is even]
JO (569)
[n is odd]
JO (570)
[n is odd]
JO (571)a
kπ sin x + n k=0
n 2k − 1 π cos nx = 2n−1 sin x + 2n
sin nx = 2n−1
n−1 -
JO (548)
JO (549)
k=1
1.393 1.
1 2k π = n−1 cos nx cos x + n 2 k=0 n 1 = n−1 (−1) 2 − cos nx 2
n−1 -
[n odd] [n even] JO (543)
2.11
n−1 -
sin x +
k=0
n−1
(−1) 2 2k π = sin nx n 2n−1 n (−1) 2 = n−1 (1 − cos nx) 2
[n odd] [n even] JO (544)
1.394
n−1 - k=0
2kπ x − 2xy cos α + n
1.395 1.
cos nx − cos ny = 2n−1
+y
n−1 - k=0
2
2
= x2n − 2xn y n cos nα + y 2n
2kπ cos x − cos y + n
JO (573)
JO (573)
42
2.
Trigonometric and Hyperbolic Functions
cosh nx − cos ny = 2
n−1
n−1 - k=0
1.396 1.
n−1 -
x2 − 2x cos
kπ +1 n
x2 − 2x cos
2kπ 2n + 1
x2 + 2x cos
2kπ 2n + 1
k=1
2.
n -
k=1
3.
n -
k=1
4.
n−1 -
x2 − 2x cos
k=0
1.396
2kπ cosh x − cos y + n
JO (538)
x2n − 1 x2 − 1
x2n+1 − 1 +1 = x−1
x2n+1 − 1 +1 = x+1 =
(2k + 1)π +1 2n
KR 58 (28.1)
KR 58 (28.2)
KR 58 (28.3)
= x2n + 1
KR 58 (28.4)
1.41 The expansion of trigonometric and hyperbolic functions in power series 1.411 1.
sin x =
∞
(−1)k
k=0
2.
sinh x =
x2k+1 (2k + 1)!
∞ x2k+1 (2k + 1)!
k=0
3.
cos x =
∞
(−1)k
k=0
4.
x2k (2k)!
∞ x2k cosh x = (2k)! k=0 ∞ 22k 22k − 1 |B2k |x2k−1 tan x = (2k)!
6.11
π2 x2 < 4 k=1 ∞ 22k 22k − 1 2x5 17 7 x3 + − x + ··· = B2k x2k−1 tanh x = x − 3 15 315 (2k)! k=1 π2 2 x < 4
7.
1 22k |B2k | 2k−1 x cot x = − x (2k)!
5.
∞
x2 < π 2
FI II 523
FI II 523a
k=1
∞
8.
coth x =
2x5 1 22k B2k 2k−1 1 x x3 + − + − ··· = + x x 3 45 945 x (2k)! k=1 2 x < π2
FI II 522a
1.414
9.
Trigonometric and hyperbolic functions: power series expansion
sec x =
∞ |E2k | k=0
(2k)!
π2 x2 < 4
x2k
43
CE 330a
∞
10.
sech x = 1 −
E2k 5x4 61x6 x2 + − + ··· = 1 + x2k 2 24 720 (2k)! k=1
11.
∞ 1 2 22k−1 − 1 |B2k |x2k−1 cosec x = + x (2k)!
π2 x2 < 4
x2 < π 2
CE 330
CE 329a
k=1
12.
∞ 7x3 31x5 1 2 22k−1 − 1 B2k 2k−1 1 1 − + ··· = − x cosech x = − x + x 6 360 15120 x (2k)! k=1 2 x < π2
JO (418)
1.412 1.
sin2 x =
∞
(−1)k+1
k=1
2.
cos x = 1 − 2
∞
22k−1 x2k (2k)!
(−1)k+1
k=1
JO (452)a
22k−1 x2k (2k)!
JO (443)
∞
3.
4.
32k+1 − 3 2k+1 1 x (−1)k+1 4 (2k + 1)! k=1 2k ∞ + 3 x2k 1 3 k 3 cos x = (−1) 4 (2k)!
sin3 x =
JO (452a)a
JO (443a)
k=0
1.413 1.
sinh x = cosec x
∞
(−1)k+1
k=1
2.
cosh x = sec x + sec x
∞
(−1)k
k=1
3.
sinh x = sec x
∞
(−1)[k/2]
k=1
4.
cosh x = cosec x
∞ k=1
1.414 1.
22k−1 x4k−2 (4k − 1)! 22k x4k (4k)!
2k−1 x2k−1 (2k − 1)!
(−1)[(k−1)/2]
2k−1 x2k−1 (2k − 1)!
2 ∞ , + n + 02 n2 + 22 . . . n2 + (2k)2 2k+2 x cos n ln x + 1 + x2 = 1 − (−1)k (2k + 2)! k=0 2 x 0]
MO 213
⎤ x+y 2 + sinh t 1 ⎢ ⎥ 2 = ln ⎣ ⎦ 4 2 2 x−y + sinh t sin 2 sin2
MO 214
1.463 x cos ϕ
∞ xn cos nϕ cos (x sin ϕ) = n! n=0
1.
e
2.
ex cos ϕ sin (x sin ϕ) =
∞ xn sin nϕ n! n=1
x2 < 1 x2 < 1
AD (6476.1)
AD (6476.2)
1.47 Series of hyperbolic functions 1.471 1.
∞ sinh kx k=1
2.
∞ cosh kx k=0
3.
k!
∞ k=0
k!
= ecosh x sinh (sinh x) .
JO (395)
= ecosh x cosh (sinh x) .
JO (394)
1 1 (2m + 1)πx (2m + 1)π π3 tanh + x tanh = (2k + 1)3 x 2 2x 16
1.472 1.
∞
pk sinh kx =
p sinh x 1 − 2p cosh x + p2
pk cosh kx =
1 − p cosh x 1 − 2p cosh x + p2
k=1
2.
∞ k=0
p2 < 1 p2 < 1
JO (396)
JO (397)a
1.48 Lobachevskiy’s “Angle of Parallelism” Π(x) 1.480 1.
Definition. Π(x) = 2 arccot ex = 2 arctan e−x
[x ≥ 0]
LO III 297, LOI 120
52
2. 1.481
Trigonometric and Hyperbolic Functions
Π(x) = π − Π(−x)
2.
Functional relations 1 sin Π(x) = cosh x cos Π(x) = tanh x
3.
tan Π(x) =
1.
[x < 0]
1.481
LO III 183, LOI 193
LO III 297 LO III 297
4.
1 sinh x cot Π(x) = sinh x
5.
sin Π(x + y) =
sin Π(x) sin Π(y) 1 + cos Π(x) cos Π(y)
LO III 297
6.
cos Π(x + y) =
cos Π(x) + cos Π(y) 1 + cos Π(x) cos Π(y)
LO III 183
LO III 297 LO III 297
1.482
Connection with the Gudermannian. π gd(−x) = Π(x) − 2 (Definite) integral of the angle of parallelism: cf. 4.581 and 4.561.
1.49 The hyperbolic amplitude (the Gudermannian) gd x 1.490 1. 2. 1.491
Definition. x gd x =
dt = 2 arctan ex − 0 cosh t gd x gd x dt = ln tan + x= cos t 2 0
cosh x = sec(gd x)
2.
sinh x = tan(gd x)
3.
ex = sec(gd x) + tan(gd x) = tan
4.
tanh x = sin(gd x)
1 x gd x tanh = tan 2 2 1 arctan (tanh x) = gd 2x 2
6.
π 4
JA
JA
Functional relations.
1.
5.
π 2
1.492 If γ = gd x, then ix = gd iγ 1.493 Series expansion.
AD (343.1), JA
π gd x + 4 2
AD (343.2), JA
=
1 + sin(gd x) cos(gd x)
AD (343.5), JA AD (343.3), JA AD (343.4), JA AD (343.6a) JA
∞
1.
x gd x (−1)k = tanh2k+1 2 2k + 1 2 k=0
JA
1.513
Series representation ∞
2.
x 1 = tan2k+1 2 2k + 1
k=0
3. 4.
53
1 gd x 2
JA
x5 61x7 x3 + − + ··· 6 24 5040 (gd x)5 61(gd x)7 (gd x)3 + + + ... x = gd x + 6 24 5040 gd x = x −
JA
+ π, gd x < 2
JA
1.5 The Logarithm 1.51 Series representation ∞
1.511
xk 1 1 1 ln(1 + x) = x − x2 + x3 − x4 + · · · = (−1)k+1 2 3 4 k k=1
[−1 < x ≤ 1] 1.512 1.
∞
(x − 1)k 1 1 (−1)k+1 ln x = (x − 1) − (x − 1)2 + (x − 1)3 − · · · = 2 3 k k=1
%
2.
3.
x−1 1 + ln x = 2 x+1 3
ln x =
x−1 1 + x 2
4.∗
ln x = lim
→0
1.513 1.
x −1
x−1 x+1
x−1 x
3
2 +
1 + 5
1 3
∞
ln
1 1+x =2 x2k−1 1−x 2k − 1
x−1 x+1
x−1 x
3
5
&
[0 < x ≤ 2]
+ ... = 2
∞ k=1
1 2k − 1
x−1 x+1
[0 < x]
k 1 x−1 + ··· = k x k=1 x ≥ 12
2k−1
∞
x2 < 1
AD (644.6)
FI II 421
k=1 ∞
2.
ln
1 x+1 =2 x−1 (2k − 1)x2k−1
x2 > 1
AD (644.9)
k=1
∞
3.
1 x = ln x−1 kxk
[x ≤ −1 or x > 1]
JO (88a)
[−1 ≤ x < 1]
JO (88b)
[−1 ≤ x < 1]
JO (102)
k=1 ∞
4.
ln
xk 1 = 1−x k k=1
∞
5.
1 xk 1−x ln =1− x 1−x k(k + 1) k=1
54
6.
The Logarithm
∞ k 1 1 1 ln = xk 1−x 1−x n n=1
1.514
x2 < 1
JO (88e)
k=1
∞
7.
xk−1 1 (1 − x)2 1 3 = + ln − 2x3 1−x 2x2 4x k(k + 1)(k + 2)
[−1 ≤ x < 1]
AD (6445.1)
k=1
1.514
1.515 1.11
2.
3.
4.
ln x + 1 + x2 = arcsinh x k=1 2 (see 1.631, 1.641, 1.642, 1.646) x ≤ 1, x cos ϕ = 1
∞ cos kϕ k x ; ln 1 − 2x cos ϕ + x2 = −2 k
1·1 2 1·1·3 4 1·1·3·5 6 x − x + x − ... ln 1 + 1 + x2 = ln 2 + 2·2 2·4·4 2·4·6·6 ∞ (2k − 1)! 2k (−1)k = ln 2 − 2x 22k (k!) k=1 2 x ≤1 1 1 1·3 ln 1 + 1 + x2 = ln x + − + − ... x 2 · 3x3 2 · 4 · 5x5 ∞ (2k − 1)! 1 (−1)k 2k−1 = ln x + + x 2 · k!(k − 1)!(2k + 1)x2k+1 k=1 2 x ≥1 ∞ 22k−1 (k − 1)!k! 2k+1 x 1 + x2 ln x + 1 + x2 = x − (−1)k (2k + 1)! k=1 2 x ≤1 √ ∞ 2 2 ln x + 1 + x2 22k (k!) 2k+1 √ x x ≤1 = (−1)k 2 (2k + 1)! 1+x
MO 98, FI II 485
JO (91)
AD (644.4)
JO (93) JO (94)
k=0
1.516 1.
∞ k k+1 k+1 (∓1) x 1 1 2 {ln (1 ± x)} = 2 k+1 n n=1
∞ k n (−1)k+1 xk+2 1 1 1 3 {ln(1 + x)} = 6 k+2 n + 1 m=1 m n=1
x2 < 1
JO (86), JO (85)
k=1
2.
x2 < 1
AD (644.14)
k=1
∞ 2k−1 2 x2k (−1)n+1 x 0] [x < 0]
NV 49 (9)
[x > 0] [x < 0]
NV 49 (11) NV 46 (4)
58
12.
1.625 1.
The Inverse Trigonometric and Hyperbolic Functions
1 x 1 = π + arctan x
arccot x = arctan
1.625
[x > 0] [x < 0]
NV 49 (12)
arcsin x + arcsin y = arcsin x 1 − y 2 + y 1 − x2 = π − arcsin x 1 − y 2 + y 1 − x2 = −π − arcsin x 1 − y 2 + y 1 − x2
xy ≤ 0 or x2 + y 2 ≤ 1
x > 0,
y > 0 and x2 + y 2 > 1
x < 0,
y < 0 and x2 + y 2 > 1
NV 54(1), GI I (880)
2.
3.
1 − x2 1 − y 2 − xy = − arccos 1 − x2 1 − y 2 − xy
arcsin x + arcsin y = arccos
√ x 1 − y 2 + y 1 − x2 arcsin x + arcsin y = arctan √ 1 − x2 1 − y 2 − xy √ x 1 − y 2 + y 1 − x2 +π = arctan √ 1 − x2 1 − y 2 − xy √ x 1 − y 2 + y 1 − x2 −π = arctan √ 1 − x2 1 − y 2 − xy
[x ≥ 0,
y ≥ 0]
[x < 0,
y < 0]
xy ≤ 0 or x2 + y 2 < 1
NV 55
x > 0,
y > 0 and x2 + y 2 > 1
x < 0,
y < 0 and x2 + y 2 > 1
NV 56
4.
arcsin x − arcsin y = arcsin x 1 − y 2 − y 1 − x2 = π − arcsin x 1 − y 2 − y 1 − x2 = −π − arcsin x 1 − y 2 − y 1 − x2
xy ≥ 0 or x2 + y 2 ≤ 1
x > 0,
y < 0 and x2 + y 2 > 1
x < 0,
y > 0 and x2 + y 2 > 1
NV 55(2)
5.
6.
7.11
arcsin x − arcsin y = arccos x 1 − x2 1 − y 2 + xy = − arccos 1 − x2 1 − y 2 + xy arccos x + arccos y = arccos xy − 1 − x2 1 − y 2 = 2π − arccos xy − 1 − x2 1 − y 2 arccos x − arccos y = − arccos xy + 1 − x2 1 − y 2 = arccos xy + 1 − x2 1 − y 2
[xy > y] [x < y]
NV 56
[x + y ≥ 0] [x + y < 0]
NV 57 (3)
[x ≥ y] [x < y]
NV 57 (4)
1.627
8.
Functional relations
x+y 1 − xy x+y = π + arctan 1 − xy x+y = −π + arctan 1 − xy
arctan x + arctan y = arctan
59
[xy < 1] [x > 0,
xy > 1]
[x < 0,
xy > 1] NV 59(5), GI I (879)
9.
x−y 1 + xy x−y = π + arctan 1 + xy x−y = −π + arctan 1 + xy
arctan x − arctan y = arctan
[xy > −1] [x > 0,
xy < −1]
[x < 0,
xy < −1] NV 59(6)
1.626 1.
2 arcsin x = arcsin 2x 1 − x2 = π − arcsin 2x 1 − x2 = −π − arcsin 2x 1 − x2
1 |x| ≤ √ 2 1 √ <x≤1 2 1 −1 ≤ x < − √ 2 NV 61 (7)
2.
3.
[0 ≤ x ≤ 1]
2 arccos x = arccos 2x2 − 1 = 2π − arccos 2x2 − 1
[−1 ≤ x < 0]
2x 1 − x2 2x +π = arctan 1 − x2 2x −π = arctan 1 − x2
2 arctan x = arctan
NV 61 (8)
[|x| < 1] [x > 1] [x < −1] NV 61 (9)
1.627 1.
2.
arctan x + arctan
arctan x + arctan
1 π = 2 x π =− 2
[x > 0] [x < 0]
1−x π = 4 1+x 3 =− π 4
GI I (878)
[x > −1] [x < −1]
NV 62, GI I (881)
60
The Inverse Trigonometric and Hyperbolic Functions
1.628
1.628 1.
arcsin
2x = −π − 2 arctan x 1 + x2 = 2 arctan x
[x ≤ −1] [−1 ≤ x ≤ 1]
= π − 2 arctan x
[x ≥ 1] NV 65
1−x = 2 arctan x 1 + x2 = −2 arctan x 2
2.
arccos
[x ≥ 0] [x ≤ 0]
NV 66
1 2x − 1 2x − 1 − arctan tan π = E (x) 1.629 2 π 2 1.631 Relations between the inverse hyperbolic functions. x 1. arcsinh x = arccosh x2 + 1 = arctanh √ x2 + 1 √ x2 − 1 2 2. arccosh x = arcsinh x − 1 = arctanh x x 1 1 3. arctanh x = arcsinh √ = arccosh √ = arccoth 2 2 x 1−x 1−x 2 4. arcsinh x ± arcsinh y = arcsinh x 1 + y ± y 1 + x2 5. arccosh x ± arccosh y = arccosh xy ± (x2 − 1) (y 2 − 1) 6.
arctanh x ± arctanh y = arctanh
GI (886)
JA
JA JA JA JA
x±y 1 ± xy
JA
1.64 Series representations 1.641 1.
2.
1 3 1·3 5 1·3·5 7 π − arccos x = x + x + x + x + ... 2 2·3 2 · 4 ·5 2 · 4 · 6 · 7 ∞ 1 1 3 2 (2k)! x2k+1 = x F , ; ;x = 2k (k!)2 (2k + 1) 2 2 2 2 k=0 2 x ≤1
arcsin x =
1 3 1·3 5 x + x − ...; arcsinh x = x − 2·3 2·4·5 ∞ (2k)! x2k+1 (−1)k = 2 2k 2 (k!) (2k + 1) k=0 = x F 12 , 12 ; 32 ; −x2
x2 ≤ 1
FI II 479
FI II 480
1.645
Series representations
61
1.642 1.
1 1 1·3 1 − + ... 2 2x2 2 · 4 4x4 ∞ (2k)!x−2k (−1)k+1 = ln 2x + 2 22k (k!) 2k k=1
arcsinh x = ln 2x +
[x ≥ 1] AD (6480.2)a
2.
arccosh x = ln 2x −
∞ (2k)!x−2k
[x ≥ 1]
2
k=1
22k (k!) 2k
AD (6480.3)a
1.643 1.
x3 x5 x7 + − + ... 3 5 7 ∞ (−1)k x2k+1 = 2k + 1
arctan x = x −
k=0
2. 1.644 1.
arctanh x = x +
x5 x3 + + ··· = 3 5
∞ k=0
x2k+1 2k + 1
k ∞ x2 x (2k)! arctan x = √ 1 + x2 k=0 22k (k!)2 (2k + 1) 1 + x2
1 1 3 x2 x , ; ; =√ F 2 2 2 1 + x2 1 + x2
x2 ≤ 1 x2 < 1
FI II 479
AD (6480.4)
2 x 1
AD (641.5)
2. 3.
∞ 2 2 22k (k!) x2k+2 x ≤1 AD (642.2), GI III (152)a (2k + 1)!(k + 1) k=0
3! 3! 1 1 1 3 (arcsin x) = x3 + 32 1 + 2 x5 + 32 · 52 1 + 2 + 2 x7 + . . . 5! 3 7! 3 5 x2 ≤ 1 2
(arcsin x) =
BR* 188, AD (642.2), GI III (153)a
62
The Inverse Trigonometric and Hyperbolic Functions
1.646
1.646 ∞
1.
arcsinh
1 (−1)k (2k)! x−2k−1 = arcosech x = 2 x 22k (k!) (2k + 1) k=0
∞
(2k)!
2.
arccosh
2 1 = arcsech x = ln − x x
3.
arcsinh
2 (−1)k+1 (2k)! 2k 1 x = arcosech x = ln + 2 x x 22k (k!) 2k
2
22k (k!) 2k k=1
x2k
x2 ≥ 1
AD (6480.5)
[0 < x ≤ 1]
AD (6480.6)
[0 < x ≤ 1]
AD (6480.7)a
∞
k=1
4. 1.647 1.
2.
arctanh
1 = arccoth x = x
∞ k=0
−(2k+1)
x 2k + 1
x2 > 1
AD (6480.8)
⎛
∗ n ∗ (−1)j−1 22j − 1 24n−2j+4 − 1 B2j−1 B4n−2j+3 ⎝2 = (2k − 1)4n+3 2 (2j)!(4n − 2j + 4)! j=1 k=1 2n+2 2 ∗ 2 ⎞ n (−1) 2 − 1 B2n+1 ⎠ + 2 [(2n + 2)!] n = 0, 1, 2, . . . , ⎛ ⎞ ∞ n−1 j ∗ ∗ k−1 4n+1 ∗ n ∗ 2 (−1) B B 2B4n (−1) B2n ⎠ (−1) sech(2k − 1) (π/2) π 2j 4n−2j + + , = 4n+3 ⎝2 (2k − 1)4n+1 2 (2j)!(4n − 2j)! (4n)! [(2n)]!2 j=1 ∞ tanh(2k − 1) (π/2)
π
4n+3
k=1
n = 1, 2, . . . (The summation term on the right is to be omitted for n = 1.) (See page xxxiii for the definition of Br∗ .)
2 Indefinite Integrals of Elementary Functions 2.0 Introduction 2.00 General remarks We omit the constant of integration in all the formulas of this chapter. Therefore, the equality sign (=) means that the functions on the left and right of this symbol differ by a constant. For example (see 201 15), we write dx = arctan x = − arctan x 1 + x2 although π arctan x = − arctan x + . 2 we !integrate certain functions, we obtain the logarithm of the absolute value (for example, ! √ " When √ dx = ln !x + 1 + x2 !). In such formulas, the absolute-value bars in the argument of the logarithm 1+x2 are omitted for simplicity in writing. In certain cases, it is important to give the complete form of the primitive function. Such primitive functions, written in the form of definite integrals, are given in Chapter 2 and in other chapters. Closely related to these formulas are formulas in which the limits of integration and the integrand depend on the same parameter. A number of formulas lose their meaning for certain values of the constants (parameters) or for certain relationships between these constants (for example, formula 2.02 8 for n = −1 or formula 2.02 15 for a = b). These values of the constants and the relationships between them are for the most part completely clear from the very structure of the right-hand member of the formula (the one not containing an integral sign). Therefore, throughout the chapter, we omit remarks to this effect. However, if the value of the integral is given by means of some other formula for those values of the parameters for which the formula in question loses meaning, we accompany this second formula with the appropriate explanation. The letters x, y, t, . . . denote independent variables; f , g, ϕ, . . . denote functions of x, y, t, . . . ; f , g , ϕ , . . . , f , g , ϕ , . . . denote their first, second, etc., derivatives; a, b, m, p, . . . denote constants, by which we generally mean arbitrary real numbers. If a particular formula is valid only for certain values of the constants (for example, only for positive numbers or only for integers), an appropriate remark is made, provided the restriction that we make does not follow from the form of the formula itself. Thus, in formulas 2.148 4 and 2.424 6, we make no remark since it is clear from the form of these formulas themselves that n must be a natural number (that is, a positive integer).
63
64
Introduction
2.01 The basic integrals
1. 2. 3. 4. 5. 6.
xn+1 (n = −1) x dx = n+1 dx = ln x x ex dx = ex ax ax dx = ln a sin x dx = − cos x cos x dx = sin x dx = − cot x sin2 x dx = tan x cos2 x n
11
7. 8.11
16. 17. 18. 19. 20. 21. 22.11 23. 24. 25. 26.
dx 1 1+x = arctanh x = ln 2 1−x 2 1−x dx √ = arcsin x = − arccos x 1 − x2 dx √ = arcsinh x = ln x + x2 + 1 x2 + 1 dx √ = arccosh x = ln x + x2 − 1 x2 − 1 sinh x dx = cosh x cosh x dx = sinh x dx = − coth x sinh2 x dx = tanh x cosh2 x tanh x dx = ln cosh x coth x dx = ln sinh x x dx = ln tanh sinh x 2
9. 10.
sin x dx = sec x cos2 x cos x dx = − cosec x sin2 x tan x dx = − ln cos x
11. 12.
cot x dx = ln sin x
13. 14. 15.
x dx = ln tan sin x 2 π x dx = ln tan + = ln (sec x + tan x) cos x 4 2 dx π = arctan x = − arccot x 2 1+x 2
General formulas
65
2.02 General formulas
1.
af dx = a f dx
[af ± bϕ ± cψ ± . . .] dx = a f dx ± b ϕ dx ± c ψ dx ± . . . d f dx = f dx f dx = f f ϕ dx = f ϕ − f ϕ dx [integration by parts] (n+1) (n) (n−1) (n−2) n (n) n+1 f ϕ dx = ϕf − ϕ f +ϕ f − . . . + (−1) ϕ f + (−1) ϕ(n+1) f dx f (x) dx = f [ϕ(y)]ϕ (y) dy [x = ϕ(y)] [change of variable]
2. 3. 4. 5. 6. 7.
11
8.
9. 10. 11. 12. 13. 14. 15.
16. 17.
(f )n f dx =
(f )n+1 n+1
For n = −1 f dx = ln f f (af + b)n+1 (af + b)n f dx = a(n + 1) √ f dx 2 af + b √ = a af + b f f ϕ−ϕf dx = ϕ2 ϕ f f ϕ − ϕ f dx = ln fϕ ϕ dx dx dx =± ∓ f (f ± ϕ) fϕ ϕ (f ± ϕ) f dx = ln f + f 2 + a f2 + a a b f dx dx dx = − (f + a)(f + b) a − b (f + a) a − b (f + b) For a = b dx dx f dx −a = (f + a)2 f +a (f + a)2 f dx dx ϕ dx = − (f + ϕ)n (f + ϕ)n−1 (f + ϕ)n qf f dx 1 arctan = 2 2 2 p +q f pq p
[n = −1]
66
Rational Functions
2.101
qf − p f dx 1 ln = q 2 f 2 − p2 2pq qf + p f dx dx = −x + 1−f 1−f 1 f dx 1 f dx f 2 dx + = f 2 − a2 2 f −a 2 f +a f f dx = arcsin a a2 − f 2 1 f f dx = ln 2 af + bf b af + b f 1 f dx = arcsec 2 2 a a f f −a (f ϕ − f ϕ ) dx f = arctan 2 2 f +ϕ ϕ 1 f −ϕ (f ϕ − f ϕ ) dx = ln f 2 − ϕ2 2 f +ϕ
18. 19. 20. 21. 22. 23. 24. 25.
2.1 Rational Functions 2.10 General integration rules F (x) , where F (x) and f (x) are polynomials with f (x) no common factors, we first need to separate out the integral part E(x) [where E(x) is a polynomial], if there is an integral part, and then to integrate separately the integral part and the remainder; thus: F (x) dx ϕ(x) = E(x) dx + dx. f (x) f (x) Integration of the remainder, which is then a proper rational function (that is, one in which the degree of the numerator is less than the degree of the denominator) is based on the decomposition of the fraction into elementary fractions, the so-called partial fractions. 2.102 If a, b, c, . . . , m are roots of the equation f (x) = 0 and if α, β, γ, . . . , μ are their corresponding ϕ(x) can be decomposed into the following multiplicities, so that f (x) = (x−a)α (x−b)β . . . (x−m)μ , then f (x) partial fractions: ϕ(x) Aα Bβ Aα−1 A1 Bβ−1 B1 = + + ... + + ... + + + ... + α α−1 β β−1 f (x) (x − a) (x − a) x − a (x − b) (x − b) x−b Mμ−1 M1 Mμ , + + ... + + μ μ−1 (x − m) (x − m) x−m 2.101
To integrate an arbitrary rational function
where the numerators of the individual fractions are determined by the following formulas: (k−1)
(a) ψ1 , (k − 1)! ϕ(x)(x − a)α ψ1 (x) = , f (x)
Aα−k+1 =
(k−1)
(b) ψ2 , (k − 1)! ϕ(x)(x − b)β ψ2 (x) = , f (x)
Bβ−k+1 =
(k−1)
..., ...,
ψm (m) , (k − 1)! ϕ(x)(x − m)μ ψm (x) = f (x)
Mμ−k+1 =
2.104
General integration rules
67
TI 51a
If a, b, . . . , m are simple roots, that is, if α = β = . . . = μ=1, then ϕ(x) A B M = + + ··· + , f (x) x−a x−b x−m where ϕ(a) ϕ(b) ϕ(m) A= , B= , ... , M= . f (a) f (b) f (m) If some of the roots of the equation f (x) = 0 are imaginary, we group together the fractions that represent conjugate roots of the equation. Then, after certain manipulations, we represent the corresponding pairs of fractions in the form of real fractions of the form M1 x + N 1 M2 x + N 2 Mp x + N p + p. 2 + ... + x2 + 2Bx + C (x2 + 2Bx + C) (x2 + 2Bx + C) g dx ϕ(x) reduces to integrals of the form 2.103 Thus, the integration of a proper rational fraction f (x) (x − a)α Mx + N or p dx. Fractions of the first form yield rational functions for α > 1 and logarithms (A + 2Bx + Cx2 ) for α = 1. Fractions of the second form yield rational functions and logarithms or arctangents: d(x − a) g g dx =g =− 1. (x − a)α (x − a)α (α − 1)(x − a)α−1 d(x − a) g dx =g = g ln |x − a| 2. x−a x−a Mx + N N B − M A + (N C − M B)x 3. p dx = p−1 (A + 2Bx + Cx2 ) 2(p − 1) (AC − B 2 ) (A + 2Bx + Cx2 ) (2p − 3)(N C − M B) dx + p−1 2(p − 1) (AC − B 2 ) (A + 2Bx + Cx2 ) dx 1 Cx + B 4. =√ arctan √ for AC > B 2 2 2 A + 2Bx + Cx2 AC − B Ac − B ! ! ! Cx + B − √B 2 − AC ! 1 ! ! √ = √ ln ! ! for AC < B 2 2 2 ! ! 2 B − AC Cx + B + B − AC (M x + N ) dx 5. A + 2Bx + Cx2 ! NC − MB M !! Cx + B ln A + 2Bx + Cx2 ! + √ = arctan √ for AC > B 2 2 2 2C C AC − B AC − B ! ! ! Cx + B − √B 2 − AC ! ! N C − M B M !! ! ! √ √ ln A + 2Bx + Cx2 ! + = ln ! ! for AC < B 2 2 2 ! ! 2C 2C B − AC Cx + B + B − AC
The Ostrogradskiy–Hermite method
ϕ(x) dx f (x) without finding the roots of the equation f (x) = 0 and without decomposing the integrand into partial fractions: 2.104
By means of the Ostrogradskiy-Hermite method, we can find the rational part of
68
Rational Functions
2.110
M N dx ϕ(x) dx = + FI II 49 f (x) D Q Here, M , N , D, and Q are rational functions of x. Specifically, D is the greatest common divisor of f (x) the function f (x) and its derivative f (x); Q = ; M is a polynomial of degree no higher than m − 1, D where m is the degree of the polynomial D; N is a polynomial of degree no higher than n − 1, where n is the degree of the polynomial Q. The coefficients of the polynomials M and N are determined by equating the coefficients of like powers of x in the following identity:
ϕ(x) = M Q − M (T − Q ) + N D f (x) where T = and M and Q are the derivatives of the polynomials M and Q. D
2.11–2.13 Forms containing the binomial a + bxk 2.110 1.
Reduction formulas for zk = a + bxk and an explicit expression for the general case. amk xn+1 zkm + xn zkm dx = xn zkm−1 dx km + n + 1 km + n + 1 p (ak)s (m + 1)m(m − 1) . . . (m − s + 1)zkm−s xn+1 = m + 1 s=0 [mk + n + 1][(m − 1)k + n + 1] . . . [(m − s)k + n + 1] (ak)p+1 m(m − 1) . . . (m − p + 1)(m − p) xn zkm−p−1 dx + [mk + n + 1][(m − 1)k + n + 1] . . . [(m − p)k + n + 1] LA 126(4)
3. 4. 5. 6. 7.∗
8.∗
km + k + n + 1 xn zkm+1 dx ak(m + 1) ak(m + 1) bkm xn+1 zkm n m − x zk dx = xn+k zkm−1 dx n+1 n+1 xn+1−k zkm+1 n+1−k − xn zkm dx = xn−k zkm+1 dx bk(m + 1) bk(m + 1) xn+1−k zkm+1 a(n + 1 − k) − xn−k zkm dx xn zkm dx = b(km + n + 1) b(km + n + 1) xn+1 zkm+1 b(km + k + n + 1) n m − x zk dx = xn+k zkm dx a(n + 1) a(n + 1) b c k−i k i b k n +n nk (−1) k! Γ a+1 n b a+1 xa+1+ib x nx + c dx = b i=0 (k − i)! Γ b + i + 1 xn zkm dx =
2.
−xn+1 zkm+1
+
xn zkm dx =
a m−i
LA 126 (6)
LA 125 (2)
LA 126 (3)
LA 126 (5)
[a, b, k ≥ 0 are all integers]
m xk(J+i+1) bm (−1)i m!J! xk + b k i=0 (m − i)!(J + i + 1)!
J=
n+1 −1 k
[a, b, k, m, n real,
k = 0,
m ≥ 0 an integer]
Forms containing the binomial a + bxk
2.114
Forms containing the binomial z1 = a + bx 2.111 1.
2.
3.8
4. 5. 6. 2.113
z1m+1 b(m + 1) For m = −1 1 dx = ln z1 z1 b n n−1 dx na xn x dx x − = m−1 z1m z1m z1 (n + 1 − m)b (n + 1 − m)b For n = m − 1, we may use the formula m−1 dx 1 xm−2 dx xm−1 x + = − m−1 z1m z1 (m − 1)b b z1m−1 For m = 1 n x dx axn−1 an−1 x (−1)n an xn a2 xn−2 − = + − . . . + (−1)n−1 + ln z1 2 3 z1 nb (n − 1)b (n − 2)b 1 · bn bn+1 n n−1 n−1 kak−1 xn−k an x dx n−1 n+1 na = (−1)k−1 + (−1) + (−1) ln z1 2 z1 (n − k)bk+1 bn+1 z1 bn+1 k=1 a x dx x = − 2 ln z1 z1 b b 2 ax a2 x dx x2 − 2 + 3 ln z1 = z1 2b b b z1m dx =
1. 2. 3. 2.114 1. 2. 3. 4.6
dx 1 =− 2 z1 bz1 x dx x 1 a 1 =− + 2 ln z1 = 2 + 2 ln z1 z12 bz1 b b z1 b x2 dx x a2 2a = − − 3 ln z1 2 2 3 z1 b b z1 b
dx 1 =− z13 2bz12 +x a , 1 x dx + = − z13 b 2b2 z12 2 2ax 3a2 1 x dx 1 = + + 3 ln z1 z13 b2 2b3 z12 b 3 3 x a 2 a2 5 a3 1 a x dx + 2 2x − 2 3x − = − 3 4 ln z1 3 2 4 z1 b b b 2 b z1 b
69
70
2.115 1. 2. 3. 4. 2.116 1. 2. 3. 4. 2.117
Rational Functions
dx 1 =− z14 3bz13 +x a , 1 x dx + 2 3 =− 4 z1 2b 6b z1 2 2 x ax a2 1 x dx + = − + z14 b b2 3b3 z13 3 3ax2 x dx 9a2 x 11a3 1 1 = + + + 4 ln z1 z14 b2 2b2 6b4 z13 b
dx 1 =− 5 z1 4bz14 +x a , 1 x dx + = − z15 3b 12b2 z14 2 2 x 1 ax x dx a2 + 2+ =− z15 2b 3b 12b3 z14 3 3 x 3ax2 x dx a2 x a3 1 + = − + + z15 b 2b2 b3 4b4 z14
1. 2. 3. 4.
dx −1 b(2 − n − m) = + xn z1m a(n − 1) (n − 1)axn−1 z1m−1 dx 1 =− m z1 (m − 1)bz1m−1 1 dx 1 + = m−1 xz1m z1 a(m − 1) a
dx xz1m−1
n−1 (−1)k bk−1 dx (−1)n bn−1 z1 = + ln xn z1 (n − k)ak xn−k an x k=1
2.118 1. 2. 3. 2.119 1.
dx 1 z1 = − ln , xz1 a x b dx 1 z1 + 2 ln =− 2 x z1 ax a x b2 1 b z1 dx =− + 2 − 3 ln 3 2 x z1 2ax a x a x
dx 1 1 z1 = − 2 ln 2 xz1 az1 a x
dx xn−1 z1m
2.115
Forms containing the binomial a + bxk
2.124
1 2b 1 dx 2b z1 + =− + 3 ln x2 z12 ax a2 z1 a x 3b2 1 dx 1 3b 3b2 z1 = − + 2 + 3 − 4 ln 2 3 2 x z1 2ax 2a x a z1 a x
2. 3. 2.121 1. 2. 3. 2.122 1. 2. 3. 2.123 1.11 2. 3. 2.124 1.
2.
3 dx bx 1 1 z1 + = − 3 ln 3 2 2 xz1 2a a z1 a x 2 1 9b dx 3b x 1 3b z1 + =− + 3 + 4 ln x2 z13 ax 2a2 a z12 a x 9b2 dx 1 2b 6b3 x 1 6b2 z1 = − + 2 + 3 + 4 − 5 ln 3 2 3 2 x z1 2ax a x a a z1 a x
11 5bx b2 x2 1 dx 1 z1 + = + 3 − 4 ln xz14 6a 2a2 a z13 a x 1 22b 10b2 x 4b3 x2 1 dx 4b z1 + = − + + + 5 ln x2 z14 ax 3a2 a3 a4 z13 a x 2 3 4 2 1 55b dx 1 5b 25b x 10b x 10b2 z1 + = − + + + − ln x3 z14 2ax2 2a2 x 3a3 a4 a5 z13 a6 x
25 13bx 7b2 x2 b3 x3 1 1 z1 dx + = + + − 5 ln xz15 12a 3a2 2a3 a4 z14 a x 2 3 2 4 3 1 125b 65b x 35b x dx 1 5b x 5b z1 − = − − − − + 6 ln 5 4 2 2 3 4 5 x z1 ax 12a 3a 2a a z1 a x 2 3 4 2 5 3 1 125b dx 1 3b 65b x 105b x 15b x 15b2 z1 = − + 2 + + + + − 7 ln 5 4 3 2 3 4 5 6 x z1 2ax a x 4a a 2a a z1 a x
Forms containing the binomial z2 = a + bx2 . b dx 1 if [ab > 0] (see also 2.141 2) = √ arctan x z2 a ab √ a + xi ab 1 √ if [ab < 0] (see also 2.143 2 and 2.1433) = √ ln 2i ab a − xi ab x dx 1 =− (see also 2.145 2, 2.145 6, and 2.18) m z2 2b(m − 1)z2m−1
71
72
Rational Functions
2.125
Forms containing the binomial z3 = a + bx3 a Notation: α = 3 b 2.125 n n−3 dx (n − 2)a xn−2 x dx x − = 1. m−1 m m z3 z3 z3 (n + 1 − 3m)b b(n + 1 − 3m) n n n+1 x dx x n + 4 − 3m x dx 2. = − m m−1 z3 3a(m − 1) 3a(m − 1)z3 z3m−1 2.126
1.
2.
3. 4. 5. 2.127
2. 3. 4. 2.128 1. 2.
√ √ (x + α)2 dx α 1 x 3 ln = + 3 arctan z3 3a 2 x2 − αx + α2 2α − x
2 √ (x + α) 2x − α α 1 √ ln 2 + 3 arctan = 3a 2 x − αx + α2 α 3 x dx 1 =− z3 3bα
2
(x + α) 1 ln − 2 x2 − αx + α2
√
(see also 2.141 3 and 2.143) 2x − α √ 3 arctan α 3 (see also 2.145 3. and 2.145 7)
2
−3
x dx 1 ln 1 + x3 α = z3 3b 3 x dx x a dx = − z3 b b z3 4 2 x dx a x dx x − = z3 2b b z3
1.
LA 133 (1)
dx x 2 = + z32 3az3 3a
x2 1 x dx = + 2 z3 3az3 3a
x3 dx x 1 =− + 2 z3 3bz3 3b
(see 2.126 2)
(see 2.126 1)
x dx z3
1 ln z3 3b (see 2.126 1)
dx z3
x2 dx 1 =− z32 3bz3
=
dx z3
(see 2.126 2)
(see 2.126 1)
dx 1 b(3m + n − 4) dx = − − m m−1 n n−3 n−1 x z3 a(n − 1) x z3m (n − 1)ax z3 dx 1 n + 3m − 4 dx = + xn z3m 3a(m − 1) xn z3m−1 3a(m − 1)xn−1 z3m−1
LA 133 (2)
Forms containing the binomial a + bxk
2.133
2.129 1. 2. 3. 2.131 1. 2. 3.
x3 1 dx ln = xz3 3a z3 b x dx 1 dx − = − x2 z3 ax a z3 dx 1 b dx =− − x3 z3 2ax2 a z3
(see 2.126 2) (see 2.126 1)
1 1 x3 dx = + ln xz32 3az3 3a2 z3 2 1 1 4bx dx 4b x dx + = − − x2 z32 ax 3a2 z3 3a2 z3 1 5bx 1 5b dx dx =− + 2 − 2 2 3 2 x z3 2ax 6a z3 3a z3
(see 2.126 2) (see 2.126 1)
Forms containing the binomial z4 = a + bx4 a a 4 α = 4 − Notation: α = b b 2.132 √ √ 2 2 + αx 2 + α 2 x α αx dx √ 1.8 ln = √ + 2 arctan 2 for ab > 0 z4 α − x2 4a 2 x2 − αx 2 + α2
x α x + α + 2 arctan for ab < 0 = ln 4a x − α α b 1 x dx 2 for ab > 0 (see = √ arctan x 2. z4 a 2 ab √ a + x2 i ab 1 √ for ab < 0 (see = √ ln 4i ab a − x2 i ab √ √ 2 x dx x2 − αx 2 + α2 1 αx 2 √ √ 3. ln = + 2 arctan 2 for ab > 0 z4 α − x2 4bα 2 x2 + αx 2 + α2
x x + α 1 − 2 arctan ln for ab < 0 =− 4bα x − α α 3 1 x dx ln z4 = 4. z4 4b 2.133 1. 2.
73
xn+1 4m − n − 5 xn dx = + m m−1 z4 4a(m − 1) 4a(m − 1)z4
(see also 2.141 4) (see also 2.143 5) also 2.145 4) also 2.145 8)
xn dx z4m−1 n n−4 dx (n − 3)a x dx xn−3 x − = m−1 z4m z4m z4 (n + 1 − 4m)b b(n + 1 − 4m)
LA 134 (1)
74
2.134
Rational Functions
1. 2. 3. 4.
x 3 dx = + z42 4az4 4a
x dx x2 1 = + z42 4az4 2a
x2 dx x3 1 = + 2 z4 4az4 4a
dx z4
2.134
(see 2.132 1)
x dx z4
(see 2.132 2)
x2 dx z4
(see 2.132 3)
x3 dx x4 1 = =− z42 4az4 4bz4
dx 1 b(4m + n − 5) 2.135 =− − m m−1 n n−1 x z4 (n − 1)a (n − 1)ax z4 For n = 1 dx dx dx 1 b = − m m−1 −3 xz4 a xz4 a x z4m 2.136 dx 1 x4 ln x ln z4 1. − = ln = xz4 a 4a 4a z4 2 b x dx dx 1 − 2. =− 2 x z4 ax a z4
dx xn−4 z4m
(see 2.132 3)
2.14 Forms containing the binomial 1 ± xn 2.141 1. 2.11
4.
dx = arctan x = − arctan 1 + x2
1 x
(see also 2.124 1)
√ 1+x x 3 1 1 dx (see also 2.126 1) = ln √ + √ arctan 1 + x3 3 2−x 3 1 − x + x2 √ √ dx 1 + x 2 + x2 x 2 1 1 √ √ √ ln arctan = + 1 + x4 1 − x2 4 2 1 − x 2 + x2 2 2
3.
dx = ln(1 + x) 1+x
(see also 2.132 1) 2.142
n
−1
2 dx 2 = − Pk cos 1 + xn n
k=0
n
2 −1 2k + 1 2k + 1 2 π + π Qk sin n n n
k=0
for n a positive even number n−3 2
=
2 1 ln(1 + x) − Pk cos n n k=0
n−3 2
2k + 1 2 π + Qk sin n n k=0
TI (43)a
2k + 1 π n for n a positive odd number TI (45)
Forms containing the binomial 1 ± xn
2.145
where
2.143 1. 2. 3. 4. 5.
75
2k + 1 1 2 Pk = ln x − 2x cos π +1 2 n x − cos 2k+1 x sin 2k+1 π π n2k+1 = arctan 2k+1n Qk = arctan 1 − x cos n π sin n π
dx = − ln(1 − x) 1−x dx 1 1+x = arctanh x [−1 < x < 1] (see also 2.141 1) = ln 2 1−x 2 1−x 1 x−1 dx = ln = − arccoth x [x > 1, x < −1] 2 x −1 2 x+1 √ √ dx 1 1 + x + x2 x 3 1 + √ arctan (see also 2.126 1) = ln 1 − x3 3 1−x 2+x 3 dx 1 1 1+x 1 + arctan x = (arctanh x + arctan x) = ln 4 1−x 4 1−x 2 2 (see also 2.132 1)
2.144
1.
n
−1
n
−1
2 2 2 2 dx 1 1+x 2k 2k ln − π + = P cos Qk sin π k 1 − xn n 1−x n n n n
k=1
k=1
for n a positive even number
1 2k + 1 π+1 , where Pk = ln x2 + 2x cos 2 n 2.
n−3
n−3
k=0
k=0
1. 2. 3.
x + cos 2k+1 n π sin 2k+1 n π
2 2 2 2 dx 1 2k + 1 2k + 1 ln(1 − x) + π + π = − P cos Qk sin k n 1−x n n n n n
1 2k 2 where Pk = ln x − 2x cos π + 1 , 2 n 2.145
Qk = arctan
TI (47)
for n a positive odd number Qk = arctan
x − cos
2k n π
sin 2k n π
x dx = x − ln(1 + x) 1+x x dx 1 = ln 1 + x2 2 1+x 2 (1 + x)2 2x − 1 1 1 x dx ln = − + √ arctan √ 1 + x3 6 1 − x + x2 3 3
(see also 2.126 2)
TI (49)
76
Rational Functions
2.146
4. 5. 6. 7. 8. 2.146 1.
x dx 1 = arctan x2 1 + x4 2 x dx = − ln(1 − x) − x 1−x x dx 1 = − ln 1 − x2 2 1−x 2 (1 − x)2 x dx 2x + 1 1 1 ln = − − √ arctan √ 1 − x3 6 1 + x + x2 3 3 x dx 1 1 + x2 = ln 1 − x4 4 1 − x2
(see also 2.126 2) (see also 2.132 2)
For m and n natural numbers.
m−1 n x 2k − 1 dx 1 mπ(2k − 1) 2 ln π + x 1 − 2x cos = − cos 1 + x2n 2n 2n 2n k=1 n 2k−1 x − cos 2n π mπ(2k − 1) 1 arctan sin + n 2n sin 2k−1 2n π k=1
[m < 2n] 2.
n
m−1
1 dx ln(1 + x) x − = (−1)m+1 2n+1 1+x 2n + 1 2n + 1 k=1 n 2k−1 x − cos 2n+1 π mπ(2k − 1) 2 arctan sin + 2k−1 2n + 1 2n + 1 sin 2n+1 π k=1
3.11
TI (44)a
2k − 1 mπ(2k − 1) 2 ln 1 − 2x cos π+x cos 2n + 1 2n + 1
[m ≤ 2n]
TI (46)a
n−1 kπ 1 1 kmπ xm−1 dx m+1 2 ln + x (−1) 1 − 2x cos = ln(1 + x) − ln(1 − x) − cos 1 − x2n 2n 2n n n k=1 n−1 kπ x − cos n 1 kmπ + arctan sin n n sin kπ n k=1
4.
[m < 2n]
TI (48)
m−1
dx x 1 ln(1 − x) =− 1 − x2n+1 2n + 1
n 2k − 1 1 mπ(2k − 1) m+1 2 +(−1) ln 1 + 2x cos π+x cos 2n + 1 2n + 1 2n + 1 k=1 n 2k−1 x + cos 2n+1 π 2 mπ(2k − 1) arctan sin +(−1)m+1 2k−1 2n + 1 2n + 1 sin 2n+1 π k=1 [m ≤ 2n]
2.147 1 xm dx 1 xm dx xm dx = + 1. 1 − x2n 2 1 − xn 2 1 + xn m−2 xm−1 x dx xm dx 1 m−1 · 2. + n =− 2n − m − 1 (1 + x2 )n−1 2n − m − 1 (1 + x2 )n (1 + x2 )
TI (50)
LA 139 (28)
Forms containing the binomial 1 ± xn
2.149
3. 4.
5. 2.148 1.
2. 3.
4.
xm−1 xm − dx = 1 + x2 m−1
77
xm−2 dx 1 + x2
m−2 xm−1 x dx 1 m−1 xm dx = − n n−1 2 2 2 2n − m − 1 (1 − x ) 2n − m − 1 (1 − x )n (1 − x ) m−1 x 1 m−1 xm−2 dx = − 2n − 2 (1 − x2 )n−1 2n − 2 (1 − x2 )n−1 m
m−1
x dx x + =− 2 1−x m−1
LA 139 (33) m−2
x dx 1 − x2
dx 1 1 2n + m − 3 − n =− m − 1 xm−1 (1 + x2 )n−1 m−1 xm (1 + x2 )
dx n xm−2 (1 + x2 )
LA 139 (29)
For m = 1 1 dx 1 dx LA 139 (31) n = n−1 + n−1 2 2 2n − 2 x (1 + x ) (1 + x ) x (1 + x2 ) For m = 1 and n = 1 x dx = ln √ 2 x (1 + x ) 1 + x2 1 dx dx = − − m 2 m−1 m−2 x (1 + x ) (m − 1)x x (1 + x2 ) dx x dx 1 2n − 3 = + FI II 40 n 2n − 2 (1 + x2 )n−1 2n − 2 (1 + x2 )n−1 (1 + x2 ) n−1 dx x (2n − 1)(2n − 3)(2n − 5) · · · (2n − 2k + 1) (2n − 3)!! arctan x = + n−1 n n−k 2 k 2 2n − 1 2 (n − 1)! (1 + x ) k=1 2 (n − 1)(n − 2) . . . (n − k) (1 + x ) TI (91)
2.149 1.
2.
3.
dx 1 2n + m − 3 n =− n−1 + m−1 2 m−1 xm (1 − x2 ) (m − 1)x (1 − x )
dx n xm−2 (1 − x2 )
LA 139 (34)
For m = 1 dx 1 dx LA 139 (36) n = n−1 + n−1 2 x (1 − x2 ) 2(n − 1) (1 − x ) x (1 − x2 ) For m = 1 and n = 1 x dx = ln √ 2 x (1 − x ) 1 − x2 x dx 1 2n − 3 dx = + LA 139 (35) n n−1 2 2 2n − 2 (1 − x ) 2n − 2 (1 − x2 )n−1 (1 − x ) n−1 1+x dx x (2n − 1)(2n − 3)(2n − 5) . . . (2n − 2k + 1) (2n − 3)!! ln + n n = n−k k 2 2n − 1 2 · (n − 1)! 1 −x (1 − x2 ) k=1 2 (n − 1)(n − 2) . . . (n − k) (1 − x ) TI (91)
78
Rational Functions
2.151
2.15 Forms containing pairs of binomials: a + bx and α + βx Notation: t = α + βx; Δ = aβ − αb z = a + bx; n+1 mΔ tm z n m − 2.151 z t dx = z n tm−1 dx (m + n + 1)b (m + n + 1)b 2.152 z bx Δ 1. dx = + 2 ln t t β β βx Δ t dx = − 2 ln z 2. z b b m m−1 tm dx t dx 1 mΔ t 2.153 = − zn (m − n + 1)b z n−1 (m − n + 1)b zn tm+1 1 (m − n + 2)β tm dx = − n−1 (n − 1)Δ z n−1 (n − 1)Δ m−1z tm 1 mβ t =− + dx (n − 1)b z n−1 (n − 1)b z n−1 1 t dx = ln 2.154 zt Δ z 1 1 (m + n − 2)b dx dx = − − 2.155 z n tm (m − 1)Δ tm−1 z n−1 (m − 1)Δ tm−1 z n 1 1 (m + n − 2)β dx = + m z n−1 (n − 1)Δ tm−1 z n−1 (n − 1)Δ t
1 a α x dx = ln z − ln t 2.156 zt Δ b β
2.16 Forms containing the trinomial a + bxk + cx2k 2.160 1. 2. 3.
2.161
Reduction formulas for Rk = a + bxk + cx2k . xm Rkn+1 (m + k + nk)b (m + 2k + 2kn)c m−1 n m+k−1 n x − Rk dx = Rk dx − x xm+2k−1 Rkn dx ma ma ma bkn xm Rkn 2ckn m−1 n m+k−1 n−1 − Rk dx = Rk dx − x x xm+2k−1 Rkn−1 dx m m m xm−2k Rkn+1 (m − 2k)a (m − k + kn)b xm−1 Rkn dx = − xm−2k−1 Rkn dx − xm−k−1 Rkn dx (m + 2kn)c (m + 2kn)c (m + 2kn)c 2kna xm Rkn bkn + = xm−1 Rkn−1 dx + xm+k−1 Rkn−1 dx m + 2kn m + 2kn m + 2kn Forms containing the trinomial R2 = a + bx2 + cx4 . b 1 2 b 1 2 − b − 4ac, g = + b − 4ac, 2 2 2 2 a h = b2 − 4ac, q = 4 , l = 2a(n − 1) b2 − 4ac , c
Notation: f =
b cos α = − √ 2 ac
Forms containing a + bx + cx2 and powers of x
2.172
1.
dx R2
2 dx dx − h >0 LA 146 (5) 2 2 cx + f cx + g
2 x2 − q 2 1 α α x2 + 2qx cos α2 + q 2 h < 0 LA 146 (8)a ln arctan = + 2 cos sin α 4cq 3 sin α 2 x2 − 2qx cos 2 + q 2 2 2qx sin α2 2 x dx cx2 + f 1 h >0 LA 146 (6) ln 2 = R2 2h cx + g 2 2 2 x − q cos α 1 h 0 = LA 146 (7) 2 2 R2 h cx + g h cx + f bcx3 + b2 − 2ac x b2 − 6ac dx bc x2 dx dx = + + R22 lR2 l R2 l R2 2 2 3 bcx + b − 2ac x (4n − 7)bc x dx 2(n − 1)h2 + 2ac − b2 dx dx = + + n 2 n−1 R2 lRn−1 l l R2 R2n−1 =
2.
3. 4. 5.
79
6.9
c h
1 (m + 2n − 3)b dx =− − xm R2n (m − 1)a (m − 1)axm−1 R2n−1
[n > 1] (m + 4n − 5)bc dx − xm−2 R2n (m − 1)a
LA 146
dx xm−4 R2n LA 147 (12)a
2.17 Forms containing the quadratic trinomial a + bx + cx2 and powers of x Notation: R = a + bx + cx2 ; Δ = 4ac − b2 2.171 am xm Rn+1 b(m + n + 1) m+1 n m−1 n − R dx = R dx − 1. x x xm Rn dx c(m + 2n + 2) c(m + 2n + 2) c(m + 2n + 2) 2. 3. 4.
TI (97)
R dx R b(n − m + 1) R dx c(2n − m + 2) R dx =− + + LA 142(3), TI (96)a m+1 m x amx am xm am xm−1 dx b + 2cx (4n − 2)c dx = + TI (94)a Rn+1 nΔRn nΔ Rn n−1 n dx (2cx + b) 2k(2n + 1)(2n − 1)(2n − 3) . . . (2n − 2k + 1)ck dx n (2n − 1)!!c = + 2 n+1 k+1 n−k n R 2n + 1 n(n − 1) · · · (n − k)Δ R n!Δ R n
n+1
n
n
k=0
2.17211
TI (96)a
√
1 −Δ − (b + 2cx) −2 b + 2cx dx √ =√ ln =√ arctanh √ R −Δ (b + 2cx) + −Δ −Δ −Δ −2 = b + 2cx b + 2cx 2 = √ arctan √ Δ Δ
for [Δ < 0] for [Δ = 0, b and c non-zero] for [Δ > 0]
80
Rational Functions
2.173
2.173 dx b + 2cx 2c dx + 1. = R2 ΔR Δ R
dx 1 b + 2cx 3c 6c2 dx 2. = + + R3 Δ 2R2 ΔR Δ2 R 2.174 1.
2. 2.175 1. 2. 3. 4. 5. 6.
m−1 m−2 xm dx dx dx xm−1 (n − m)b (m − 1)a x x = − − + n n−1 n n R (2n − m − 1)cR (2n − m − 1)c R (2n − m − 1)c R For m = 2n − 1, this formula is inapplicable. Instead, we may use 2n−1 x dx 1 x2n−3 dx a x2n−3 dx b x2n−2 dx = − − Rn c Rn−1 c Rn c Rn
1 b x dx dx = ln R − R 2c 2c R x dx b 2a + bx dx − =− R2 ΔR Δ R x dx 2a + bx 3b(b + 2cx) 3bc dx − 2 =− − R3 2ΔR2 2Δ2 R Δ R 2 x dx x b b2 − 2ac dx = − 2 ln R + R c 2c 2c2 R 2 2 ab + b − 2ac x 2a dx x dx + = R2 cΔR Δ R 2 2 ab + b − 2ac x 2ac + b2 (b + 2cx) x dx + = + R3 2cΔR2 2cΔ2 R
7.
8.
9. 2.176
1.
(see 2.172)
2.177
(see 2.172)
(see 2.172) (see 2.172) (see 2.172) (see 2.172) 2ac + b2 Δ2
b b2 − 3ac x3 dx x2 bx b2 − ac dx = − 2 + ln R − 3 3 R 2c c 2c 2c R
dx R
(see 2.172)
(see 2.172) 2 a 2ac − b + b 3ac − b x b 6ac − b2 x dx 1 dx − = 2 ln R + 2 2 2 R 2c c ΔR 2c Δ R
3
3
x dx =− R3 dx xm Rn
(see 2.172)
=
2
2
2
x abx 2a + + c cΔ cΔ
−1 (m − 1)axm−1 Rn−1
1 x2 b dx = ln − xR 2a R 2a
dx R
1 3ab − 2R2 2cΔ
(see 2.172)
dx (see 2.173 1) R2 dx dx b(m + n − 2) c(m + 2n − 3) − − m−1 n m−2 a(m − 1) x R a(m − 1) x Rn (see 2.172)
2.177
Quadratic trinomials and binomials
2.
3.
1 1 x2 dx + = ln 2 2 xR 2a R 2aR
5.
dx 1 b 1 1 x − = + 2 + 3 ln xR3 4aR2 2a R 2a R 2a
b − 2 2a
2ac 1+ Δ
dx R
(see 2.172) b b dx dx dx − 2 − 3 R3 2a R2 2a R (see 2.172, 2.173)
dx b 1 b − 2ac dx x = − 2 ln − + (see 2.172) x2 R 2a R ax 2a2 R 2
b − 3ac (b + 2cx) dx a + bx 1 b4 b x2 6b2 c 6c2 dx − + − = − ln − + 2 2 3 2 2 3 2 x R a R a xR a ΔR Δ a a a R
6. 7.
b(b + 2cx) 1− Δ
2
4.
81
2
2
dx 1 3b =− − x2 R3 axR2 a
5c dx − xR3 a
(see 2.172) dx R3
(see 2.173 and 2.177 3)
b 3ac − b2 ac − b2 x2 b 1 dx dx =− + 2 − ln + 3 3 2 3 x R 2a R a x 2ax 2a R
8.
9.
(see 2.172) 2
3b dx 1 1 3b 2c 9bc dx dx + = − + − + x3 R2 2ax2 2a2 x R a2 a xR2 2a2 R2
dx = x3 R3
−1 2b + 2 2ax2 a x
1 + R2
2
6b 3c − a2 a
(see 2.173 1 and 2.177 2) 10bc dx dx + 2 xR3 a R3 (see 2.173 2 and 2.177 3)
2.18 Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx Notation: R = a + bx + cx2 ; B = bβ − 2cα; 1. 2.
z = α + βx;
A = aβ 2 − αbβ + cα2 ;
Δ = 4ac − b2
(m − 1)A R dx − z z m−2 Rn dx (m + 2n + 1)c n n−1 Rn dx R dx R 1 2nA = − − zm (m − 2n − 1)β z m−1 (m − 2n − 1)β 2 zm nB Rn−1 dx − ; LA 184 (4)a (m − 2n − 1)β 2 z m−1 n n n+1 R −β (m − n − 2)B R dx (m − 2n − 3)c R dx = − − LA 148 (5) m−1 (m − 1)A z (m − 1)A z m−1 (m − 1)A z m−2 Rn Rn−1 dx Rn−1 dx 1 nB 2nc =− + + LA 418 (6) m−1 2 m−1 2 (m − 1)β z (m − 1)β z (m − 1)β z m−2 (m + n)B βz m−1 Rn+1 − z R dx = (m + 2n + 1)c (m + 2n + 1)c m
n
m−1
n
82
Algebraic Functions
3.
z m−1 β (m − n)B z m dx = − n n−1 R (m − 2n + 1)c R (m − 2n + 1)c b + 2cx z 2(m − 2n + 3)c − (n − 1)Δ Rn−1 (n − 1)Δ m
=
2.201
z m−1 dx (m − 1)A − n R (m − 2n + 1)c
m
Bm z dx − Rn−1 (n − 1)Δ
z m−2 dx Rn LA 147 (1)
z
m−1
dx Rn−1 LA 148 (3)
4.3
1 β (m + n − 2)B dx =− − m n m−1 n−1 z R (m − 1)A z R (m − 1)A
=
1 β B − 2(n − 1)A z m−1 Rn−1 2A
dx (m + 2n − 3)c − m−1 n z R (m − 1)A
(m + 2n − 3)β dx + z m−1 Rn 2(n − 1)A
2
dx z m−2 Rn LA 148 (7)
dx z m Rn−1 LA 148 (8)
For m = 1 and n = 1 β z2 B dx dx = ln − zR 2A R 2A R For A = 0 1 β (m + 2n − 2)c dx dx =− − z m Rn (m + n − 1)B z m Rn−1 (m + n − 1)B z m−1 Rn
LA 148 (9)
2.2 Algebraic Functions 2.20 Introduction
r
s αx + β αx + β 2.201 The integrals R x, , , . . . dx, where r, s, . . . are rational numbers, γx + δ γx + δ can be reduced to integrals of rational functions by means of the substitution αx + β = tm , FI II 57 γx + δ where m is the common denominator of the fractions r, s, . . . . p 2.202 Integrals of the form xm (a + bxn ) dx,∗ where m, n, and p are rational numbers, can be
expressed in terms of elementary functions only in the following cases: (a)
When p is an integer; then, this integral takes the form of a sum of the integrals shown in 2.201;
(b)
is an integer: by means of the substitution xn = z, this integral can be transformed m+1 1 to the form (a + bz)p z n −1 dz, which we considered in 2.201; n
(c)
When
When
m+1 n
m+1 n
+ p is an integer; by means of the same substitution xn = z, this integral can be
p m+1 a + bz 1 z n +p−1 dz, considered in 2.201; reduced to an integral of the form n z
For reduction formulas for integrals of binomial differentials, see 2.110. ∗ Translator:
The authors term such integrals “integrals of binomial differentials.”
2.214
Forms containing the binomial a + bxk and
2.21 Forms containing the binomial a + bxk and
√
√
x
x
Notation: z1 = a + bx. dx bx 2 √ = √ arctan [ab > 0] 2.211 a √ z1 x ab a − bx + 2i xab 1 [ab < 0] = √ ln z1 i ab m√ m m+1 √ (−1)k ak xm−k x x dx m+1 a √ dx = 2 x + (−1) 2.212 z1 (2m − 2k + 1)bk+1 bm+1 z1 x k=0
(see 2.211) 2.213
√ √ dx x dx 2 x a √ 1. − (see 2.211) = z1 b b z1 x √ a √ x x dx x a2 dx √ − 2 2 x+ 2 (see 2.211) 2. = z1 3b b b z1 x 2
2√ √ x xa x x dx a2 a3 dx √ − 2 + 3 2 x− 3 (see 2.211) 3. = z1 5b 3b b b z1 x √ dx x 1 dx √ √ 4. = (see 2.211) + az1 2a z1 x z12 x √ √ x dx x 1 dx √ 5. (see 2.211) =− + 2 z1 bz1 2b z1 x √ √ √ x x dx x dx 2x x 3a 6. = − (see 2.213 5) z12 bz1 b z12 2
√ √ 2√ x 5ax 2 x 5a2 x dx x x dx − 2 = + 2 (see 2.213 5) 7. 2 z1 3b 3b z1 b z12
√ 1 dx 3 3 dx √ √ 8. = (see 2.211) + x + 2az12 4a2 z1 8a2 z1 x z13 x
√ √ x dx 1 1 1 dx √ 9. (see 2.211) = − + x + 3 2 z1 2bz1 4abz1 8ab z1 x √ √ √ x x dx x dx 2x x 3a 10. =− + (see 2.213 9) z13 bz12 b z13 2
√ √ 2√ x 5ax 2 x 15a2 x dx x x dx + = − (see 2.213 9) 11. 3 2 2 2 z1 b b z1 b z13 a a 4 2 , α = 4 − . Notation: z2 = a + bx , α = b b % √ √ & + , 2 a x + α 2x + α 1 α 2x dx √ = √ ln >0 + arctan 2 2.214 √ z2 α −x b z2 x bα3 2 √ √
+ , a x 1 α − x 0 , 0] [a < 0]
(see 2.224 4) (see 2.224 4) (see 2.224 4)
85
86
2.226 1. 2. 3.
Algebraic Functions
√ 3 √ z dx z dx √ = + a 2 z + a2 x 3 x z √ √ 3 √ 3 z dx z5 z dx 3b + =− 2 x ax 2a x
√ √ 3 √ 3 1 z dx z dx b 3b2 5 =− + 2 z + 2 x3 2ax2 4a x 8a x
2.226
(see 2.224 4) (see 2.226 1)
(see 2.226 1) 2.227
m−1 dx 2 1 dx √ √ = √ + m m m−k k a xz z (2k + 1)a z z x z
(see 2.224 4)
k=0
2.228 1. 2. 2.229 1. 2. 3.
√ b dx z dx √ √ − = − 2 ax 2a x z x z
√ 1 dx 3b 3b2 dx √ √ = − + 2 z+ 2 2ax2 4a x 8a x3 z x z
(see 2.224 4) (see 2.224 4)
dx 1 2 dx √ = √ + √ (see 2.224 4) a x z a z x z3
1 dx 3b 3b 1 dx √ √ = − − 2 √ − 2 (see 2.224 4) 2 3 ax a 2a z x z x z
1 15b2 15b2 1 5b dx dx √ √ = − √ + + + 2ax2 4a2 x 4a3 8a3 x z z x3 z 3 (see 2.224 4)
Cube root 2.231 1.
2.
3.
4.
√ 3 (−1)k nk z n−k ak 3 z 3(m+1)+1 3n − 3k + 3(m + 1) + 1 bn+1 k=0 n (−1)k n z n−k ak 3 xn dx k √ √ = 3 3n − 3k − 3(m − 1) − 2 bn+1 3 z 3(m−1)+2 z 3m+2 k=0 √ n √ 3 (−1)k n z n−k ak 3 z 3(m+1)+2 3 n k z 3m+2 x dx = 3n − 3k + 3(m + 1) + 2 bn+1 k=0 n (−1)k n z n−k ak xn dx 3 k √ √ = 3 3 3m+1 n+1 3n − 3k − 3(m − 1) − 1 z b z 3(m−1)+1 n √ 3 n 3m+1 z x dx =
k=0
Forms containing the binomial a + bxk and
2.236
√
x
1 z n+ 3 3n − 3m + 4 b z n dx z n dx √ √ = − + 3 m−1 (m − 1)ax 3(m − 1) a xm−1 3 z 2 xm x2 For m = 1 n−1 n dx 3z n z z dx √ √ √ = + a 3 3 3 2 2 x z (3n − 2) z x z2 √ √ 3 z dx dx 33z 1 √ 6. = + 3 n (3n − 1)az a xz n xz n z 2 √ √ √ √ 3 z− 3a √ 33z 3 1 dx √ √ √ ln − 3 arctan √ 2.232 = √ 3 3 3 3 x z+23a x z2 a2 2 5.
2.233 1. 2. 3.
√ 3
√ z dx dx √ =33z+a (see 2.232) 3 x x z2 √ √ 3 b√ dx z dx z3z b 3 √ + = − z + (see 2.232) x2 ax a 3 x 3 z2
√ 3 √ z dx 1 b b2 √ b2 dx 3 3 √ = − + z − z − z 3 2 2 2 x 2ax 3a x 3a 9a x 3 z 2 √ 3
4. 5. 2.234
z 2b dx dx √ √ − =− 3 3 2 ax 3a z x z2 1 5b √ 5b2 dx dx 3 √ √ = − + z + 3 2ax2 6a2 x 9a2 x 3 z 2 x3 z 2 x2
(see 2.232) (see 2.232) (see 2.232)
√ 3 z n dx zn z2 3n − 3m + 5 b z n dx √ √ 1. = − + 3 (m − 1)axm−1 3(m − 1) a xm−1 3 z xm z 2 For m = 1: n n−1 z dx dx 3z n z √ √ √ 2. = + a x3z (3n − 1) 3 z x3z √ √ 3 3 3 z2 z 2 dx 1 dx √ = + 3. 3 n n (3n − 2)az a xz n xz z √ √ √ √ 3 z− 3a √ 33z 1 dx 3 √ = √ √ √ ln + 3 arctan √ 2.235 3 3 3 x3z x z+23a a2 2 2.236 1. 2.
√ 3
3√ z 2 dx dx 3 √ = z2 + a x 2 x3z √ √ 3 3 b√ z 2 dx z5 2b dx 3 2+ √ + = − z 2 x ax a 3 x3z
(see 2.235) (see 2.235)
87
88
Algebraic Functions
√ 3 3.
4. 5.
z 2 dx 1 b b2 √ b2 dx 3 5/3 2− √ = − + − z z x3 2ax2 6a2 x 6a2 9a x 3 z
dx √ x2 3 z dx √ 3 z
x3
√ 3 b z2 dx √ − =− ax 3a x 3 z 1 2b √ 2b2 dx 3 √ = − + 2 z+ 2 2 2ax 3a x 9a x3z
2.24 Forms containing Notation: z = a + bx, 2.241 1. 2.
2.241
(see 2.235) (see 2.235) (see 2.235)
√ a + bx and the binomial α + βx
t = α + βx,
Δ = aβ − bα.
m−1 n √ z m tn dx t dx 2 (2m − 1)Δ z √ √ tn+1 z m−1 z + = (2n + 2m + 1)β (2n + 2m + 1)β z z
n m k n n−k k √ k z k−p ap n α β t z dx p √ = 2 z 2m+1 (−1) p 2k − 2p + 2m + 1 k bk+1 p=0 z k=0
2.242 1.
11
2. 3.
4. 5. 6.
7. 8.
√ z 2√z 2α z t dx √ = +β −a b 3 b2 z 2
√ √ z 2√z z 2 z 2 2α2 z t2 dx 2 2 √ = + 2αβ −a − za + a + β b 3 b2 5 3 b3 z
√ √ √ 2 z z z2 2 z 2 t3 dx 2α3 z 2 2 2 √ = + 3α β −α − za + a + 3αβ 2 b 3 b √ 5 3 b3 z 3 2 z 2 z 3z a − + za2 − a3 +β α 7 5 b4 √ √ z a 2 z3 2α z 3 tz dx √ = +β − 3b 5 3 b2 z √ √ 2
√ z z a 2 z3 2za a2 2 z 3 2α2 z 3 t2 z dx 2 √ + 2αβ − − + = +β 3b 5 3 b2 7 5 3 b3 z √ √ 2
√ z z a 2 z3 2za a2 2 z 3 t3 z dx 2α3 z 3 2 2 √ = + 3α β − − + + 3αβ 3b 5 3 b2 7 5 3 b3 z √ 3
2 2 3 z 3z a 3za a 2 z3 − + − +β 3 9 7 5 3 b4 √ √ z a 2 z5 2α z 5 tz 2 dx √ +β − = 5b 7 5 b2 z √ √ 2
√ z z a 2 z5 2za a2 2 z 5 2α2 z 5 t2 z 2 dx 2 √ + 2αβ − − + = +β 5b 7 5 b2 9 7 5 b3 z
LA 176 (1)
2.244
Forms with
√ a + bx and α + βx
89
√ √ 2
√ z z a 2 z5 2za a2 2 z 5 t3 z 2 dx 2α3 z 5 2 2 √ + 3α β − − + = + 3αβ 5b 7 5 b2 9 7 5 b3 z √ 3
2 2 3 5 z 3z a 3za a 2 z − + − +β 3 11 9 7 5 b4 √ √ 3 z a 2 z7 2α z 7 tz dx √ +β − = 7b 9 7 b2 z √ √ 2
√ 2 3 z z a 2 z7 2za a2 2 z 7 2α2 z 7 t z dx 2 √ + 2αβ − − + = +β 7b 9 7 b2 11 9 7 b3 z √ √ √
3 3 z z2 a 2 z7 2za a2 2 z 7 t z dx 2α3 z 7 2 2 √ + 3α β − − + = + 3αβ 7b 9 7 b2 11 9 7 b3 z √ 3
2 2 3 7 z 3z a 3za a 2 z − + − +β 3 13 11 9 7 b4
9.
10. 11. 12.
2.243
1.
tn dx tn+1 √ 2 (2n − 2m + 3)β tn dx √ √ = z − (2m − 1)Δ z m (2m − 1)Δ zm z z m−1 z tn √ tn−1 dx 2nβ 2 √ z+ =− m (2m − 1)b z (2m − 1)b z m−1 z LA 176 (2)
2. 2.244
1. 2. 3. 4. 5. 6. 7.
2 tn dx √ =√ 2m−1 zm z z
k n n−k k k z k−p ap n a β p (−1) p 2k − 2p − 2m + 1 k bk+1 p=0
k=0
t dx 2a 2β(z + a) √ =− √ + √ z z b z b2 z
2 − 2za − a t dx 2α 4αβ(z + a) √ =− √ + √ √ + z z b z b2 z b3 z 2 z2 2 3 z3 2 2 3 6αβ 2β 3 3 2 − 2za − a − z a + 3za + a 3 5 t dx 2α 6α β(z + a) √ =− √ + √ √ √ + + z z b z b2 z b3 z b4 z 2β z − a3 2a t dx √ √ √ = − − z2 z 3b z 3 b2 z 3 a2 2 2 a z 2β + 2az − 2 2 4αβ z − 3 3 2a t dx √ =− √ − √ √ + 3 2 3 3 3 z2 z 3b z b z b z 3 2 2 6αβ z 2 + 2za − a3 2β 3 z3 − 3z 2 a − 3za2 + 6α2 β z − a3 2α3 t3 dx √ =− √ − √ √ √ + + z2 z 3b z 3 b2 z 3 b3 z 3 b4 z 3 2β z3 − a5 2α t dx √ =− √ − √ z3 z 5b z 5 b2 z 5 2
2
2β 2
z2 3
a3 3
90
Algebraic Functions
8. 9.
2.245 1.
2.
2
t dx √ z3 z t3 dx √ z3 z
2za a2 2 2 z 2β − + 4αβ 3 − 3 5 2α √ √ =− √ − − 5 2 5 3 5 5b z b z b z 2za a2 2 2 z a 2 z 6αβ − + 3 6α β 3 − 5 3 5 2α √ √ =− √ − − 5 2 5 3 5 5b z b z b z 3 2β 3 z 3 + 3z 2 a − za2 + a5 √ + b4 z 5 a 5
m−1 z m dx z m−1 √ dx 2 (2m − 1)Δ z √ =− √ z− (2n − 2m − 1)β tn−1 (2n− 2m − 1)β tn z tn z z m−1 z m−1 √ 1 (2m − 1)b √ dx =− z + (n − 1)β tn−1 2(n − 1)β tn−1 zm m √ z 1 (2n − 2m − 3)b z dx √ =− z− n−1 n−1 (n − 1)Δ t 2(n − 1)Δ t z ⎡ m √ 1 1 z dz √ = −z m z ⎣ n−1 n (n − 1)Δ t t z n−1 k=2
−
4.
z
2
+
3.
2.245
(2n − 2m − 3)(2n − 2m − 5) . . . (2n − 2m − 2k + 1)b 2k−1 (n − 1)(n − 2) . . . (n − k)Δk
k=0
2.246
2.247
√ √ 1 dx β z − βΔ √ √ √ ln √ [βΔ > 0] t z βΔ β z + βΔ √ 2 β z =√ arctan √ [βΔ < 0] −βΔ −βΔ √ 2 z [Δ = 0] =− bt m dx βm 2 β k−1 z k dx √ √ = m−1 √ + + m k m Δ (2m − 2k + 1) Δ z tz z z t z k=1
(see 2.246) 2.248 1.
k−1
(2n − 2m − 3)(2n − 2m − 5) . . . (−2m + 3)(−2m + 1)bn−1 2n−1 · (n − 1)!Δn
For n = 1 m zm 2 Δ z m−1 dx z dx √ = √ √ + (2m − 1)β z β t z t z m m−1 Δk z m−k Δm z dx dx √ =2 √ √ + k+1 m (2m − 2k − 1)β β t z z t z
LA 176 (3)
2 β dx √ = √ + Δ tz z Δ z
dx √ t z
(see 2.246)
⎤ 1 ⎦
tn−k
z m dx √ t z
2.248
Forms with
2. 3.
4. 5. 6.
7.
8. 9.
10. 11. 12. 13. 14. 15. 16.
2 2β β2 dx √ √ √ = + + Δ2 tz 2 z 3Δz z Δ2 z
√ a + bx and α + βx
dx √ t z
2 2β 2β 2 β3 dx √ + 3√ + 3 √ = √ + 3 2 2 Δ tz z 5Δz z 3Δ z z Δ z √
b z dx √ =− − Δt 2Δ z
t2
91
(see 2.246)
dx √ t z (see 2.246)
dx √ t z
dx 1 3b 3bβ √ = − √ − 2√ − 2Δ2 t2 z z Δt z Δ z
(see 2.246)
dx √ t z
1 5b 5bβ 5bβ 2 dx √ √ √ √ = − − − − 2Δ3 t2 z 2 z Δtz 2 z 3Δ2 z z Δ3 z
(see 2.246)
dx √ t z
(see 2.246) dx 1 7b 7bβ 7bβ 7bβ 3 dx √ − 4√ − √ √ =− √ − √ − 2Δ4 t z t2 z 3 z Δtz 2 z 5Δ2 z 2 z 3Δ3 z z Δ z 2
√ √ dx 3b2 z 3b z dx √ √ + = − + 2Δt2 4Δ2 t 8Δ2 t z t3 z dx 1 5b 15b2 15b2 β √ √ √ √ = − + + + 8Δ3 t3 z z 2Δt2 z 4Δ2 t z 4Δ3 z
(see 2.246) (see 2.246)
dx √ t z
(see 2.246) √ 2 dx 1 7b z 35b 35b2 β 35b2 β 2 dx √ + √ + √ + √ √ =− √ + 4 3 2 2 2 2 4 8Δ t z z 2Δt z z 4Δ tz z 12Δ z z 4Δ z t z (see 2.246) dx 1 9b 63b2 21b2 β 63b2 β 2 63b2 β 3 √ √ √ √ √ √ = − + + + + + 8Δ5 t3 z 3 z 2Δt2 z 2 z 4Δ2 tz 2 z 20Δ3 z 2 z 4Δ4 z z 4Δ5 z (see 2.246) √ Δ 2 z z dx dx √ = √ + (see 2.246) β β t z t z √ √ 2Δ z z 2 dx 2z z Δ2 dx √ = √ + (see 2.246) + 2 2 3β β β t z t z √ √ √ 2Δz z 2z 2 z z 3 dx 2Δ2 z Δ3 dx √ = √ + (see 2.246) + + 5β 3β 2 β3 β3 t z t z √ √ z dx b z b z z dx √ √ =− + + (see 2.246) 2 Δt βΔ 2β t z t z √ √ √ bz z 3b z z2 z z 2 dx 3bΔ dx √ √ + + = − (see 2.246) + Δt βΔ β2 2β 2 t z t2 z
dx √ t z
92
Algebraic Functions
17.
18.3 19.
20.
2.249
√ √ √ √ bz 2 z 5bz z z3 z 5bΔ z 5Δ2 b z 3 dx dx √ √ =− + + + + 2 3 3 2 Δt βΔ 3β β 2β t z t z
z dx √ t3 z z 2 dx √ t3 z
(see 2.246) √ √ 2 2 b z z z bz z b dx √ − =− (see 2.246) + + 2Δt2 4Δ2 t 4βΔ2 8βΔ t z √ √ √ √ b2 z z 3b2 z2 z bz 2 z 3b2 z dx √ + + =− + + 2 2 2 2 2 2Δt 4Δ t 4βΔ 4β Δ 8β t z √
(see 2.246) √ √ √ √ √ 3 2 2 2 z dx 3b z z 15b2 z z z 3bz z 5b z z 15b2 Δ dx √ √ + + = − + + + 2 2 2 2 3 3 3 2Δt Δ t 4βΔ 4β Δ 4β 8β t z t z 3
3
(see 2.246) 2.249 1.
2.
√ z 2 (2n + 2m − 3)β dx dx √ = √ + m n z m−1 z (2m − 1)Δ tn−1 z (2m − 1)Δ z m tn z t √ z 1 (2n + 2m − 3)b dx √ =− − m n−1 n−1 (n − 1)Δ z t 2(n − 1)Δ t zm z ⎡ √ 1 z ⎣ −1 dx √ = m n−1 m n z (n − 1)Δ t z t z
LA 177 (4)
⎤ k−1 (2n + 2m − 3)(2n + 2m − 5) . . . (2n + 2m − 2k + 1)b 1 + (−1)k · n−k ⎦ 2k−1 (n − 1)(n − 2) . . . (n − k)Δk t k=2 n−1 (2n + 2m − 3)(2n + 2m − 5) . . . (−2m + 3)(−2m + 1)b dx √ +(−1)n−1 n−1 n−1 m 2 (n − 1)!Δ tz z n−1
For n = 1 1 2 β dx dx √ √ √ = + m m−1 m−1 (2m − 1)Δ z Δ tz z t z z z
2.25 Forms containing
√
a + bx + cx2
Integration techniques 2.251
It is possible to rationalize the integrand in integrals of the form
R x, a + bx + cx2 dx by
using one or more of the following three substitutions, known as the “Euler substitutions”: √ √ a + bx + cx2 = xt ± a for a > 0; 1. √ √ a + bx + cx2 = t ± x c for c > 0; 2. c (x − x1 ) (x − x2 ) = t (x − x1 ) when x1 and x2 are real roots of the equation a + bx + cx2 = 0. 3.
2.252
Forms containing
√
a + bx + cx2
93
2.252
Besides Euler substitutions, there is also the following method of calculating integrals of the the form R x, a + bx + cx2 dx. By removing the irrational expressions in the denominator and performing simple algebraic operations, we can reduce the integrand to the sum of some rational function P (x) √ 1 , where P 1 (x) and P 2 (x) are both polynomials. of x and an expression of the form P 2 (x) a + bx + cx2 P 1 (x) By separating the integral portion of the rational function from the remainder and decomposing P 2 (x) the latter into partial fractions, we can reduce the integral of these partial fractions to the sum of integrals, each of which is in one of the following three forms: P (x) dx √ , where P (x) is a polynomial of some degree r; 1. a + bx + cx2 dx √ 2. ; k (x + p) a + bx + cx2
(M x + N ) dx a b , a1 = , b 1 = 3. . m c c (a + βx + x2 ) c (a1 + b1 x + x2 ) In more detail: P (x) dx dx √ 1. = Q(x) a + bx + cx2 + λ √ , where Q(x) is a polynomial of 2 a + bx + cx a + bx + cx2 degree (r − 1). Its coefficients, and also the number λ, can be calculated by the method of undetermined coefficients from the identity
2.
3.
1 P (x) = Q (x) a + bx + cx2 + Q(x)(b + 2cx) + λ LI II 77 2 P (x) dx Integrals of the form √ (where r ≤ 3) can also be calculated by use of formulas a + bx + cx2 2.26. P (x) dx √ , where the degree n of the polynomial P (x) is Integrals of the form k (x + p) a + bx + cx2 1 lower than k can, by means of the substitution t = , be reduced to an integral of the form x + p P (t) dt . (See also 2.281). a + βt + γt2 (M x + N ) dx can be calculated by the following Integrals of the form m (α + βx + x2 ) c (a1 + b1 x + x2 ) procedure: • If b1 = β, by using the substitution x=
< 2 t − 1 (a1 − α) − (αb1 − a1 β) (β − b1 )
a1 − α + βb1 t+1
β − b1
94
Algebraic Functions
2.260
P (t) dt , where P (t) m + p) c (t2 + q) P (t) dt is a polynomial of degree no higher than 2m − 1. The integral can be 2 + p)m (t t2 + q t dt dt reduced to the sum of integrals of the forms and . k k 2 2 2 (t + p) t +q (t + p) t2 + q P (t) dt • If b1 = β, we can reduce it to integrals of the form by means of the m 2 c (t2 + q) (t + p) b1 substitution t = x + . 2 t dt can be evaluated by means of the substitution t2 + q = The integral k 2 (t + p) c (t2 + q) u2 . t dt = can be evaluated by means of the substitution The integral k 2 2 2 t +q (t + p) c (t + q) υ (see also 2.283). FI II 78-82 we can reduce this integral to an integral of the form
2.26 Forms containing
(t2
√ a + bx + cx2 and integral powers of x
Notation: R = a + bx + cx2 , Δ = 4ac − b2 For simplified formulas for the case b = 0, see 2.27. 2.260 √ √ √ (2m + 2n + 1)b xm−1 R2n+3 m 2n+1 1. x − R dx = xm−1 R2n+1 dx (m + 2n + 2)c 2(m + 2n + 2)c √ (m − 1)a − xm−2 R2n+1 dx (m + 2n + 2)c
2. 3.
TI (192)a
2cx + b √ 2n+1 2n + 1 Δ √ 2n−1 R R R2n+1 dx = + dx TI (188) 4(n + 1)c 8(n + 1) c √ √ n−1 (2n + 1)(2n − 1) . . . (2n − 2k + 1) Δ k+1 R (2cx + b) n n−k−1 R + R2n+1 dx = R 4(n + 1)c 8k+1 n(n − 1) . . . (n − k) c k=0 n+1 Δ (2n + 1)!! dx √ + n+1 8 (n + 1)! c R √
TI (190)
2.26111
For n = −1 √ 1 dx 2 cR + 2cx + b √ = √ ln √ c R Δ
2cx + b 1 √ = √ arcsinh c Δ 1 = √ ln(2cx + b) c
2cx + b −1 = √ arcsin √ −c −Δ
[c > 0]
TI (127)
[c > 0,
Δ > 0]
DW
[c > 0,
Δ = 0]
DW
[c < 0,
Δ < 0]
TI (128)
2.263
2.262 1. 2. 3.
Forms containing
√ Δ dx (2cx + b) R √ + R dx = 4c 8c R √ √ 3 (2cx + b)b √ R bΔ x R dx = − R− 3c 8c2 16c2 2
√ √ x 5b 5b x2 R dx = − R3 + − 4c 24c2 16c2
√
4.
5.
6.
2.263 1.
2
x 7bx 7b 2a − + − 2 3 2 5c 40c 48c 15c 3
7b 3ab Δ dx √ − − 2 32c3 8c 8c R
R3 dx =
2
R 3Δ + 8c 64c2
95
(see 2.261) dx √ (see 2.261) R √ 2
5b a (2cx + b) R a Δ dx √ + − 4c 4c 16c2 4c 8c R
√
R3 −
√ 3Δ2 (2cx + b) R + 128c2
(see 2.261) √
7b3 3ab (2cx + b) R − 2 32c3 8c 4c
(see 2.261) dx √ R
(see 2.261) √
√ b √ 3 R5 3Δb √ 3Δ2 b dx √ − (2cx + b) R + R − x R3 dx = 5c 16c2 128c3 256c3 R
(see 2.261)
√ 3 √ x 7b2 R 7b √ 5 3Δ √ a b 2 3 x R dx = − + R + − R 2x + 6c 60c2 24c2 6c c 8 64c
2 Δ2 7b dx √ −a + 3 4c 256c R (see 2.261) 2
√ x 3bx 3b2 2a √ 5 − x3 R3 dx = + − R 2 3 7c 28c 40c 35c2 √ 3
3b 3Δ √ R3 ab b + − − 2 R 2x + 16c3 4c c 8 64c
2 3Δ2 b 3b dx √ −a − 4c 512c4 R (see 2.261)
8.
√ x3 R dx =
√
7.
√ a + bx + cx2 and xn
xm dx xm−1 (2m − 2n − 1)b √ √ = − 2n+1 2n−1 2(m − 2n)c R (m − 2n)c R
(m − 1)a xm−1 dx √ − 2n+1 (m − 2n)c R
xm−2 dx √ R2n+1 TI (193)a
2.
For m = 2n x2n dx x2n−1 b 1 x2n−1 x2n−2 √ √ √ √ =− − dx + dx 2c c R2n+1 (2n − 1)c R2n−1 R2n+1 R2n−1
TI (194)a
96
Algebraic Functions
dx
√ R2n+1
3. 4.
dx √ R2n+1
2.264
dx 2(2cx + b) 8(n − 1)c √ √ = + 2n−1 (2n − 1)Δ (2n − 1)Δ R R2n−1 n−1 ck k 8k (n − 1)(n − 2) . . . (n − k) 2(2cx + b) √ 1+ = R (2n − 3)(2n − 5) . . . (2n − 2k − 1) Δk (2n − 1)Δ R2n−1 k=1
[n ≥ 1] . 2.264
1.
3. 4.
5. 6. 7. 8.
TI (191)
(see 2.261)
√ x dx b R dx √ = √ (see 2.261) − c 2c R R 2
2 x 3b x dx 3b √ a dx √ √ = (see 2.261) − 2 R+ − 2 2c 4c 8c 2c R R 2 3
3 x 5b 5bx x dx 5b2 2a √ 3ab dx √ √ = − + 3− 2 R− − 2 2 3 3c 12c 8c 3c 16c 4c R R
2.
dx √ R
TI (189)
(see 2.261)
dx 2(2cx + b) √ √ = 3 Δ R R x dx 2(2a + bx) √ √ =− 3 Δ R R 2 Δ − b2 x − 2ab 1 x dx dx √ √ √ + (see 2.261) =− 3 c cΔ R R R 3 cΔx2 + b 10ac − 3b2 x + a 8ac − 3b2 3b x dx dx √ √ − 2 √ = 2c c2 Δ R R R3
√ 2n+1 R 2.265 xm
(see 2.261) √ √ 2n+1 R2n+3 R (2n − 2m + 5)b dx = − + dx (m − 1)axm−1 √ 2(m − 1)a xm−1 R2n+1 (2n − m + 4)c dx + (m − 1)a xm−2 TI (195)
For√m = 1 √ √ 2n−1 R2n+1 R2n+1 b √ 2n−1 R dx = + dx R dx + a x 2n + 1 2 x For 0] [a < 0,
TI (137)
Δ < 0]
TI (138)
[a < 0] [a > 0,
97
LA 178 (6)a
Δ > 0]
DW
[a > 0] [a > 0,
Δ = 0]
[a = 0,
b = 0]
LA 170 (16)
[a > 0, Δ < 0]
2.267 √ b dx R dx √ dx √ + √ (see 2.261 and 2.266) = R+a 1. x x R 2 R √ √ dx b R dx R dx √ √ 2. + c (see 2.261 and 2.266) + = − 2 x x 2 x R R For a = 0 √ √ dx bx + cx2 2 bx + cx2 √ + c dx = − (see 2.261) 2 x x bx + cx2 2
√ √ 1 b c R dx b dx √ − 3. =− + R− x3 2x2 4ax 8a 2 x R (see 2.266)
4.
5.
For a = 0 < 3 √ 2 2 (bx + cx2 ) bx + cx dx = − x3 3bx3 √ √ b 12ac − b2 2bcx + b2 + 8ac √ R3 R3 dx dx 2 √ + √ dx = + R+a x 3 8c 16c x R R (see 2.261 and 2.266) √ √ 3 √ 3 4ac + b2 R5 R cx + b √ 3 3 dx dx 3 √ + √ + dx = − R + (2cx + 3b) R + ab 2 x ax a 4 2 8 x R R (see 2.261 and 2.266) For a = 0 < 3 (bx + cx2 ) x2
< =
3
(bx + cx2 ) 2x
+
3b 3b2 bx + cx2 + 4 8
dx √ bx + cx2
(see 2.261)
98
6.
Algebraic Functions
2.268
√ √ 3 1 R b bcx + 2ac + b2 √ 3 3 bcx + 2ac + b2 √ 5 dx = − + 2 R + R + R 2 x3 2ax2 4ax 4a 4a 3 dx dx 3 √ + bc √ + 4ac + b2 8 x R 2 R (see 2.261 and 2.266) For a = 0 < 3 (bx + cx2 ) x3
dx =
dx 2b 3bc √ c− bx + cx2 + x 2 bx + cx2 (see 2.261)
2.268
xm
dx 1 √ √ =− 2n+1 m−1 2n−1 R (m − 1)ax R (2n + 2m − 3)b (2n + m − 2)c dx dx √ √ − − 2(m − 1)a (m − 1)a xm−1 R2n+1 xm−2 R2n+1 TI (196)
For m = 1 For a = 0
2.269
1.
3.
√
dx R2n+1
+
1 a
dx √ x R2n−1
dx 2 < < =− 2n+1 2n−1 xm (bx + cx2 ) (2n + 2m − 1)bxm (bx + cx2 ) dx (4n + 2m − 2)c < − (2n + 2m − 1)b 2n+1 xm−1 (bx + cx2 ) (cf. 2.265)
dx √ x R
(see 2.266)
√ dx b dx R √ √ =− − (see 2.266) 2 ax 2a x R x R For a = 0
2 2c dx 1 √ = bx + cx2 − 2+ 2 3 bx b x x2 bx + cx2 2
√ 3b 1 dx 3b c dx √ √ = − + R + − 2ax2 4a2 x 8a2 2a x3 R x R
2.
dx 1 b √ √ = − 2n+1 2n−1 2a x R (2n − 1)a R
(see 2.266)
4.
For a = 0
2 4c 8c2 dx 1 √ = bx + cx2 − 3+ 2 2− 3 5 bx 3b x 3b x x3 bx + cx2 2 bcx − 2ac + b2 1 dx dx √ √ √ + =− (see 2.266) 3 a x R aΔ R x R For a = 0
TI (199)
2.271
Forms containing
5.11
dx
< 3 x (bx + cx2 )
2 = 3
4c 8c2 x 1 − + 2 + 3 bx b b
3b A dx √ = −√ − 2 2 3 R 2a x R
dx √ x R
where A =
6.
√
a + cx2 and xn
99
1 √ bx + cx2
b 10ac − 3b2 c 8ac − 3b2 x 1 − − − ax a2 Δ a2 Δ
(see 2.266)
For a = 0
1 dx 8c2 2 2c 16c3 x 1 < √ = − 2+ 2 − 3 − 4 5 bx b x b b 3 bx + cx2 x2 (bx + cx2 ) dx √ 3 x R3 bc 15b2 − 52ac x 15b2 − 12ac 15b4 − 62acb2 + 24a2 c2 1 1 5b dx √ + √ − = − 2+ 2 − 3 3 3 ax 2a x 2a Δ 2a Δ 8a 2 R x R (see 2.266) For a = 0
1 64c3 2 8c 16c2 128c4 x 1 dx < √ = + − 3+ 2 2− 3 + 4 5 7 bx 5b x 5b x 5b 5b 3 bx + cx2 x3 (bx + cx2 )
2.27 Forms containing
√ a + cx2 and integral powers of x
√ Notation: u = a + cx2 . √ 1 I1 = √ ln x c + u c 1 c = √ arcsin x − a −c √ u− a 1 √ I2 = √ ln 2 a √ u+ a a−u 1 = √ ln √ 2 a a + u 1 1 c a 1 =√ − arcsec x − = √ arccos a x c −a −a 2.271 1. 2. 3. 4.
[c > 0] [c < 0 and a > 0] [a > 0 and c > 0] [a > 0 and c > 0] [a < 0 and c > 0]
1 5 5 5 5 xu + axu3 + a2 xu + a3 I1 6 24 16 16 3 1 3 u3 dx = xu3 + axu + a2 I1 4 8 8 1 1 u dx = xu + aI1 2 2 dx = I1 u u5 dx =
DW DW DW DW
100
Algebraic Functions
5. 6.
dx u2n+1
7. 2.272
dx 1x = u3 au
n−1 1 (−1)k n − 1 ck x2k+1 = n a 2k + 1 k u2k+1
x2 u3 dx = x2 u dx =
2. 3. 4. 5. 6. 7.
4. 5. 6.
DW
1 x2 x dx = − + I1 u3 cu c
DW
x2 1 x3 dx = u5 3 au3
DW
n−2 x2 dx 1 (−1)k n − 2 ck x2k+3 = n−1 u2n+1 a 2k + 3 k u2k+3 k=0
3
x dx 1 a =− + u2n+1 (2n − 3)c2 u2n−3 (2n − 1)c2 u2n−1
x4 u dx =
3.
DW
DW
DW
1 xu 1 a x2 dx = − I1 u 2 c 2c
x4 u3 dx =
2.
1 axu3 1 a2 xu 1 a3 1 xu5 − − − I1 6 c 24 c 16 c 16 c
1 axu 1 a2 1 xu3 − − I1 4 c 8 c 8 c
1.
7.
k=0
1 x dx =− 2n+1 u (2n − 1)cu2n−1
2.273
DW
1.
2.272
axu5 1 x3 u5 a2 xu3 3a3 xu 3a4 − + + + I1 8 c 16c2 64c2 128c2 128c2
axu3 1 x3 u3 a2 xu a3 − + + I1 2 2 6 c 8c 16c 16c2
DW
DW DW
1 x3 u 3 axu 3 a2 x4 dx = − + I1 u 4 c 8 c2 8 c2
DW
3 a x4 1 xu ax dx = + 2 − I1 3 2 u 2c c u 2 c2
DW
1 x3 x4 x 1 dx = − 2 − + 2 I1 5 u c u 3 cu3 c
DW
x4 1 x5 dx = u7 5 au5
DW
n−3 x4 dx 1 (−1)k n − 3 ck x2k+5 = n−2 u2n+1 a 2k + 5 k u2k+5 k=0
2.275
Forms containing
8. 2.274
x6 u3 dx = x6 u dx =
2. 3. 4. 5. 6. 7. 8. 9. 2.275
1. 2. 3. 4. 5. 6. 7. 8.
a + cx2 and xn
1 2a a2 x5 dx = − + − u2n+1 (2n − 5)c3 u2n−5 (2n − 3)cu2n−3 (2n − 1)c3 u2n−1
1.
√
101
DW
ax3 u5 1 x5 u5 a2 xu5 a3 xu3 3a4 xu 3 a5 − + − − − I1 2 3 3 3 10 c 16c 32c 128c 256c 256 c3
5 ax3 u3 1 x5 u3 5a2 xu3 5a3 xu 5 a4 − + − − I1 8 c 48 c2 64c3 128c3 128 c3
1 x5 u 5 ax3 u x6 5 a2 xu 5 a3 dx = − + − I1 u 6 c 24 c2 16 c3 16 c3
DW
5 ax3 15 a2 x 15 a2 x6 1 x5 − − + dx = I1 3 2 u 4 cu 8 c u 8 c3 u 8 c3
DW
x6 1 x5 10 ax3 5 a2 x 5 a dx = + + − I1 u5 2 cu3 3 c2 u3 2 c3 u3 2 c3
DW
x6 23 x5 7 ax3 a2 x 1 dx = − − − + 3 I1 7 5 2 5 3 5 u 15 cu 3c u c u c
DW
x6 1 x7 dx = 9 u 7 au7
DW
n−4 x6 dx 1 (−1)k n − 4 ck x2k+7 = u2n+1 an−3 2k + 7 k u2k+7 k=0
7
x dx 1 3a 3a2 a3 = − + − + u2n+1 (2n − 7)c4 u2n−7 (2n − 5)c4 u2n−5 (2n − 3)c4 u2n−3 (2n − 1)c4 u2n−1
DW
u5 1 u5 dx = + au3 + a2 u + a3 I2 x 5 3
DW
u3 u3 dx = + au + a2 I2 x 3 u dx = u + aI2 x dx = I2 xu n−1 1 dx 1 = I + 2 2n+1 n xu a (2k + 1)an−k u2k+1 k=0 5 5 15 u u5 15 dx = − + cxu3 + acxu + a2 I1 2 x x 4 8 8 3 3 u 3 3 u dx = − + cxu + aI1 x2 x 2 2 u u dx = − + cI1 x2 x
DW DW DW
DW DW DW
102
Algebraic Functions
9. 2.276
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 2.277
1. 2. 3. 4.
dx 1 = − n+1 x2 u2n+1 a
2.276
n u (−1)k+1 n k x 2k−1 + c x 2k − 1 k u
k=1
5 u5 u5 5 5 dx = − 2 + cu3 + acu + a2 cI2 3 x 2x 6 2 2
DW
u3 3 u3 3 dx = − 2 + cu + acI2 3 x 2x 2 2 u c u dx = − 2 + I2 x3 2x 2 dx u c =− − I2 x3 u 2ax2 2a 3c 3c dx 1 − − =− I2 x3 u3 2ax2 u 2a2 u 2a2 5 c dx 1 5 c 5 c − =− − − I2 x3 u5 2ax2 u3 6 a2 u 3 2 a3 u 2 a3
DW DW DW DW DW
u5 au3 2acu c2 xu 5 + + acI1 dx = − − x4 3x3 x 2 2
DW
u3 u3 cu + cI1 dx = − 3 − 4 x 3x x
DW
u u3 dx = − x4 3ax3 dx 1 = n+2 x4 u2n+1 a
DW
cu (−1)k u3 − 3 + (n + 1) + 3x x 2k − 3 n+1
k=2
n+1 k
ck
x 2k−3
u
u3 u3 3 cu3 3 c2 u 3 2 + c I2 dx = − 4 − + 5 2 x 4x 8 ax 8 a 8
DW
u u 1 cu 1 c2 I2 dx = − − − x5 4x4 8 ax2 8 a
DW
u dx 3 cu 3 c2 = − + + I2 x5 u 4ax4 8 a2 x2 8 a2
DW
5 c 15 c2 15 c2 dx 1 + + + =− I2 5 3 4 2 2 3 x u 4ax u 8 a x u 8 a u 8 a3 DW
2.278
1. 2.
u3 u5 dx = − x6 5ax5
DW
u u3 2 cu3 dx = − + 6 5 x 5ax 15 a2 x3
DW
2.284
Forms containing
3. 4.
√
a + bx + cx2 and polynomials
103
2 cu3 c2 u u5 − DW − 5+ 5x 3 x3 x
n+2 n + 2 c2 u (−1)k n + 2 k x 2k−5 1 1 n + 2 cu3 u5 + = n+3 − 5 + − c 2 a 5x 3 1 x3 x 2k − 5 k u
1 dx = 3 6 x u a dx x6 u2n+1
2.28 Forms containing
k=3
√ a + bx + cx2 and first- and second-degree polynomials
Notation: R = a + bx + cx2 See also 2.252 tn−1 dt dx 3 √ =− 2.281 (x + p)n R c + (b − 2pc)t + (a − bp + cp2 ) t2 t= 2.282 1.
3
√ dx x dx dx R dx √ = c √ + (b − cp) √ + a − bp + cp2 x+p R R (x + p) R
2. 3. 4. 5.
[x + p > 0]
1 1 dx dx dx √ = √ + √ q − p p − q (x + p)(x + q) R (x + p) R (x + q) R √ √ √ 1 1 R dx R dx R dx = + (x + p)(x + q) q−p x+p p−q x+q √ √ √ R dx (x + p) R dx = R dx + (p − q) x+q x+q (rx + s) dx s − pr s − qr dx dx √ = √ + √ q − p p − q (x + p)(x + q) R (x + p) R (x + q) R
2.283
1 >0 x+p
(Ax + B) dx A √ = n c (p + R) R
2Bc − Ab du n + 2c (p + u2 )
n−1 dυ 1 − cυ 2 n , 2 b − cpυ 2 p+a− 4c
√ b + 2cx √ . where u = R and υ = 2c R 2Bc − Ab A Ax + B √ dx = I1 + I2 , 2.284 c (p + R) R c2 p [b2 − 4(a + p)c] where . R 1 [p > 0] I1 = √ arctan p p √ √ 1 −p − R √ = √ [p < 0] ln √ 2 −p −p + R
104
Algebraic Functions
b + 2cx p √ b2 − 4(a + p)c R b + 2cx p √ = − arctan b2 − 4(a + p)c R √ √ 4(a + p)c − b2 R + p(b + 2cx) 1 √ ln = √ 2i 4(a + p)c − b2 R − p(b + 2cx) √ √ b2 − 4(a + p)c R − −p(b + 2cx) 1 √ ln = √ 2i b2 − 4(a + p)c R + −p(b + 2cx)
I2 = arctan
2.290
2 p b − 4(a + p)c > 0,
p 0,
p>0
2 p b − 4(a + p)c < 0,
p>0
2 p b − 4(a + p)c < 0,
p1 1. = − F α, 2 2 a a a sin x − 1 √ √ 2−1 2−1 a a 1 − a E α, 2. a2 sin2 x − 1 dx = F α, a a a 2 a >1 √ r 2 a2 − 1 a2 − 1 dx 1 Π α, 2 2 , 3. = a (r2 − 1) a (r − 1) a 1 − r2 sin2 x a2 sin2 x − 1 2 a > 1, r2 > 1 2 sin x dx α a >1 4. =− a a2 sin2 x − 1 2 cos x dx 1 5. a >1 = ln a sin x + a2 sin2 x − 1 2 2 a a sin x − 1 2 dx cos x a >1 6. = − arctan sin x a2 sin2 x − 1 a2 sin2 x − 1 √ a2 − 1 sin x + a2 sin2 x − 1 1 dx ln √ = √ 7. 2 a2 − 2 cos x a2 sin2 x − 1 a2 − 1 sin x − a2 sin2 x − 1 2 a >1 √ a2 − 1 + a2 sin2 x − 1 tan x dx 1 8. ln √ = √ 2 a2 − 1 a2 sin2 x − 1 a2 − 1 − a2 sin2 x − 1 2 a >1
2 1 cot x dx a >1 = − arcsin 9. a sin x a2 sin2 x − 1 2.611 To calculate integrals of the type R sin x, cos x, a2 sin2 x − 1 dx for a2 > 1,
BY (285.00)a
BY (285.06)a
BY (285.02)a
we may use
formulas 2.583 and 2.584. In doing so, we should follow the procedure outlined below: (1)
In the right members of these formulas, the following functions should be replaced with integrals equal to them:
206
Trigonometric Functions
2.611
F (x, k)
should be replaced with
E (x, k)
should be replaced with
1 − ln (k cos x + Δ) k
should be replaced with
1 arcsin (k sin x) k
should be replaced with
1 Δ − cos x ln 2 Δ + cos x
should be replaced with
1 Δ + k sin x ln 2k Δ − k sin x
should be replaced with
1 Δ + k ln 2k Δ − k
should be replaced with
1 1−Δ ln 2 1+Δ
should be replaced with
(2)
dx Δ Δdx
sin x dx Δ cos x dx Δ dx Δ sin x dx Δ cos x tan x dx Δ cot x dx Δ
2 Then, on both sides of the equations, we should replace Δ with i a2 sin2 x − 1, k with a and k 2 with 1 − a .
(3)
Both sides of theresulting equations should be multiplied by i, as a result of which only real functions a2 > 1 should appear on both sides of the equations.
(4)
The integrals on the right sides of the equations should be replaced with their values found from formulas 2.599.
Examples: 1.
We rewrite equation 2.584 4 in the form sin2 x dx 1 1 dx = 2 − 2 i a2 sin2 x − 1 dx, a i a2 sin2 x − 1 i a2 sin2 x − 1 a from which we get √ 2−1 a sin2 x dx 1 dx 1 a2 sin2 x − 1 dx = − E α, = 2 + a a a a2 sin2 x − 1 a2 sin2 x − 1 2 a >1
2.
We rewrite equation 2.584 58 as follows: 2a4 a2 − 2 sin2 x − 3a2 − 5 a2 dx 1]
√ 2 a sin x a −1 2 2 a cos x − 1 dx = a E arcsin √ , a a2 − 1
√ 2 a sin x 1 a −1 [a > 1] − F arcsin √ , a a a2 − 1
2.616
Elliptic and pseudo-elliptic integrals
11
2.616
Integrals of the form R sin x, cos x, 1 − p2 sin x . Notation: α = arcsin 1 − p2 sin2 x 1.
1, |x| > π] , + π |x| < , n > 0 2
|E2k |xn+2k+1 (n + 2k + 1)(2k)!
GU (333)(9b) GU (333)(10b)
∞ 1 |E2k |x2k−n+1 dx n |En−1 | = [1 − (−1) ln x + ] n x cos x 2 (n − 1)! (2k − n + 1) · (2k)! k=0 k= n−1 2
+ π, |x| < 2
GU (333)(11b)
222
Trigonometric Functions
7. 8.
2.644
∞ 22k xn+2k−1 n xn dx n n−1 = −x x B2k cot x + + n (−1)k 2 n−1 (n + 2k − 1)(2k)! sin x k=1 [|x| < π, n > 1]
GU (333)(8c)
n
xn
n+1 cot x 2 n n dx =− n + Bn+1 ln x − [1 − (−1)n ] (−1) 2 2 n+1 x (n + 1)x (n + 1)! sin x ∞ n (−1)k (2x)2k − n+1 B2k 2 (2k − n − 1)(2k)! k=1 k= n+1 2
9.
xn dx = xn tan x + n cos2 x
10.
xn
∞ k=1
(−1)k
2
2k
22k − 1 xn+2k−1 B2k (n + 2k − 1) · (2k)! + n > 1,
1.
2.
3.
GU (333)(9c)
|x|
1] 4.
5. 6.
m
xp dx k sinn−2k x k=0 m xp cos x cos2m+1 x k m dx = dx xp (−1) k sinn x sinn−2k x k=0 p−1 cos x dx xp x p xp n = − + sin x (n − 1) sinn−1 x n − 1 sinn−1 x xp
cos2m x dx = sinn x
m
(−1)k
(see 2.643 2)
GU (333)(12)
(see 2.643 1)
(see 2.645 6) [n > 1]
(see 2.643 1)
GU (333)(13)
224
Trigonometric Functions
2.646
7. 8. 2.646
x x cos x x + ln tan dx = − sin x 2 sin2 x x x π x sin x dx = − ln tan + 2 cos x cos x 2 4
p
1.
x tan x dx =
∞
(−1)
k=1
xp cot x dx =
2.
(−1)k
k=0
3.
∞
22k−1 − 1 B2k xp+2k (p + 2k) · (2k)!
k+1 2
2k
22k B2k xp+2k (p + 2k)(2k)!
xp tan2 x dx = x tan x + ln cos x −
x2 2
x cot2 x dx = −x cot x + ln sin x −
x2 2
4. 2.647
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
+ p ≥ −1, [p ≥ 1,
|x|
0)
TI (603)
238
Logarithms and Inverse-Hyperbolic Functions
2.721
2.72–2.73 Combinations of logarithms and algebraic functions 2.721
xn lnm x dx =
1.
m xn+1 lnm x − n+1 n+1
xn lnm−1 x dx
(see 2.722)
For n = −1 m ln x dx lnm+1 x 2. = x m+1 For n = −1 and m = −1 dx = ln (ln x) 3. x ln x m xn+1 lnm−k x (−1)k (m + 1)m(m − 1) · · · (m − k + 1) 2.722 xn lnm x dx = m+1 (n + 1)k+1
TI (604)
k=0
2.723 1. 2. 3. 2.724
ln x 1 x ln x dx = x − n + 1 (n + 1)2 2 ln x 2 ln x 2 − xn ln2 x dx = xn+1 + n + 1 (n + 1)2 (n + 1)3 3 ln x 3 ln2 x 6 ln x 6 − xn ln3 x dx = xn+1 + − n + 1 (n + 1)2 (n + 1)3 (n + 1)4 n
1.
2. 2.725 1. 2.
n+1
xn dx xn+1 n+1 m =− m−1 + m − 1 (ln x) (m − 1) (ln x)
TI 375
TI 375
xn dx m−1
(ln x)
For m = 1 n x dx = li xn+1 ln x 1 (a + bx)m+1 dx (a + bx)m+1 ln x − (m + 1)b x m m m−k k k+1 b x 1 m m+1 m+1 k a (a + bx) ln x − (a + bx) ln x dx = −a (m + 1)b (k + 1)2
(a + bx)m ln x dx =
k=0
For m = −1, see 2.727 2. 2.726
(a + bx)2 a2 1 − 1. (a + bx) ln x dx = ln x − ax + bx2 2b 2b 4
b2 x3 1 abx2 2 3 3 2 2. (a + bx) ln x dx = (a + bx) − a ln x − a x + + 3b 2 9
TI 374
2.731
Combinations of logarithms and algebraic functions
239
1 3 2 2 1 2 3 1 3 4 4 4 3 (a + bx) − a ln x − a x + a bx + ab x + b x (a + bx) ln x dx = 4b 4 3 16 3
3. 2.727 8
1.
2.8 3. 4. 5.
2.728 1. 2.9 2.729
1 dx ln x dx ln x = + − (a + bx)m b(m − 1) (a + bx)m−1 x(a + bx)m−1 For m = 1 1 1 ln(a + bx) dx ln x dx = ln x ln(a + bx) − (see 2.728 2) a + bx b b x 1 x ln x dx ln x + ln =− 2 (a + bx) b(a + bx) ab a + bx 1 x ln x dx ln x 1 + ln =− + (a + bx)3 2b(a + bx)2 2ab(a + bx) 2a2 b a + bx
√ √ (a + bx)1/2 − a1/2 ln x dx 2 √ = [a > 0] (ln x − 2) a + bx − 2 a ln x1/2 a + bx b √ √ a + bx 2 (ln x − 2) a + bx + 2 −a arctan [a < 0] = b −a
m+1 dx 1 x xm+1 ln(a + bx) − b m+1 a + bx
bx bx ln(a + bx) = ln a ln x + Φ − , 2, 1 [a > 0] x a a
xm ln(a + bx) dx =
m+1 1 (−a)m+1 1 (−1)k xm−k+2 ak−1 m+1 x ln(a + bx) dx = − ln(a + bx) + x m+1 bm+1 m+1 (m − k + 2)bk−1 k=1 ax 1 2 a2 1 x2 − x ln(a + bx) dx = x − 2 ln(a + bx) − 2 b 2 2 b 3 3 ax2 a2 x 1 3 a 1 x 2 − + 2 x ln(a + bx) dx = x + 3 ln(a + bx) − 3 b 3 3 2b b ax3 a2 x2 1 4 a4 a3 x 1 x4 x3 ln(a + bx) dx = − + − x − 4 ln(a + bx) − 4 b 4 4 3b 2b2 b3 ⎧ ⎨ 1 x x2n ln x2 + a2 dx = x2n+1 ln x2 + a2 + (−1)n 2a2n+1 arctan 2n + 1 ⎩ a ⎫ n (−1)n−k 2n−2k 2k+1 ⎬ a x −2 ⎭ 2k + 1 m
1. 2. 3. 4. 2.731
TI 376
k=0
240
Logarithms and Inverse-Hyperbolic Functions
x2n+1 ln x2 + a
2.7327
2
dx =
2.732
⎧ ⎨
1 x2n+2 + (−1)n a2n+2 ln x2 + a2 2n + 2 ⎩ ⎫ n+1 ⎬ (−1)n−k a2n−2k+2 x2k + ⎭ k k=1
2.733
1. 2. 3. 4. 5. 2.734
x ln x2 + a2 dx = x ln x2 + a2 − 2x + 2a arctan a 2 1 x2 + a2 ln x2 + a2 − x2 x ln x + a2 dx = 2 2 3 2 1 3 2 x 2 2 2 2 3 x ln x + a dx = x ln x + a − x + 2a x − 2a arctan 3 3 a x4 1 4 + a2 x2 x − a4 ln x2 + a2 − x3 ln x2 + a2 dx = 4 2 2 5 2 2 3 2 1 5 2 x 4 2 2 4 5 x ln x + a dx = x ln x + a − x + a x − 2a x + 2a arctan 5 5 3 a ! ! x2n ln !x2 − a2 ! dx
DW DW DW DW DW
! ! n ! 2 ! !x + a! 1 2! 2n+1 2n−2k 2k+1 ! ! ! a −2 ln x − a + a ln ! x x x − a! 2k + 1 k=0 1 ! 2 ! ! n+1 2n+2 ! 2 1 2n+1 2 2n+2 2 2n−2k+2 2k 2.735 x a x ln !x − a ! − ln !x − a ! dx = −a x 2n + 2 k 1 = 2n + 1
2n+1
k=1
2.736
1. 2. 3. 4. 5.
! ! ! !x + a! ! ! ! ! ln !x2 − a2 ! dx = x ln !x2 − a2 ! − 2x + a ln !! x − a! ! ! ! ! 1 2 x − a2 ln !x2 − a2 ! − x2 x ln !x2 − a2 ! dx = 2 ! !
! 2 ! 2 !x + a! ! ! 2 3 1 2 2! 3 2! 2 3 ! ! ! ! x ln x − a dx = x ln x − a − x − 2a x + a ln ! 3 3 x − a!
! 2 ! ! x4 ! 2 1 4 3 2! 4 2! 2 2 ! ! x − a ln x − a − −a x x ln x − a dx = 4 2 ! !
! 2 ! 2 !x + a! ! ! 2 5 2 2 3 1 4 2! 5 2! 4 5 ! ! ! ! x ln x − a dx = x ln x − a − x − a x − 2a x + a ln ! 5 5 3 x − a!
DW DW DW DW DW
2.74 Inverse hyperbolic functions 2.741 x x 1. arcsinh dx = x arcsinh − x2 + a2 a a
DW
2.813
Arcsines and arccosines
2.
arccosh
3. 4. 2.742 1. 2.
x x dx = x arccosh − x2 − a2 a a x 2 = x arccosh + x − a2 a
241
+ , x arccosh > 0 a + , x arccosh < 0 a
x a x dx = x arctanh + ln a2 − x2 a a 2 x a x arccoth dx = x arccoth + ln x2 − a2 a a 2 arctanh
x2 a2 + 2 4
DW
DW DW
x x 2 − x + a2 a 4 2
x a2 x x x 2 − x arccosh dx = x − a2 arccosh − a 4
a 4 22 x a2 x x 2 = − x − a2 arccosh + 2 4 a 4 x x arcsinh dx = a
DW
arcsinh
DW
+ , x arccosh > 0 a + , x arccosh < 0 a DW
2.8 Inverse Trigonometric Functions 2.81 Arcsines and arccosines n/2
n x n x n−2k arcsin dx = x (−1)k · (2k)! arcsin 2k a a k=0
(n+1)/2
n x n−2k+1 k−1 2 2 + a −x (−1) · (2k − 1)! arcsin 2k − 1 a k=1 n/2
n x n x n−2k 2.812 arccos dx = x (−1)k · (2k)! arccos 2k a a k=0
(n+1)/2
n x n−2k+1 k 2 2 + a −x (−1) · (2k − 1)! arccos 2k − 1 a 2.811
k=1
2.813 x x + a2 − x2 arcsin dx = sign(a) x arcsin 1.11 a |a|
2 x 2 x x arcsin − 2x 2.9 dx = x arcsin + 2 a2 − x2 arcsin a |a| |a| ⎡
3
2 x 3 x x 3. arcsin dx = sign(a) ⎣x arcsin + 3 a2 − x2 arcsin a |a| |a| ⎤ x − 6 a2 − x2 ⎦ − 6x arcsin |a|
242
Inverse Trigonometric Functions
2.814
2.814 x x 1. arccos dx = x arccos − a2 − x2 a a 2 x x 2 x 2. arccos dx = x arccos − 2 a2 − x2 arccos − 2x a a a 3 3 x x 2 x x arccos 3. dx = x arccos − 3 a2 − x2 arccos − 6x arccos + 6 a2 − x2 a a a a
2.82 The arcsecant, the arccosecant, the arctangent, and the arccotangent 2.821 x a x 1. arccosec dx = arcsin dx = x arcsin + a ln x + x2 − a2 a x 2 a = x arcsin − a ln x + x2 − a2 x 2.
a a x arcsec dx = arccos dx = x arccos − a ln x + x2 − a2 a x x a = x arccos − a ln x + x2 − a2 x
+ a π, 0 < arcsin < x 2 , + π a − < arcsin < 0 2 x DW
+ a π, 0 < arccos < x 2 , + π a − < arccos < 0 2 x DW
2.822 x a x 1.8 arctan dx = x arctan − ln a2 + x2 a a 2 x a x 2. arccot dx = x arccot − ln a2 + x2 a a 2 1 2 x ax x 9 x + a2 arctan − 3. x arctan dx = a 2 a 2 2 ax πx 1 2 x x + − x + a2 arctan 4.9 x arccot dx = a 2 4 2 a ax2 1 x x 1 5.9 x2 arctan dx = x3 arctan + a3 ln x2 + a2 − a 3 a 6 6 1 ax2 πx3 x x 1 + 6.9 x2 arccot dx = − x3 arctan − a3 ln x2 + a2 + a 3 a 6 6 6
2.83 Combinations of arcsine or arccosine and algebraic functions
DW DW
n+1 dx xn+1 x 1 x x √ arcsin − 2.831 x arcsin dx = (see 2.263 1, 2.264, 2.27) 2 a n + 1 a n + 1 − x2 an+1 n+1 dx x x 1 x x √ arccos + 2.832 xn arccos dx = (see 2.263 1, 2.264, 2.27) 2 a n+1 a n+1 a − x2 arccos x arcsin x dx and dx) cannot be expressed as a 1. For n = −1, these integrals (that is, x x finite combination of elementary functions. n
2.838
Arcsine or arccosine and algebraic functions
2. 2.8339
1. 2. 3. 4. 5. 6. 2.834
π 1 arccos x dx = − ln − x 2 x
243
arcsin x dx x
x2 a2 x 2 x 2 − + a −x arcsin 2 4 |a| 4 1 2 πx2 x 2 x x 2 2 − sign(a) 2x − a arcsin + a −x x arccos dx = a 4 4 |a| 4 3 x x 1 2 x arcsin + x + 2a2 a2 − x2 x2 arcsin dx = sign(a) a 3 |a| 9 3 3 x πx x 1 2 x 2 2 2 2 − sign(a) arcsin + x + 2a a −x x arccos dx = a 6 3 |a| 9 4
x 3a4 1 x x − + x 2x2 + 3a2 a2 − x2 x3 arcsin dx = sign(a) arcsin a 4 32 |a| 32 % & 4 4 4 8x − 3a πx x 1 x x3 arccos dx = − sign(a) arcsin + x 2x2 + 3a2 a2 − x2 a 8 32 |a| 32
x x arcsin dx = sign(a) a
√
a2 − x2 x √ 1 x 1 a + a2 − x2 x 1 arccos dx = − arccos − ln 2. 2 x a x a a x . 2 arcsin x 2 (a − b)(1 − x) arcsin x 2.835 − √ a > b2 dx = − arctan 2 2 2 (a + bx) b(a + bx) b a − b (a + b)(1 + x) 2 (a + b)(1 + x) + (b − a)(1 − x) 1 arcsin x − √ =− a < b2 ln b(a + bx) b b2 − a2 (a + b)(1 + x) − (b − a)(1 − x) √ 1 c + 1x arcsin x x arcsin x √ √ + dx = − [c > −1] arctan 2.8368 2 2) 2 2 2c (1 + cx 2c c + 1 1 (1 + cx ) √ −x 1 − x2 + x −(c + 1) 1 arcsin x + ln √ [c < −1] =− 2 2c (1 + cx ) 4c −(c + 1) 1 − x2 − x −(c + 1) 1.
1 x 1 a+ 1 x arcsin dx = − arcsin − ln 2 x a x a a
2.837 x arcsin x √ dx = x − 1 − x2 arcsin x 1. 2 1−x x arcsin x x x2 1 2 √ 2. − dx = 1 − x2 arcsin x + (arcsin x) 4 2 4 1 − x2 3 2x 1 2 x arcsin x x3 √ + − x +2 3. dx = 1 − x2 arcsin x 9 3 3 1 − x2 2.838 arcsin x x arcsin x 1 < 1. dx = √ + ln 1 − x2 2 2 3 1−x (1 − x2 )
244
Inverse Trigonometric Functions
2.
2.841
arcsin x 1 1−x x arcsin x < dx = √ + ln 2 2 1+x 3 1−x (1 − x2 )
2.84 Combinations of the arcsecant and arccosecant with powers of x 2.841
1.
x arcsec
2.
3.
a 1 2 a arccos dx = x arccos − a x2 − a2 x 2 x a 1 2 2 2 x arccos + a x − a = 2 x
x dx = a
+ a π, 0 < arccos < x 2, +π a < arccos < π 2 x DW
a 1 x a a 2 a3 2 3 2 2 2 ln x + x − a x arcsec dx = arccos dx = x arccos − x x − a − a x 3 x 2 2 + a π, 0 < arccos < x 2
1 a a 2 a3 3 2 2 2 = ln x + x − z x arccos + x x − a + 3 x 2 2 +π , a < arccos < π 2 x
DW
a 1 2 x a x arcsin + a x2 − a2 x arccosec dx = arcsin dx = a x 2 x a 1 2 2 2 x arcsin − a x − a = 2 x
+ a π, 0 < arcsin < x 2 , + π a − < arcsin < 0 2 x DW
2.85 Combinations of the arctangent and arccotangent with algebraic functions
2.851 2.852
3. 2.853 1. 2.
xn+1 x a x dx = arctan − a n+1 a n+1
xn arccot
xn+1 x a x dx = arccot + a n+1 a n+1
1.
2.
xn arctan
xn+1 dx a2 + x2 xn+1 dx a2 + x2
For n = −1 arctan x dx cannot be expressed as a finite combination of elementary functions. x arccot x π arctan x dx = ln x − dx x 2 x
1 2 x ax x dx = x + a2 arctan − a 2 a 2 1 2 x ax x x + a2 arccot + x arccot dx = a 2 a 2 x arctan
2.859
Arctangent and arccotangent with algebraic functions
245
ax2 x3 x a3 2 x dx = arctan + ln x + a2 − a 3 a 6 6 3 3 x x a ax2 πx3 x ln x2 + a2 + + x2 arccot dx = − arctan − 4.9 a 3 a 6 6 6 1 x 1 a2 + x2 x 1 ln arctan dx = − arctan − 2.854 2 a x x2
a 2a x arctan x 1 β − αx α + βx √ arctan x 2.855 dx = − ln (α + βx)2 α2 + β 2 α + βx 1 + x2 2.856 1 ln 1 + x2 dx x arctan x 1 2 1. − arctan x ln 1 + x dx = 1 + x2 2 2 1 + x2 2 1 x arctan x 1 2 2. dx = x arctan x − ln 1 + x2 − (arctan x) 2 1+x 2 2 3 1 x arctan x x arctan x 1 2 x + 1 + x arctan x − 3. dx = − dx 2 1+x 2 2 1 + x2 3.
x2 arctan
9
(see 2.8511)
x 1 2 − x arctan x + (arctan x) 3 2
TI (689) TI (405)
x arctan x 1 2 dx = − x2 + ln 1 + x2 + 2 1+x 6 3 & % n x (2n − 2k)!!(2n − 1)!! arctan x dx 1 (2n − 1)!! arctan x arctan x 2.857 + n+1 = (2n)!!(2n − 2k + 1)!! (1 + x2 )n−k+1 2 (2)!! (1 + x2 ) k=1 n 1 (2n − 1)!!(2n − 2k)!! 1 + 2 (2n)!!(2n − 2k + 1)!!(n − k + 1) (1 + x2 )n−k+1 k=1 √ √ 2 x arctan x x √ 2.858 dx = − 1 − x2 arctan x + 2 arctan √ − arcsin x 1 − x2 1 − x2 arctan x a + bx2 x arctan x 1 < 2.859 [a < b] dx = √ − √ arctan 2 b − a√ 3 a b−a a a + bx √ (a + bx2 ) x arctan x a + bx2 − a − b 1 √ = √ + √ [a > b] ln √ 2a a − b a a + bx2 a + bx2 + a − b 4.
4
3
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3–4 Definite Integrals of Elementary Functions 3.0 Introduction 3.01 Theorems of a general nature 3.011 Suppose that f (x) is integrable† over the largest of the intervals (p, q), (p, r), (r, q). Then (depending on the relative positions of the points p, q, and r) it is also integrable over the other two intervals, and we have q
r
f (x) dx = p
q
f (x) dx + p
f (x) dx.
FI II 126
r
3.012 The first mean-value theorem. Suppose (1) that f (x) is continuous and that g(x) is integrable over the interval (p, q), (2) that m ≤ f (x) ≤ M , and (3) that g(x) does not change sign anywhere in the interval (p, q). Then, there exists at least one point ξ (with p ≤ ξ ≤ q) such that q q f (x)g(x) dx = f (ξ) g(x) dx. FI II 132 p
p
3.013 The second mean-value theorem. If f (x) is monotonic and non-negative throughout the interval (p, q), where p 0, . . . , pn > 0, q1 > 0, q2 > 0, . . . , qn > 0, p1 + p2 + · · · + pn > r > 0]
= [p1 > 0,
···
4.647
exp
0≤x21 +x22 +···+x2n ≤1
p1 x1 + p2 x2 + · · · + pn xn x21 + x22 + · · · + x2n =
∞ ∞
4.6488 0
0
m
∞
··· 0
FI III 493
dx1 dx2 . . . dxn √ 2 πn n
n (p21 + p22 + · · · + p2n ) 4
exp − x1 + x2 + · · · + xn +
λn+1 x1 x2 . . . xn 1
− 12
I n2 −1
<
p21 + p22 + · · · + p2n FI III 495
−1
2
−1
n
×xc1n+1 x2n+1 . . . xnn+1 n 1 =√ (2π) 2 e−(n+1)λ n+1
−1
dx1 dx2 · · · dxn
FI III 496
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5 Indefinite Integrals of Special Functions 5.1 Elliptic Integrals and Functions Notation: k =
√ 1 − k 2 (cf. 8.1).
5.11 Complete elliptic integrals 5.111
K(k)k2p+3 dk =
1.
2.
1 (2p + 3)2
+ , 2 4(p + 1)2 K(k)k2p+1 dk + k 2p+2 E(k) − (2p + 3) K(k)k
⎧ ⎨ 1 E(k)k2p+3 dk = 2 4(p + 1)2 E(k)k2p+1 dk 4p + 16p + 15 ⎩ − E(k)k2p+2
+
BY (610.04)
⎫ ⎬ 2 2 (2p + 3)k − 2 − k 2p+2 k K(k) ⎭ ,
BY (611.04)
5.112 1.
⎡ ⎤ ∞ 2 πk ⎣ [(2j)!] k 2j ⎦ K(k) dk = 1+ 4 4j 2 j=1 (2j + 1)2 (j!) ⎡
2.6
E(k) dk =
3.
∞
2
BY (610.00)
⎤
2j
[(2j)!] k πk ⎣ ⎦ 1− 2 − 1) 24j (j!)4 2 (4j j=1
K(k)k dk = E(k) − k K(k) 2
BY (611.00)
BY (610.01)
4. 5.
, 1 + 2 1 + k 2 E(k) − k K(k) 3 , 1 + 2 4 + k 2 E(k) − k 4 + 3k 2 K(k) K(k)k3 dk = 9 E(k)k dk =
619
BY (611.01) BY (610.02)
620
Elliptic Integrals and Functions
5.113
6. 7. 8.
, 1 + 2 4 + k 2 + 9k 4 E(k) − k 4 + 3k 2 K(k) BY 611.02) 45 , 1 + 2 64 + 16k2 + 9k 4 E(k) − k 64 + 48k 2 + 45k 4 K(k) K(k)k5 dk = BY (610.03) 225 , 1 + 2 64 + 16k 2 + 9k 4 + 225k 6 E(k) − k 64 + 48k 2 + 45k 4 K(k) E(k)k5 dk = 1575 E(k)k3 dk =
BY (611.03)
9. 10. 11. 12. 13. 5.113
E(k) K(k) dk = − 2 k k + , E(k) 1 2 k dk = K(k) − 2 E(k) k2 k E(k) dk = k K(k) k 2 , E(k) 1 + 2 2 2 k dk = − 2 E(k) + k K(k) k4 9k 3 k E(k) dk = K(k) − E(k) k 2
dk = − E(k) k + , dk 2 2 E(k) − k K(k) = 2 E(k) − k K(k) k dk 2 = −k K(k) 1 + k 2 K(k) − E(k) k , dk 1+ 2 E(k) − k K(k) [K(k) − E(k)] 2 = k k + , dk 1 2 E(k) − k K(k) 2 2 = [K(k) − E(k)] k k k + , dk E(k) 2 1 + k2 E(k) − k K(k) 4 = kk k 2 [K(k) − E(k)]
1. 2. 3. 4. 5. 6.
5.114 5.115 1. 2. 3.
k K(k) dk E(k) −
k 2
1 2 = 2 k K(k) − E(k) K(k)
π 2 , r2 , k k dk = k 2 − r2 Π , r , k − K(k) + E(k) 2 2 + π , π 2 K(k) − Π , r2 , k k dk = k2 K(k) − k 2 − r2 Π ,r ,k 2 2 π π 2 2 E(k) 2 2 , r ,r ,k Π + Π , k k dk = k − r 2 2 2 k Π
π
BY (612.05) BY (612.02) BY (612.01) BY (612.03) BY (612.04)
BY (612.06) BY (612.09) BY (612.12) BY (612.07)
BY (612.13)
BY (612.11)
BY (612.14) BY (612.15) BY (612.16)
5.124
Elliptic integrals
621
5.12 Elliptic integrals
2 + F (x, k) dx [F (x, k)] π, = 0<x≤ 2 2 0 1 − k 2 sin2 x x 2 [E (x, k)] 5.12211 E (x, k) 1 − k 2 sin2 x dx = 2 0 5.123 x 1 F (x, k) sin x dx = − cos x F (x, k) + arcsin (k sin x) 1. k 0 .
x 1 1 − k 2 sin2 x 1 1 2. F (x, k) cos x dx = sin x F (x, k) + arccosh − arccosh 2 k k k k 0 x
5.121
5.124 1.
2.
3.∗
, 1+ k sin x 1 − k 2 sin2 x+arcsin (k sin x) 2k 0 ⎡ x 1 ⎣ k cos x 1 − k 2 sin2 x E (x, k) cos x dx = sin x E (x, k) + 2k 0 ⎤ .
2 2 1 1 − k sin x 2 2 ⎦ − k arccosh − k + k arccosh k k 2
5.∗
BY (630.11)
BY (630.21)
a
x E(x) dx π = 2√ 2 2 2 2 2 4 (k + k x ) a − x
BY (630.12)
BY (630.22)
√ a 1 − a2
a2 E (λ, k)
1−a
2
F (λ, k)
+ 2 + 2 3/2 3/2 (k 2 + k 2 a2 ) k (k 2 + k 2 a2 ) (k 2 + k 2 a2 )
a λ = arcsin √ [0 < a < 1, 0 < k < 1] k = 1 − k2 k 2 + k 2 a2 √ a F (φ, k) x E(x) dx π a2 E (φ, k) a 1 − a2 + √ = + √ 3/2 2 2 2 4 k 2 (k 2 − a2 ) k 2 k 2 − a2 a2 − x2 0 (k − x ) k 2 (k 2 − a2 ) a [0 < a < k < 1] φ = arcsin k π/2 E (x, k ) sin x cos x dx 0 1 − k 2 cosh2 v sin2 x 1 − k 2 sin2 x
tanh v 1 π tanh v π − [F (φ, k) − E (φ, k)] = 2 E(k ) arctanh − k sinh v cosh v 2 2
k tanh v 2 φ = arcsin [0 < tanh v < k < 1] k = 1−k k 0
4.∗
BY (630.32)
x
E (x, k) sin x dx = − cos x E (x, k)+
BY (630.01)
622
6.∗
Elliptic Integrals and Functions
5.124
π/2
E (x, k) sin x cos x dx 2 2 ψ sin2 x 1 − k cos 1 − k 2 sin2 x
tan ψ π tan ψ 1 π 2 2 1 − 1 − k cos ψ = 2 − E (β, k) + E(k) arctan k sin ψ cos ψ k 2 2 1 − k 2 cos2 ψ
+ π, tan ψ 0 < k < 1, 0 < ψ < k = 1 − k2 β = arctan k 2 π/2 E (x, k ) sin x cos x dx 0 1 + k 2 sinh2 μ sin2 x 1 − k 2 sin2 x < −1 π 2 2 = 2 E(k ) arctanh (k tanh μ) − F (φ, k) − E (φ, k) + tanh μ 1 + k sinh μ k sinh μ cosh 2 μ
k2 BY (630.13)
x
Π x, α2 , k cos x dx = sin x Π x, α2 , k − f − f0
0
% & 2 1 − α2 α2 − k 2 + 1 − α2 sin2 x 2k 2 − α2 − α2 k 2 f= arctan 2 (1 − α2 ) (α2 − k 2 ) 2α2 (1 − α2 ) (α2 − k 2 ) cos x 1 − k2 sin2 x for 1 − α2 α2 − k 2 > 0; % 2 2 α2 − 1 α2 − k 2 + 1 − α2 sin x α2 + α2 k 2 − 2k 2 1 = ln 1 − α2 sin2 x 2 (α2 − 1) (α2 − k 2 ) & 2α2 (α2 − 1) (α2 − k 2 ) cos x 1 − k 2 sin2 x + 1 − α2 sin2 x for 1 − α2 α2 − k 2 < 0, f0 is the value of f at x = 0 BY (630.23) where
1
Integration with respect to the modulus 2 5.126 F (x, k)k dk = E (x, k) − k F (x, k) + 1 − k 2 sin2 x − 1 cot x BY (613.01) , + 1 2 1 + k 2 E (x, k) − k F (x, k) + 5.127 E (x, k)k dk = 1 − k 2 sin2 x − 1 cot x BY (613.02) 3 5.128 Π x, r2 , k k dk = k 2 − r2 Π x, r2 , k − F (x, k) + E (x, k) + 1 − k 2 sin2 x − 1 cot x BY (613.03)
5.13 Jacobian elliptic functions 5.131 1.
⎡ 1 ⎣ sn m+1 u cn u dn u + (m + 2) 1 + k2 sn m u du = sn m+2 u du m+1 ⎤ − (m + 3)k2 sn m+4 u du⎦ SI 259, PE(567)
624
Elliptic Integrals and Functions
⎡
cn m u du =
2.
1 ⎣ − cn m+1 u sn u dn u (m + 1)k 2
+ (m + 2) 1 − 2k
2
cn m+2 u du + (m + 3)k 2
⎤ cn m+4 u du⎦ PE (568)
⎡
dn m u du =
3.
5.132
1 ⎣k 2 dn m+1 u sn u cn u (m + 1)k 2
+ (m + 2) 2 − k
2
⎤ dn m+2 u du − (m + 3) dn m+4 u du⎦ PE (569)
By using formulas 5.131, we can reduce the integrals (for m = 1) dn m u du to the integrals 5.132, 5.133 and 5.134. 5.132 sn u du = ln 1. sn u cn u + dn u dn u − cn u = ln sn u 1 du k sn u + dn u = ln 2. cn u k cn u 1 du k sn u − cn u = arctan 3. dn u k k sn u + cn u 1 cn u = arccos k dn u 1 cn u + ik sn u = ln ik dn u 1 k sn u = arcsin k dn u "
5.133 1.
2.
"
sn m u du,
"
cn m u du, and
H 87(164) SI 266(4) SI 266(5) H 88(166) JA SI 266(6) JA
1 ln (dn u − k cn u) k dn u − k 2 cn u 1 = arccosh 2 k 1−k
dn u − cn u 1 = arcsinh k ; k 1 − k2 1 = − ln (dn u + k cn u) k 1 cn u du = arccos (dn u) ; k i = ln (dn u − ik sn u) ; k 1 = arcsin (k sn u) k sn u du =
H 87(161) JA JA SI 365(1) H 87(162) SI 265(2)a, ZH 87(162) JA
5.137
Jacobian elliptic functions
625
3.
dn u du = arcsin (sn u) ; = am u = i ln (cn u − i sn u)
5.134 1. 2. 3.
H 87(163) SI 266(3), ZH 87(163)
1 [u − E (am u, k)] k2 , 1 + 2 cn 2 u du = 2 E (am u, k) − k u k dn 2 u du = E (am u, k) sn 2 u du =
5.135 1 dn u + k sn u du = ln 1. cn u k cn u 1 dn u + k = ln 2k dn u − k i sn u ik − k cn u du = ln 2. dn u kk dn u 1 k cn u = arccot kk k 1 − dn u cn u du = ln 3. sn u sn u 1 1 − dn u = ln 2 1 + dn u 1 1 − k sn u cn u du = − ln 4. dn u k dn u 1 + k sn u 1 ln = 2k 1 − k sn u 1 1 + sn u dn u du = ln 5. cn u 2 1 − sn u 1 + sn u = ln cn u 1 1 − cn u dn u du = ln 6. sn u 2 1 + cn u
PE (564) PE (565) PE (566)
SI 266(7) H 88(167) SI 266(8)
SI 266(10) H 88(168) SI 266(9)
H 88(172) JA H 87(170)
5.136 1 1. sn u cn u du = − 2 dn u k 2. sn u dn u du = − cn u 3. cn u dn u du = sn u 5.137 1.
1 dn u sn u du = 2 2 cn u k cn u
H 88(173)
626
Elliptic Integrals and Functions
2. 3. 4. 5. 6.
5.138
1 cn u sn u du = − 2 dn 2 u k dn u dn u cn u du = − sn 2 u sn u sn u cn u du = dn 2 u dn u cn u dn u du = − sn 2 u sn u sn u dn u du = cn 2 u cn u
H 88(175) H 88(174) H 88(177) H 88(176) H 88(178)
5.138 sn u cn u du = ln 1. sn u dn u dn u 1 sn u dn u du = 2 ln 2. cn u dn u cn u k sn u dn u du = ln 3. sn u cn u cn u
H 88(183) H 88(182) H 88(184)
5.139 cn u dn u du = ln sn u 1.11 sn u 1 sn u dn u du = ln 2. cn u cn u 1 sn u cn u du = − 2 ln dn u 3. dn u k
H 88(179) H 88(180) H 88(181)
5.14 Weierstrass elliptic functions The invariants g1 and g2 used below are defined in 8.161. 5.141 ℘(u) du = − ζ(u)
1.
2. 3. 4.8
1 1 ℘ (u) + g2 u 6 12 1 3 1 ℘ (u) − g2 ζ(u) + g3 u ℘3 (u) du = 120 20 10 1 du σ(u − v) = 2u ζ(v) + ln ℘(u) − ℘(v) ℘ (v) σ(u + v) ℘2 (u) du =
5.
H 120(192) H 120(193)
[℘(v) = e1 , e2 , e3 ]
(see 8.162) H 120(194)
au αδ − βγ α℘(u) + β σ(u + v) du = + 2 − 2u ζ(v) ln γ℘(u) + δ γ γ ℘ (v) σ(u − v) −1
where v = ℘
−δ γ
H 120(195)
5.221
The exponential integral function and powers
627
5.2 The Exponential Integral Function 5.21 The exponential integral function
∞
5.211
Ei(−βx) Ei(−γx) dx = x
1 1 + β γ
Ei[−(β + γ)x]
−x Ei(−βx) Ei(−γx) −
e−γx e−βx Ei(−γx) − Ei(−βx) β γ [Re(β + γ) > 0]
NT 53(2)
5.22 Combinations of the exponential integral function and powers 5.221
∞
1. x
k=0
∞
2. x
3.∗ 4.∗
n−1 1 Ei[−a(x + b)] (−1)n Ei[−a(x + b)] e−ab (−1)n−k−1 ∞ e−ax + dx = n − dx k+1 xn+1 x bn n n bn−k x x Ei[−a(x + b)] dx = x2
1 1 + x b
[a > 0,
Ei[−a(x + b)] −
b > 0]
NT 52(3)
e−ab Ei(−ax) b [a > 0, b > 0]
NT 52(4)
1 xe−ax x2 Ei(−ax) + 2 e−ax + [a > 0] 2 2a 2a ∞ n!e−ax (ax)k xn+1 Ei(−ax) + xn Ei(−ax) dx = n+1 (n + 1)an+1 k! x Ei(−ax) dx =
k=0
5.∗
6.∗
7.∗
8.∗
[a > 0] x Ei(−ax)e−bx dx =
[a > 0, 2 Ei(−ax)e−ax − Ei(−2ax) Ei2 (−ax) dx = x Ei2 (−ax) + a
x Ei2 (−ax) dx =
x2 2 Ei (−ax) + 2
Ei(−ax) dx = u Ei(−au) +
0
1 x + a2 a
b > 0]
[a > 0] Ei(−ax)e−ax −
1 1 Ei(−2ax) + 2 e−2ax a2 a
[a > 0] u
0
9.∗
1 1 x 1 e−(a+b)x Ei [−(a + b)x] − 2 Ei(−ax)e−bx − Ei(−ax)e−bx − 2 b b b b(a + b)
∞
e
−au
a
−1
[a > 0]
2
x x a + b2 b2 ab a2 Ei − dx = ln a − ln b − x Ei − ln(a + b) − a b 2 2 2 2 [a > 0,
b > 0]
628
10.
The Sine Integral and the Cosine Integral
∗
∞
0
5.231
x x ab 2 2 3 3 3 3 2 a + b ln(a + b) − a ln a − b ln b − a − ab + b x Ei − Ei − dx = a b 3 a+b 2
[a > 0,
b > 0]
5.23 Combinations of the exponential integral and the exponential 5.231
x
ex Ei(−x) dx = − ln x − C + ex Ei(−x)
1.
ET II 308(11)
0
1.
x
e−βx Ei(−αx) dx = −
0
1 β
β e−βx Ei(−αx) + ln 1 + − Ei[−(α + β)x] α
ET II 308(12)
5.3 The Sine Integral and the Cosine Integral 5.31
1.
cos αx ci(βx) dx =
sin αx ci(βx) dx = −
2. 5.32
1.
cos αx si(βx) dx =
1.
cos αx si(βx) si(αx + βx) − si(αx − βx) + α 2α
NT 49(1) NT 49(2)
NT 49(3) NT 49(4)
1 (si(αx + βx) + si(αx − βx)) ci(αx) ci(βx) dx = x ci(αx) ci(βx) + 2α 1 1 1 + (si(αx + βx) + si(βx − αx)) − sin αx ci(βx) − sin βx ci(αx) 2β α β NT 53(5)
1 (si(αx + βx) + si(αx − βx)) 2β 1 1 1 − (si(αx + βx) + si(βx + αx)) + cos αx si(βx) + cos βx si(αx) 2α α β
si(αx) si(βx) dx = x si(αx) si(βx) −
2.
3.
cos αx ci(βx) ci(αx + βx) + ci(αx − βx) + α 2α
sin αx si(βx) ci(αx + βx) − ci(αx − βx) + α 2α
sin αx si(βx) dx = −
2. 5.33
sin αx ci(βx) si(αx + βx) + si(αx − βx) − α 2α
NT 54(6)
1 cos αx ci(βx) α
1 1 1 1 1 − sin βx si(αx) − + − ci(αx + βx) − ci(αx − βx) β 2α 2β 2α 2β
si(αx) ci(βx) dx = x si(αx) ci(βx) +
NT 54(10)
5.54
Combinations of the exponential integral and the exponential
5.34
∞
si[a(x + b)]
1. x
∞
2.
ci[a(x + b)] x
dx = x2 dx = x2
1 1 + x b 1 1 + x b
si[a(x + b)] −
cos ab si(ax) + sin ab ci(ax) b [a > 0,
ci[a(x + b)] +
629
b > 0]
NT 52(6)
sin ab si(ax) − cos ab ci(ax) b [a > 0,
b > 0]
NT 52(5)
5.4 The Probability Integral and Fresnel Integrals
5.41
11
5.42 5.43
e−α x Φ(αx) dx = x Φ(αx) + √ α π cos2 αx2 √ S (αx) dx = x S (αx) + α 2π sin2 αx2 C (αx) dx = x C (αx) − √ α 2π 2
2
NT 12(20)a NT 12(22)a NT 12(21)a
5.5 Bessel Functions Notation: Z and Z denote any of J, N , H (1) , H (2) . In formulae 5.52–5.56, Z p (x) and Zp (x) are arbitrary Bessel functions of the first, second, or third kinds. ∞ J p+2k+1 (x) JA, MO 30 5.51 J p (x) dx = 2 k=0
5.52
1. 2.11 5.5310
xp+1 Z p (x) dx = xp+1 Z p+1 (x)
WA 132(1)
x−p Z p+1 (x) dx = −x−p Z p (x)
WA 132(2)
p2 − q 2 α2 − β 2 x − Z p (αx) Zq (βx) dx x = αx Z p+1 (αx) Zq (βx) − βx Z p (αx) Zq+1 (βx) − (p − q) Z p (αx) Zq (βx) = βx Z p (αx) Zq−1 (βx) − αx Z p−1 (αx) Zq (βx) + (p − q) Z p (αx) Zq (βx) JA, MO 30, WA 134(7)
5.54 1.
10
αx Z p+1 (αx) Zp (βx) − βx Z p (αx) Zp+1 (βx) α2 − β 2 βx Z p (αx) Zp−1 (βx) − αx Z p−1 (αx) Zp (βx) = α2 − β 2
x Z p (αx) Zp (βx) dx =
WA 134(8)
2.
2
x [Z p (αx)] dx =
x2 2 [Z p (αx)] − Z p−1 (αx) Z p+1 (αx) 2
WA 135(11)
630
3.
Bessel Functions
∗
x Z p (ax) Zp (ax) dx = 10
5.55
5.55
x4 2 Z p (ax) Zp (ax) − Z p−1 (ax) Zp+1 (ax) − Z p+1 (ax) Zp−1 (ax) 4
Z p (αx) Zq+1 (αx) − Z p+1 (αx) Zq (αx) Z p (αx) Zq (αx) 1 Z p (αx) Zq (αx) dx = αx + x p2 − q 2 p+q Z p−1 (αx) Zq (αx) − Z p (αx) Zq−1 (αx) Z p (αx) Zq (αx) = αx − p2 − q 2 p+q WA 135(13)
5.56
1.
Z 1 (x) dx = − Z 0 (x)
JA
x Z 0 (x) dx = x Z 1 (x)
JA
2.
6–7 Definite Integrals of Special Functions 6.1 Elliptic Integrals and Functions Notation: k =
√ 1 − k 2 (cf. 8.1).
6.11 Forms containing F (x, k)
π/2
6.111
F (x, k) cot x dx = 0
6.112
π/2
1. 0
F (x, k)
2 π sin x cos x 1 √ − K(k) ln K(k ) dx = 4k 1 − k sin2 x (1 − k) k 16k
F (x, k)
sin x cos x 1 dx = − 2 ln k K(k) 2k 1 − k 2 sin2 x
0
π/2
3. 0
6.113
π/2
1.
F (x, k )
0
π/2
2.
F (x, k) 0
BI (350)(1)
√ (1 + k) k π sin x cos x 1 K(k) ln + K(k ) F (x, k) dx = 4k 2 16k 1 + k sin2 x
π/2
2.
π 1 K(k ) + ln k K(k) 4 2
2 sin x cos x dx 1 √ K(k ) ln = 4(1 − k) (1 + k) k cos2 x + k sin2 x
BI (350)(6)
BI (350)(7)
BI (350)(2)a, BY(802.12)a
BI (350)(5)
sin x cos x dx · 1 − k 2 sin2 t sin2 x 1 − k 2 sin2 x =−
, + 1 π F (t, k) tan t) − K(k) arctan (k k 2 sin t cos t 2 BI (350)(12)
6.114
6.115
dx 1 K(k) K 1 − tan2 u cot2 v = 2 cos u sin v u sin2 x − sin2 u sin2 v − sin2 x 2 BI (351)(9) k = 1 − cot2 u · cot2 v √ 1 (1 + k) k π x dx 1 K(k) ln + K(k ) F (arcsin x, k) = 2 1 + kx 4k 2 16k 0 (cf. 6.112 2) BI (466)(1) v
F (x, k) 1]
(see 6.147)
[0 < a < 1]
p2 > 1
BY (615.14) LO I 252 FI II 489
6.16 The theta function 6.161 1.
∞
0
2. 0
∞
s xs−1 ϑ2 0 | ix2 dx = 2s 1 − 2−s π − 2 Γ 12 s ζ(s) s xs−1 ϑ3 0 | ix2 − 1 dx = π − 2 Γ 12 s ζ(s)
[Re s > 2]
ET I 339(20)
[Re s > 2]
ET I 339(21)
634
Elliptic Integrals and Functions
∞
3. 0
6.162
1 xs−1 1 − ϑ4 0 | ix2 dx = 1 − 21−s π − 2 s Γ 12 s ζ(s) [Re s > 2]
∞
4. 0
ET I 339(22)
1 xs−1 ϑ4 0 | ix2 + ϑ2 0 | ix2 − ϑ3 0 | ix2 dx = − (2s − 1) 21−s − 1 π − 2 s Γ 12 s ζ(s) ET I 339(24)
6.162
∞
1.11
e−ax ϑ4
0
∞
2.
e−ax ϑ1
0
∞
3.11
e−ax ϑ2
0
∞
4.11 0
6.16310
∞
1. 0
!
! iπx √ √ l ! dx = √ cosh b a cosech l a ! l2 a
[Re a > 0, !
√ √ bπ !! iπx l dx = − √ sinh b a sech l a ! 2 2l l a
|b| ≤ l]
ET I 224(1)a
[Re a > 0, !
√ √ (l + b)π !! iπx l dx = − √ sinh b a sech l a ! l2 2l a
|b| ≤ l]
ET I 224(2)a
[Re a > 0, !
√ √ (l + b)π !! iπx l dx = √ cosh b a cosech l a ! 2 2l l a
|b| ≤ l]
ET I 224(3)a
[Re a > 0,
|b| ≤ l]
ET I 224(4)a
√ √ 1 √ √ √ e−(a−μ)x ϑ3 (π μx |iπx ) dx = √ coth a + μ + coth a − μ 2 a [Re a > 0]
∞
2.10 0
e−ax ϑ3
bπ 2l
∞
ϑ3 (iπkx | iπx) e−(k
2
+l2 )x
dx =
ET I 224(7)a
sinh 2l l (cosh 2l − cos 2k)
1 ϑ4 0 | ie2x + ϑ2 0 | ie2x − ϑ3 0 | ie2x e 2 x cos(ax) dx 0 1 1 1 1 1 +ia = − 1 1 − 2 2 −ia π − 4 − 2 ia Γ 14 + 12 ia ζ 12 + ia 22 2 ET I 61(11) [a > 0] ∞ 1 6.165 e 2 x ϑ3 0 | ie2x − 1 cos(ax) dx 0 + − 1 ia− 1 1 , 2 2 1 1 1 2 4 Γ = 1 + a π ζ ia + + ia + 4 2 4 2 1 + 4a2 [a > 0] ET I 61(12) 6.16411
6.165
Generalized elliptic integrals
635
6.1710 Generalized elliptic integrals 1.
Set
π
Ωj (k) ≡
−(j+ 12 ) 1 − k 2 cos φ dφ,
0
j! (4m + 2j)! π αm (j) = (64)m (2j)! (2m + j)!
1 m!
2 ,
π λ= 2
(2j + 1)k 2 , 1 − k2
then
⎡
π 1 1 −j 4m 2 −1 ⎣ 1−k erf λ + (2j + 1) αm (j)k = 1+ 2 Ωj (k) = (2j + 1)k 2 2 2k m=0
2 1 1 13 2 2 −λ2 −2 √ λe × erf λ − − (2j + 1) 16 + 2 + 4 1+ λ 3 12 k k π ⎤
2 2 4 2 λe−λ + . . .⎦ × erf λ − √ 1 + λ2 + λ4 3 15 π
∞
while for large λ lim Ωj (k) =
j→∞
2.
−j π k2 1 − k2 (2j + 1)
1 1 1 4 13 −1 −2 × 1 + (2j + 1) + 1 + 2 − (2j + 1) 1+ + ... 2 2k 3 16k 2 16k 4
Set Rμ (k, α, δ) =
π
cos2α−1 (θ/2) sin2δ−2α−1 (θ/2) dθ
, μ+ 1 [1 − k 2 cos θ] 2 0 < k < 1, Re δ > Re α > 0, Re μ > −1/2, (−1)ν 2ν μ + 12 ν Γ(α) Γ (δ − α + ν) Mν (μ, α, δ) = , ν! Γ(δ + ν) with (λ)ν = Γ(λ + ν)/ Γ(λ), and ν 2 μ + 12 ν Γ(α + ν) Γ (δ − α) , Wν (μ, α, δ) = ν! Γ(δ + ν) 0
then: • for small k:
Rμ (k, α, δ) = 1 − k
1 2 −(μ+ 2 )
−(μ+ 12 ) = 1 + k2
∞ ν=0 ∞ ν=0
ν k2 / 1 − k2 Mν (μ, α, δ) ν k2 / 1 + k2 W ν (μ, α, δ),
636
The Exponential Integral Function and Functions Generated by It
6.211
• for k 2 close to 1: Rμ (k, α, δ) 2 α−δ δ−α−μ− 12 2k 1 − k2 = Γ(δ − α) Γ μ + α − δ + 12 Γ μ + 12 + μ+ 12 , × Γ δ − α − μ − 12 Γ(α) Γ δ − μ − 12 2k 2 Re μ + α − δ + 12 not an integer , + 1 = 2μ+ 2 k 2μ+1 Γ μ + 12 Γ(1 − α) ×
∞ n=0
α−δ+μ−n+ 1 2 Γ (δ − α + n) Γ(1 − α + n) Γ α − δ + μ − n + 12 n! 2k 2 / 1 − k 2 α − δ + μ + 12 = m, with m a non-negative integer
6.2–6.3 The Exponential Integral Function and Functions Generated by It 6.21 The logarithm integral
1
li(x) dx = − ln 2
6.211
BI (79)(5)
0
6.212
1
1. 0
1
2. 0
1 li x dx = 0 x 1 li(x)xp−1 dx = − ln(p + 1) p
1
3.
li(x) 0
∞
4.
li(x) 1
6.213 1. 2. 3. 4. 5.
dx 1 = ln(1 − q) xq+1 q dx 1 = − ln(q − 1) xq+1 q
1 π 1 a ln a − li sin (a ln x) dx = x 1 + a2 2 0 ∞
1 π 1 + a ln a li sin (a ln x) dx = − 2 x 1+a 2 1 1
1 π 1 a li ln a + cos (a ln x) dx = − x 1 + a2 2 0 ∞
1 π 1 ln a − a li cos (a ln x) dx = x 1 + a2 2 1 1 ln 1 + a2 dx = li(x) sin (a ln x) x 2a 0
BI (255)(1)
[p > −1]
BI (255)(2)
[q < 1]
BI (255)(3)
[q > 1]
BI (255)(4)
[a > 0]
BI (475)(1)
[a > 0]
BI (475)(9)
[a > 0]
BI (475)(2)
[a > 0]
BI (475)(10)
[a > 0]
BI(479)(1), ET I 98(20)a
1
6.216
The logarithm integral
6. 7. 8. 9. 10. 11.
1
arctan a dx =− x a 0 1 dx 1 π li(x) sin (a ln x) 2 = a ln a + [a > 0] x 1 + a2 2 0 ∞ dx 1 π − a ln a [a > 0] li(x) sin (a ln x) 2 = x 1 + a2 2 1 1 π dx 1 ln a − a [a > 0] li(x) cos (a ln x) 2 = x 1 + a2 2 0 ∞ π dx 1 ln a + a [a > 0] li(x) cos (a ln x) 2 = − x 1 + a2 2 1
1 a a 1 p−1 2 2 ln (1 + p) + a − p arctan li(x) sin (a ln x) x dx = 2 a + p2 2 1+p 0 li(x) cos (a ln x)
1
li(x) cos (a ln x) xp−1 dx = −
12. 0
6.214 1. 2. 6.215
p a + ln (1 + p)2 + a 1+p 2
li
1
xp−1 dx = −2 1 ln x
li(x) 0
2
BI (479)(3) BI (479)(13)
BI (479)(4) BI (479)(14)
BI (477)(1)
[p > 0]
BI (477)(2)
[0 < p < 1]
BI (340)(1)
[0 < p < 1]
BI (340)(9)
1
π √ arcsinh p = −2 p
π √ ln p+ p+1 p
[p > 0]
BI (444)(3)
[1 > p > 0]
BI (444)(4)
ax 1 = − Γ(p) x p
[0 < p ≤ 1]
BI (444)(1)
dx π Γ(p) =− 2 x sin pπ
[0 < p < 1]
BI (444)(2)
dx . = −2 1 xp+1 ln x
p−1 1 li(x) ln x 0 p−1 1 1 li(x) ln x 0
li(x) .
1
2.
2.
[p > 0]
BI (479)(2)
1
0
1.
1 a2 + p2
a arctan
p−1 1 1 dx = −π cot pπ · Γ(p) ln x x 0 ∞
1 π p−1 Γ(p) li dx = − (ln x) x sin pπ 1
1.
6.216
637
π √ arcsin p p
638
The Exponential Integral Function and Functions Generated by It
6.221
6.22–6.23 The exponential integral function
6.221 6.222
6.223
1 − eαp NT 11(7) 0 α
∞ 1 1 ln q ln p + − Ei(−px) Ei(−qx) dx = ln(p + q) − p q p q 0 [p > 0, q > 0] FI II 653, NT 53(3) ∞ Γ(μ) Ei(−βx)xμ−1 dx = − μ [Re β ≥ 0, Re μ > 0] μβ 0 p
Ei(αx) dx = p Ei(αp) +
NT 55(7), ET I 325(10)
6.224
∞
1.
Ei(−βx)e−μx dx = −
0
μ 1 ln 1 + μ β
= −1/β
[Re(β + μ) ≥ 0,
μ > 0]
[μ = 0] FI II 652, NT 48(8)
∞
2.
Ei(ax)e−μx dx = −
0
μ
1 ln −1 μ a
[a > 0,
Re μ > 0,
μ > a]
ET I 178(23)a, BI (283)(3)
6.225
∞
1.
Ei −x
2
e
−μx2
0
∞
2. 0
6.226 1. 2. 3.
2 px2 π √ arcsin p Ei −x e dx = − p
∞
∞
4. 0
1. 0
∞
[Re μ > 0]
BI (283)(5), ET I 178(25)a
[1 > p > 0]
NT 59(9)a
1 2 √ Ei − e−μx dx = − K 0 ( μ) 4x μ 0 ∞ 2
a 2 √ Ei e−μx dx = − K 0 (a μ) 4x μ 0
∞ π 1 √ −μx2 Ei (− μ) Ei − 2 e dx = 4x μ 0
6.227
π π √ √ arcsinh μ = − ln dx = − μ+ 1+μ μ μ
1 Ei − 2 4x
e
−μx2 +
Ei(−x)e−μx x dx =
[Re μ > 0]
MI 34
[a > 0,
MI 34
Re μ > 0]
[Re μ > 0]
MI 34
1 π √ √ √ √ 2 4x dx = [cos μ ci μ − sin μ si μ] μ
1 1 − 2 ln(1 + μ) μ(μ + 1) μ
[Re μ > 0]
MI 34
[Re μ > 0]
MI 34
6.241
The sine integral and cosine integral functions
∞
2. 0
e−ax Ei(ax) eax Ei(−ax) − dx = 0 x−b x+b = π 2 e−ab
[a > 0,
b < 0]
[a > 0,
b > 0]
639
ET II 253(1)a
6.228
∞
1.
Ei(−x)ex xν−1 dx = −
0
∞
2.
Ei(−βx)e
−μx ν−1
x
0
6.231
1. 2. 6.233
[0 < Re ν < 1]
Γ(ν) μ dx = − 2 F 1 1, ν; ν + 1; ν(β + μ)ν β+μ [|arg β| < π, Re(β + μ) > 0,
ET II 308(13)
Re ν > 0]
√ dx 1 1 √ √ √ √ Ei − 2 exp −μx2 + 2 = 2 π (cos μ si μ − sin μ ci μ) 2 4x 4x x 0 [Re μ > 0] ∞ −x −μx 1 [a < 1, Re μ > 0] Ei(−a) − Ei −e e dx = γ(μ, a) μ − ln a
6.229
6.232
π Γ(ν) sin νπ
∞
b2 ∞ ln 1 + 2 a Ei(−ax) sin bx dx = − 2b 0 ∞ b 1 Ei(−ax) cos bx dx = − arctan b a 0
∞
1.
Ei(−x)e−μx sin βx dx = −
0
∞
2.
Ei(−x)e−μx cos βx dx = −
0
1 β 2 + μ2
β2
1 + μ2
MI 34 MI 34
[a > 0,
b > 0]
BI (473)(1)a
[a > 0,
b > 0]
BI (473)(2)a
β β ln (1 + μ)2 + β 2 − μ arctan 2 1+μ [Re μ > |Im β|] μ β ln (1 + μ)2 + β 2 + β arctan 2 1+μ [Re μ > |Im β|]
ET II 308(14)
BI (473)(7)a
BI (473)(8)a
∞
Ei(−x) ln x dx = C + 1
6.234
NT 56(10)
0
6.24–6.26 The sine integral and cosine integral functions 6.241
∞
1.
si(px) si(qx) dx =
π 2p
[p ≥ q]
BI II 653, NT 54(8)
ci(px) ci(qx) dx =
π 2p
[p ≥ q]
FI II 653, NT 54(7)
0
2.
0
∞
640
The Exponential Integral Function and Functions Generated by It
∞
3. 0
2 2
2 p − q2 p+q 1 1 ln ln si(px) ci(qx) dx = + 4q p−q 4p q4 1 = ln 2 q
6.242
[p = q] [p = q] FI II 653, NT 54(10, 12)
∞
6.242 0
6.243
1 ci(ax) 2 2 dx = − [si(aβ)] + [ci(aβ)] β+x 2
2. 6.244
∞
1.8 2.
x dx π = − ci(pq) q 2 − x2 2
[p > 0,
q > 0]
BI (255)(6)
[p > 0,
q > 0]
BI (255)(7)
[p > 0,
q > 0]
BI (255)(8)
[a > 0,
0 < Re μ < 1]
∞
ci(px) 0
∞
1.
q2
dx π Ei(−pq) = 2 +x 2q
dx π si(pq) = q 2 − x2 2q
si(ax)xμ−1 dx = −
0
ET II 253(2)
si(px)
0
2.
[a > 0]
BI (255)(6)
ci(px)
6.246
ET II 253(3)
q > 0]
∞
1.
b > 0]
[p > 0,
0
[a > 0,
x dx π = Ei(−pq) q 2 + x2 2
∞
8
6.245
ET II 224(1)
si(px) 0
|arg β| < π]
∞
si (a|x|) sign x dx = π ci (a|b|) −∞ x − b ∞ ci (a|x|) dx = −π sign b · si (a|b|) −∞ x − b
1.
[a > 0,
Γ(μ) μπ sin μ μa 2
NT 56(9), ET I 325(12)a ∞
2.
ci(ax)xμ−1 dx = −
0
Γ(μ) μπ cos μaμ 2
[a > 0,
0 < Re μ < 1] NT 56(8), ET I 325(13)a
6.247
∞
μ 1 arctan μ β 0 . ∞ 1 μ2 −μx ci(βx)e dx = − ln 1 + 2 μ β 0
1.
2. 6.248 1.
8
si(βx)e−μx dx = −
∞
si(x)e 0
−μx2
1 π x dx = Φ −1 √ 4μ 2 μ
[Re μ > 0]
NT 49(12), ET I 177(18)
[Re μ > 0]
NT 49(11), ET I 178(19)a
[Re μ > 0]
MI 34
6.253
The sine integral and cosine integral functions
1 π Ei − ci(x)e [Re μ > 0] μ 4μ 0
2 2
2 ∞+ 2 π , −μx μ2 μ π 1 1 si x + e S dx = + C − − 2 μ 4 2 4 2 0
2.
6.249
6.251 1. 2.
641
∞
−μx2
1 dx = 4
1 2 √ si e−μx dx = kei (2 μ) x μ 0 ∞
1 2 √ ci e−μx dx = − ker (2 μ) x μ 0
6.252 1.
[Re μ > 0]
ME 26
[Re μ > 0]
MI 34
[Re μ > 0]
MI 34
∞
∞
0
π 2p π =− 4p
sin px si(qx) dx = −
=0
MI 34
∞
2.6
cos px si(qx) dx = −
0
=
1 ln 4p
1 q
p+q p−q
p2 > q 2 p2 = q 2 p2 < q 2
FI II 652, NT 50(8)
2
p = 0,
p2 = q 2
[p = 0] FI II 652, NT 50(10)
∞
3.
sin px ci(qx) dx = −
0
1 ln 4p
p2 −1 q2
2
=0
[p = 0]
π 2p π =− 4p
0
cos px ci(qx) dx = −
=0
6.253 0
p2 = q 2
FI II 652, NT 50(9) ∞
4.
p = 0,
∞
m
m+1
p2 > q
2
p2 = q 2 p2 < q 2
FI II 654, NT 50(7)
π r +r si(ax) sin bx dx = − 2 2 1 − 2r cos x + r 4b(1 − r) (1 −mr ) m+1 π 2 + 2r − r − r =− 4b(1 − r) (1 − r2 ) πrm+1 =− 2 2b(1 − r) (1 − r ) π 1 + r − rm+1 =− 2b(1 − r) (1 − r2 )
[b = a − m] [b = a + m] [a − m − 1 < b < a − m] [a + m < b < a + m + 1] ET I 97(10)
642
The Exponential Integral Function and Functions Generated by It
6.254 1.∗
∞
0
1 1 dx 1 = ci(x) sin x L2 − L2 − x 2 2 2 2
where L2 (x) is the Euler dilogarithm defined as L2 (z) = −
2.11
log(1 − t) dt and this in turn can t 0 be expressed as L2 (z) = Φ(z, 2, 1) in terms of the Lerch function defined in 9.550, with z real. ∞+ π a π, dx = ln H(a − b) si(ax) + cos bx · 2 x 2 b 0 [a > 0, b > 0, H(x) is the Heaviside step function] ET I 41(11)
6.255 1.
∞
[cos ax ci (a|x|) + sin (a|x|) si (a|x|)]
−∞
z
dx = −π [sign b cos ab si (a|b|) − sin ab ci (a|b|)] x−b [a > 0]
∞
2. −∞
[sin ax ci (a|x|) − sign x cos ax si (a|x|)]
6.256 1.
∞
0
∞
2 π si (x) + ci2 (x) cos ax dx = ln(1 + a) a 2
[si(x) cos x − ci(x) sin x] dx =
0
3.∗
∞
si2 (x) cos(ax) dx =
0
4.∗ 6.257
∞
6.258 1.
∞
0
2. 0
ET II 253(5)
[a > 0]
π 2
π log(1 + a) 2a
π log(1 + a) 2a 0 ∞ √ π a sin bx dx = − J 0 2 ab si x 2b 0 ci2 (x) cos(ax) dx =
ET II 253(4)
dx = −π [sin (a|b|) si (a|b|) + cos ab ci (a|b|)] x−b [a > 0]
2.∗
6.254
[0 ≤ a ≤ 2] [0 ≤ a ≤ 2] [b > 0]
+ π, dx si(ax) + sin bx 2 2 x + c2 π −bc e = [Ei(bc) − Ei(−ac)] + ebc [Ei(−ac) − Ei(−bc)] 4c π = e−bc [Ei(ac) − Ei(−ac)] 4c
ET I 42(18)
[0 < b ≤ a,
c > 0]
[0 < a ≤ b,
c > 0] BI (460)(1)
∞
+ π, x dx si(ax) + cos bx 2 2 x + c2 π −bc e =− [Ei(bc) − Ei(−ac)] + ebc [Ei(−bc) − Ei(−ac)] 4 π = e−bc [Ei(−ac) − Ei(ac)] 4
[0 < b ≤ a,
c > 0]
[0 < a ≤ b,
c > 0]
BI (460)(2, 5)
6.262
6.259
The sine integral and cosine integral functions
∞
si(ax) sin bx
1. 0
dx π Ei(−ac) sinh(bc) = x2 + c2 2c π = e−cb [Ei(−bc) + Ei(bc) − Ei(−ac) − Ei(ac)] 4c π + Ei(−bc) sinh(bc) 2c
ci(ax) sin bx 0
∞
ci(ax) cos bx 0
4.∗
x dx π = − sinh(bc) Ei(−ac) 2 +c 2 π π = − sinh(bc) Ei(−bc) + e−bc [Ei(−bc) + Ei(bc) 2 4 − Ei(−ac) − Ei(ac)]
x2
dx x2 + c2 π cosh bc Ei(−ac) = 2c π −bc = e [Ei(ac) + Ei(−ac) − Ei(bc)] + ebc Ei(−bc) 4c
5.∗
[0 < a ≤ b,
c > 0]
[0 < b ≤ a,
c > 0]
[0 < a ≤ b,
c > 0]
[0 < b ≤ a,
c > 0]
[0 < a ≤ b,
c > 0]
BI (460)(4), ET I 41(15) ∞
[ci(x) sin x − Si(x) cos x] sin x
0
c > 0]
BI (460)(3)a, ET I 97(15)a
3.
[0 < b ≤ a,
ET I 96(8) ∞
2.
643
2 x dx 1 Ei(a)e−a − Ei(−a)ea = 2 +x 8
a2
[a real] ∞
[ci(x) sin x − Si(x) cos x]
2
0
2 π x dx π 3 e−|a| sinh(a) − Ei(a)e−a − Ei(−a)ea = 2 +x 8a 8|a|
a2
[a real] 6.261
∞
1.
−px
si(bx) cos axe 0
a p2 + (a + b)2 1 2bp ln dx = − + p arctan 2 2 (a2 + p2 ) 2 p2 + (a − b)2 b − a2 − p2 [a > 0,
∞
2.
si(βx) cos axe−μx dx = −
0
1.
∞
−μx
ci(bx) sin axe 0
p > 0]
ET I 40(8)
Re μ > |Im β|]
ET I 40(9)
μ − ai μ + ai arctan β β − 2(μ + ai) 2(μ − ai)
arctan
[a > 0, 6.262
b > 0,
1 dx = 2 2 (a + μ2 )
2 2 μ + b2 − a2 + 4a2 μ2 2aμ a μ arctan 2 − ln μ + b 2 − a2 2 b4 [a > 0, b > 0, Re μ > 0] ET I 98(16)a
644
The Exponential Integral Function and Functions Generated by It
∞
2.
ci(bx) cos axe−px dx =
0
3.
⎧ ⎨p
−1 ln 2 (a2 + p2 ) ⎩ 2
, + 2 b2 + p2 − a2 + 4a2 p2 b4
[a > 0, b > 0, 2 (μ − ai)2 (μ + ai) ∞ ln 1 + − ln 1 + β2 β2 − ci(βx) cos axe−μx dx = 4(μ + ai) 4(μ − ai) 0 [a > 0,
6.263
⎫ ⎬
+ a arctan
2ap b2 + p2 − a2 ⎭
Re p > 0]
Re μ > |Im β|]
ET I 41(16)
ET I 41(17)
6.263 π − μ ln μ [ci(x) cos x + si(x) sin x] e dx = 2 1 + μ2 0 π ∞ − μ + ln μ −μx [si(x) cos x − ci(x) sin x] e dx = 2 1 + μ2 0 ∞ ln 1 + μ2 [sin x − x ci(x)] e−μx dx = 2μ2 0
1.
2. 3. 6.264
∞
−μx
−
[Re μ > 0]
ME 26a, ET I 178(21)a
[Re μ > 0]
ME 26a, ET I 178(20)a
[Re μ > 0]
ME 26
∞
si(x) ln x dx = C + 1
1.
NT 46(10)
0
∞
2.
ci(x) ln x dx = 0
π 2
NT 56(11)
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions 6.271
∞
μ+1 1 1 ln = arccoth μ 2μ μ − 1 μ 0 ∞ 1 ln μ2 − 1 2.11 chi(x)e−μx dx = − 2μ 0
∞ 2 1 1 π Ei 6.27211 chi(x)e−px dx = 4 p 4p 0 6.273 ∞ ln μ [cosh x shi(x) − sinh x chi(x)] e−μx dx = 2 1.11 μ −1 0 ∞ μ ln μ 2.11 [cosh x chi(x) + sinh x shi(x)] e−μx dx = 1 − μ2 0 1.
shi(x)e−μx dx =
[Re μ > 1]
MI 34
[Re μ > 1]
MI 34
[p > 0]
MI 35
[Re μ > 0]
MI 35
[Re μ > 2]
MI 35
6.284
The probability integral
∞
11
6.274
[cosh x shi(x) − sinh x chi(x)] e
−μx2
0
1 π 4μ 1 e Ei − μ 4μ [Re μ > 0]
MI 35
ln μ2 − 1 −μx [x chi(x) − sinh x] e dx = − [Re μ > 1] 2μ2
0 ∞ 1 1 π 1 −μx2 [cosh x chi(x) + sinh x shi(x)] e x dx = exp Ei − 8 μ3 4μ 4μ 0 [Re μ > 0]
6.275 6.276
6.277
1 dx = 4
645
∞
∞
1.
[chi(x) + ci(x)] e
−μx
0
∞
2.
ln μ4 − 1 dx = − 2μ
[chi(x) − ci(x)] e−μx dx =
0
μ2 + 1 1 ln 2 2μ μ − 1
MI 35
MI 35
[Re μ > 1]
MI 34
[Re μ > 1]
MI 35
6.28–6.31 The probability integral 6.281 1.
∞
6
[1 − Φ(px)] x
2q−1
0
Γ q + 12 √ dx = 2 πqp2q
[Re q > 0,
Re p > 0] NT 56(12), ET II 306(1)a
∞
2.6 0
b 2b b 1 − Φ atα ± α dt = √ t π a
1−α 2α
+
, K 1+α (2ab) ± K 1−α (2ab) e±2ab 2α
2α
[a > 0, 6.282
∞
1.
Φ(qt)e−pt dt =
0
2
p p 1 1−Φ exp p 2q 4q 2
b > 0,
+ Re p > 0,
α = 0]
|arg q|
0,
Re β < Re α]
[Re p > 0,
Re(q + p) > 0]
ET II 307(5)
EH II 148(12)
∞
6.284 0
√ q 1 √ 1−Φ e−px dx = e−q p p 2 x
+ Re p > 0,
|arg q|
0]
s2 4a2
s2 Ei − 2 4a + Re s > 0,
MI 37
|arg a|
Re μ2 ,
Re ν > 0
ET II 306(2)
& e
x2 2
xν−1 dx = 2 2 −1 sec ν
νπ ν Γ 2 2 [0 < Re ν < 1]
6.287
∞
1.
Φ(βx)e−μx x dx = 2
0
∞
2.
β 2μ μ + β 2
[1 − Φ(βx)] e−μx x dx = 2
0
1 2μ
β 1− μ + β2
Re μ > − Re β 2 ,
ET I 325(9)
Re μ > 0
ME 27a, ET I 176(4)
Re μ > − Re β 2 ,
Re μ > 0
NT 49(14), ET I 177(9)
r 1 r 1 A B Q(rA) Q(rB) dr = − 3.∗ I= exp α arctan + β arctan B = A 2 σ2 4 2π αB βA −∞ σ 1 1 1 B=A = − α arctan 4 π α ∞ 2 x σ 2 A2 σ2 B 2 1 1 Q(x) = √ , α= e−t /2 dt = , β= , 1 − erf √ 2 2 2 1+σ A 1 + σ2 B 2 2π x 2 ∞ 2 ai 6.288 a > 0, Re μ > Re a2 Φ(iax)e−μx x dx = MI 37a 2 2μ μ − a 0 6.289 ∞ + 2 2 2 β π, 2 2 Re 1. Φ(βx)e(β −μ )x x dx = μ > Re β , |arg μ| < 2μ (μ2 − β 2 ) 4 0
2. 0
∞
ET I 176(5) ∞
[1 − Φ(βx)] e(β
2
−μ2 )x2
x dx =
1 2μ(μ + β)
+ Re2 μ > Re β 2 ,
arg μ
−a > 0, 2(μ + a) μ + b 0 2 ∞ μ μ i 1 −(μx+x2 ) + Ei − x dx = √ Φ(ix)e [Re μ > 0] 4 π μ 4 0 ∞ 2 2 arctan μ 1 1 [1 − Φ(x)] e−μ x x2 dx = √ − μ3 μ2 (μ2++ 1) 2 π 0 π, |arg μ| < 4 √ ∞ μ+1+1 1 −μx2 dx = ln √ Φ(x)e = arccoth μ + 1 x 2 μ+1−1 0 [Re μ > 0]
3.
6.291 6.292
6.293 6.294
∞
∞
1. 0
∞
2. 0
6.295
647
√ 2 Φ b − ax e−(a+μ)x x dx =
2 2 β 1 exp(−2βμ) 1−Φ e−μ x x dx = x 2μ2
+ π |arg β| < , 4
2 2 dx 1 = − Ei(−2μ) 1−Φ e−μ x x x
+ π, |arg μ| < 4
b > a]
ME 27
MI 37
MI 37
MI 37a
|arg μ|
0 4 6.297
∞ 2 2 1 β √ exp [−2 (βγ + β μ)] 1. 1 − Φ γx + e(γ −μ)x x dx = √ √ x 2 μ μ+γ 0
2. 0
∞
∞
[Re β > 0, Re μ > 0]
2 b + 2ax e−bμ 1−Φ exp − μ2 − a2 x2 + ab x dx = 2x 2μ(μ + a) [a > 0,
b > 0,
Re μ > 0]
MI 37
MI 37
MI 37
MI 38a
ET I 177(12)a
MI 38
648
The Exponential Integral Function and Functions Generated by It
∞
3. 0
6.298
2 b + 2ax2 b − 2ax2 1 −ab + 1−Φ 1−Φ e eab e−μx x dx = exp −b a2 + μ 2x 2x μ
MI 38 [a > 0, b > 0, Re μ > 0]
∞ 2 2 b − 2ax2 b + 2ax2 1 √ exp (−b μ) 2 cosh ab − e−ab Φ − eab Φ e−(μ−a )x x dx = 2 2x 2x μ − a 0 MI 38 [a > 0, b > 0, Re μ > 0] ∞ + , 1 2 exp 12 a2 K ν a2 cosh(2νt) exp (a cosh t) [1 − Φ (a cosh t)] dt = 2 cos(νπ) 0 Re a > 0, − 12 < Re ν < 12
6.298
6.299
b2 1 [1 − Φ(ax)] sin bx dx = [a > 0, b > 0] 1 − e− 4a2 b 0 √ √ ∞ b + a2 + a 2b a 2b 1 2 √ + 2 arctan ln Φ(ax) sin bx dx = √ b − a2 4 2πb b + a2 − a 2b 0 [a > 0, b > 0]
6.311 6.312
∞
6.313
α ⎞ 12 ,− 12 + 1 √ 1 α2 + β 2 2 − α sin(βx) 1 − Φ αx dx = − ⎝ 2 2 2 ⎠ β α +β
∞
⎛ √ cos(βx) 1 − Φ αx dx = ⎝
0
[Re α > |Im β|] 2.
0
∞
1. 0
ET I 96(3)
ET II 307(6)
α ⎞ + ,− 12 1 2 2 2 2 ⎠ α + β + α α2 + β 2 1 2
[Re α > |Im β|] 6.314
ET I 96(4)
⎛
∞
1.
ET II 308(10)
ET II 307(7)
, + , + 1 1 a sin(bx) 1 − Φ dx = b−1 exp −(2ab) 2 cos (2ab) 2 x
[Re a > 0, b > 0]
∞ + , + , 1 1 a cos(bx) 1 − Φ dx = −b−1 exp −(2ab) 2 sin (2ab) 2 x 0
ET II 307(8)
2.
[Re a > 0, 6.315
∞
1.
ν−1
x 0
2. 0
∞
b > 0]
ET II 307(9)
Γ 1 + 12 ν β ν+1 ν 3 ν+3 β2 , + 1; , ; − sin(βx) [1 − Φ(αx)] dx = √ F 2 2 2 2 2 2 4α2 π(ν + 1)αν+1 1
xν−1 cos(βx) [1 − Φ(αx)] dx =
Γ 2 + 12 ν √ πναν
2F 2
Re ν > −1]
ν ν+1 1 ν β2 , ; , + 1; − 2 2 2 2 2 4α [Re α > 0,
[Re α > 0,
Re ν > 0]
ET II 307(3)
ET II 307(4)
6.323
Fresnel integrals
∞
3. 0
∞
4.
1 b2 1 b2 [1 − Φ(ax)] cos bx · x dx = 2 exp − 2 − 2 1 − exp − 2 2a 4a b 4a
[Φ(ax) − Φ(bx)] cos px
0
∞
5. 0
649
2
1 dx p = Ei − 2 x 2 4b
√ 1 1 x− 2 Φ a x sin bx dx = √ 2 2πb
− Ei
[a > 0,
p2 4a2
b > 0]
ET I 40(5)
[a > 0, b > 0, p > 0] & % √ & √ b + a 2b + a2 a 2b √ ln + 2 arctan 2 b − a2 b − a 2b + a
%
[a > 0,
∞ 1 2 x b π b2 x 2 2 √ √ e e 1−Φ 1−Φ sin bx dx = 2 2 2 0 [b > 0] √ ∞ 2 2 i π − b22 e 4a e−a x Φ(iax) sin bx dx = [b > 0] a 2 0 ∞ 2 2 2 1 − e−p − √ (1 − Φ(p)) [1 − Φ(x)] si(2px) dx = πp π 0 [p > 0]
b > 0]
ET I 40(6)
ET I 96(3)
6.316
6.3176 6.318
ET I 96(5) ET I 96(2)
NT 61(13)a
6.32 Fresnel integrals 6.321
∞
1. 0
∞
2. 0
6.322 1. 2.
1 − S (px) x2q−1 dx = 2 1 − C (px) x2q−1 dx = 2
√ 2q + 1 π 2 Γ q + 12 sin 4 √ 2q 4 πqp √ 2q + 1 π 2 Γ q + 12 cos 4 √ 2q 4 πqp
p p2 1 p2 −C cos + sin 4 2 2 4 0
∞ 2 p 1 p 1 p2 −S C (t)e−pt dt = cos − sin p 4 2 2 4 0
∞
S (t)e−pt dt =
1 p
6.323 1. 0
∞
√ S t e−pt dx =
0 < Re q < 32 ,
p>0
0 < Re q < 32 ,
p>0
p 1 −S 2 2 p 1 −C 2 2
NT 56(14)a
NT 56(13)a
MO 173a MO 172a
12 p2 + 1 − p 2p p2 + 1
EF 122(58)a
650
The Gamma Function and Functions Generated by It
∞
2.
√ C t e−pt dt =
0
6.324 1. 2. 6.325
12 p2 + 1 + p 2p p2 + 1
1 1 + sin p2 − cos p2 − S (x) sin 2px dx = 2 4p 0 ∞ 1 1 − sin p2 − cos p2 − C (x) sin 2px dx = 2 4p 0
∞
∞
1. 0
∞
2.
6.324
EF 122(58)a
√ π −5 2 2 S (x) sin b x dx = b =0
[p > 0]
NT 61(11)a
0 < b2 < 1 2 b >1
ET I 98(21)a
√
0
NT 61(12)a
2 2
C (x) cos b2 x2 dx =
[p > 0]
π −5 2 2 b
0 < b2 < 1 2 b >1
=0
ET I 42(22)
6.326
∞
∞
1. 0
2. 0
π 1/2 1 1 + sin p2 − cos p2 − S (x) si(2px) dx = (S (p) + C (p) − 1) − 2 8 4p [p > 0] 1 π 1/2 1 − sin p2 − cos p2 − C (x) si(2px) dx = (S (p) − C (p)) − 2 8 4p [p > 0]
NT 61(15)a
NT 61(14)a
6.4 The Gamma Function and Functions Generated by It 6.41 The gamma function
∞
11
6.411
−∞
Γ(α + x) Γ(β − x) dx = −iπ21−α−β Γ(α + β) [Re(α + β) < 1 and either Im α < 0 < Im β or Im β < 0 < Im α ] ET II 297(1)
= iπ2
1−α−β
Γ(α + β) [Re(α + β) < 1,
Im α < 0,
Im β < 0] ET II 297(2)
=0 [Re(α + β) < 1,
Im α > 0,
Im β > 0] ET II 297(3)
6.415
The gamma function
i∞
6.412 −i∞
Γ(α + s) Γ(β + s) Γ(γ − s) Γ(δ − s) ds = 2πi
651
Γ(α + γ) Γ(α + δ) Γ(β + γ) Γ(β + δ) Γ(α + β + γ + δ) [Re α, Re β, Re γ, Re δ > 0] ET II 302(32)
6.413
∞
1.
√ 2
|Γ(a + ix) Γ(b + ix)| dx =
0
2.
√ !2 1 1 ! Γ b − a − π Γ(a) Γ a + Γ(a + ix) 2 2 ! ! ! Γ(b + ix) ! dx = 2 Γ(b) Γ b − 12 Γ(b − a)
1.
2. 3.
5.
6.
b > 0]
0 Im δ and the minus sign if Im γ < Im δ.] ET II 300(19) 1 ∞ π exp ± 2 π(δ − γ)i Γ(α − β − γ + x + 1) dx = Γ(β + γ − 1) Γ 12 (α + β) Γ 12 (γ − δ + 1) −∞ Γ(α + x) Γ(β − x) Γ(γ + x) [Re(β + γ) > 1, δ = α − β − γ + 1, Im δ = 0. The sign is plus in the argument if the ET II 300(20) exponential for Im δ > 0 and minus for Im δ < 0.] ∞ Γ(α + β + γ + δ − 3) dx = Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) −∞ [Re(α + β + γ + δ) > 3]
6.415
ET II 302(27)
Re(α − β) < −1]
2 dx = [Re(α + β) > 1] Γ(α + x) Γ(β − x) Γ(α + β − 1) −∞ ∞ Γ(γ + x) Γ(δ + x) dx = 0 Γ(α + x) Γ(β + x) −∞ [Re(α + β − γ − δ) > 1, Im γ, Im δ > 0]
4.
∞
Γ(α + x) dx = 0 −∞ Γ(β + x)
[a > 0,
∞ !!
0
6.414
π Γ(a) Γ a + 12 Γ(b) Γ b + 12 Γ(a + b) 2 Γ a + b + 12
−∞
1. −∞
R(x) dx Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x)
ET II 300(21)
1 Γ(α + β + γ + δ − 3) R(t) dt = Γ(α + β − 1) Γ(β + γ − 1) Γ(γ + δ − 1) Γ(δ + α − 1) 0 [Re(α + β + γ + δ) > 3, R(x + 1) = R(x)] ET II 301(24)
652
The Gamma Function and Functions Generated by It
R(t) cos 12 π(2t + α − β) dt R(x) dx
= 0 α+β γ+δ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ Γ Γ Γ(α + δ − 1) 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2, R(x + 1) = −R(x)]
2.
6.421
1
∞
ET II 301(25)
6.42 Combinations of the gamma function, the exponential, and powers 6.421
∞
1. −∞
Γ(α + x) Γ(β − x) exp [2(πn + θ)xi] dx = 2πi Γ(α + β)(2 cos θ)−α−β exp[(β − α)iθ]
% Re(α + β) < 1,
2.
3.
4.
6.422
−
π π 0 n an integer, ηn (ξ) = sign 12 − n if 12 − n Im ξ < 0 ET II 298(7)
∞
eπicx dx =0 −∞ Γ(α + x) Γ(β − x) Γ(γ + kx) Γ(δ − kx) [Re(α + β + γ + δ) > 2, c and k are real,
∞
i∞
|c| > |k| + 1]
ET II 301(26)
Γ(α + x) exp[(2πn + π − 2θ)xi] dx −∞ Γ(β + x) (2 cos θ)β−α−1 = 2πi sign n + 12 exp[−(2πn + π − θ)αi + θi(β − 1)] Γ(β − α) + , π π Re(β − α) > 0, − < θ < , n is an integer, n + 12 Im α < 0 ET II 298(8) 2 2 ∞ Γ(α + x) exp[(2πn + π − 2θ)xi] dx = 0 −∞ Γ(β + x) + , π π Re(β − α) > 0, − < θ < , n is an integer, n + 12 Im α > 0 ET II 297(6) 2 2
1. −i∞
Γ(s − k − λ) Γ λ + μ − s + 12 Γ λ − μ − s + 12 z s ds
Re(k + λ) < 0,
1
z − k − μ Γ 12 − k + μ z λ e 2 W k,μ (z) Re λ > |Re μ| − 12 , |arg z| < 32 π ET II 302(29)
= 2πi Γ
2
γ+i∞
2. γ−i∞
Γ(α + s) Γ(−s) Γ(1 − c − s)xs ds = 2πi Γ(α) Γ(α − c + 1)Ψ(α, c; x) − Re α < γ < min (0, 1 − Re c) , − 32 π < arg x < 32 π
EH I 256(5)
6.422
The gamma function, the exponential, and powers
γ+i∞
3.
Γ(−s) Γ(β + s)ts ds = 2πi Γ(β)(1 + t)−β
653
[0 > γ > Re(1 − β),
|arg t| < π]
γ−i∞
∞i
4.
Γ −∞i
t−p 2
i∞
5.
Γ(s) Γ −i∞
6.
EH I 256, BU 75
√ t−p−2 1 2 2 z t dt = 2πie 4 z Γ(−p) D p (z) Γ(−t) |arg z| < 34 π, p is not a positive integer
2ν +
1 4
− s Γ 12 ν −
1 4
−s
z2 2
WH
s ds
1 1 1 3 2 = 2πi · 2 4 − 2 ν z − 2 e 4 z Γ 12 ν + 14 Γ 12 ν − 14 D ν (z) |arg z| < 34 π, ν = 12 , − 12 , − 32 , . . . EH II 120
c+i∞
3 c−i∞
1
1 −s 1 −1 Γ 2 ν + 12 s Γ 1 + 12 ν − 12 s ds = 4πi J ν (x) 2x [x > 0, − Re ν < c < 1]
−c+i∞
7. −c−i∞
ν+2s 1 Γ(−ν − s) Γ(−s) − 12 iz ds = −2π 2 e 2 iνπ H (1) ν (z) + π |arg(−iz)| < , 2
EH II 21(34)
, 0 < Re ν < c EH II 83(34)
−c+i∞
8. −c−i∞
Γ(−ν − s) Γ(−s)
1 ν+2s 1 ds = 2π 2 e− 2 iνπ H (2) ν (z) 2 iz + π |arg(iz)| < , 2
, 0 < Re ν < c EH II 83(35)
9. 10.
1 ν+2s i∞ x ds = 2πi J ν (x) Γ(−s) 2 [x > 0, Re ν > 0] EH II 83(36) Γ(ν + s + 1) −i∞ i∞ 5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (−2iz)s ds = −π 2 e−i(z−νπ) sec(νπ)(2z)−ν H (1) ν (z) −i∞ |arg(−iz)| < 32 π, 2ν = ±1, ±3 . . . EH II 83(37)
11.
5 Γ(−s) Γ(−2ν − s) Γ ν + s + 12 (2iz)s ds = π 2 ei(z−νπ) sec(νπ)(2z)−ν H (2) ν (z) −i∞ |arg(iz)| < 32 π, 2ν = ±1, i∞
±3 . . . EH II 84(38)
i∞
12.
Γ(s) Γ −i∞
1 2
3 3 1 − s − ν Γ 12 − s + ν (2z)s ds = 2 2 π 2 iz 2 ez sec(νπ) K ν (z) |arg z| < 32 π, 2ν = ±1,
±3, . . . EH II 84(39)
− 12 +i∞
13. − 12 −i∞
Γ(−s) 2s x ds = 4π s Γ(1 + s)
∞
2x
J 0 (t) dt t
[x > 0]
MO 41
654
The Gamma Function and Functions Generated by It
14.
6.423
i∞
Γ(α + s) Γ(β + s) Γ(−s) Γ(α) Γ(β) (−z)s ds = 2πi F (α, β; γ; z) Γ(γ + s) Γ(γ) −i∞
[For arg(−z) < π, the path of integration must separate the poles of the integrand at the points s = 0, 1, 2, 3, . . . from the poles s = −α − n and s = −β − n (for n = 0, 1, 2, . . . ).]
15.
16.
δ+i∞
Γ(α + s) Γ(−s) 2πi Γ(α) (−z)s ds = 1 F 1 (α; γ; z) Γ(γ + s) Γ(γ) δ−i∞ + π , π − < arg(−z) < , 0 > δ > − Re α, γ = 0, 1, 2, . . . 2 2 & % 2 i∞ + 1 1 , Γ 12 − s 1 z s ds = 2πiz 2 2π −1 K 0 4z 4 − Y 0 4z 4 Γ(s) −i∞
EH I 62(15), EH I 256(4)
[z > 0] i∞ Γ λ + μ − s + 12 Γ λ − μ − s + 12 s z z ds = 2πiz λ e− 2 W k,μ (z) Γ(λ − k − s + 1) −i∞ +
ET II 303(33)
17.
Re λ > |Re μ| − 12 ,
18.
Γ(k − λ + s) Γ λ + μ − s + Γ μ − λ + s + 12 + −i∞ i∞
1 2
z s ds = 2πi
Re(k − λ) > 0,
m -
i∞
19. −i∞
j=1 q j=m+1
Γ (bj − s)
n -
Γ (1 − aj + s)
j=1
Γ (1 − bj + s)
p j=n+1
Γ (aj − s)
|arg z|
− 12 ,
π, 2
|arg z|
0, j = 1, . . . , m ET II 303(34)
6.423
∞
1.
e−αx
0
2. 3.
∞
dx = ν e−α Γ(1 + x)
dx = eβα ν e−α , β Γ(x + β + 1) 0 ∞ xm dx = μ e−α , m Γ(m + 1) e−αx Γ(x + 1) 0
MI 39, EH III 222(16)
e−αx
MI 39, EH III 222(16)
[Re m > −1]
MI 39, EH III 222(17)
6.433
Gamma functions and trigonometric functions
4.
6.424
∞
655
xm dx = enα μ e−α , m, n Γ(m + 1) MI 39, EH III 222(17) Γ(x + n + 1) 0 α+β−2 θ 1 ∞ 2 cos 1 R(x) exp[(2πn + θ)xi] dx 2 = exp θ(β − α)i R(t) exp(2πnti) dt Γ(α + x) Γ(β − x) Γ(α + β − 1) 2 −∞ 0 [Re(α + β) > 1, −π < θ < π, n is an integer, R(x + 1) = R(x)] ET II 299(16) e−αx
6.43 Combinations of the gamma function and trigonometric functions 6.431
−∞
1. −∞
sin rx dx = Γ(p + x) Γ(q − x)
2 cos
r p+q−2 r(q − p) sin 2 2 Γ(p + q − 1)
=0
[|r| > π] [r is real;
r p+q−2 r(q − p) 2 cos cos cos rx dx 2 2 = Γ(p + q − 1) −∞ Γ(p + x) Γ(q − x)
2.
[|r| < π]
Re(p + q) > 1]
dx sin(mπx) =0 sin(πx) Γ(α + x) Γ(β − x) −∞
[m is an even integer] α+β−2
2 Γ(α + β − 1)
[m is an odd integer] [Re(α + β) > 1]
ET II 300(22)
, +π (β − α) cos cos πx dx
2
= α+β γ+δ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ 2Γ Γ Γ(α + δ − 1) 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]
ET II 301(23)
2.
ET II 298(11, 12)
, +π (β − α) sin sin πx dx
2
= α+β γ+δ Γ(α + x) Γ(β − x) Γ(γ + x) Γ(δ − x) −∞ 2Γ Γ Γ(α + δ − 1) 2 2 [α + δ = β + γ, Re(α + β + γ + δ) > 2]
1.
MO 10a, ET II 299(13, 14)
∞
= 6.433
MO 10a, ET II 298(9, 10)
[|r| > π] [r is real;
6.432
Re(p + q) > 1]
∞
=0
[|r| < π]
∞
∞
656
The Gamma Function and Functions Generated by It
6.441
6.44 The logarithm of the gamma function∗ 6.441
p+1
1.
ln Γ(x) dx = p
1
2.
1 ln 2π + p ln p − p 2 1
ln Γ(1 − x) dx =
ln Γ(x) dx = 0
0 1
3.
ln Γ(x + q) dx = 0
z
FI II 784
1 ln 2π 2
1 ln 2π + q ln q − q 2
FI II 783
[q ≥ 0]
NH 89(17), ET II 304(40)
z(z + 1) z ln 2π − + z ln Γ(z + 1) − ln G(z + 1), 2 2
∞
z z k z(z + 1) Cz 2 - z2 2 where G(z + 1) = (2π) exp − 1+ − exp −z + 2 2 k 2k
4.
ln Γ(x + 1) dx = 0
WH
k=1
n
5.
ln Γ(α + x) dx = 0
n−1 k=0
1 1 (a + k) ln(a + k) − na + n ln(2π) − n(n − 1) 2 2 [a ≥ 0;
1
6.442
[a > 0;
ET II 304(41)
exp(2πnxi) ln Γ(a + x) dx = (2πni)−1 [ln a − exp(−2πnai) Ei(2πnai)]
0
6.443
n = 1, 2, . . .]
n = ±1, ±2, . . .]
ET II 304(38)
1
1 [ln(2πn) + C] NH 203(5), ET II 304(42) 2πn 0
1 1 1 1 π 1 + 2 1 + + ··· + ln Γ(x) sin(2n + 1)πx dx = ln + (2n + 1)π 2 3 2n − 1 2n + 1 0
1.
ln Γ(x) sin 2πnx dx =
2.
ET II 305(43)
1
3.
ln Γ(x) cos 2πnx dx = 0
4.
1
8 0
1
5.
1 4n
2 ln Γ(x) cos(2n + 1)πx dx = 2 π
NH 203(6), ET II 305(44)
%
∞
ln k 1 (C + ln 2π) + 2 2 2 (2n + 1) 4k − (2n + 1)2
& NH 203(6)
k=2
sin(2πnx) ln Γ(a + x) dx = −(2πn)−1 [ln a + cos(2πna) ci(2πna) − sin(2πna) si(2πna)]
0
[a > 0; 6.
1
n = 1, 2, . . .]
ET II 304(36)
cos(2πnx) ln Γ(a + x) dx = −(2πn)−1 [sin(2πna) ci(2πna) + cos(2πna) si(2πna)]
0
[a > 0;
n = 1, 2, . . .]
ET II 304(37)
∗ Here, we are violating our usual order of presentation of the formulas in order to make it easier to examine the integrals involving the gamma function.
6.457
The incomplete gamma function
657
6.45 The incomplete gamma function 6.451 1. 2. 6.452
∞
1 Γ(β)(1 + α)−β α 0 ∞ 1 1 e−αx Γ(β, x) dx = Γ(β) 1 − α (α + 1)β 0
∞
1. 0
e−αx γ(β, x) dx =
[β > 0]
MI 39
[β > 0]
MI 39
2 x2 1 e−μx γ ν, 2 dx = 2−ν−1 Γ(2ν)e(aμ) D −2ν (2aμ) 8a μ |arg a|
− , 2
Re μ > 0 ET I 179(36)
∞
2.
e−μx γ
0
1 x2 , 4 8a2
√ 2 2 a dx = √ e(aμ) K 14 a2 μ2 μ 3 4
+ π |arg a| < , 4
, Re μ > 0
+ , a 1 1 π √ |arg a| < , Re μ > 0 dx = 2a 2 ν μ 2 ν−1 K ν (2 μa) e−μx Γ ν, x 2
2 0 ∞ √ 1 1 α α −βx − 2 ν ν − 2 ν−1 e γ ν, α x dx = 2 α β Γ(ν) exp D −ν √ 8β 2β 0 [Re β > 0, Re ν > 0]
6.453 6.454
ET I 179(35)
∞
ET I 179(32)
ET II 309(19), MI 39a
6.455
∞
1.
αν Γ(μ + ν) β 1, μ + ν; μ + 1; F 1 2 μ(α + β)μ+ν α+β [Re(α + β) > 0, Re μ > 0, Re(μ + ν) > 0]
ET II 309(16)
αν Γ(μ + ν) α γ(ν, αx) dx = 2 F 1 1, μ + ν; ν + 1; ν(α + β)μ+ν α+β [Re(α + β) > 0, Re β > 0, Re(μ + ν) > 0]
ET II 308(15)
xμ−1 e−βx Γ(ν, αx) dx =
0
∞
2.
x 0
6.456 1. 2. 6.457 1. 2.
μ−1 −βx
e
√ √ γ (2ν, α) 1 1 e−αx (4x)ν− 2 γ ν, dx = π 1 4x αν+ 2 0
√ √ ∞ π Γ (2ν, α) 1 −αx ν− 12 e (4x) Γ ν, dx = 1 4x αν+ 2 0
∞
√ √ γ (2ν + 1, α) 1 √ γ ν + 1, e dx = π 1 4x x αν+ 2 0
√ ∞ √ Γ (2ν + 1, α) (4x)ν 1 e−αx √ Γ ν + 1, dx = π 1 4x x αν+ 2 0
∞
ν −αx (4x)
MI 39a MI 39a
MI 39 MI 39
658
The Gamma Function and Functions Generated by It
∞
6.458
1−2ν
x
2
exp αx
0
6.458
2
3 1 b b 2 −ν ν−1 2 sin(bx) Γ ν, αx dx = π 2 α Γ 2 − ν exp D 2ν−2 1 8α (2α) 2 3π , 0 < Re ν < 1 |arg α| < 2 ET II 309(18)
6.46–6.47 The function ψ(x)
x
6.461
ψ(t) dt = ln Γ(x) 1
6.462
1
ψ(α + x) dx = ln α 0 ∞ x−α [C + ψ(1 + x)] = −π cosec(πα) ζ(α) 0 1 e2πnxi ψ(α + x) dx = e−2πnαi Ei(2πnαi)
6.463 6.464
[α > 0]
ET II 305(1)
[1 < Re α < 2]
ET II 305(6)
[α > 0;
n = ±i, ±2, . . .]
ET II 305(2)
0
6.465 1.
1
8 0
% & ∞ ln k 2 C + ln 2π + 2 ψ(x) sin πx dx = − π 4k 2 − 1 k=2
(see 6.443 4)
1
2. 0
1 ψ(x) sin(2πnx) dx = − π 2 ∞
6.466
[n = 1, 2, . . .]
NH 204 ET II 305(3)
−1 [ψ(α + ix) − ψ(α − ix)] sin xy dx = iπe−αy 1 − e−y
0
[α > 0, 6.467 1.
y > 0]
ET I 96(1)
1
sin(2πnx) ψ(α + x) dx = sin(2πnα) ci(2πnα) + cos(2πnα) si(2πnα)
0
[α ≥ 0;
n = 1, 2, . . .]
ET II 305(4)
1
cos(2πnx) ψ(α + x) dx = sin(2πnα) si(2πnα) − cos(2πnα) ci(2πnα)
2. 0
[α > 0;
1
6.468 0
6.469 1.
1 ψ(x) sin2 πx dx = − [C + ln(2π)] 2
1
ψ(x) sin πx cos πx dx = −
0
2.8
1
ET II 305(5) NH 204
π 4
n 1 − n2 1 n−1 = ln 2 n+1
ψ(x) sin πx sin(nπx) dx = 0
n = 1, 2, . . .]
NH 204
[n is even] [n > 1 is odd] NH 204(8)a
6.511
6.471
Bessel functions
∞
1.
x−α [ln x − ψ(1 + x)] dx = π cosec(πα) ζ(α)
659
[0 < Re α < 1]
ET II 306(7)
0
∞
2.
x−α [ln(1 + x) − ψ(1 + x)] dx = π cosec(πα) ζ(α) − (α − 1)−1
0
[0 < Re α < 1] ∞
[ψ(x + 1) − ln x] cos(2πxy) dx =
3. 0
6.472
∞
1.
ET II 306(8)
1 [ψ(y + 1) − ln y] 2
ET II 306(12)
x−α (1 + x)−1 − ψ (1 + x) dx = −πα cosec(πα) ζ(1 + α) − α−1
0
[|Re α| < 1] ∞
2.
ET II 306(9)
x−α x−1 − ψ (1 + x) dx = −πα cosec(πα) ζ(1 + α)
0
[−2 < Re α < 0]
∞
6.473
x−α ψ (n) (1 + x) dx = (−1)n−1
0
π Γ(α + n) ζ(α + n) Γ(α) sin πα [n = 1, 2, . . . ;
ET II 306(10)
0 < Re α < 1] ET II 306(11)
6.5–6.7 Bessel Functions 6.51 Bessel functions 6.511
∞
J ν (bx) dx =
1. 0
∞
2. 0
1 b
νπ 1 Y ν (bx) dx = − tan b 2
[Re ν > −1,
b > 0]
[|Re ν| < 1,
b > 0]
ET II 22(3)
WA 432(7), ET II 96(1)
a
3.
J ν (x) dx = 2 0
a
J 12 (t) dt = 2 S
0
J − 12 (t) dt = 2 C
0
6.
ET II 333(1)
WA 599(4)
√ a
a
J 0 (x) dx = a J 0 (a) + 0
[Re ν > −1]
√ a
a
5.
J ν+2k+1 (a)
k=0
4.
∞
WA 599(3)
πa [J 1 (a) H0 (a) − J 0 (a) H1 (a)] 2 [a > 0]
ET II 7(2)
660
Bessel Functions
6.512
a
J 1 (x) dx = 1 − J 0 (a)
7.
[a > 0]
ET II 18(1)
0
∞
8.
J 0 (x) dx = 1 − a J 0 (a) +
a
πa [J 0 (a) H1 (a) − J 1 (a) H0 (a)] 2 [a > 0]
ET II 7(3)
[a > 0]
ET II 18(2)
∞
9.
J 1 (x) dx = J 0 (a) a
b
10.
Y ν (x) dx = 2 a
11.
a
I ν (x) dx = 2
∞
∞
0
6.512 1.11
ET II 339(46)
∞
[Re ν > −1]
(−1)n I ν+2n+1 (a)
ET II 364(1)
n=0
K 0 (ax) =
π 2a
[a > 0]
K 20 (ax) =
π2 4a
[a > 0]
0
13.∗
[Y ν+2n+1 (b) − Y ν+2n+1 (a)]
n=0
0
12.∗
∞
μ+ν+1
∞ Γ b2 μ+ν+1 ν−μ+1 2 ν −ν−1
F , ; ν + 1; 2 J μ (ax) J ν (bx) dx = b a μ−ν+1 2 2 a 0 Γ(ν + 1) Γ 2 [a > 0, b > 0, Re(μ + ν) > −1, b < a. For a > b, the positions of μ and ν should be reversed.]
∞
2.7 0
β2 Γ(ν) β F ν, −n; ν − n; 2 J ν+n (αt) J ν−n−1 (βt) dt = ν−n α n! Γ(ν − n) α 1 = (−1)n 2α =0
0
β αν 1 = 2β =0
0
[0 < β = α] [0 < α < β]
∞
MO 50
ν−1
J ν (αx) J ν−1 (βx) dx =
4.
[0 < β < α]
[Re(ν) > 0] ∞
3.8
ET II 48(6)
ν−n−1
[β < α] [β = α] [β > α]
[Re ν > 0] WA 444(8), KU (40)a
2 2b J ν+2n+1 (ax) J ν (bx) dx = bν a−ν−1 P (ν,0) 1− 2 [Re ν > −1 − n, 0 < b < a] n a =0
[Re ν > −1 − n,
0 < a < b] ET II 47(5)
6.513
Bessel functions
∞
5.
661
1 2a
J ν+n (ax) Y ν−n (ax) dx = (−1)n+1
0
Re ν > − 12 ,
a > 0,
n = 0, 1, 2, . . . ET II 347(57)
∞ b2 b−1 ln 1 − 2 J 1 (bx) Y 0 (ax) dx = − π a 0 a ∞ 2 J ν (x) J ν+1 (x) dx = [J ν+n+1 (a)]
6. 7.
0
8.
9
9.∗ 10.∗ 6.513
ET II 21(31)
[Re ν > −1]
ET II 338(37)
n=0 ∞
1 δ(b − a) a 0
∞ b2 1 ln 1 + 2 K 0 (ax) J 1 (bx) = 2b a 0
∞ b2 1 K 0 (ax) I 1 (bx) = − ln 1 − 2 2b a 0
[n = 0, 1, . . .]
k J n (ka) J n (kb) dk =
JAC 110
[a > 0,
b > 0]
[a > 0,
b > 0]
1 + ν + 2μ ∞ 2 2
[J μ (ax)] J ν (bx) dx = a2μ b−2μ−1 1 + ν − 2μ 2 0 [Γ(μ + 1)] Γ 2 ⎡ ⎛
1.
[0 < b < a]
Γ
1− ⎢ ⎜ 1 − ν + 2μ 1 + ν + 2μ ⎜ , ; μ + 1; ×⎢ ⎣F ⎝ 2 2 [Re ν + Re 2μ > −1,
∞
2. 0
b−1 Γ [J μ (ax)] K ν (bx) dx = 2 2
2μ + ν + 1 2
∞
3. 0
4. 0
∞
⎞⎤2 4a2 1 − 2 ⎟⎥ b ⎟⎥ ⎠⎦ 2
0 < 2a < b]
ET II 52(33)
&2
% 2μ − ν + 1 4a2 −μ 1+ 2 Γ P 1 ν− 1 2 2 2 b [2 Re μ > |Re ν| − 1, Re b > 2|Im a|] ET II 138(18)
ν + 2μ + 1 eμπi Γ 4a2 4a2 2 −μ −μ
P 1 ν− 1 I μ (ax) K μ (ax) J ν (bx) dx = 1 + 2 Q 1 ν− 1 1+ 2 2 2 2 2 ν − 2μ + 1 b b bΓ 2 [Re a > 0, b > 0, Re ν > −1, Re(ν + 2μ) > −1] ET II 65(20) νπ π sec J μ (ax) J −μ (ax) K ν (bx) dx = P μ1 ν− 1 2 2 2b 2
2 4a 4a2 1+ 2 1 + 2 P −μ 1 1 2 ν− 2 b b [|Re ν| < 1, Re b > 2|Im a|] ET II 138(21)
662
Bessel Functions
6.514
1 + ν + 2μ % &2 ∞ Γ e 4a2 2 2 −μ
[K μ (ax)] J ν (bx) dx = 1+ 2 Q 1 ν− 1 2 2 1 + ν − 2μ b 0 bΓ 2 Re a > 0, b > 0, Re 12 ν ± μ > − 12 2μπi
5.
z
J μ (x) J ν (z − x) dx = 2
6. 0
∞
(−1)k J μ+ν+2k+1 (z)
[Re μ > −1,
ET II 66(28)
Re ν > −1]
WA 414(2)
k=0 z
7.
J μ (x) J −μ (z − x) dx = sin z
[−1 < Re μ < 1]
WA 415(4)
J μ (x) J 1−μ (z − x) dx = J 0 (z) − cos(z)
[−1 < Re μ < 2]
WA 415(4)
0
8.
z
0
9.
∗
∞
1 b
b 2 = arcsin πb 2a
J 20 (ax) J 1 (bx) =
0
6.514
∞
Jν
1.
a x
0
∞
2.
Jν
Jν
∞
0
ET II 57(9)
b > 0,
− 12 < Re ν
0, Re b > 0, |Re ν| < 52
a x
J ν (bx) dx = −
√ π √ , 2b−1 + K 2ν 2 ab − Y 2ν 2 ab π 2 a > 0, b > 0,
|Re ν|
0,
b > 0,
|Re ν|
− 12
a
Yν
6.
b > 0,
ET II 141(31)
4.
a > 0,
√ 2 √ −1 2ab Y 2ν 2 ab + K 2ν Y ν (bx) dx = b x π a > 0,
0
ET II 110(12) ∞
3.
[2a > b > 0]
a
0
√ J ν (bx) dx = b−1 J 2ν 2 ab
[b > 2a > 0]
a
1 √ √ 1 1 1 K ν (bx) dx = −b−1 e 2 νπi K 2ν 2e 4 πi ab − b−1 e− 2 νπi K 2ν 2e− 4 πi ab x a > 0, Re b > 0, |Re ν| < 52 ET II 143(37)
6.516
Bessel functions
∞
7.
Kν
a x
0
∞
Kν
a x
0
∞
1.
Jμ
a
0
∞
2. 0
√ √ 3νπ 3νπ sin ker2ν 2 ab + cos kei2ν 2 ab 2 2 Re a > 0, b > 0, |Re ν| < 12 ET II 113(28)
8. 6.515
Y ν (bx) dx = −2b
−1
663
∞
3. 0
x
√ K ν (bx) dx = πb−1 K 2ν 2 ab
Yμ
a x
[Re a > 0,
Re b > 0]
√ √ K 0 (bx) dx = −2b−1 J 2μ 2 ab K 2μ 2 ab
[a > 0, Re b > 0] + a ,2 1 √ √ 1 K 0 (bx) dx = 2πb−1 K 2μ 2e 4 πi ab K 2μ 2e− 4 πi ab Kμ x H (1) μ
2
a x
H (2) μ
ET II 146(54)
2
a x
ET II 143(42)
[Re a > 0, Re b > 0] ET II 147(59) √ √ J 0 (bx) dx = 16π −2 b−1 cos μπ K 2μ 2eπi/4 a b K 2μ 2e−πi/4 a b , + π |arg a| < , b > 0, |Re μ| < 14 4 ET II 17(36)
6.516
∞
1.
√ J 2ν a x J ν (bx) dx = b−1 J ν
0
∞
2.
a2 4b
√ J 2ν a x Y ν (bx) dx = −b−1 Hν
0
3. 0
4.10
5.
∞
√ π J 2ν a x K ν (bx) dx = b−1 I ν 2
2
a 4b
b > 0,
Re ν > − 12
ET II 58(16)
2
a 4b
a > 0,
− Lν
2
a 4b
a > 0,
b > 0,
Re ν > − 12
ET II 111(18)
Re b > 0, Re ν > − 12 ET II 144(45) 2
2
∞ √ a a 1 1 J −ν Y 2ν a x J ν (bx) dx = J ν cot(2πν) − cosec(2πν) b 4b 2b 4b 0
a4 3 3 23ν−3 a2−2ν bν−2 1 , − ν; Γ ν − 1; − F 1 2 2 2 2 64b2 π 3/2 [a > 0, b > 0] MC ∞ √ Y 2ν a x Y ν (bx) dx 0 2
2
2 a a a b−1 = sec(νπ) J −ν + cosec(νπ) H−ν − 2 cot(2νπ) Hν 2 4b 4b 4b a > 0, b > 0, |Re ν| < 12 ET II 111(19)
664
Bessel Functions
⎡ 2
2
−1 √ a a πb ⎣ cosec(2νπ) L−ν Y 2ν a x K ν (bx) dx = − cot(2νπ) Lν 2 4b 4b 0 ⎤ 2
2
a a ⎦ sec(νπ) Kν − tan(νπ) I ν − 4b π 4b ET II 144(46) Re b > 0, |Re ν| < 12 2
2 ∞ √ a a 1 K 2ν a x J ν (bx) dx = πb−1 sec(νπ) H−ν − Y −ν 4 4b 4b 0 Re a > 0, b > 0, Re ν > − 12
6.
7.
∞
ET II 70(22) ∞
√ K 2ν a x Y ν (bx) dx 2
2
2 a a a 1 −1 = − πb sec(νπ) J −ν − cosec(νπ) H−ν + 2 cosec(2νπ) Hν 4 4b 4b 4b Re a > 0, b > 0, |Re ν| < 12 ET II 114(34)
∞
√ K 2ν a x K ν (bx) dx =
∞
√ πb−1 I 2ν a x K ν (bx) dx = 2
8. 0
9. 0
10. 0
6.517
z
J0
0
0
J 2ν (2z sin x) dx =
π 2 J (z) 2 ν
π/2
2.
π 2 J (z) 2 ν
0
2 π Jν (z) + Nν2 (z) 8 cos νπ
J 2ν (2z cos x) dx =
π/2
1.
MO 48 2
K 2ν (2z sinh x) dx =
2
2
2
a a a π Kν + L−ν − Lν 4b 2 sin(νπ) 4b 4b 1 Re b > 0, |Re ν| < 2 ET II 147(63) 2 2
a a Iν + Lν 4b 4b Re b > 0, Re ν > − 12 ET II 147(60)
πb−1 4 cos(νπ)
z 2 − x2 dx = sin z
∞
6.518 6.519
6.517
0
− 12 < Re ν
0,
Re ν > − 12 Re ν > − 12
1 2
MO 45
WH
WA 42(1)a
6.52 Bessel functions combined with x and x2 6.521
1.
1
β J ν−1 (β) J ν (α) − α J ν−1 (α) J ν (β) α2 − β 2 α J ν (β) J ν (α) − β J ν (α) J ν (β) = β 2 − α2
x J ν (αx) J ν (βx) dx = 0
[α = β,
ν > −1]
[α = β,
ν > −1] WH
Bessel functions combined with x and x2
6.522
2.
∞
10
x K ν (ax) J ν (bx) dx = 0
bν aν (b2 + a2 )
665
[Re a > 0,
b > 0,
Re ν > −1] ET II 63(2)
∞ π(ab)−ν a2ν − b2ν x K ν (ax) K ν (bx) dx = 2 sin(νπ) (a2 − b2 ) 0
3.
[|Re ν| < 1,
Re(a + b) > 0] ET II 145(48)
ν a −1 λ x J ν (λx) K ν (μx) dx = μ2 + λ2 + λa J ν+1 (λa) K ν (μa) − μa J ν (λa) K ν+1 (μa) μ 0
4.
5.∗
[Re ν > −1] ∞
x K 1 (ax) =
π 2a2
[a > 0]
x K 20 (ax) =
1 2a2
[a > 0]
0
6.∗
∞
0
7.
∗
∞
b a (a2 + b2 )
x K 1 (ax) J 1 (bx) = 0
8.∗
∞
x K 0 (ax) I 0 (bx) = 0
9.∗
∞
x K 1 (ax) I 1 (bx) = 0
10.
∗
∞
∞
12.∗
∞
[a > b > 0]
b a (a2 − b2 )
[a > b > 0]
π 2a3
[a > 0]
x2 K 1 (ax) =
2 a3
[a > 0]
0
x2 K 0 (ax) J 1 (bx) =
0
13.∗
∞
x2 K 1 (ax) J 0 (bx) =
0
14.
∗
∞
x2 K 0 (ax) I 1 (bx) =
0
15.∗
∞
x2 K 1 (ax) I 0 (bx) =
0
6.522
Notation: 1 =
1.8 0
∞
b > 0]
1 − b2
a2
x2 K 0 (ax) =
0
11.∗
[a > 0,
2b 2
[a > 0,
2
[a > b > 0]
2
[a > b > 0]
2
[a > b > 0]
(a2 + b2 ) 2a (a2
+ b2 ) 2b
(a2
− b2 ) 2a
(a2
− b2 )
ET II 367(26)
b > 0]
, , 1 + 1 + (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2
2 x [J μ (ax)] K ν (bx) dx = Γ μ + 12 ν + 1 Γ μ − 12 ν + 1 b−2 + − 1 1 , −μ + 1 , 2 −2 2 2 −2 2 1 + 4a P 1 + 4a × 1 + 4a2 b−2 2 P −μ b b 1 1 − ν ν 2
[Re b > 2|Im a|,
2
2 Re μ > |Re ν| − 2]
ET II 138(19)
666
Bessel Functions
∞
2. 0
∞
3.11 0
6.522
2e2μπi Γ 1 + 12 ν + μ x [K μ (ax)] J ν (bx) dx = 1 b 4a2 + b2 2 Γ 12 ν − μ 2 b−2 ) Q −μ 2 b−2 ) (1 + 4a (1 + 4a × Q −μ 1 1 2ν 2 ν−1 b > 0, Re a > 0, Re 12 ν ± μ > −1 2
ν ν −ν x K 0 (ax) J ν (bx) J ν (cx) dx = r1−1 r2−1 (r2 − r1 ) (r2 − r1 ) = ν 2 1 2 , 2 ( 2 − 1 ) + 2 2 2 2 r1 = a + (b − c) , r2 = a + (b + c) , c > 0, Re ν > −1,
ET II 66(27)a
Re a > |Im b|
,
ET II 63(6) ∞
4.10
− 1 x I 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 b2 + 2a2 c2 + 2b2 c2 2
0
[Re b > Re a,
5.10
c > 0]
ET II 16(27)
alternatively, with a and c interchanged ∞ 1 x I 0 (cx) K 0 (bx) J 0 (ax) dx = 2 [Re b > Re c, a > 0] − 21 0 2 ∞ − 1 x J 0 (ax) K 0 (bx) J 0 (cx) dx = a4 + b4 + c4 − 2a2 c2 + 2a2 b2 + 2b2 c2 2 0
alternatively, with a and b interchanged ∞ 1 x J 0 (bx) K 0 (ax) J 0 (cx) dx = 2 2 − 21 0 ∞ x J 0 (ax) Y 0 (ax) J 0 (bx) dx = 0
6.
c > 0]
[Re a > |Im b|,
c > 0]
− 12
[0 < 2a < b < ∞] ET II 15(21)
∞
7. 0
− 1 3+ν 3−ν x J μ (ax) J μ+1 (ax) K ν (bx) dx = Γ μ + Γ μ+ b−2 1 + 4a2 b−2 2 2 2 + , 1 1 + , 1 1 2 ν− 2 2 ν− 2 ×P −μ 1 + 4a2 b−2 P −μ−1 1 + 4a2 b−2 [Re b > 2|Im a|,
9.
0
ET II 138(20)
x K μ− 12 (ax) K μ+ 12 (ax) J ν (bx) dx + 1 , −μ− 12 + 1 , 2e2μπi Γ 12 ν + μ + 1 −μ+ 12 2 −2 2 2 −2 2 1 + 4a 1 + 4a =− Q b b Q 1 1 1 1 1 ν− 2 ν− 2 b Γ 12 ν − μ b2 + 4a2 2 2 2 b > 0, Re a > 0, Re ν > −1, |Re μ| < 1 + 12 Re ν ET II 67(29)a
0
2 Re μ > |Re ν| − 3]
∞
8.
8
ET II 15(25)
[0 < b < 2a]
= −2π −1 b−1 b2 − 4a2
0
[Re b > |Im a|,
∞
− 1 x I 12 ν (ax) K 12 ν (ax) J ν (bx) dx = b−1 b2 + 4a2 2 [b > 0,
Re a > 0,
Re ν > −1] ET II 65(16)
Bessel functions combined with x and x2
6.522
∞
10.
x J 12 ν (ax) Y
0
1 2ν
(ax) J ν (bx) dx =0 = −2π −1 b−1 b2 − 4a2
− 12
[a > 0,
Re ν > −1,
0 < b < 2a]
[a > 0,
Re ν > −1,
2a < b < ∞] ET II 55(48)
∞
11.8
x J 12 (ν+n) (ax) J 12 (ν−n) (ax) J ν (bx) dx
0
= 2π
−1 −1
b
4a − b 2
1 2 −2
Tn
b 2a
=0 0
∞
8 0
14.
8
Re ν > −1,
0 < b < 2a]
[a > 0,
Re ν > −1,
2a < b]
x I 12 (ν−μ) (ax) K 12 (ν+μ) (ax) J ν (bx) dx = 2−μ a−μ b−1 b2 + 4a2 [b > 0,
13.
[a > 0,
ET II 52(32) ∞
12.
667
Re a > 0,
Re ν > −1,
− 12
+ 1 ,μ b + b2 + 4a2 2
Re(ν − μ) > −2] μ
ν
ET II 66(23)
ν−μ
μ−ν
(cos ψ) (sin ϕ) (sin ψ) (cos ϕ) x J μ (xa sin ϕ) K ν−μ (ax cos ϕ cos ψ) J ν (xa sin ψ) dx = 2 2 2 a 1 − sin ϕ ,sin ψ + π π ET II 64(10) a > 0, 0 < ϕ < , 0 < ψ < , Re μ > −1, Re ν > −1 2 2
∞
x J μ (xa sin ϕ cos ψ) J ν−μ (ax) J ν (xa cos ϕ sin ψ) dx 0 μ
15.10
16.10
17.11
−ν
−μ
−1
= −2π −1 a−2 sin(μπ) (sin ϕ) (sin ψ) (cos ϕ) (cos ψ) [cos(ϕ + ψ) cos(ϕ − ψ)] + , π a > 0, 0 < ϕ < , 0 < ψ < 12 π, Re ν > −1 ET II 54(39) 2 ∞ 23ν (abc)ν Γ ν + 12 ν+1 x J ν (bx) K ν (ax) J ν (cx) dx = √ 2ν+1 π ( 22 − 21 ) 0 [Re a > |Im b|, c > 0] ∞ 1 3ν ν 2 (abc) Γ ν + 2 xν+1 I ν (cx) K ν (bx) J ν (ax) dx = √ 2ν+1 π ( 22 − 21 ) 0 [Re b > |Im a| + |Im c|] ∞ tν−μ−ρ+1 J μ (ct) J ν (bt) K ρ (at) dt 0 μ−ν+ρ−1 1 1+2ν−2ρ 2 1 − x2 22 − x2 x 21+ν−μ−ρ dx = μ ν ρ μ−ν c b a Γ (μ − ν + ρ) 0 (b2 − x2 ) , , 1 + 1 + 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0] ν
668
Bessel Functions
18.
∞
11
6.523
tμ−ν+ρ+1 J μ (ct) J ν (bt) K ρ (at) dt ν−μ−ρ−1 1 1+2μ+2ρ 2 1 − x2 22 − x2 x 21+μ−ν+ρ aρ dx = μ ν ν−μ c b Γ (ν − μ − ρ) 0 (c2 − x2 ) , , + + 1 1 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 [Re a > |Im b|, c > 0]
0
∞
6.523 0
+ −1 2 −1 , b ln x 2π −1 K 0 (ax) − Y 0 (ax) K 0 (bx) dx = 2π −1 a2 + b2 + b − a2 a [Re b > |Im a|, Re(a + b) > 0] ET II 145(50)
6.524
∞
1. 0
x J 2ν (ax) J ν (bx) Y ν (bx) dx = 0
= −(2πab)−1 ∞
2.
2
x [J 0 (ax) K 0 (bx)] dx = 0
1 π − arcsin 8ab 4ab
b −a b 2 + a2 2
2
0 < a < b,
Re ν > − 12
0 < b < a,
Re ν > − 12
1.10
ET II 352(14)
[a > 0, 6.525
b > 0]
ET II 373(9)
, , 1 + 1 + (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2 ∞ ,− 32 + 2 2 a + b2 + c2 − 4a2 c2 x2 J 1 (ax) K 0 (bx) J 0 (cx) dx = 2a a2 + b2 − c2 Notation: 1 =
0
[c > 0,
Re b ≥ |Im a|,
Re a > 0] ET II 15(26)
2.10
alternatively, with a and b interchanged ∞ 2b a2 + b2 − c2 2 x J 1 (bx) K 0 (ax) J 0 (cx) dx = [Re a > |Im b|, Re b > 0, 3 ( 22 − 21 ) 0 ∞ ,− 32 + 2 2 a + b2 + c2 − 4a2 b2 x2 I 0 (ax) K 1 (bx) J 0 (cx) dx = 2b b2 + c2 − a2
c > 0]
0
∞
3.10
x2 I 0 (cx) K 0 (bx) J 0 (ax) dx =
2b a2 + b2 − c2
0
6.526
1. 0
∞
2
x J 12 ν ax
−1
J ν (bx) dx = (2a)
( 22 − 21 ) J 12 ν
b2 4a
3
[Re b > |Re a|,
c > 0]
[Re a > |Im b|,
c > 0]
ET II 16(28)
[a > 0,
b > 0,
Re ν > −1]
ET II 56(1)
Bessel functions combined with x and x2
6.527
∞
2. 0
x J 12 ν ax2 Y ν (bx) dx −1
= (4a)
∞
0
∞
4.
xY 0
2
x J 12 ν ax
3.
1 2ν
xY 0
1 2ν
b2 4a
− tan
νπ
∞
6. 0
∞
7. 0
ax2 J ν (bx) dx = −(2a)−1 H 12 ν
1 2ν
J 12 ν 2 [a > 0,
ET II 140(27)
2
b 4a
[a > 0,
∞
8. 0
2
2 2 b b π x K 12 ν ax J ν (bx) dx = I 12 ν − L 12 ν 4a 4a 4a [Re a > 0, π ⎣ cosec(νπ) L− 12 ν x K 12 ν ax2 Y ν (bx) dx = 4a
x K 12 ν ax2 K ν (bx) dx
2. 3.
∞
Re ν > −1]
b > 0,
ET II 68(9)
⎡
π = 8a
1.
Re ν > −1]
Re b > 0,
2
2
2 νπ νπ b b b π H− 12 ν J − 12 ν cos − sin − H 12 ν 4a sin(νπ) 2 4a 2 4a 4a [a > 0, Re b > 0, |Re ν| < 1] ET II 141(28)
νπ 2
I 12 ν
b2 4a
b2 4a
− cot(νπ) L 12 ν
νπ 1 K 12 ν − sec π 2
[Re a > 0,
6.527
Re ν > −1]
ax2 K ν (bx) dx
− tan
2
νπ b2 b H− 12 ν + sec 4a 2 4a b > 0, Re ν > −1] ET II 109(9)
2
2 b b π νπ H− 1 ν K ν (bx) dx = − Y − 12 ν 2 4a 4a 8a cos 2 [a > 0, Re b > 0,
=
ET II 61(35) ∞
5.
Y
669
b > 0,
b2 4a
⎤
b2 ⎦ 4a
|Re ν| < 1]
ET II 112(25)
2
2
2
b b b νπ 1 1 1 K 2ν sec + π cosec(νπ) L− 2 ν − L2ν 2 4a 4a 4a [Re a > 0, |Re ν| < 1] ET II 146(52)
1 x2 J 2ν (2ax) J ν− 12 x2 dx = a J ν+ 12 a2 2 0 ∞ 1 x2 J 2ν (2ax) J ν+ 12 x2 dx = a J ν− 12 a2 2 0 ∞ 1 x2 J 2ν (2ax) Y ν+ 12 x2 dx = − a Hν− 12 a2 2 0
a > 0,
Re ν > − 12
[a > 0,
Re ν > −2]
ET II 355(35)
[a > 0,
Re ν > −2]
ET II 355(36)
ET II 355(33)
670
Bessel Functions
6.528 0
6.529
∞
1. 0
2.
∞
x K 14 ν
x2 4
I 14 ν
x2 4
J ν (bx) dx = K 14 ν
x2 4
√ √ 2a 1 x J ν 2 ax K ν 2 ax J ν (bx) dx = b−2 e− b 2
6.528
b2 I 14 ν 4 [b > 0,
ν > −1]
[Re a > 0,
b > 0,
MO 183a
Re ν > −1] ET II 70(23)
a
x J λ (2a) I λ (2x) J μ 2 a2 − x2 I μ 2 a2 − x2 dx
0
=
a2λ+2μ+2 2 Γ(λ + 1) Γ(μ + 1) Γ(λ + μ + 2)
λ+μ+1 λ+μ+3 ; λ + 1, μ + 1, λ + μ + 1, ; −a4 × 1F 4 2 2 [Re λ > −1, Re μ > −1] ET II 376(31)
6.53–6.54 Combinations of Bessel functions and rational functions 6.531 1.10
∞
πν a2 b 2 1 2−ν 2+ν = −π J ν (ab) cot(πν) cosec(πν) − π J −ν (ab) cosec2 (πν) + cot , ; − 1; F 1 2 ν 2 2 2 4
2 2 a b 3−ν 3+ν ab πν , ; − + 2 tan 1 F 2 1; ν −1 2 2 4 2 [Re ν < 1, arg a = π, b > 0] MC ∞ Y ν (bx) 2 dx = π cot(νπ) [Y ν (ab) + Eν (ab)] + Jν (ab) + 2 [cot(νπ)] [Jν (ab) − J ν (ab)] x−a 0 [b > 0, a > 0, |Re ν| < 1] 0
2.
Y ν (bx) dx x+a
ET II 98(9) ∞
3. 0
, + 1 1 π2 K ν (bx) 2 dx = [cosec(νπ)] I ν (ab) + I −ν (ab) − e− 2 iνπ Jν (iab) − e 2 iνπ J−ν (iab) x+a 2 [Re b > 0, |arg a| < π, |Re ν| < 1] ET II 128(5)
6.532
1.11 0
∞
νπ , 1+ , J ν (x) i+ π S 0,ν (ia) − e−iνπ/2 K ν (a) = i s 0,ν (ia) + sec I ν (a) dx = 2 2 x +a a a 2 2 [Re a > 0,
Re ν > −1]
6.536
Bessel functions and rational functions
∞
2. 0
⎡
νπ Y ν (x) π 1 ⎣ 1 tan I ν (ab) − K ν (ab) − dx = νπ x2 + a2 2a 2 a cos 2 ⎤ νπ
b sin 2 2 3−ν 3+ν a b ⎦ 2 , ; + 1 F 2 1; 1 − ν2 2 2 4 [b > 0,
∞
3. 0
∞
4. 0
∞
5. 0
∞
6. 0
2. 3.11
3b
4.
Re a > 0,
|Re ν| < 1]
ET II 99(13)
νπ νπ Y ν (bx) π J ν (ab) + tan tan [Jν (ab) − J ν (ab)] − Eν (ab) − Y ν (ab) dx = 2 2 x −a 2a 2 2 [b > 0, a > 0, |Re ν| < 1] ET II 101(21)
6.533 1.
671
x J 0 (ax) dx = K 0 (ak) x2 + k 2
[a > 0,
Re k > 0]
WA 466(5)
Y 0 (ax) K 0 (ak) dx = − x2 + k 2 k
[a > 0,
Re k > 0]
WA 466(6)
J 0 (ax) π [I 0 (ak) − L0 (ak)] dx = 2 2 x +k 2k
[a > 0,
Re k > 0]
WA 467(7)
z
J p+q (z) dx = x p 0
z 1 1 J p+q (z) J p (x) J q (z − x) dx = + x z−x p q z 0 ∞ + a, dx b [J 0 (ax) − 1] J 1 (bx) 2 = − 1 + 2 ln x 4 b 0 a2 =− 4b J p (x) J q (z − x)
[Re p > 0,
Re q > −1]
WA 415(3)
[Re p > 0,
Re q > 0]
WA 415(5)
[0 < b < a] [0 < a < b] ET II 21(28)a
⎧
b 1 1 b2 ⎪ ⎪ ∞ , ; 2, − 1 [0 < b < a] ⎨ F 2 1 dx 2a 2 2 a2 = [J 0 (ax) − 1] J 1 (bx) 2 ⎪ x 0 ⎪2 E b −1 [0 < a < b] ⎩ π a2 ∞ dx =0 [0 < a < b] [1 − J 0 (ax)] J 0 (bx) x 0 a = ln [0 < b < a] b
ET II 14(16)
6.534 6.535 6.536
∞
x3 J 0 (x) 1 1 dx = K 0 (a) − π Y 0 (a) 4 4 2 4 0 ∞ x − a x 2 [J ν (x)] dx = I ν (a) K ν (a) 2 2 0 ∞x 3+ a x J 0 (bx) dx = ker(ab) x4 + a4 0
[a > 0]
ET II 340(5)
[Re a > 0,
b > 0,
Re ν > −1]
|arg a| < 14 π
ET II 342(26)
ET II 8(9), MO 46a
672
Bessel Functions
∞
6.537 0
6.538
∞
1. 0
x2 J 0 (bx) 1 dx = − 2 kei(ab) 4 4 x +a a
+ b > 0,
|arg a|
0,
∞
2.8
x−1 J ν+2n+1 (x) J ν+2m+1 (x) dx = 0
0
b > 0]
ET II 21(30)
[m = n with m, n integers, ν > −1]
= (4n + 2ν + 2)−1
[m = n,
ν > −1] EH II 64
6.539 1. 2.
π Y ν (b) Y ν (a) − 2 = 2 J ν (b) J ν (a) a x [J ν (x)] b J ν (b) dx π J ν (a) − = 2 2 Y ν (a) Y ν (b) a x [Y ν (x)]
b
dx
[J ν (x) = 0 for x ∈ [a, b]]
ET II 338(41)
[Y ν (x) = 0 for x ∈ [a, b]] ET II 339(49)
3. a
6.541
b
J ν (a) Y ν (b) π dx = ln x J ν (x) Y ν (x) 2 J ν (b) Y ν (a)
∞
x J ν (ax) J ν (bx)
1. 0
2.8 0
ET II 339(50)
dx = I ν (bc) K ν (ac) x2 + c2 = I ν (ac) K ν (bc)
[0 < b < a,
Re c > 0,
Re ν > −1]
[0 < a < b,
Re c > 0,
Re ν > −1] ET II 49(10)
∞
dx x1−2n J ν (ax) J ν (bx) 2 x + c2 2 2 p n−1−p 2 2 k &
n % ν n−1 a c /4 b c /4 π 1 1 b = − 2 I ν (bc) K ν (ac) − c 2 a sin(πν) p=0 p! Γ(1 − ν + p) k! Γ(1 − ν + k) k=0
=
−
1 c2
n %
& ν n−1 a2 c2 /4 p n−1−p b2 c2 /4 k 1 b I ν (bc) K ν (ac) − 2ν a p!(1 − ν)p k!(1 + ν)k p=0
[0 < b < a]
k=0
[n = 1, 2, . . . ,
Re ν > n − 1,
Re c > 0,
0 < b < a]
6.544
Bessel functions and rational functions
3.
∞
8 0
673
xα−1 1 c 2ρ−α J (cx) J (cx) dx = μ ν ρ 2 2 (x2 + z 2 ) (μ + ν + α)/2 − ρ, 1 + 2ρ − α ×Γ (μ⎛− ν − α)/2 + ρ + 1, (μ + ν − α)/2 + ρ + 1, (ν − μ − α)/2 + ρ + 1 α μ+ν+α μ−ν−α 1−α + ρ, 1 − + ρ, ρ; ρ + 1 − ,ρ + 1 + , 2 2 2 2 ⎞ μ+ν−α ν − μ − α 2 2 ⎠ z α−2ρ cz μ+ν ρ+1+ ,ρ + 1 + ;c z + , 2 2 2 2 ⎛ ρ − (α + μ + ν)/2, (α + μ + ν) /2 ⎝1 + μ + ν , 1 + μ + ν Γ 3F 4 ρ, μ + 1, ν + 1 2 2 ⎞ α+μ+ν α+μ+ν ;1 − ρ + , μ + 1, ν + 1, μ + ν + 1; c2 z 2 ⎠ 2 2 × 3F 4 ⎝
a1 , . . . , ap Γ (a1 ) . . . Γ (ap ) , c > 0, Re z > 0, Re(α + μ + ν) > 0; Re(α − 2ρ) > 1 Γ = b1 , . . . , bq Γ (b1 ) . . . Γ (bq ) ν ∞ J ν (ax) Y ν (bx) − J ν (bx) Y ν (ax) π b 6.542 [0 < b < a] ET II 352(16) dx = − 2 2 2 a 0 x [J ν (bx)] + [Y ν (bx)]
∞ 1 1 x dx 6.543 J μ (bx) cos (ν − μ)π J ν (ax) − sin (ν − μ)π Y ν (ax) = I μ (br) K ν (ar) 2 2 2 x + r2 0 [Re r > 0, 6.544
∞
Jν
1.
√
√ x dx 2 a 2 a 1 2 K 2ν √ =− Yν − Y 2ν √ x b x2 a π b b a > 0, b > 0,
∞
2. 0
√
a x dx 2 a 1 √ J Jν = Jν 2ν x b x2 a b
∞
3.
Jν
4.
|Re ν|
0,
b > 0,
Re ν > − 12
ET II 57(10)
√
2 a − 1 iπ 1 − 1 iνπ e 2 K 2ν √ e 4 a b Re b > 0, a > 0, |Re ν| < 12
√
√ x dx 2 a 2 a 2 π Jν = K 2ν √ + Y 2ν √ x b x2 aπ 2 b b a > 0, b > 0,
ET II 142(32)
a
|Re ν|
0, a > 0, |Re ν| < 12
5.
∞
Yν 0
√
x dx 2 a 1 iπ 1 1 iνπ = e2 K 2ν √ e 4 + Kν x b x2 a b
a
0
Re μ > |Re ν| − 2]
a
0
a ≥ b > 0,
ET II 143(38)
674
Bessel Functions
∞
6.
Kν 0
6.551
√
√ x dx 1 i 1 νπi πi 2 a − 12 νπi − 14 πi 2 a 2 4 √ √ Jν = K 2ν e K 2ν e e −e x b x2 a b b Re a > 0, b > 0, |Re ν| < 52
a
ET II 70(19)
∞
7.
Kν 0
√
√ x dx 3 2 a 2 a 3 2 Yν πν kei2ν √ πν ker2ν √ − cos = sin x b x2 a 2 2 b b Re a > 0, b > 0, |Re ν| < 52
a
ET II 113(29)
√
∞ a x dx 2 a π √ Kν K Kν = 2ν 2 x b x a b 0
8.
[Re a > 0,
Re b > 0]
ET II 146(55)
6.55 Combinations of Bessel functions and algebraic functions 6.55110
1 1/2
1.
x 0
∞
2. 1
√ −3/2 Γ 34 + 12 ν J ν (xy) dx = 2y Γ 14 + 12 ν +y −1/2 ν − 12 J ν (y) S −1/2,ν−1 (y) − J ν−1 (y) S 1/2,ν (y) y > 0, Re ν > − 32
x1/2 J ν (xy) dx = y −1/2 J ν−1 (y) S 1/2,ν (y) + 12 − ν J ν (y) S −1/2,ν−1 (y) [y > 0]
6.552
∞
1.
J ν (xy) 0
dx 1/2 a2 )
(x2 +
= I ν/2
1
2 ay
K ν/2
1
2 ay
ET II 22(2)
[Re a > 0,
y > 0,
Re ν > −1]
ET II 23(11), WA 477(3), MO 44 ∞
2.
Y ν (xy) 0
ET II 21(1)
dx (x2
+
1/2 a2 )
=−
1 sec 12 νπ K ν/2 12 ay K ν/2 12 ay + π sin 12 νπ I ν/2 12 ay π [y > 0, Re a > 0, |Re ν| < 1] ET II 100(18)
∞
3.
K ν (xy) 0
dx (x2
+
π sec 12 νπ 8 2
1/2 a2 )
=
J ν/2
1
2 ay
2
+ Y ν/2 12 ay
[Re a > 0,
2
Re y > 0,
|Re ν| < 1] ET II 128(6)
1
4.
J ν (xy) 0
5.
dx (1 −
1
Y 0 (xy) 0
6.
dx (1 −
∞
J ν (xy) 1
1/2 x2 )
1/2 x2 )
dx 1/2
(x2 − 1)
=
2 π J ν/2 12 y 2
[y > 0,
=
π J 0 12 y Y 0 12 y 2
[y > 0]
ET II 102(26)a
[y > 0]
ET II 24(23)a
=−
π J ν/2 12 y Y ν/2 12 y 2
Re ν > −1]
ET II 24(22)a
6.561
Bessel functions and powers
∞
7.
Y ν (xy) 1
dx 1/2
(x2 − 1)
=
675
2 2 π J ν/2 12 y − Y ν/2 12 y 4 [y > 0]
∞
6.553
x−1/2 I ν (x) K ν (x) K μ (2x) dx =
0
Γ
1 4
ET II 102(27)
+ 12 μ Γ 14 − 12 μ Γ 14 + ν + 12 μ Γ 14 + ν − 12 μ 3 4 Γ 4 + ν + 12 μ Γ 34 + ν − 12 μ |Re μ| < 12 , 2 Re ν > |Re μ| − 12 ET II 372(2)
6.554
∞
1.
x J 0 (xy) 0
x J 0 (xy) 0
dx (1 −
∞
3.
x J 0 (xy) 1
x J 0 (xy) 0
5.
∞
11
[y > 0]
ET II 7(6)a
= a−1 e−ay
[y > 0,
Re a > 0]
1 ν √ 2a π dx = J ν (ak) K ν (ak) 2ν (2k) Γ ν + 12 a > 0,
|arg k| >
3/2 a2 )
ν+1/2
(x4 + 4k 4 )
0
ET II 7(4)
= y −1 cos y
− 1)
xν+1 J ν (ax)
Re a > 0]
ET II 7(5)a
1/2
+
[y > 0,
= y −1 sin y
dx (x2
= y −1 e−ay
[y > 0]
1/2 x2 )
dx (x2
∞
4.
1/2
(a2 + x2 )
1
2.
dx
ET II 7(7)a
π 4,
Re ν > − 12
WA 473(1)
∞
6.555 0
∞
6.556 0
a x1/2 J 2ν−1 ax1/2 Y ν (xy) dx = − 2 Hν−1 2y
2
a 4y
a > 0,
+ a a 1/2 , dx π √ Y ν/2 J ν a x2 + 1 = − J ν/2 2 2 2 x2 + 1
y > 0,
Re ν > − 12
ET II 111(17)
[Re ν > −1,
a > 0]
MO 46
6.56–6.58 Combinations of Bessel functions and powers 6.561
1
1. 0
1
2. 0
3. 0
1
1 xν J ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [J ν (a) Hν−1 (a) − Hν (a) J ν−1 (a)] Re ν > − 12 1 xν Y ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [Y ν (a) Hν−1 (a) − Hν (a) Y ν−1 (a)] Re ν > − 12 1 xν I ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [I ν (a) Lν−1 (a) − Lν (a) I ν−1 (a)] Re ν > − 12
ET II 333(2)a
ET II 338(43)a
ET II 364(2)a
676
Bessel Functions
1
4. 0
1
5.
6.561
1 xν K ν (ax) dx = 2ν−1 a−ν π 2 Γ ν + 12 [K ν (a) Lν−1 (a) + Lν (a) K ν−1 (a)] Re ν > − 12 xν+1 J ν (ax) dx = a−1 J ν+1 (a)
ET II 367(21)a
[Re ν > −1]
ET II 333(3)a
0
1
6.
xν+1 Y ν (ax) dx = a−1 Y ν+1 (a) + 2ν+1 a−ν−2 π −1 Γ(ν + 1)
0
1
7.
xν+1 I ν (ax) dx = a−1 I ν+1 (a)
[Re ν > −1]
ET II 339(44)a
[Re ν > −1]
ET II 365(3)a
[Re ν > −1]
ET II 367(22)a
0
1
8.
xν+1 K ν (ax) dx = 2ν a−ν−2 Γ(ν + 1) − a−1 K ν+1 (a)
0
1
x1−ν J ν (ax) dx =
9. 0
1
x1−ν Y ν (ax) dx =
10. 0
1
11.
a
ν−2
2ν−1
Γ(ν)
− a−1 J ν−1 (a)
aν−2 cot(νπ) − a−1 Y ν−1 (a) 2ν−1 Γ(ν)
x1−ν I ν (ax) dx = a−1 I ν−1 (a) −
0
1
12.
ET II 333(4)a
[Re ν < 1]
ET II 339(45)a
aν−2 Γ(ν)
ET II 365(4)a
2ν−1
x1−ν K ν (ax) dx = 2−ν aν−2 Γ(1 − ν) − a−1 K ν−1 (a)
0
1
xμ J ν (ax) dx =
13.7 0
2μ Γ
[Re ν < 1]
ν+μ+1
aμ+1 Γ
2 + a−μ {(μ + ν − 1) J ν (a) S μ−1,ν−1 (a) − J ν−1 (a) S μ,ν (a)} ν−μ+1 2
1 1 1 ∞ μ μ −μ−1 Γ 2 + 2 ν + 2 μ x J ν (ax) dx = 2 a Γ 12 + 12 ν − 12 μ 0
14.
∞
15. 0
16. 0
∞
ET II 367(23)a
[a > 0,
Re(μ + ν) > −1]
− Re ν − 1 < Re μ < 12 ,
1 −μ−1 Γ 12 + 12 ν + 12 μ μ μ x Y ν (ax) dx = 2 cot 2 (ν + 1 − μ)π a Γ 12 + 12 ν − 12 μ |Re ν| − 1 < μ < 12 , xμ K ν (ax) dx = 2μ−1 a−μ−1 Γ
1+μ+ν 1+μ−ν Γ 2 2 [Re (μ + 1 ± ν) > 0,
ET II 22(8)a
a>0
EH II 49(19)
a>0
ET II 97(3)a
Re a > 0] EH II 51(27)
6.564
Bessel functions and powers
∞
17. 0
∞
18. 0
Γ 12 q + 12 J ν (ax) dx = ν−q q−ν+1 xν−q 2 a Γ ν − 12 q + 12 Γ Y ν (x) dx = xν−μ
1 2
677
−1 < Re q < Re ν −
1 2
WA 428(1), KU 144(5)
+ 12 μ Γ 12 + 12 μ − ν sin 12 μ − ν π 2ν−μ π |Re ν| < Re(1 + μ − ν) < 32
WA 430(5)
1
x2m+n+1/2 K n+1/2 (αx) dx =
19. 0
6.562
∞
xμ Y ν (bx)
0
∞
2. 0
STR
k=0
1.
n π (n + k)! γ(2m + n − k + 1, α) 2 k!(n − k)! α2m+n+3/2 2k
dx = (2a)μ π −1 sin 12 π(μ − ν) Γ 12 (μ + ν + 1) Γ 12 (1 + μ − ν) S −μ,ν (ab) x+a −2 cos 12 π (μ − ν) Γ 1 + 12 μ + 12 ν Γ 1 + 12 μ − 12 ν S −μ−1,ν (ab) ET II 98(8) b > 0, |arg a| < π, Re (μ ± ν) > −1, Re μ < 32
πk ν xν J ν (ax) dx = [H−ν (ak) − Y −ν (ak)] x+k 2 cos νπ
− 12 < Re ν < 32 ,
a > 0,
|arg k| < π
WA 479(7) ∞
3.
dx x+a
μ − ν a2 b 2 μ+ν μ−2 ,1 − ; Γ 12 (μ + ν) Γ 12 (μ − ν) b−μ 1 F 2 1; 1 − =2 2 2 4
3 − μ + ν a2 b 2 3 − μ − ν μ−3 1−μ 1 1 , ; Γ 2 (μ − ν − 1) Γ 2 (μ + ν − 1) ab −2 1 F 2 1; 2 2 4
xμ K ν (bx)
0
−πaμ cosec[π(μ − ν)] {K ν (ab) + π cos(μπ) cosec[π(ν + μ)] I ν (ab)} [Re b > 0,
∞
6.563
x−1 J ν (bx)
0
6.564
1.
∞
ν+1
x 0
|arg a| < π,
Re μ > |Re ν| − 1]
ET II 127(4)
dx πa−μ−1 = (x + a)1+μ sin[( ⎧ + ν − μ)π] Γ(μ + 1) ν+2m ∞ ⎨ (−1)m 12 ab Γ( + ν + 2m) × ⎩ m! Γ(ν + m + 1) Γ ( + ν − μ + 2m) m=0 ⎫ μ+1−+m ⎬ 1 ∞ 1 Γ(μ + m + 1) sin 2 ( + ν − μ − m)π 2 ab − 1 m! Γ (μ + ν − + m + 3) Γ 12 (μ − ν − + m + 3) ⎭ 2 m=0 ET II 23(10), WA 479 b > 0, |arg a| < π, Re( + ν) > 0, Re( − μ) < 52
dx J ν (bx) √ = 2 x + a2
2 ν+ 1 a 2 K ν+ 12 (ab) πb
Re a > 0,
b > 0,
−1 < Re ν
0,
b > 0,
Re ν > − 12
ET II 23(16)
6.565
∞
1. 0
∞
2.
−ν− 12 ab ab Γ(ν + 1) Iν x−ν x2 + a2 J ν (bx) dx = 2ν a−2ν bν Kν Γ(2ν + 1) 2 2 Re a > 0, b > 0, Re ν > − 12 −ν− 12 xν+1 x2 + a2
0
∞
3.
−ν− 32 xν+1 x2 + a2
0
WA 477(4), ET II 23(17)
Re a > 0,
√ bν π J ν (bx) dx = ν+1 ab 2 ae Γ ν + 32 [Re a > 0,
b > 0,
Re ν > − 12
ET II 24(18)
b > 0,
Re ν > −1] ET II 24(19)
∞
4.
J ν (bx)xν+1 (x2 +
0
√ ν−1 πb J ν (bx) dx = ν ab 2 e Γ ν + 12
∞
5. 0
μ+1 a2 )
dx =
aν−μ bμ K ν−μ (ab) Γ(μ + 1) −1 < Re ν < Re 2μ + 32 ,
2μ
a > 0,
b>0
MO 43
μ xν+1 x2 + a2 Y ν (bx) dx = 2ν−1 π −1 a2μ+2 (1 + μ)−1 Γ(ν)b−ν
a2 b 2 −1 × 1 F 2 1; 1 − ν, 2 + μ; − 2μ aμ+ν+1 [sin(νπ)] 4 × Γ(μ + 1)b−1−μ [I μ+ν+1 (ab) − 2 cos(μπ) K μ+ν+1 (ab)] [b > 0,
ET II 100(19)
μ 2μ a1+μ−ν b−1−μ π I −1−μ+ν (ab) cot[π(μ − ν)] cosec(πμ) x1−ν x2 + a2 Y ν (bx) dx = Γ(−μ) 2μ a1+μ−ν b−1−μ π − I 1+μ−ν (ab) cosec[π(μ − ν)] cosec(πν) Γ(−μ)
2−1−ν a2+2μ bν a2 b 2 + cos(πν) Γ(−μ) 1 F 2 1; 2 + μ, 1 + ν; 4 (1 + μ)π 2 MC Re ν < 1, Re(ν − 2μ) > −3, arg a = π, b > 0
∞
μ x1+ν x2 + a2 K ν (bx) dx = 2ν Γ(ν + 1)aν+μ+1 b−1−μ S μ−ν,μ+ν+1 (ab)
0
7.
−1 < Re ν < −2 Re μ]
∞
6.10
Re a > 0,
0
[Re a > 0,
Re b > 0,
Re ν > −1] ET II 128(8)
6.567
Bessel functions and powers
8.
∞
11
μ+1
(x2 + k 2 )
0
6.566
x−1 J ν (ax)
∞
1.
xμ Y ν (bx)
0
∞
2.
x2
xν+1 J ν (ax)
0
0
WA 477(1)
dx = 2μ−2 π −1 b1−μ + a2 , +π (μ − ν + 1) Γ 12 μ + 12 ν − 12 Γ 12 μ − 12 ν − 12 × cos 2
μ + 1 − ν a2 b 2 μ+1+ν ,2 − ; × 1 F 2 1; 2 − 2 2 4 , + , + π π 1 μ−1 cosec (μ + ν + 1) cot (μ − ν + 1) I ν (ab) − πa 2 2 , +π 2 −aμ−1 cosec (μ − ν + 1) K ν (ab) 2 b > 0, Re a > 0, |Re ν| − 1 < Re μ < 52 ET II 100(17)
x2
dx = bν K ν (ab) + b2
a > 0,
Re b > 0,
−1 < Re ν
0,
Re b > 0,
Re ν > − 12
WA 468(9) ∞
4.
x−ν K ν (ax)
0
aν k +ν−2μ−2 Γ 12 + 12 ν Γ μ + 1 − 12 − 12 ν dx = 2ν+1 Γ(μ + 1) Γ(ν + 1)
+ν +ν a2 k 2 ; − μ, ν + 1; × 1F 2 2 2 4 a2μ+2− Γ 12 ν + 12 − μ − 1
+ 1 1 22μ+3− Γ μ + 2 + ν − 2 2
ν + a2 k 2 ν− × 1 F 2 μ + 1; μ + 2 + ,μ + 2 − ; 2 2 4 a > 0, − Re ν < Re < 2 Re μ + 72 , Re k > 0
EH II 96(58) ∞
3.
679
2
dx π [Hν (ab) − Y ν (ab)] = ν+1 x2 + b2 4b cos νπ a > 0,
Re b > 0,
Re ν
0,
Re b > 0,
Re ν > − 52
WA 468(11)
6.567
1. 0
1
μ xν+1 1 − x2 J ν (bx) dx = 2μ Γ(μ + 1)b−(μ+1) J ν+μ+1 (b) [b > 0,
Re ν > −1,
Re μ > −1] ET II 26(33)a
680
Bessel Functions
1
2.
μ xν+1 1 − x2 Y ν (bx) dx
0
= b−(μ+1) 2μ Γ(μ + 1) Y μ+ν+1 (b) + 2ν+1 π −1 Γ(ν + 1) S μ−ν,μ+ν+1 (b) [b > 0,
3. 4.
6.567
Re μ > −1,
Re ν > −1]
μ 21−ν S ν+μ,μ−ν+1 (b) [b > 0, Re μ > −1] x1−ν 1 − x2 J ν (bx) dx = bμ+1 Γ(ν) 0 1 μ x1−ν 1 − x2 Y ν (bx) dx = b−(μ+1) 21−ν π −1 cos(νπ) Γ (1 − ν)
ET II 103(35)a
1
ET II 25(31)a
× S μ+ν,μ−ν+1 (b) − 2 cosec(νπ) Γ(μ + 1) J μ−ν+1 (b)
0
μ
[b > 0,
1 1−ν
5.
x
2 μ
1−x
K ν (bx) dx = 2
0
−ν−2 ν
−1
b (μ + 1)
Re μ > −1,
Γ(−ν) 1 F 2
Re ν < 1]
b2 1; ν + 1, μ + 2; 4
ET II 104(37)a
+π2μ−1 b−(μ+1) cosec (νπ) Γ(μ + 1) I μ−ν+1 (b) 6. 7.
8.
9.
10.
11.
12. 13.
1
[Re μ > −1,
Re ν < 1]
π Hν− 12 (b) [b > 0] 2b 0 1 + , π dx cosec(νπ) cos(νπ) J ν+ 12 (b) − H−ν− 12 (b) x1+ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν > −1] 1 + , π dx cot(νπ) Hν− 12 (b) − Y ν− 12 (b) − J ν− 12 (b) x1−ν Y ν (bx) √ = 2b 1 − x2 0 [b > 0, Re ν < 1] 2 1 ν− 12 √ b xν 1 − x2 J ν (bx) dx = 2ν−1 πb−ν Γ ν + 12 J ν 2 0 b > 0, Re ν > − 12
1 1 √ −ν b b 1 ν 2 ν− 2 ν−1 x 1−x Y ν (bx) dx = 2 πb Γ ν + Jν Yν 2 2 2 0 b > 0, Re ν > − 12
1 ν− 12 √ b b 1 xν 1 − x2 K ν (bx) dx = 2ν−1 πb−ν Γ ν + Iν Kν 2 2 2 0 Re ν > − 12
2 1 1 √ −ν b 1 ν 2 ν− 2 −ν−1 x 1−x I ν (bx) dx = 2 πb Γ ν + Iν 2 2 0
1 ν−1 1 1 b −ν− 2 − ν sin b xν+1 1 − x2 J ν (bx) dx = 2−ν √ Γ 2 π 0 b > 0, |Re ν| < 12 x1−ν J ν (bx) √
dx = 1 − x2
ET II 129(12)a ET II 24(24)a
ET II 102(28)a
ET II 102(30)a
ET II 24(25)a
ET II 102(31)a
ET II 129(10)a ET II 365(5)a
ET II 25(27)a
6.571
Bessel functions and powers
√ −ν b b b b 1 x x −1 Y ν (bx) dx = 2 πb Γ ν + Jν J −ν −Yν Y −ν 2 2 2 2 2 1 |Re ν| < 12 , b > 0 ET II 103(32)a
∞ 2 ν− 12 b 2ν−1 1 xν x2 − 1 K ν (bx) dx = √ b−ν Γ ν + Kν 2 2 π 1 Re b > 0, Re ν > − 12 ET II 129(11)a
∞ −ν− 12 √ 1 b b − ν Jν x−ν x2 − 1 J ν (bx) dx = −2−ν−1 πbν Γ Yν 2 2 2 1 b > 0, |Re ν| < 12 ET II 25(26)a
∞ ν− 12 1 2ν + ν cos b x−ν+1 x2 − 1 J ν (bx) dx = √ b−ν−1 Γ 2 π 1 b > 0, |Re ν| < 12 ET II 25(28)
14.
15.
16.
17.8
6.568
∞
∞
1.
ν
2
ν− 12
xν Y ν (bx)
0
xμ Y ν (bx)
0
ν−2
dx π = aν−1 J ν (ab) x2 − a2 2
a > 0,
b > 0,
− 12 < Re ν
0, b > 0, |Re ν| − 1 < Re μ < 52
ET II (101)(25)
1
xλ (1 − x)μ−1 J ν (ax) dx
6.569 0
=
6.571
681
∞
1.
+
0
2. 0
∞
+
x2 + a2
12
Γ(μ) Γ(1 + λ + ν)2−ν aν Γ(ν + 1) Γ(1 + λ + μ + ν)
λ+1+ν λ+2+ν λ+1+μ+ν λ+2+μ+ν a2 , ; ν + 1, , ;− × 2F 3 2 2 2 2 4 [Re μ > 0, Re(λ + ν) > −1] ET II 193(56)a
,μ ab ab dx ± x J ν (bx) √ = aμ I 12 (ν∓μ) K 12 (ν±μ) 2 2 x2 + a 2 Re a > 0, b > 0, Re ν > −1, Re μ < 32
ET II 26(38)
,μ 1 dx x2 + a2 2 − x Y ν (bx) √ 2 2 x + a
ab ab ab ab = aμ cot(νπ) I 12 (μ+ν) K 12 (μ−ν) − cosec(νπ) I 12 (μ−ν) K 12 (μ+ν) 2 2 2 2 Re a > 0, b > 0, Re μ > − 32 , |Re ν| < 1 ET II 104(40)
682
Bessel Functions
∞
3.
+
6.572
∞
−μ
x
1.
,μ dx + x K ν (bx) √ 2 2 x + a
ab ab ab ab π2 μ a cosec(νπ) J 12 (ν−μ) = Y − 12 (ν+μ) − Y 12 (ν−μ) J − 12 (ν+μ) 4 2 2 2 2 [Re a > 0, Re b > 0] ET II 130(15)
x2 + a2
0
+
2
12
x +a
2
12
,μ +a
0
6.572
Γ 1+ν−μ dx 2 W 12 μ, 12 ν (ab) M − 12 μ, 12 ν (ab) J ν (bx) √ = ab Γ(ν + 1) x2 + a2 [Re a > 0, b > 0, Re(ν − μ) > −1] ET II 26(40)
∞
2. 0
+ ,μ 1 dx x−μ x2 + a2 2 + a K ν (bx) √ 2 x + a2
1+ν−μ 1−ν−μ Γ 2 2 W 12 μ, 12 ν (iab) W 12 μ, 12 ν (−iab) = 2ab Re b > 0, Re μ + |Re ν| < 1] ET II 130(18), BU 87(6a) Γ
[Re a > 0,
∞
3.
x−μ
+
x2 + a2
12
−a
,μ
0
6.573
∞
1.
xν−M +1 J ν (bx)
0
k -
dx + a2
Γ 1+ν+μ ν−μ 1 2 tan π M 12 μ, 12 ν (ab) = − W − 12 μ, 12 ν (ab) ab Γ(ν + 1) 2
ν−μ π W 12 μ, 12 ν (ab) + sec 2 Re a > 0, b > 0, |Re ν| < 12 + 12 Re μ ET II 105(42)
Y ν (bx) √
x2
J μi (ai x) dx = 0
i=1 %
ai > 0,
k
M=
k
μi &
i=1
ai < b < ∞,
−1 < Re ν < Re M +
1 2k
−
1 2
ET II 54(42)
i=1
2. 0
∞
xν−M −1 J ν (bx) %
k -
J μi (ai x) dx = 2ν−M −1 b−ν Γ(ν)
i=1
ai > 0,
k i=1
ai < b < ∞,
k -
aμi i , Γ (1 + μi ) i=1 &
0 < Re ν < Re M + 12 k +
3 2
M=
k
μi
i=1
WA 460(16)a, ET II 54(43)
6.576
6.574
Bessel functions and powers
683
ν+μ−λ+1 ∞ α Γ 2
J ν (αt) J μ (βt)t−λ dt = −ν + μ + λ + 1 0 2λ β ν−λ+1 Γ Γ(ν + 1) 2
ν+μ−λ+1 ν−μ−λ+1 α2 , ; ν + 1; 2 ×F 2 2 β [Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < α < β] ν
1.8
2.
WA 439(2)a, MO 49
If we reverse the positions of ν and μ and at the same time reverse the positions of α and β, the function on the right-hand side of this equation will change. Thus, the right-hand side represents α α that is not analytic at = 1. a function of β β For α = β, we have the following equation:
ν+μ−λ+1 ∞ αλ−1 Γ(λ) Γ 2
J ν (αt) J μ (αt)t−λ dt = −ν + μ + λ + 1 ν + μ + λ +1 ν−μ+λ+1 0 λ 2 Γ Γ Γ 2 2 2 [Re(ν + μ + 1) > Re λ > 0, α > 0] MO 49, WA 441(2)a
3.∗
If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right-hand side of equation 6.574 1 (or 6.574 3) vanishes. The cases in which the hypergeometric function F in 6.574 3 (or 6.574 1) can be reduced to an elementary function are then especially important.
μ+ν−λ+1 ∞ βν Γ 2
J ν (αt) J μ (βt)t−λ dt = ν − μ + λ+1 0 2λ αμ−λ+1 Γ Γ(ν + 1) 2
ν + μ − λ + 1 −ν + μ − λ + 1 β2 , ; μ + 1; 2 ×F 2 2 α [Re(ν + μ − λ + 1) > 0, Re λ > −1, 0 < β < α] MO 50, WA 440(3)a If μ − ν + λ + 1 (or ν − μ + λ + 1) is a negative integer, the right-hand side of equation 6.754 1 (or 6.574 3) vanishes. The cases in which the hypergeometric function F in 6.754 3 (or 6.574 1) can be reduced to an elementary function are then especially important.
6.575 1.11
∞
J ν+1 (αt) J μ (βt)tμ−ν dt = 0
0
= 2. 0
∞
2 ν−μ
[α < β]
α −β β Γ(ν − μ + 1) 2
2ν−μ αν+1 √
μ
[α ≥ β] [Re(ν + 1) > Re μ > −1]
MO 51
J ν (x) J μ (x) π Γ(ν + μ) dx = ν+μ xν+μ 2 Γ ν + μ + 12 Γ ν + 12 Γ μ + 12 [Re(ν + μ) > 0]
KU 147(17), WA 434(1)
684
6.576
Bessel Functions
∞
1.
xμ−ν+1 J μ (x) K ν (x) dx =
0
2.11
1 2
6.576
Γ(μ − ν + 1)
[Re μ > −1,
Re(μ − ν) > −1]
1−λ ν ν ∞ a b Γ ν+ 2
x−λ J ν (ax) J ν (bx) dx = 1+λ 0 λ 2ν−λ+1 2 (a + b) Γ(ν + 1) Γ 2
1 4ab 1−λ , ν + ; 2ν + 1; ×F ν + 2 2 (a + b)2 [a > 0, b > 0, 2 Re ν + 1 > Re λ > −1]
ET II 370(47)
ET II 47(4)
ν−λ+μ+1 ν−λ−μ+1 ∞ b Γ Γ 2 2 x−λ K μ (ax) J ν (bx) dx = λ+1 ν−λ+1 2 a Γ(1 + ν) 0
ν−λ+μ+1 ν−λ−μ+1 b2 , ; ν + 1; − 2 ×F 2 2 a [Re (a ± ib) > 0, Re(ν − λ + 1) > |Re μ|] EH II 52(31), ET II 63(4), WA 449(1) ν
3.
∞
4.
−λ
x 0
∞
5.
−λ
x 0
∞
6.
1−λ+μ+ν 1−λ−μ+ν 2−2−λ a−ν+λ−1 bν Γ K μ (ax) K ν (bx) dx = Γ 2 Γ(1 − λ)
2
1−λ+μ−ν 1−λ−μ−ν ×Γ Γ 2 2
1−λ+μ+ν 1−λ−μ+ν b2 , ; 1 − λ; 1 − 2 ×F 2 2 a [Re a + b > 0, Re λ < 1 − |Re μ| − |Re ν|] ET II 145(49), EH II 93(36) − 12 λ + 12 μ + 12 ν Γ 12 − 12 λ − 12 μ + 12 ν K μ (ax) I ν (bx) dx = 2λ+1 Γ(ν + 1)a−λ+ν+1
2 1 1 − λ + 12 μ + 12 ν, 12 − 12 λ − 12 μ + 12 ν; ν + 1; ab 2 ×F 2 2 [Re (ν + 1 − λ ± μ) > 0, a > b] EH II 93(35) bν Γ
[a > b, 7.
0
∞
2
π(ν − μ − λ) ∞ −λ 2 sin x K μ (ax) I ν (bx) dx π 2 0 Re λ > −1, Re (ν − λ + 1 ± μ) > 0] (see 6.576 5)
x−λ Y μ (ax) J ν (bx) dx =
0
8
1
xμ+ν+1 J μ (ax) K ν (bx) dx = 2μ+ν aμ bν
EH II 93(37)
Γ(μ + ν + 1) μ+ν+1
(a2 + b2 )
[Re μ > |Re ν| − 1,
Re b > |Im a|]
ET 137(16), EH II 93(36)
6.578
Bessel functions and powers
6.577 1.8
∞
[a > 0, 2.
∞
dx = (−1)n cν−μ+2n I μ (ac) K ν (bc) x2 + c2 Re c > 0, 2 + Re μ − 2n > Re ν > −1 − n, n ≥ 0 an integer]
ET II 49(13)
dx = (−1)n cμ−ν+2n I ν (bc) K μ (ac) + c2 Re ν − 2n + 2 > Re μ > −n − 1, n ≥ 0 an integer]
ET II 49(15)
xν−μ+1+2n J μ (ax) J ν (bx)
0
8
b > a,
xμ−ν+1+2n J μ (ax) J ν (bx)
0
[b > 0, 6.578
∞
1. 0
∞
2.
∞
3. 0
∞
4. 0
a > b,
x2
2−1 aλ bμ c−λ−μ− Γ λ+μ+ν+ 2
x−1 J λ (ax) J μ (bx) J ν (cx) dx = λ+μ−ν+ Γ(λ + 1) Γ(μ + 1) Γ 1 − 2
λ+μ−ν+ λ+μ+ν+ a2 b 2 , ; λ + 1, μ + 1; 2 , 2 ×F4 2 2 c c 5 Re(λ + μ + ν + ) > 0, Re < , a > 0, b > 0, c > 0, c > a + b ET II 351(9) 2 x−1 J λ (ax) J μ (bx) K ν (cx) dx
+λ+μ−ν +λ+μ+ν 2−2 aλ bμ c−−λ−μ Γ = Γ Γ(λ + 1) Γ(μ + 1) 2 2
+λ+μ−ν +λ+μ+ν a2 b2 , ; λ + 1, μ + 1; − 2 , − 2 ×F4 2 2 c c [Re( + λ + μ) > |Re ν|, Re c > |Im a| + |Im b|] ET II 373(8)
0
685
∞
5.
xλ−μ−ν+1 J ν (ax) J μ (bx) J λ (cx) dx = 0 Re λ > −1, Re(λ − μ − ν) < 12 ,
c > b > 0,
0 b > 0,
0 −1, Re(μ + ν) > −1 2bcu = a2 + b2 + c2 , WA 452(2), ET II 64(12)
∞
− 1 μ− 1 μ+ 1 1 1 1 xμ+1 I ν (ax) K μ (bx) J ν (cx) dx = √ a−μ−1 bμ c−μ−1 e−(μ− 2 ν+ 4 )πi v 2 + 1 2 4 Q ν− 12 (iv), 2 2π [Re b > |Re a| + |Im c|; Re ν > −1, Re(μ + ν) > −1] ET II 66(22) 2acv = b2 − a2 + c2
686
Bessel Functions
8.
∞
x1−μ J μ (ax) J ν (bx) J ν (cx) dx 1 1 2 −μ μ− 1 2 −μ a (bc)μ−1 (sinh u) 2 sin[(μ − ν)π]e(μ− 2 )πi Q ν− = 1 (cosh u) 3 2 π 1 1 −μ μ− 12 μ−1 2 −μ (sin v) P ν− 1 (cos v) = √ a (bc) 2 2π
11 0
[a > b + c] [|b − c| < a < b + c] [0 < a < |b − c|]
=0
9.
6.578
2bc cosh u = a − b − c , 2bc cos v = b + c − a , ∞ J ν (ax) J ν (bx) J ν (cx)x1−ν dx = 0 2
0
2
2
2
2
ν−1
2
b > 0,
c > 0;
Re ν > −1, Re μ > − 12
[0 < c ≤ |a − b| or c ≥ a + b]
2ν−1
Δ 2 [|a − b| < c < a + b] (abc)ν Γ ν + 12 Γ 12 a > 0, b > 0, c > 0; Re ν > − 12 Δ = 14 [c2 − (a − b)2 ] [(a + b)2 − c2 ], =
10.11
(Δ > 0 is equal to the area of a triangle whose sides are a, b, and c.) √ ν ∞ πc Γ(ν + μ + 1) Γ(ν − μ + 1) −ν− 12 ν+1 x K μ (ax) K μ (bx) J ν (cx) dx = P μ− 1 (u) 1 ν+ 1 2 0 23/2 (ab)ν+1 (u2 − 1) 2 4 2 2 2 2abu = a + b + c , Re(a + b) > |Im c|, Re (ν ± μ) > −1, Re ν > −1
∞
11.11 0
∞
12.8
ET II 67(30)
1 ν+ 1 (ab)−ν−1 cν e−(ν+ 2 )πi Q μ− 21 (u) 2 xν+1 K μ (ax) I μ (bx) J ν (cx) dx = 2abu = a2 + b2 + c2 √ 1 ν+ 1 2π (u2 − 1) 2 4 [Re a > |Re b| + |Im c|; Re ν > −1, Re(μ + ν) > −1] ET II 66(24)
2
xν+1 [J ν (ax)] Y ν (bx) dx = 0
0
−ν− 12 23ν+1 a2ν b−ν−1 2 1 b − 4a2 = √ πΓ 2 −ν
0 < b < 2a,
|Re ν|
0,
2a < b < ∞,
0 < b < 2a |Re ν|
0,
π > ϕ > 0, 2
2ν Γ(μ + ν + 1) (sin ϕ) aν+2 0 −1 ET II 64(11)
6.581
Bessel functions and powers
∞
15.
ν+1
x
J ν (ax) K ν (bx) J ν (cx) dx =
0
∞
16.8
23ν (abc)ν Γ ν + 12
,ν+ 12 √ + 2 2 π (a + b2 + c2 ) − 4a2 c2 Re b > |Im a|, 3ν
xν+1 I ν (ax) K ν (bx) J ν (cx) dx =
0
687
ν
1 2
c > 0,
Re ν > − 12
ET II 63(8)
2 (abc) Γ ν + ,ν+ 12 √ + 2 2 π (b − a2 + c2 ) + 4a2 c2 Re b > |Re a| + |Im c|;
Re ν > − 12
ET II 65(18)
6.579 1.
∞
x2ν+1 J ν (ax) Y ν (ax) J ν (bx) Y ν (bx) dx
a2ν Γ(3ν + 1) 3 a2 1 1 = 3 F ν + 2 , 3ν + 1; 2ν + 2 ; b2 2πb4ν+2 Γ 2 − ν Γ 1 2ν + 2 EH II 94(45), ET II 352(15) 0 < a < b, − 3 < Re ν < 12
0
∞
2. 0
∞
3.
x2ν+1 J ν (ax) K ν (ax) J ν (bx) K ν (bx) dx
2ν−3 a2ν Γ ν+1 Γ ν + 12 Γ 3ν+1 a4 3ν + 1 2 2 √ 4ν+2 ; 2ν + 1; 1 − 4 = F ν + 12 , 2 b πb Γ(ν + 1) 0 < a < b, Re ν > − 13 ET II 373(10) 4
x1−2ν [J ν (ax)] dx =
0
∞
4.
2
Γ(ν) Γ(2ν) 2 2π Γ ν + 12 Γ(3ν) 2
x1−2ν [J ν (ax)] [J ν (bx)] dx =
0
[Re ν > 0]
a2ν−1 Γ(ν) F 2πb Γ ν + 12 Γ 2ν + 12
ET II 342(25)
1 a2 1 ν, − ν; 2ν + ; 2 2 2 b ET II 351(10)
6.581
a λ−1
x
1. 0
∞ (−1)m Γ(λ + μ + m) Γ(λ + m) J λ+μ+ν+2m (a) J μ (x) J ν (a − x) dx = 2 m! Γ(λ) Γ(μ + m + 1) m=0 [Re(λ + μ) > 0, Re ν > −1] λ
ET II 354(25) a
2.8
xλ−1 (a − x)−1 J μ (x) J ν (a − x) dx
0
=
∞ 2λ (−1)m Γ(λ + μ + m) Γ(λ + m) (λ + μ + ν + 2m) J λ+μ+ν+2m (a) aν m=0 m! Γ(λ) Γ(μ + m + 1)
[Re(λ + μ) > 0,
a
0
ET II 354(27)
Γ μ + 12 Γ ν + 12 μ+ν+ 1 2 J a x (a − x) J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 1) Re μ > − 12 , Re ν > − 12 μ
3.
Re ν > 0]
ν
ET II 354(28), EH II 46(6)
688
Bessel Functions
a
x (a − x) μ
4.
ν+1
0
5.
a
Γ μ + 12 Γ ν + 32 μ+ν+ 3 2 J a J μ (x) J ν (a − x) dx = √ μ+ν+ 12 (a) 2π Γ(μ + ν + 2) Re ν > −1, Re μ > − 12
xμ (a − x)−μ−1 J μ (x) J ν (a − x) dx =
0
∞
6.582
−μ
xμ−1 |x − b|
0
∞
6.583
6.582
ET II 354(29)
1
2μ Γ μ + 2 Γ(ν − μ) μ √ a J ν (a) π Γ(μ + ν + 1) Re ν > Re μ > − 12
ET II 355(30)
1 K μ (|x − b|) K ν (x) dx = √ (2b)−μ Γ 12 − μ Γ(μ + ν) Γ(μ − ν) K ν (b) π b > 0, Re μ < 12 , Re μ > |Re ν|
xμ−1 (x + b)−μ K μ (x + b) K ν (x) dx =
√
0
ET II 374(14)
π Γ(μ + ν) Γ(μ − ν) K ν (b) 2μ bμ Γ μ + 12 [|arg b| < π, Re μ > |Re ν|] ET II 374(15)
6.584 1.8
∞ x−1
+
, πi H (1) axeπi H (1) ν (ax) − e ν m+1
(x2 − r2 ) m = 0, 1, 2, . . . ,
0
∞
2.8 0
∞
3. 0
4. 0
∞
m + , d πi (1) −2 r H (ar) ν 2m m! r dr Im r > 0, a > 0, |Re ν| < Re < 2m + 72
dx =
WA 465
x−1 1 1 cos ( − ν)π J ν (ax) + sin ( − ν)π Y ν (ax) m+1 dx 2 2 (x2 + k 2 )
m −2 d (−1)m+1 k K ν (ak) = m 2 · m! k dk m = 0, 1, 2, . . . , Re k > 0, a > 0, |Re ν| < Re < 2m + 72 WA 466(2) x1−ν dx am K ν+m (ak) {cos νπ J ν (ax) − sin νπ Y ν (ax)} = m+1 2m · m!k ν+m (x2 + k 2 ) m = 0, 1, 2, . . . , Re k > 0, a > 0, −2m −
1
3 2
< Re ν < 1
WA 466(3)
x−1 − 12 ν − μ π J ν (ax) + sin 12 − 12 ν − μ π Y ν (ax) μ+1 dx (x2 + k 2 ) ⎡ ν
1 ak Γ 12 + 12 ν +ν +ν a2 k 2 πk−2μ−2 2 ⎣ 1 ; − μ, ν + 1; = 1F 2 2 sin νπ · Γ(μ + 1) Γ(ν + 1) Γ 2 + 12 ν − μ 2 2 4 ⎤ 1 −ν 1
Γ 2 − 12 ν −ν −ν a2 k 2 ⎦ 2 ak ; − μ, 1 − ν; − 1F 2 2 2 4 Γ(1 − ν) Γ 12 − 12 ν − μ WA 407(1) a > 0, Re k > 0, |Re ν| < Re < 2 Re μ + 72
cos
2
6.591
Powers and Bessel functions of complicated arguments
∞
5.8 0
689
⎤⎧ ⎡ ⎛ ⎞ ⎤ ⎨ 1 ⎣ J μj (bn x)⎦ cos ⎣ ⎝ + μj − ν ⎠ π ⎦ J ν (ax) ⎩ 2 j=1 j ⎫ ⎡ ⎛ ⎞ ⎤ ⎬ x−1 1 + sin ⎣ ⎝ + μj − ν ⎠ π ⎦ Y ν (ax) dx ⎭ x2 + k 2 2 j ⎡ ⎤ n = −⎣ I μj (bn k)⎦ K ν (ak)k −2 ⎡
n -
j=1
⎡ ⎣Re k > 0,
a>
⎛
|Re bj |,
Re ⎝ +
j
⎞
⎤
μj ⎠ > |Re ν|⎦
WA 472(9)
j
6.59 Combinations of powers and Bessel functions of more complicated arguments 6.591
∞
1. 0
1
x2ν+ 2 J ν+ 12
a x
K ν (bx) dx =
√ √ √ 1 2πb−ν−1 aν+ 2 J 1+2ν 2ab K 1+2ν 2ab [a > 0,
∞
0
1
x2ν+ 2 Y ν+ 12
a x
K ν (bx) dx =
√ √ √ 1 2πb−ν−1 aν+ 2 Y 2ν+1 2ab K 2ν+1 2ab [a > 0,
0
∞
4. 0
1
x2ν+ 2 K ν+ 12
a
x−2ν+ 2 J ν− 12 1
x
K ν (bx) dx =
√
∞
5.
−2ν+ 12
x 0
6. 0
Re ν > −1]
2πb−ν−1 aν+ 2 1
1 √ 1 √ K 2ν+1 e 4 iπ 2ab K 2ν+1 e− 4 iπ 2ab
[Re a > 0, Re b > 0] ET II 146(56) √ √ 1 K ν (bx) dx = 2πbν−1 a 2 −ν K 2ν−1 2ab x + √ √ , 2ab + cos(νπ) Y 2ν−1 2ab × sin(νπ) J 2ν−1
a
[a > 0,
Re b > 0,
ET II 143(41) ∞
3.
Re ν > −1] ET II 142(35)
2.
Re b > 0,
Y ν− 12
Re b > 0,
x−2ν+ 2 J 12 −ν 1
ET II 142(34)
√ π ν−1 1 −ν K ν (bx) dx = − b a 2 sec(νπ) K 2ν−1 2ab x 2 √ √ , + 2ab − J 1−2ν 2ab × J 2ν−1
a
[a > 0, ∞
Re ν < 1]
a x
Re ν < 1]
ET II 143(40)
J ν (bx) dx
1 1 = − i cosec(2νπ)bν−1 a 2 −ν e2νπi J 1−2ν (u) J 2ν−1 (v) − e−2νπi J 2ν−1 (u) J 1−2ν (v) , + 2 1 1 1 1 ET II 58(12) u = 12 ab 2 e 4 πi , v = 12 ab 2 e− 4 πi , a > 0, b > 0, − 12 < Re ν < 3
690
Bessel Functions
∞
7. 0
x−2ν+ 2 K ν− 12 1
a x
Y ν (bx) dx =
6.592
√ √ √ 1 2πbν−1 a 2 −ν Y 2ν−1 2ab K 2ν−1 2ab b > 0, Re a > 0, Re ν > 16 ET II 113(30)
∞ aν− bν Γ 12 μ + 12 − 12 ν b −1 x J μ (ax) J ν dx = 2ν−+1 x 2 Γ(ν + 1) Γ 12 μ + 12 ν − 12 + 1 0
ν+μ− a2 b 2 ν−μ− + 1, + 1; × 0 F 3 ν + 1, 16 2 2 aμ bμ+ Γ 12 ν − 12 μ − 12 + 2μ++1 2 Γ(μ + 1) Γ 12 μ + 12 ν + 12 + 1
ν+μ+ a2 b 2 μ−ν+ + 1, + 1; × 0 F 3 μ + 1, 2 2 16 3 a > 0, b > 0, − Re μ + 2 < Re < Re ν + 32
8.
6.592
∞
1.
x (1 − x) λ
μ−1
0
2.10
1
WA 480(1)
√ Γ(μ) Γ λ + 1 + 12 ν −ν ν Y ν a x dx = 2 a cot(νπ) Γ(1 + ν) Γ λ + 1 + μ + 12 ν
a2 1 1 × 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + 2 ν; − 4 Γ(μ) Γ λ + 1 − 12 ν ν −ν −2 a cosec(νπ) Γ(1 − ν) Γ λ + 1 + μ − 12 ν
a2 × 1 F 2 λ − 12 ν + 1; 1 − ν, λ + 1 + μ − 12 ν; − 4 Re λ > −1 + 12 |Re ν|, Re μ > 0 ET II 197(76)a
√ xλ (1 − x)μ−1 K ν a x dx
0
=2
+2
−ν−1 −ν
a
Γ(ν) Γ(μ) Γ λ + 1 − 12 ν a2 1 1 1 F 2 λ + 1 − 2 ν; 1 − ν, λ + 1 + μ − 2 ν; 4 Γ λ + 1 + μ − 12 ν
Γ(−ν) Γ λ + 1 + 12 ν Γ(μ) 1 a2 1 a 1 F 2 λ + 1 + 2 ν; 1 + ν, λ + 1 + μ + ν; 2 4 Γ λ + 1 + μ + 12 ν
−1−ν ν
2ν−1 = ν Γ(μ) G 21 13 a
a2 4
!ν
! −λ !2 ! ν, 0, ν − λ − μ 2 OB 159 (3.16)
Re λ > −1 + 12 |Re ν|,
Re μ > 0
ET II 198(87)a
6.592
Powers and Bessel functions of complicated arguments
3.
∞
11
x (x − 1) λ
μ−1
1
∞
4.
1
5.
x− 2 (1 − x)− 2 1
1
0
1
6.
x− 2 (1 − x)− 2 1
1
0
7.
1
x− 2 (1 − x)− 2
0
8. 9.
a2 4
!
!0 ! ! −μ, λ + 1 ν, λ − 1 ν Γ(μ) 2 2 a > 0, 0 < Re μ < 34 − Re λ
√ xλ (x − 1)μ−1 K ν a x dx = Γ(μ)22λ−1 a−2λ G 30 13
1
√ J ν a x dx = 22λ a−2λ G 20 13
1
1
691
ET II 205(36)a
!
a !! 0 4 ! −μ, 12 ν + λ, − 12 ν + λ 2
[Re a > 0, Re μ > 0] 2 √ 1 a J ν a x dx = π J 1 ν [Re ν > −1] 2 2 2 √ 1 a I ν a x dx = π I 1 [Re ν > −1] ν 2 2
+ a , a √ 1 a 1 νπ I ν2 + I − ν2 K ν2 K ν a x dx = π sec 2 2 2 2 2
ET II 209(60)a ET II 194(59)a
ET II 197(79)
[|Re ν| < 1] ET II 198(85)a , + 2 √ 1 1 a x− 2 (x − 1)− 2 K ν a x dx = K ν2 [Re a > 0] ET II 208(56)a 2 1
1 + + a ,2 a ,2 √ − 12 − 12 x (1 − x) Y ν a x dx = π cot(νπ) J ν2 − cosec(νπ) J − ν2 2 2 0
∞
∞
[|Re ν| < 1]
10. 1
a > 0,
0 < Re μ
0, 0 < Re μ < 12 Re ν + 34 ET II 205(35)a
∞
12.
√ 1 x− 2 ν (x − 1)μ−1 K ν a x dx = Γ(μ)2μ a−μ K ν−μ (a)
1
[Re a > 0, ∞
13. 1
√ 1 x− 2 ν (x − 1)μ−1 Y ν a x dx = 2μ a−μ Y ν−μ (a) Γ(μ) a > 0,
Re μ > 0]
0 < Re μ
0, ∞
Im a > 0]
ET II 206(45)a
Im a < 0]
ET II 207(48)a
√ 1 (2) a x dx = 2μ a−μ H ν−μ (a) Γ(μ) x− 2 ν (x − 1)μ−1 H (2) ν
1
[Re μ > 0,
692
Bessel Functions
1
16. 0
17. 0
1
6.593
√ 1 22−ν a−μ s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 J ν a x dx = Γ(ν) [Re μ > 0]
ET II 194(64)a
√ 1 22−ν a−μ cot(νπ) s μ+ν−1,μ−ν (a) x− 2 ν (1 − x)μ−1 Y ν a x dx = Γ(ν) −2μ a−μ cosec(νπ) J μ−ν (a) Γ(μ) [Re μ > 0,
2
√ a 1 b > 0, Re ν > − 12 x J 2ν−1 a x J ν (bx) dx = ab−2 J ν−1 2 4b 0 2
2 ∞ √ √ a a πa x J 2ν−1 a x K ν (bx) dx = 2 I ν−1 − Lν−1 4b 4b 4b 0 Re b > 0, Re ν > − 12
6.593 1. 2.
Re ν < 1]
6.594 1.
∞√
∞
0
√ √ √ 1 x I 2ν−1 a x J 2ν−1 a x K ν (bx) dx = π2−ν a2ν−1 b−2ν− 2 J ν− 12 ν
[Re b > 0, ∞
2.
a2 2b
ET II 196(75)a
ET II 58(15)
ET II 144(44)
Re ν > 0]
ET II 148(65)
√ √ xν I 2ν−1 a x Y 2ν−1 a x K ν (bx) dx
0
√ −ν−1 2ν−1 −2ν− 1 2 cosec(νπ) π2 a b 2
2
2 a a a × H 12 −ν + cos(νπ) J ν− 12 + sin(νπ) Y ν− 12 2b 2b 2b [Re b > 0, Re ν > 0] ET II 148(66) ∞ √ √ xν J 2ν−1 a x K 2ν−1 a x K ν (bx) dx 0 2
2 a a 2 −ν−2 2ν−1 −2ν− 12 =π 2 a b cosec(νπ) H 12 −ν − Y 12 −ν 2b 2b [Re b > 0, Re ν > 0] ET II 148(67) =
3.
6.595 1.
∞
ν+1
x
J ν (cx)
0
n -
zi−μi J μi (ai zi ) dx = 0 i=1 % ai > 0,
Re bi > 0,
< zi = x2 + b2i & n n 1 1 n+ > Re ν > −1 ai < c; Re μi − 2 2 i=1 i=1 EH II 52(33), ET II 60(26)
2. 0
∞
xν−1 J ν (cx)
< −μi bi J μi (ai bi ) zi−μi J μi (ai zi ) dx = 2ν−1 Γ(ν)c−ν zi = x2 + b2i i=1 % i=1 & n n 1 3 n+ > Re ν > 0 ai < c, Re μi + ai > 0, Re bi > 0, 2 2 i=1 i=1 n -
n -
EH II 52(34), ET II 60(27)
6.596
6.596
Powers and Bessel functions of complicated arguments
∞
1. 0
693
x2μ+1 2μ Γ(μ + 1) J ν α x2 + z 2 dx = μ+1 ν−μ−1 J ν−μ−1 (αz) ν α z (x2 + z 2 )
1 1 ν− α > 0, Re > Re μ > −1 2 4
√ ∞ α α J ν α t2 + 1 π √ Y ν2 [Re ν > −1, dt = − J ν2 2 2 2 t2 + 1 0 ∞ x2μ+1 2μ Γ(μ + 1) K ν α x2 + z 2 dx = K ν−μ−1 (αz) ν αμ+1 z ν−μ−1 (x2 + z 2 ) 0
WA 457(5)
2. 3.
4.8
∞
J ν (βx) 0
√ J μ−1 α x2 + z 2 (x2 + z 2 )
1 1 2 μ+ 2
[α > 0, xν+1 dx =
αμ−1 z ν K ν (βz) 2μ−1 Γ(μ) [α < β,
α > 0]
MO 46
Re μ > −1]
WA 457(6)
Re(μ + 2) > Re ν > −1]
√ ∞ J μ α x2 + z 2 ν−1 2ν−1 Γ(ν) J μ (αz) J ν (βx) x dx = μ βν zμ (x2 + z 2 ) 0 [Re(μ + 2) > Re ν > 0,
ET II 59(19)
5.8
β > α > 0] WA 459(12)
√ ∞ J μ α x2 + z 2 ν+1 J ν (βx) dx μ x (x2 + z 2 ) 0
6.6
=0
μ−ν−1 βν α2 − β 2 = μ J μ−ν−1 z α2 − β 2 α z
8.8
[α > β > 0]
[Re μ > Re ν > −1] WA 415(1) √ μ−ν−1 ∞ K μ α x2 + z 2 ν+1 α2 + β 2 βν 2 + β2 z J ν (βx) x dx = K α μ−ν−1 μ αμ z (x2 + z 2 ) 0 + π, α > 0, β > 0, Re ν > −1, |arg z| < KU 151(31), WA 416(2) 2
ν ∞ μ 1 π u 2 2 − 2 ν+1 2 2 x −y J ν (ux) K μ v x − y x dx = exp −iπ μ − ν − · μ 2 2 v 0 &μ−ν−1 %√ 2 2 u +v (2) · H μ−ν−1 y u2 + v 2 y + , 1α 1α 2 1 Re μ < 1, Re ν > −1, u > 0, v > 0, y > 0; x − y 2 2 = e 2 απi y 2 − n2 2 if x < y
7.8
[0 < α < β]
694
Bessel Functions
9.
∞
8
6.597
− μ v x2 + y 2 x2 + y 2 2 xν+1 dx J ν (ux) H (2) μ
0
uν = μ v ⎡ ⎣ Re μ > Re ν > −1,
u > 0,
arg v 2 − u
2 σ
%√ &μ−ν−1 v 2 − u2 (2) H μ−ν−1 y v 2 − u2 y [u < v]
v > 0,
y > 0; ,
arg
v 2 − u2 = 0, for v > u ⎤
= −πσ for v < u, where σ =
μ−ν−1 ⎦ 1 or σ = 2 2 MO 43
∞
10.8 0
∞
11.8 0
xν−1 2ν−1 Γ(ν) J μ (αz) J μ (γz) J ν (βx) J μ α x2 + z 2 J μ γ x2 + z 2 dx = μ β ν zμ zμ (x2 + z 2 ) α > 0; β > α + γ; γ > 0, Re 2μ + 52 > Re ν > 0 WA 459(14) n n < −μ −nμ x J μ (αk x) J ν (βt)tν−1 J μ αk t2 + x2 dt = 2ν−1 β −ν Γ(ν) (t2 + x2 ) k=1 k=1 % &
n 1 1 x > 0, α1 > 0, α2 > 0, . . . , αn > 0, β > αk ; Re nμ + n + > Re ν > 0 2 2 k=1
MO 43
√ a2 + x2 2ν−2 Γ ν − 12 8 √ Hν (2a) 12. x dx = Re ν > 12 WA 457(8) ν ν+1 2 2 2a π (a + x ) 0 ∞ + 1 , − 1 μ −1 6.597 tν+1 J μ b t2 + y 2 2 t2 + y 2 2 t2 + β 2 J ν (at) dt 0 + 1 , − 1 μ = β ν J μ b y 2 − β 2 2 y 2 − β 2 2 K ν (aβ)
∞J2 μ
[a ≥ b,
1
6.598
Re β > 0,
−1 < Re ν < 2 + Re μ]
EH II 95(56)
√ √ − 1 (ν+μ+1) μ ν x 2 (1 − x) 2 J μ a x J ν b 1 − x dx = 2aμ bν a2 + b2 2 J ν+μ+1 a2 + b 2
0
[Re ν > −1,
Re μ > −1]
EH II 46a
6.61 Combinations of Bessel functions and exponentials 6.611 1. 0
∞
e−αx J ν (βx) dx =
β −ν
+ ,ν α2 + β 2 − α α2 + β 2
[Re ν > −1,
Re (α ± iβ) > 0] EH II 49(18), WA 422(8)
6.611
Bessel functions and exponentials
∞
2. 0
− 1 e−αx Y ν (βx) dx = α2 + β 2 2 cosec(νπ)
+ + ,−ν ,ν 1 1 × β ν α2 + β 2 2 + α cos(νπ) − β −ν α2 + β 2 2 + α [Re α > 0,
∞
3.
695
e−αx K ν (βx) dx =
0
β > 0,
sin(νθ) π β sin(νπ) sin θ
|Re ν| < 1]
α cos θ = ; β
MO 179, ET II 105(1)
θ→
π 2
for β → ∞
ET II 131(22) −ν ν π cosec(νπ) −ν = α2 − β 2 + α β α + α2 − β 2 − β ν 2 α2 − β 2 [|Re ν| < 1, Re(α + β) > 0] ET I 197(24), MO 180
∞
4.8 0
+ ,ν β −ν α − α2 − β 2 e−αx I ν (βx) dx = α2 − β 2
[Re ν > −1,
Re α > |Re β|] MO 180, ET I 195(1)
5.
⎧ ⎡ 2ν ⎤⎫ ⎪ ⎪ 2 + β2 ⎨ ⎬ α α + i ⎥ ⎢ 1± cos(νπ) − ⎦ ⎣ 2ν ⎪ ⎪ sin(νπ) b ⎩ ⎭
ν ∞ α2 + β 2 − α e−αx H (1,2) (βx) dx = ν β ν α2 + β 2 0
(2) [−1 < Re ν < 1; a plus sign corresponds to the function H (1) ν , a minus sign to the function H ν .]
⎧ ⎡ ⎤⎫ . 2 ⎬ ⎨ ∞ α α 1 2i (1) ⎦ e−αx H 0 (βx) dx = 1 − ln ⎣ + 1 + ⎭ π β β α2 + β 2 ⎩ 0
MO 180, ET I188(54, 55)
6.
7.
[Re α > |Im β|] ⎧ ⎡ ⎤⎫ . 2 ⎬ ∞ ⎨ α ⎦ α 1 2i (2) e−αx H 0 (βx) dx = 1 + ln ⎣ + 1 + 2 2 ⎭ ⎩ π β β α +β 0
MO 180, ET I 188(53)
[Re α > |Im β|]
MO 180, ET I 188(53)
[Re α > |Im β|]
MO 47, ET I 187(44)
∞
8. 0
9.11 0
∞
e−αx Y 0 (βx) dx =
π
−2 α2 + β 2
ln
α+
α2 + β 2 β
α β
arccos e−αx K 0 (βx) dx = β 2 − α2 1 = ln 2 α − β2
[Re(α + β) > 0] WA 424, ET II 131(22)
. α + β
α2 −1 β2
[Re(α + β) > 0] MO 48
696
Bessel Functions
10.10
b
∞
α dα a
dk J 1 (kα)e
−k|β|
b
=
0
1−
a
|β| α2 + β 2
6.612
dα (see 3.241 6)
6.612 1.
∞
e−2αx J 0 (x) Y 0 (x) dx =
+ − 1 , K α α2 + 1 2 1
π (α2 + 1) 2 ∞ 1 , 1 + e−2αx I 0 (x) K 0 (x) dx = K 1 − α2 2 2 % 0
1 & 1 2 1 K 1− 2 = 2α α
[Re α > 0]
ET II 347(58)
0
2.
∞
3. 0
4.
6.
[1 < α < ∞]
−αx
2
2
ET II 370(48)
2
α2 + β 2 + γ 1 e−αx J ν (βx) J ν (γx) dx = √ Q ν− 12 2βγ π γβ Re (α ± iβ ± iγ) > 0, γ > 0,
∞
2β
Re ν > − 12
WA 426(2), ET II 50(17)
K α2 + 4β 2 α2 + 4β 2
2 ∞ 2α + β 2 K √ 2β 2 − 2 α2 + β 2 E √ 2β 2 α +β α +β e−2αx J12 (βx) dx = 2 2 2 πβ α +β 0 ∞ r2 e−3x I l (x) I m (x) I n (x) dx = r1 g + 2 + r3 π g 0 where √
11 3−1 2 1 g= Γ Γ2 96π 3 24 24 e
0
5.
[0 < α < 1]
and
[J 0 (βx)] dx =
π
MO 178
WA 428(3)
6.614
Bessel functions and exponentials
(lmn) 000 100 110 111 200 210 211 220 221 222 300 310 311 320 321 322 330 331 332 333 400 410 411 420 421 422 430 431
∞
11
6.613
e
r1 1 1 5/12 − 1/8 10/3 3/8 − 2/3 73/36 − 15/16 5/8 35/2 − 79/36 − 11/4 319/48 − 125/36 35/16 50/3 − 35/3 35/9 − 35/16 994/9 − 515/16 − 9/2 12907/120 − 229/16 35/3 2641/48 − 1505/36 −xz
0
6.614
∞
1.
e 0
−αx
J
ν+ 12
r2 0 0 − 1/2 3/4 2 − 9/4 2 − 29/6 21/8 − 27/20 21 − 85/6 21/2 − 119/8 269/30 − 213/40 − 1046/25 148/5 − 1012/105 1587/280 542/3 − 879/8 357/5 − 13903/10 1251/40 − 1024/35 − 28049/200 118051/1050 x2 2
dx =
r3 0 − 1/3 0 0 −2 1/3 0 0 0 0 −13 4 − 2/3 − 1/3 0 0 0 0 0 0 −92 115/3 −12 −6 1 0 1/3 0
(lmn) 432 433 440 441 442 443 444 500 510 511 520 521 522 530 531 532 533 540 541 542 543 544 550 551 552 553 554 555
r1 525/32 − 595/72 6025/36 − 29175/224 2975/48 − 539/32 77/8 9287/12 − 189029/180 275/4 2897/16 − 937/12 509/8 3589/18 − 1329/8 2555/36 − 2233/48 18471/32 − 1390/3 7777/32 − 5621/72 1155/32 197045/108 − 12023/8 1683/2 − 5159/16 24563/312 − 9251/208
697
r2 − 4617/112 8809/420 − 620161/1470 131379/400 − 31231/200 119271/2800 − 186003/7700 3005/2 − 138331/50 5751/10 − 15123/20 27059/30 − 4209/28 − 1993883/3075 297981/700 − 187777/1050 164399/1400 − 28493109/19600 286274/245 − 1715589/2800 4550057/23100 − 560001/6160 − 101441689/22050 18569853/4900 − 5718309/2695 2504541/3080 − 1527851/77000 12099711/107800
π πi Γ(ν + 1) √ D −ν−1 ze 4 i D −ν−1 ze− 4 π
r3 0 0 0 0 0 0 0 − 2077/3 348 −150 − 229/3 24 0 0 − 4/3 0 0 − 1/3 0 0 0 0 0 0 0 0 0 0
[Re ν > −1]
2
2
√ β β π β β2 J ν β x dx = exp − I 12 (ν−1) − I 12 (ν+1) 4 α3 8α 8α 8α 2 1 = e−β /4α α
MO 122
[ν = 0] MO 178
2. 0
1 β β e− 2 α Γ(ν + 1) M 12 ,ν e−αx Y 2ν 2 βx dx = √ cot(νπ) − cosec(νπ) W Γ(2ν + 1) α αβ
β 1 2 ,nu α
[Re α > 0,
∞ β e Γ(ν + 1) M − 12 ,ν e−αx I 2ν 2 βx dx = √ Γ(2ν + 1) α αβ 0
|Re ν| < 1]
ET I 188(50)a
[Re α > 0,
Re ν > −1]
ET I 197(20)a
3.
∞
1 β 2 α
698
Bessel Functions
∞
4. 0
6.615
1 β β e2 α e−αx K 2ν 2 βx dx = √ Γ(ν + 1) Γ(1 − ν) W − 12 ,ν α 2 αβ
[Re α > 0, |Re ν| < 1] ET I 199(37)a
√ β2 β2 β2 π β 5. e−αx K 1 β x dx = exp MO 181 K1 − K0 3 8 α 8α 8α 8α 0
∞ √ √ 2βγ 1 β2 + γ2 6.615 e−αx J ν 2β x J ν 2γ x dx = I ν exp − [Re ν > −1] α α α 0
∞
MO 178
6.616 1. 2. 3.
∞
∞
, + 1 e−αx J 0 β x2 + 2γx dx = exp γ α − α2 + β 2 MO 179 α2 + β 2 0 ∞ 1 e−αx J 0 β x2 − 1 dx = exp − α2 + β 2 MO 179 α2 + β 2 1 √ ∞ 2 2 eiα r +x (1) itx 2 2 e H 0 r α − t dt = −2i √ 2 2 −∞ + , r + x 0 ≤ arg α2 − t2 < π, 0 ≤ arg α < π; r and x are real MO 49
4.
e −∞
1
5.3 −1
∞
1. 0
√ 2 2 e−iα r +x 2 2 r α − t dt = 2i √ 2 2 + r +x −π < arg α2 − t2 ≤ 0,
, r and x are real
−π < arg α ≤ 0,
[a > 0, ∞
0
(2) H0
∞
2. 0
b > 0]
∞ + , P n (x) e−xy J 0 y 1 − x2 /(α + y) dy = n! n+1 α n=0
K q−p (2z sinh x) e(p+q)x dx =
π2 [J p (z) Y q (z) − J q (z) Y p (z)] 4 sin[(p − q)π] [Re z > 0, −1 < Re(p − q) < 1] MO 44
π ∂ Y p (z) ∂ J p (z) −2px − Y p (z) K 0 (2z sinh x) e dx = − J p (z) 4 ∂p ∂p [Re z > 0]
6.618
1. 0
MO 49
−1/2 e−ax I 0 b 1 − x2 dx = 2 a2 + b2 sinh a2 + b2
6.8 6.617
−itx
∞
2
√ 2 β π β2 e−αx J ν (βx) dx = √ exp − I 12 ν 8α 8α 2 α
[Re α > 0,
MO 44
β > 0,
Re ν > −1] WA 432(5), ET II 29(8)
6.621
Bessel functions, exponentials, and powers
∞
2.
e
−αx2
0
699
2
2
√ νπ β β π β2 νπ 1 I 12 ν K 12 ν Y ν (βx) dx = − √ exp − tan + sec 8α 2 8α π 2 8α 2 α [Re α > 0, β > 0, |Re ν| < 1]
2
2
∞ νπ √π 2 β β 1 √ exp e−αx K ν (βx) dx = sec K 12 ν 4 2 8α 8α α 0 [Re α > 0,
WA 432(6), ET II 106(3)
3.
∞
4. 0
∞
5.
|Re ν| < 1] EH II 51(28), ET II 132(24)
2
2
√ β β π −αx2 e I ν (βx) dx = √ exp I 12 ν 8α 8α 2 α
[Re ν > −1,
Re α > 0]
EH II 92(27)
e−αx J μ (βx) J ν (βx) dx 2
Γ μ+ν+1 2 =2 α β Γ(μ + 1) Γ(ν + 1)
ν+μ+1 ν+μ+2 ν+μ+1 β2 , , ; μ + 1, ν + 1, ν + μ + 1; − × 3F 3 2 2 2 α [Re(ν + μ) > −1, Re α > 0] EH II 50(21)a
0
−ν−μ−1 − ν+μ+1 2
ν+μ
6.62–6.63 Combinations of Bessel functions, exponentials, and powers 6.621
Notation: , 1 + (a + ρ)2 + z 2 − (a − ρ)2 + z 2 , 1 = 2
1.
∞
e−αx J ν (βx)xμ−1 dx
0
=
ν
Γ(ν + μ) F
αμ Γ(ν + 1)
=
β 2α
2 =
, 1 + (a + ρ)2 + z 2 + (a − ρ)2 + z 2 2
ν+μ ν+μ+1 β2 , ; ν + 1; − 2 2 2 α
WA 421(2)
ν
1 −μ
Γ(ν + μ) ν−μ+1 ν−μ β2 β2 2 , + 1; ν + 1; − 2 F 1+ 2 αμ Γ(ν + 1) α 2 2 α
β 2α
WA 421(3)
ν =
0,
ν+μ
(α2 + β 2 )
2
Re (α + iβ) > 0,
+ − 1 μ 1, 2 2 −2 = α2 + β 2 2 Γ(ν + μ) P −ν μ−1 α α + β [α > 0,
β > 0,
Re (α − iβ) > 0] WA 421(3)
Re(ν + μ) > 0] ET II 29(6)
700
Bessel Functions
∞
2.
e−αx Y ν (βx)xμ−1 dx
6.621
ν
0
β 2
= cot νπ
0,
Re (α ± iβ) > 0] WA 421(4)
Re μ > |Re ν|] ET II 105(2)
∞
3.
μ−1 −αx
x
e
0
∞
4.
√ 1 α−β π(2β)ν Γ(μ + ν) Γ(μ − ν) 1 K ν (βx) dx = F μ + ν, ν + ; μ + ; (α + β)μ+ν 2 2 α+β Γ μ + 12 [Re μ > |Re ν|, Re(α + β) > 0] ⎡
m+1 −αx
x
e
m+1 −ν
J ν (βx) dx = (−1)
β
0
5.10
e−zx J 1 (ax) J 1/2 (ρx) x−3/2 dx
8.10
9.10
ET II 131(23)a, EH II 50(26)
⎦
Re ν > −m − 2]
ET II 28(3)
< 1 a2 arcsin ρ2 − 21 − ρ +z 2 2 [arg a > 0, arg ρ > 0, arg z > 0] ∞ < 2 1 −zx −1/2 2 2 e J 1 (ax) J 1/2 (ρx) x dx = ρ − ρ − 1 a πρ 0 [arg a > 0, arg ρ > 0, arg z > 0] ∞ 2 1 a2 − 21 1 e−zx J 1 (ax) J 1/2 (ρx) x1/2 dx = a πρ 22 − 21 0 [arg a > 0, arg ρ > 0, arg z > 0] ∞ 2 21 ρ2 − 21 e−zx J 1 (ax) J 3/2 (ρx) x1/2 dx = π ρ3/2 a ( 22 − 21 ) 0 [arg a > 0, arg ρ > 0, arg z > 0]
∞ < 1 1 1 2 2 − 2 e−zx J 1 (ax) J 3/2 (ρx) x−3/2 dx = √ arcsin a a − 1 1 a 2π ρ3/2 a 0 [arg a > 0, arg ρ > 0, arg z > 0] =
7.10
α2 + β 2 − α α2 + β 2 [β > 0,
∞
0
6.10
dm+1 ⎣ dαm+1
ν ⎤
1 a
2 πρ
1 2
0, arg ρ > 0, arg z > 0]
e−zx J 1 (ax) J 5/2 (ρx) x−3/2 dx
[arg a > 0, ∞
12.10
−1/2
⎡ 2
7a 1 5a2 21 1 ⎣ 41 1 2 2 =√ −a z − − 8 4 8 2π ρ5/2 a a2 − 21 ⎤
< 4 1 3 2 2 1 2 2 3a ⎦ 1 2 + 22 1 a2 − 21 + arcsin a z + a ρ − − 2 1 a 2 2 8
0
−zx
701
∞
arg ρ > 0,
arg z > 0]
e−zx J 1 (ax) J 5/2 (ρx) x−5/2 dx ⎧ + , 2 2 5/2 2 5/2 ⎨ − ρ − 2 ρ 1 1 ρ2 z2 3a 1 1 2 + za arcsin − − =√ 15 a 8 2 2 2π ρ5/2 a ⎩ ⎫ 2 < ρ 3a2 z2 21 z 3 a2 1 ⎬ 2 2 − + − + z 1 a − 1 + 2 8 6 4 3 a2 − 21 ⎭
[arg a > 0,
arg ρ > 0,
arg z > 0]
22 − ρ2 2 2 3/2 a ρ 2 π ( 2 − 21 ) 42 0 [arg a > 0, arg ρ > 0, arg z > 0] % 2 3/2 & ∞ ρ − 21 ρ2 − 21 2 ρ3/2 2 −zx −1/2 − + e J 2 (ax) J 3/2 (ρx) x dx = π a2 3 ρ 3ρ3 0 [arg a > 0, arg ρ > 0, arg z > 0] ∞ e−zx J 3 (ax) J 1/2 (ρx) x−1/2 dx 0
< 2 2 2 1 2 2 2 2 2 2 2 = ρ 3a − 4ρ + 12z − ρ − 1 12 2 − 16ρ + 4 1 − 3a πρ 3a3 [arg a > 0, arg ρ > 0, arg z > 0] ∞ e−zx J 3 (ax) J 3/2 (ρx) x1/2 dx 0 % 2 3/2 & ρ − 21 ρ2 − 21 a 22 − a2 2 3/2 4 2 ρ − + − 2 = π a3 3 ρ 3ρ2 ( 2 − 21 ) 32 [arg a > 0, arg ρ > 0, arg z > 0] %< & 2 ∞ 2 2 4 3/2 2ρ − 4ρ − 2 ρ 1 1 e−zx J 3 (ax) J 3/2 (ρx) x−1/2 dx = 22 − ρ2 − 8z π 3a3 ρ4 0 [arg a > 0, arg ρ > 0, arg z > 0] e−zx J 2 (ax) J 3/2 (ρx) x1/2 dx =
702
Bessel Functions
18.
∞
10 0
∞
19.10 0
6.622
e−zx J 3 (ax) J 3/2 (ρx) x−3/2 dx
2 < 4 2 24ρ 8 21 a2 41 2 ρ3/2 4 2 2 2 2 − 2 ρ − + − + − + 4z − ρ a = 1 π 3a3 5 ρ 5 5ρ ρ 5ρ3 [arg a > 0, arg ρ > 0, arg z > 0] e−zx J 3 (ax) J 3/2 (ρx) x−5/2 dx ⎧
2 ρ3/2 ⎨ 2 4 2 4z 3 =− ρ − z + a π 3a3 ⎩ 5 3 < 32 2 12 2 4 2 2 41 a4 21 a2 21 61 2 2 2 + + + 2 − ρ a + ρ − 1 − 2 + 2 + 15⎫ 5 3 5ρ 16ρ4 24ρ4 30ρ4
ρ ⎬ a6 − arcsin 16ρ3 2 ⎭ [arg a > 0,
6.622
∞
1. 0
2.
3.8
dx = ln 2α J 0 (x) − e−αx x
arg ρ > 0,
arg z > 0]
[α > 0]
NT 66(13)
∞ i(u+x)
e π (1) J 0 (x) dx = i H 0 (u) u+x 2 0 μ− 12 ∞ 2 −(μ− 12 )πi Q ν− 12 (cosh α) −x cosh α μ−1 e e I ν (x)x dx = 1 π 0 sinhμ− 2 α [Re(μ + ν) > 0,
MO 44
Re (cosh α) > 1] WA 388(6)a
6.623
∞
1.
e 0
−αx
(2β)ν Γ ν + 12 J ν (βx)x dx = √ ν+ 1 π (α2 + β 2 ) 2
ν
Re ν > − 12 ,
Re α > |Im β|
WA 422(5)
∞ 2α(2β)ν Γ ν + 32 e−αx J ν (βx)xν+1 dx = √ ν+ 3 0 π (α2 + β 2 ) 2
2.
∞
3. 0
e−αx J ν (βx)
dx = x
[Re ν > −1,
WA 422(6)
ν α2 + β 2 − α νβ ν [Re ν > 0;
6.624 1. 0
∞
Re α > |Im β|]
Re α > |Im β|]
(cf. 6.611 1)
⎫ ⎤ .
2 ⎬ α α 1 α xe−αx K 0 (βx) dx = 2 ln ⎣ + − 1⎦ − 1 2 ⎭ α − β ⎩ α2 − β 2 β β ⎧ ⎨
WA 422(7)
⎡
MO 181
6.625
Bessel functions, exponentials, and powers
2.
∞√
−αx
xe
0
∞
3.
e−tz(z
2
K ± 12 (βx) dx = −1/2
−1)
π 1 2β α + β
∞
4.
e−tz(z
2
e−tz(z
2
−1/2
−1)
(z 2
I −μ (t)tν dt =
0
∞
5.
−1 2
−1)
I μ (t)tν dt =
0
∞
6.
MO 181
Γ(ν − μ + 1)
K μ (t)tν dt =
0
− 12 (ν+1)
− 1)
Γ(−ν − μ) 1
(z 2 − 1) 2
ν
P μν (z)
Γ(ν + μ + 1) − 12 (ν+1)
(z 2 − 1)
eiμπ Q μν (z)
∞
7. 0
[Re (ν ± μ) > −1]
EH II 57(7)
[Re(ν + μ) < 0]
EH II 57(8)
[Re(ν + μ) > −1]
EH II 57(9)
P −μ ν (z)
e−t cos θ J μ (t sin θ) tν dt = Γ(ν + μ + 1) P −μ ν (cos θ)
0
Re(ν + μ) > −1,
EH II 57(10)
∞ (2b) Γ ν + 12 J ν (bx)xν 1 √ dx = 1 πx e −1 π 2 2 2 ν+ 2 n=1 (n π + b )
1 λ−ν−1
x
1.
0 ≤ θ < 12 π
ν
[Re ν > 0, 6.625
703
μ−1 ±iαx
(1 − x)
e
0
|Im b| < π]
WA 423(9)
2−ν αν Γ(λ) Γ(μ) 1 J ν (αx) dx = 2 F 2 λ, ν + ; λ + μ, 2ν + 1; ±2iα Γ(λ + μ) Γ(ν + 1) 2
[Re λ > 0, Re μ > 0] ET II 194(58)a
1 (2α)ν Γ(μ) Γ ν + 12 1 ν μ−1 ±iαx x (1 − x) e J ν (αx) dx = √ 1 F 1 ν + ; μ + 2ν + 1; ±2iα 2 π Γ(μ + 2ν + 1) 0 Re μ > 0, Re ν > − 12 ET II 194(57)a
1 (2α)ν Γ ν + 12 Γ(μ) 1 ν μ−1 ±αx x (1 − x) e J ν (αx) dx = √ 1 F 1 ν + ; μ + 2ν + 1; ±2α 2 π Γ(μ + 2ν + 1) 0 Re μ > 0, Re ν > − 12
2.
3.
BU 9(16a), ET II 197(77)a
1 ν 1 α Γ (λ + ν) Γ(μ) λ−1 μ−1 ±αx x (1 − x) e I ν (αx) dx = 2 Γ(ν + 1) Γ(λ + μ + ν) 0
1 × 2 F 2 ν + , λ + ν; 2ν + 1, μ + λ + ν; ±2α 2 [Re μ > 0, Re(λ + ν) > 0] ET II 197(78)a
4.
1
xμ−κ (1 − x)2κ−1 I μ−κ
5. 0
1 x 1 1 Γ(2κ) xz e− 2 xz dx = √ e 2 z −κ− 2 M κ,u (z) 2 π Γ(1 + 2μ) Re κ − 12 − μ < 0, Re κ > 0 BU 129(14a)
704
Bessel Functions
∞
6.
−λ
x
μ−1 −αx
(x − 1)
1
∞
7.
e
6.626
!1
! − λ, 0 (2α)λ Γ(μ) 21 2 ! √ I ν (αx) dx = G 23 2α ! π −μ, ν − λ,1−ν − λ 0 < Re μ < 2 + Re λ, Re α > 0
√ x−λ (x − 1)μ−1 e−αx K ν (αx) dx = Γ(μ) π(2α)λ G 30 23
1
∞
8. 1
∞
9.
! 1
! 0, − λ 2 ! 2α ! −μ, ν − λ, −ν − λ
[Re μ > 0, Re α > 0] (2α) Γ 2 − μ + ν Γ(μ) −ν μ−1 −αx √ x (x − 1) e I ν (αx) dx = π Γ(1 − μ + 2ν)
1 − μ + ν; 1 − μ + 2ν; −2α × 1F 1 2 0 < Re μ < 12 + Re ν, Re α > 0 ν−μ
1
[Re μ > 0, ∞
10.
x−μ− 2 (x−1)μ−1 e−αx K ν (αx) dx = 1
ET II 207(50)a
ET II 208(55)a
ET II 207(49)a
√ 1 1 π Γ(μ)(2α)− 2 μ− 2 e−α W − 12 μ,ν− 12 μ (2α)
x−ν (x − 1)μ−1 e−αx K ν (αx) dx =
1
Re α > 0]
ET II 208(53)a
√ 1 π Γ(μ)(2α)− 2 e−α W −μ,ν (2α)
1
[Re μ > 0,
ET II 207(51)a Re α > 0] 1 −1/2 −ax −1/2 2 sinh a − a a2 + b2 1 − x2 xe I 1 b 1 − x2 dx = sinh a2 + b2 b −1
11.3
[a > 0, 6.626 1.11
∞
xλ−1 e−αx J μ (βx) J ν (γx) dx =
0
∞
2.
e 0
−2αx
ν+μ
J ν (βx) J μ (βx)x
∞ β μ γ ν −ν−μ −λ−μ−ν Γ(λ + μ + ν + 2m) 2 α Γ(ν + 1) m! Γ(μ + m + 1) m=0
m 2 γ β2 × F −m, −μ − m; ν + 1; 2 − 2 β 4α [Re(λ + μ + ν) > 0, Re (α ± iβ ± iγ) > 1] EH II 48(15)
Γ ν + μ + 12 β ν+μ √ dx = π3 π2 cosν+μ ϕ cos(ν − μ)ϕ dϕ × ν+μ 2 2 2 2 + β 2 cos2 ϕ α 0 (α + β cos ϕ) Re α > |Im β|, Re(ν + μ) > − 12
3.
K e−2αx J 0 (βx) J 1 (βx)x dx =
√
β α2 +β 2
−E
√
WA 427(1)
β α2 +β 2
α2 + β 2
∞ β β α 1 1 −2αx e I 0 (βx) I 1 (βx)x dx = E − K 2 − β2 2πβ α α α α 0 0
4.
∞
b > 0]
2πβ
[Re α > Re β]
WA 427(2)
WA 428(5)
6.628
5.
10
Bessel functions, exponentials, and powers
∞
∞
1 a μ−ν−2n−1 ρ ν xν−μ+2n e−zx J μ (αx) J ν (ρx) dx = √ a π 2 0 ∞ Γ ν +n+q+ 1 ν − μ + n + 12 q 1 2 × Γ μ − ν − n + 12 q=0 q! Γ ν + q + 12
q 1 /ρ dx z2 √ ×a−2q x2ν+2q ρ2 + 2 1 − x2 0 + , 1 − x μ > ν + 2n, n = 0, 1, . . . , ν > − 12 where 1 = 12 (a + ρ)2 + z 2 − (a − ρ)2 + z 2
6.627 0
6.628
∞
1. 0
2.
3.
705
x−1/2 −x πea K ν (a) e K ν (x) dx = √ x+a a cos(νπ)
|arg a| < π,
|Re ν|
−1, 0 < β < WA 424(4) 2
1 u xu π ixu B(2ν, 2μ − 2ν + 1) e 2 e 2 (1 − x)2ν−1 xμ−ν J μ−ν dx = 22(ν−μ) e 2 (μ−ν)i 1 M ν,μ (u) 2 Γ(μ − ν + 1) uν+ 2 0 MO 118a
4.8 0
e−x cos β Y ν (x sin β) xμ dx = −
e−x cosh α I ν (x sinh α) xμ dx = Γ(ν + μ + 1) P −ν μ (cosh α) Re(μ + ν) > −1,
|Im α| < 12 π
WA 423(1) ∞
5.
e−x cosh α K ν (x sinh α) xμ dx =
0
sin μπ Γ(μ − ν + 1) Q νμ (cosh α) sin(ν + μ)π [Re(μ + 1) > |Re ν|]
∞
6.
e−x cosh α I ν (x)xμ−1 dx =
0
WA 423(2)
ν− 12 μ− 12
(cosh α) Q cos νπ sin(μ + ν)π π (sinh α)μ− 12 2 [Re(μ + ν) > 0,
Re (cosh α) > 1] WA 424(6)
7.
∞
e 0
−x cosh α
μ−1
K ν (x)x
dx =
1
−μ
2 P ν− 1 (cosh α) π 2 Γ(μ − ν) Γ(μ + ν) μ− 1 2 (sinh α) 2 [Re μ > |Re ν|,
Re (cosh α) > −1] WA 424(7)
706
Bessel Functions
∞
8
6.629
x−1/2 e−xα cos ϕ cos ψ J μ (αx sin ϕ) J ν (αx sin ψ) dx
0
1 = Γ μ + ν + 12 α− 2 P −μ (cos ϕ) P −ν 1 (cos ψ) ν− 12 μ− 2 π 1 π 0 < ϕ < , 0 < ψ < , Re(μ + ν) > − ET II 50(19) 2 2 2
α > 0, 6.631
∞
1.
6.629
μ −αx2
x e
1
1 1 2ν + 2μ + 2 1 2ν+1 α 2 (μ+ν+1) Γ(ν +
βν Γ
J ν (βx) dx =
0
1
1)
1F 1
ν+μ+1 β2 ; ν + 1; − 2 4α
1 2
BU 8(15)
2
β β2 = exp − M 12 μ, 12 ν 1 μ 8α 4α 2 βα Γ(ν + 1) [Re α > 0, 1 2ν + 2μ +
Γ
Re(μ + ν) > −1]
EH II 50(22), ET II 30(14), BU 14(13b)
∞
2.
xμ e−αx Y ν (βx) dx 2
ν−μ β2 = −α π exp − β sec 2 8α
2
1 1 1 Γ 2 + 2μ + 2ν ν−μ β sin π M 12 μ, 12 ν × +W Γ(1 + ν) 2 4α
0
− 12 μ −1
Re μ > |Re ν| − 1,
[Re α > 0,
∞
3.
xμ e−αx K ν (βx) dx = 2
0
∞
4.11
1 − 1 μ −1 α 2 β Γ 2
xν+1 e−αx J ν (βx) dx = 2
0
∞
5. 0
∞
6.
7. 0
∞
1+ν+μ 2
β (2α)ν+1
xν+1 e±iαx J ν (βx) dx = 2
xe−αx J ν (βx) dx = 2
√ πβ 8α
3 2
β > 0]
[Re μ > |Re ν| − 1]
2
[Re α > 0,
exp −
2
β 8α
I 12 ν− 12
ET II 106(4)
ET II 132(25)
WA 431(4), ET II 29(10)
[Re α > 0, Re ν > 0]
ν+1 β2 π− exp ±i 2 4α α > 0, −1 < Re ν < 12 ,
Re ν > −1]
β2 ν−1 −αx2 ν−1 −ν x e J ν (βx) dx = 2 β 1 − γ ν, 4α ν
1 1 2 μ, 2 ν
β2 4α
2
2
β 1−ν+μ β Γ exp W − 12 μ, 12 ν 2 8α 4α
βν β exp − ν+1 (2α) 4α
0
2
β 8α
− I 12 ν+ 12
[Re α > 0,
2
ET II 30(11)
β>0
ET II 30(12)
β 8α
Re ν > −2]
ET II 29(9)
6.633
Bessel functions, exponentials, and powers
8.
1
n+1 −αx2
x
e
0
∞
9.
& % n 1 α −α I n (2αx) dx = I r (2α) e −e 4α r=−n %
x1−n e−αx I n (2αx) dx = 2
1
∞
10. 0
707
[n = 0, 1, . . .] &
ET II 365(8)a
[n = 1, 2, . . .]
ET II 367(20)a
n−1 1 eα − e−α I r (2α) 4α r=1−n
√ 2 1 n! e−x x2n+μ+1 J μ 2x z dx = e−z z 2 μ Lμn (z) 2
[n = 0, 1, . . . ;
n + Re μ > −1] BU 135(5)
∞
6.632
+ 1 , − 1 1 x− 2 exp − x2 + a2 − 2ax cos ϕ 2 x2 + a2 − 2ax cos ϕ 2 K ν (x) dx
0
= πa− 2 sec(νπ) P ν− 12 (− cos ϕ) K ν (a) |arg a| + |Re ϕ| < π, |Re ν| < 12 ET II 368(32) 1
6.633
∞
1.
λ+1 −αx2
x
e
0
2.
3.
4.
m μ+ν+λ+2 ∞ 2 Γ m + 12 ν + 12 μ + 12 λ + 1 β μ γ ν α− β2 J μ (βx) J ν (γx) dx = ν+μ+1 − 2 Γ(ν + 1) m=0 m! Γ(m + μ + 1) 4α
γ2 × F −m, −μ − m; ν + 1; 2 β [Re α > 0, Re(μ + ν + λ) > −2, β > 0, γ > 0] EH II 49(20)a, ET II 51(24)a
αβ 1 α2 + β 2 e J p (αx) J p (βx)x dx = 2 exp − Ip 2 42 22 0 + , π Re p > −1, |arg | < , α > 0, β > 0 KU 146(16)a, WA 433(1) 4
∞ 1 1 − 3 ν− 1 1 2ν+1 −αx2 2 2 x e J ν (x) Y ν (x) dx = − √ α exp − W 12 ν, 12 ν 2α α 2 π 0 Re α > 0, Re ν > − 12 ET II 347(59) 2
∞ 2 β − γ2 βγ 1 exp xe−αx I ν (βx) J ν (γx) dx = Jν 2α 4α 2α 0
∞
∞
5. 0
−2 x2
[Re α > 0, xλ−1 e−αx J μ (βx) J ν (βx) dx
Re ν > −1]
ET II 63(1)
2
+ 12 μ + 12 ν =2 α β Γ(μ + 1) Γ(ν + 1) ν μ 1 ν μ ν+μ+λ β2 + + , + + 1, ; μ + 1, ν + 1, μ + ν + 1; − × 3F 3 2 2 2 2 2 2 α [Re(ν + λ + μ) > 0, Re α > 0] WA 434, EH II 50(21) −ν−μ−1 − 12 (ν+λ+μ) ν+μ Γ
1
2λ
708
Bessel Functions
∞
6.634
x2
xe− 2a [I ν (x) + I −ν (x)] K ν (x) dx = aea K ν (a)
6.634
[Re a > 0,
−1 < Re ν < 1]
0
ET II 371(49)
6.635 1.
∞
x−1 e− x J ν (βx) dx = 2 J ν α
2αβ K ν 2αβ
0
[Re α > 0,
∞
2.
x−1 e− x
α
β > 0]
ET II 30(15)
Y ν (βx) dx = 2 Y ν 2αβ K ν 2αβ
0
∞
3.
ET II 106(5) [Re α > 0, β > 0]
1 + , + , 12 √ √ α 2 x−1 e− x −βx J ν (γx) dx = 2 J ν 2α β2 + γ2 − β 2α β2 + γ2 + β Kν
0
[Re α > 0,
Re β > 0,
γ > 0] ET II 30(16)
∞
6.636 0
√ 1 1 1 √ 1 1 1 2 − 12 −α x x e J ν (βx) dx = √ Γ ν + 12 D −ν− 12 2− 2 αe 4 πi β − 2 D −ν− 12 2− 2 αe− 4 πi β − 2 πβ Re α > 0, β > 0, Re ν > − 12
ET II 30(17)
6.637 1.
∞
+ 2 − 1 1 , β + x2 2 exp −α β 2 + x2 2 J ν (γx) dx
+
+ , , 1 1 1 2 1 2 β α + γ 2 2 − α K 12 ν β α + γ2 2 + α = I 12 ν 2 2 [Re α > 0, Re β > 0, γ > 0, Re ν > −1] ET II 31(20)
∞
+ 2 − 1 1 , β + x2 2 exp −α β 2 + x2 2 Y ν (γx) dx
+ νπ , 1 1 2 2 2 β α +γ K1 = − sec +α 2 2ν 2
+ + , νπ ,
1 1 1 2 1 2 1 K 12 ν β α + γ 2 2 + α + sin β α + γ2 2 − α × I 12 ν π 2 2 2 [Re α > 0, Re β > 0, γ > 0, |Re ν| < 1] ET II 106(6)
∞
+ 2 − 1 1 , x + β 2 2 exp −α x2 + β 2 2 K ν (γx) dx
νπ 2 2 1 , 1 , 1 + 1 + 1 2 2 2 2 K 12 ν β α+ α −γ β α− α −γ = sec K 12 ν 2 2 2 2 [Re α > 0, Re β > 0, Re(γ + β) > 0, |Re ν| < 1] ET II 132(26)
0
2. 0
3. 0
6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers
6.641 0
∞√
−αx
xe
2
J ± 14 x
2
2 √ α α πα dx = H∓ 14 − Y ∓ 14 4 4 4
MI 42
6.645
6.642
Bessel functions of complicated arguments, exponentials, and powers
∞
1.10
x−1 e−αx Y ν
0
∞
2. 0
6.643
∞
x
1. 0
2.
3.
∞
∞
√ √ 2 dx = 2 K ν 2 a Y ν 2 a x
√ √ 2 x−1 e−αx H (1,2) α Kν α dx = H (1,2) ν ν x
μ− 12 −αx
e
709
J 2ν
[Re a > 0]
MC MI 44, EH II 91(26)
2
√ Γ μ + ν + 12 − β2 −μ β 2α e 2β x dx = α M μ,ν β Γ(2ν + 1) α Re μ + ν + 12 > 0
√ Γ μ+ν+ 1 xμ− 2 e−αx I 2ν 2β x dx = Γ(2ν + 1)
1 2
BU 14(13a), MI 42a
2
β2 β β −1 e 2α α−μ M −μ,ν α 0 Re μ + ν + 12 > 0 MI 45
∞ 1 1 2 √ Γ μ + ν + 2 Γ μ − ν + 2 β2 −μ 1 β e 2α α W −μ,ν xμ− 2 e−αx K 2ν 2β x dx = 2β 0 α Re μ + ν + 12 > 0 , (cf. 6.631 3)
4.
√ β2 1 xn+ 2 ν e−αx J ν 2β x dx = n!β ν e− α α−n−ν−1 Lνn
0
exp − β √ 8α 1 π x− 2 e−αx Y 2ν β x dx = − α cos(νπ)
2
β α
MI 47a
[n + ν > −1]
MO 178a
2
2 β β 1 sin(νπ) I ν + Kν 8α π 8α 0 |Re ν| < 12 MI 44 1 1
∞ m− 2 √ 1 1 1 Γ(m + 1) 1 2 6. x 2 m e−αx K m 2 x dx = e 2α W − 12 (m+1),− 12 m MI 48a 2α α α 0
∞ √ a2 b 1 a2 β −βx 6.644 e J 2ν 2a x J ν (bx) dx = exp − 2 Jν 2 2 + b2 2 β + b β β + b2 0 Re β > 0, b > 0, Re ν > − 12
∞
5.
2
ET II 58(17)
6.645
∞
1. 1
2. 1
∞
2 − 1 1 1 x − 1 2 e−αx J ν β x2 − 1 dx = I 12 ν α2 + β 2 + α α2 + β 2 − α K 12 ν 2 2 2 − 1 ν− 1 12 ν −αx 2 ν 2 β α + β 2 2 4 K ν+ 12 x −1 e J ν β x2 − 1 dx = α2 + β 2 π
MO 179a
MO 179a
710
Bessel Functions
1
3.3
−1
1 − x2
−1/2
6.646
2 cosh a2 + b2 − cosh a e−ax I 1 b 1 − x2 dx = b [a > 0,
6.646
∞
1. 1
2.
x−1 x+1
12 ν
b > 0]
ν exp − α2 + β 2 β −αx 2 e J ν β x − 1 dx = α2 + β 2 α + α2 + β 2
[Re ν > −1] EF 89(52), MO 179 ν
1ν ∞ exp − α2 − β 2 x − 1 2 −αx 2 β e I ν β x − 1 dx = x+1 α2 − β 2 α + α2 − β 2 1
∞
3.7 b
[Re ν > −1,
ν/2
α > β]
MO 180
+ 1/2 , Γ(ν + 1) ν −bx x e K ν a t2 − b 2 Γ(−ν, bx) − y ν ebs Γ(−ν, by) dt = ν 2sa 1/2 2 [Re(p + a) > 0, |Re(ν)| < 1] . where x = p − s, y = p + s, s = p − a2
e−pt
t−b t+b
ME 39a
6.647
∞
1.
x−λ− 2 (β + x) 1
λ− 12
e−αx K 2μ
+ , x(β + x) dx
0
=
1 1 αβ 1 e 2 Γ 2 − λ + μ Γ 12 − λ − μ W λ,μ (z1 ) W λ,μ (z2 ) β z1 = 12 β α + α2 − 1 ,
|arg β| < π, 2.
∞
(α + x)− 2 x− 2 e−x cosh t K ν 1
1
, + x(α + x) dx
Re α > −1,
z2 = 12 β α − α2 − 1 Re λ + |Re μ| < 12 ET II 377(37)
νπ 1 1 t 1 −t 1 α cosh t 2 e αe K 12 ν αe = sec K 12 ν 2 2 4 4 [−1 < Re ν < 1] ET II 377(36) α , + 1 1 3.11 xλ− 2 (α − x)−λ− 2 e−x sinh t I 2μ x(α − x) dx 0 1 1
1 t 1 −t −(α/2) sinh t 2 Γ 2 + λ + μ Γ 2 − λ + μ αe M −λ,μ αe =e M λ,μ 2 2 2 α [Γ(2μ + 1)] Re μ > |Re λ| − 12 ET II 377(32)
∞ ν + 12 , α + βex 2 2 α dx = 2 K ν+ (α) K ν− (β) 6.648 ex K + β + 2αβ cosh x 2ν αex + β −∞ [Re α > 0, Re β > 0] ET II 379(45) 0
6.651
6.649
Bessel and exponential functions and powers
∞
1.
K μ−ν (2z sinh x) e(ν+μ)x dx =
0
0
∞
3.
J ν+μ (2x sinh t) e(ν−μ)t dt = K ν (x) I μ (x) Re(ν − μ) < 32 , Y ν−μ (2x sinh t) e−(ν+μ)t dt =
0
π2 [J ν (z) Y μ (z) − J μ (z) Y ν (z)] 4 sin[(ν − μ)π] [Re z > 0, −1 < Re(ν − μ) < 1] MO 44
∞
2.
711
∞
4.
K 0 (2z sinh x) e−2νx dx = −
0
π 4
Re(ν + μ) > −1,
x>0
EH II 97(68)
1 {I μ (x) K ν (x) − cos[(ν − μ)π] I ν (x) K μ (x)} sin[π(μ − ν)] |Re(ν − μ)| < 1, Re(ν + μ) > − 12 , x > 0 EH II 97(73)
J ν (z)
∂ Y ν (z) ∂ J ν (z) − Y ν (z) ∂ν ∂ν
6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers 6.651
∞
1.
xλ+ 2 e− 4 α 1
1
2
x2
Iμ
0
1
4α
2 2
x
3 1 = √ 2λ+1 β −λ− 2 2π
J ν (βx) dx 2 !
β !! 1 − μ, 1 + μ 21 G 23 2α2 ! h, 12 , k + π |arg α| < , 4
∞
2. 0
h= β > 0,
3. 0
∞
+ 12 λ + 12 ν,
k=
3 4
− 32 − Re(2μ + ν) < Re λ < 0
+ 12 λ − 12 ν , ET II 68(8)
1 1 2 2 xλ+ 2 e− 4 α x K μ 14 α2 x2 J ν (βx) dx !
π λ+1 −λ− 3 12 β 2 !! 1 − μ, 1 + μ 2 = 2 β G 23 2 2α2 ! h, 12 , k +
3 4
x2μ−ν+1 e− 4 αx I μ 1
2
1
2 4 αx
h= |arg α|
− 32
3
,4
+ 12 λ − 12 ν ET II 69(15)
J ν (βx) dx
β ν−2μ−1 1 1 1 β2 μ−ν+ 12 − 12 1F 1 =2 +μ + μ; − μ + ν; − (πα) Γ 2 2 2 2α Γ 12 − μ + ν ET II 68(6) Re α > 0, β > 0, Re ν > 2 Re μ + 12 > − 12
712
Bessel Functions
∞
4.
x2μ+ν+1 e− 4 α 1
2
x2
∞
2
x
J ν (βx) dx
2 2 αx
β2 3 ;− 2 2 2α
ET II 69(13)
K ν (βx) dx
2
2
β β 2 −μ− 32 − 12 μ− 12 ν− 14 1 √ β α Γ(2μ + ν + 1) Γ μ + 2 exp = W k,m 8α 4α π 2k = −3μ − ν − 12 , 2m = μ + ν + 12 Re α > 0, Re μ > − 12 , Re (2μ + ν) > −1 ET II 146(53)
0
2 2
1
x2μ+ν+1 e− 2 αx I μ 1
4α
√ μ −2μ−2ν−2 ν Γ (1 + 2μ + ν) 1 F 1 1 + 2μ + ν; μ + ν + = π2 α β Γ μ + ν + 32 |arg α| < 14 π, Re ν > −1, Re(2μ + ν) > −1, β > 0
0
5.
1
Kμ
6.652
μ− 12
∞
6.
− 14 αx2
xe 0
J 12 ν
1
2
4 βx
1 2 βγ 2 αγ 2 2 −2 J ν (γx) dx = 2 α + β exp − 2 J 12 ν α + β2 α2 + β 2 [γ > 0, Re α > |Im β|, Re ν > −1] ET II 56(2)
∞
7. 0
∞
8.
xe− 4 αx I 12 ν 1
2
x1−ν e− 4 α 1
2
x2
1
2 J ν (βx) dx = 4 αx
Iν
0
∞
9.
x−ν−1 e− 4 α 1
2
x2
1
1 πα 2
2 2 J ν (βx) dx = 4α x
I ν+1
1
0
4α
2 2
x
− 12
β2 β −1 exp − 2α [Re α > 0,
β > 0,
Re ν > −1] ET II 67(3)
2 β ν−1 π α
J ν (βx) dx =
β β2 exp − 2 D −2ν α 4α |arg α| < 14 π, β > 0,
Re ν > − 12
ET II 67(1)
β 2 ν β2 β exp − 2 D −2ν−3 π 4α α |arg α| < 14 π, Re ν > −1,
β>0
ET II 67(2)
∞
6.652
2ν −
x e 0
6.653
∞
1. 0
x2 8
+αx
Iν
x2 8
2
α Γ(4ν + 1) e 2 W 3 1 α2 dx = 4ν 2 Γ(ν + 1) αν+1 − 2 ν, 2 ν Re ν + 14 > 0
ab dx 1 2 1 2 a +b = 2 I ν (a) K ν (b) exp − x − Iν 2 2x x x = 2 K ν (a) I ν (b) [Re ν > −1]
2. 0
∞
MI 45
[0 < a < b] [0 < b < a] WA 482(2)a, EH II 53(37), WA 482(3)a
zw dx 1 2 1 z + w2 K ν = 2 K ν (z) K ν (w) exp − x − 2 2x x x [|arg z| < π, |arg w| < π, arg(z + w)] < 14 π
WA 483(1), EH II 53(36)
6.662
Bessel, hyperbolic, and exponential functions
6.655
√ √ 1 β2 x e Kν ME 39 dx = 4πα− 2 K 2ν β α 8x 0
∞ − 1 α2 x α2 β √ x β 2 + x2 2 exp − 2 Jν J ν (γx) dx = γ −1 e−βγ J 2ν (2α γ) 2 2 2 β +x β +x 0 Re β > 0, γ > 0, Re ν > − 12
6.654
713
∞
2
− 12 − β 8x −αx
ET II 58(14)
6.656
∞
1.
+ , 1 e−(ξ−z) cosh t J 2ν 2(zξ) 2 sinh t dt = I ν (z) K ν (ξ)
0
Re ν > − 12 ,
Re(ξ − z) > 0
EH II 98(78)
∞
2.
+
,
e−(ξ+z) cosh t K 2ν 2(zξ) 2 sinh t dt = 1
0
1 K ν (z) K ν (ξ) sec(νπ) 2 |Re ν| < 12 ,
2 1 1 Re z 2 + ξ 2 ≥ 0 EH II 98(79)
6.66 Combinations of Bessel, hyperbolic, and exponential functions Bessel and hyperbolic functions 6.661
∞
sinh(ax) K ν (bx) dx =
1. 0
∞
2. 0
π cosec 2
νπ 2
sin ν arcsin ab √ b 2 − a2 [Re b > |Re a|,
π cos ν arcsin ab νπ cosh(ax) K ν (bx) dx = 2 b2 − a2 cos 2
|Re ν| < 2] ET II 133(32)
[Re b > |Re a|,
|Re ν| < 1] ET II 134(33)
6.662
Notation: , 1 + 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2
1.
10 0
∞
K(k) cosh(βx) K 0 (αx) J 0 (γx) dx = √ u+v
2 =
, 1 + (b + c)2 + a2 + (b − c)2 + a2 2
< 2 2 2 2 2 2 (α + β + γ ) − 4α β + α2 − β 2 − γ 2
< 1 2 2 2 2 2 2 (α + β + γ ) − 4α β − α2 + β 2 + γ 2 v= 2
1 u= 2
k 2 = v(u + v)−1
[Re α > |Re β|,
γ > 0]
ET II 15(23)
714
Bessel Functions
6.663
alternatively, with a = γ, b = β, c = α, ∞ K(k) cosh(bx) K 0 (cx) J 0 (ax) dx = 2 2 − 21 0 k2 = 2.
∞
10 0
22 − c2 , 22 − 21
[Re c > |Re b|,
a > 0]
K(k) sn u dn u sinh(βx) K 1 (αx) J 0 (γx) dx = a u E(k) − K(k) E(u) + cn u
+ −1 ,1 2 2 2 2 2 2 2 2 2 2 2 2 α +β +γ cn u = 2γ − 4α β −α +β +γ −1
1 k = 2 2
,− 12 + 2 2 2 2 2 2 2 2 2 α +β +γ − 4α β 1− α −β −γ [Re α > |Re β|,
γ > 0]
ET II 15(24)
alternatively, with a = γ, b = β, c = α, ∞ K(k) sn u dn u −1 sinh(bx) K 1 (cx) J 0 (ax) dx = c u E(k) − K(k) E(u) + cn u 0 2 2 2 − c a , k2 = 22 [Re c > |Re b|, cn 2 u = 2 2 − c 2 2 − 21 6.663
∞
1.
K ν±μ (2z cosh t) cosh [(μ ∓ ν) t] dt =
0
2.
Y μ+ν (2z cosh t) cosh[(μ − ν)t] dt =
0
0
∞
5. 0
1.
π [J μ (z) Y ν (z) + J ν (z) Y μ (z)] 4 EH II 97(65)
1 J μ+ν (2z sinh t) cosh[(μ − ν)t] dt = [I ν (z) K μ (z) + I μ (z) K ν (z)] 2 Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,
z>0
1 J μ+ν (2z sinh t) sinh[(μ − ν)t] dt = [I ν (z) K μ (z) − I μ (z) K ν (z)] 2 Re(ν + μ) > −1, |Re(μ − ν)| < 32 ,
z>0
∞
J 0 (2z sinh t) sinh(2νt) dt = 0
EH II 96(64)
[z > 0] ∞
4.
6.664
π [J μ (z) J ν (z) − Y μ (z) Y ν (z)] 4
J μ+ν (2z cosh t) cosh[(μ − ν)t] dt = −
0
WA 484(1), EH II 54(39)
[z > 0] ∞
3.
1 K μ (z) K ν (z) 2 [Re z > 0]
∞
a > 0]
sin(νπ) 2 [K ν (z)] π
|Re ν| < 34 ,
z>0
EH II 97(71)
EH II 97(72)
EH II 97(69)
6.668
Bessel, hyperbolic, and exponential functions
∞
715
cos(νπ) 2 [K ν (z)] |Re ν| < 34 , z > 0 EH II 97(70) π 0 ∞ 1 ∂ K ν (z) ∂ I ν (z) 1 2 − K ν (z) Y 0 (2z sinh t) sinh(2νt) dt = I ν (z) − cos(νπ) [K ν (z)] π ∂ν ∂ν π 0 |Re ν| < 34 , z > 0 EH II 97(75) ∞ 2 π Jν2 (z) + Nν2 (z) K 0 (2z sinh t) cosh 2νt dt = [Re z > 0] MO 44 8 0
∞ 1 1 1 Γ +μ−ν Γ − μ − ν W ν,μ (iz) W ν,μ (−iz) K 2μ (z sinh 2t) coth2ν t dt = 4z 2 2+ 0 , π |arg z| ≤ , |Re μ| + Re ν < 12 2
2. 3.
4. 5.
Y 0 (2z sinh t) cosh(2νt) dt = −
MO 119
∞
6.
cosh(2μx) K 2ν (2a cosh x) dx = 0
1 K μ+ν (a) K μ−ν (a) 2
∞
6.665
sech x cosh(2λx) I 2μ (a sech x) dx = 0
[Re a > 0] Γ 12 + λ + μ Γ 12 − λ + μ
ET II 378(42)
M λ,μ (a) M −λ,μ (a) 2 2a [Γ(2μ + 1)] |Re λ| − Re μ < 12 ET II 378(43)
Bessel, hyperbolic, and algebraic functions ∞ ∞ 2 xν+1 sinh(αx) cosech(πx) J ν (βx) dx = (−1)n−1 nν+1 sin(nα) K ν (nβ) 6.666 π n=1 0 [|Re α| < π, Re ν > −1] ET II 41(3), WA 469(12)
6.667 1.
3
2.
√
a2 − x2 sinh t I 2ν (x) 1 t 1 −t π √ ae I ν ae dx = I ν 2 2 a2 − x2 0 2 Re ν > − 12 ET II 365(10) √ a cosh a2 − x2 sinh t K 2ν (x) π2 √ cosec(νπ) I −ν aet I −ν ae−t − I ν aet I ν ae−t dx = 2 2 4 a −x 0 |Re ν| < 12 ET II 367(25)
a
cosh
Exponential, hyperbolic, and Bessel functions 6.668
Notation: , 1 + 1 = (b + c)2 + a2 − (b − c)2 + a2 , 2
2 =
, 1 + (b + c)2 + a2 + (b − c)2 + a2 2
716
Bessel Functions
1.
10
2.10
6.669
∞
∞
0
∞
2. 0
4.
1
−1
e−αx sinh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 − r1 ) 2 (r2 + r1 ) 2 0 r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α, ∞ 1 e−cx sinh(bx) J 0 (ax) dx = 2 2 − 21 0 [Re c > |Re b|, a > 0] ∞ 1 1 −1 e−αx cosh(βx) J 0 (γx) dx = (αβ) 2 r1−1 r2−1 (r2 − r1 ) 2 (r2 + r1 ) 2 0 r1 = γ 2 + (β − α)2 , r2 = γ 2 + (β + α)2 , [Re α > |Re β|, γ > 0] alternatively, with a = γ, b = β, c = α, ∞ 2 e−cx cosh(bx) J 0 (ax) dx = 2 2 − 21 0 [Re c > |Re b|, a > 0]
1.
3.
6.669
1
ET II 12(52)
ET II 12(54)
2λ , + 1 Γ 12 − λ + μ 1 x M −λ,μ α2 + β 2 2 − β e−β cosh x J 2μ (α sinh x) dx = coth 2 α Γ(2μ + 1) , + 1 × W λ,μ α2 + β 2 2 + β Re β > |Re α|, Re(μ − λ) > − 12 BU 86(5b)a, ET II 363(34) 2λ 1 x e−β cosh x Y 2μ (α sinh x) dx coth 2 sec[(μ + λ)π] W λ,μ α2 + β 2 + β W −λ,μ α2 + β 2 − β =− α tan[(μ + λ)π] Γ 12 − λ + μ W λ,μ α2 + β 2 + β M −λ,μ α2 + β 2 − β − α Γ(2μ + 1) Re β > |Re α|, Re λ < 12 − |Re μ| ET II 363(35)
2ν √ 1 1 x e− 2 (a1 a2 )t cosh x coth K 2μ (t a1 a2 sinh x) dx 2 0 Γ 12 + μ − ν Γ 12 − μ − ν W ν,μ (a1 t) W ν,μ (a2 t) = √ 2t a1 a2 + √ √ 2, 1 ± 2μ , Re t ( a1 + a2 ) > 0 BU 85(4a) Re ν < Re 2 ∞ + x ,2ν Γ 12 + μ − ν √ − 12 (a1 a2 )t cosh x coth W ν,μ (a1 t) M ν,μ (a2 t) e I 2μ (t a1 a2 sinh x) dx = √ 2 0 t a1 a2 Γ(1 + 2μ) 1 BU 86(5c) Re 2 + μ − ν > 0, Re μ > 0, a1 > a2
∞
∞
5.
e −∞
2νs− x−y 2 tanh s
√
Γ 12 + μ + ν Γ 12 + μ − ν xy ds = I 2μ M ν,μ (x) M −ν,μ (y) √ 2 cosh s cosh s xy [Γ(1 + 2μ)] Re ±ν + 12 + μ > 0 BU 83(3a)a
6.671
Bessel and trigonometric functions
∞
6.
e −∞
2νs− x+y 2 tanh s
717
√
Γ 12 + μ + ν Γ 12 + μ − ν xy ds = J 2μ M ν,μ (x) M ν,μ (y) √ 2 cosh s cosh s xy [Γ(1 + 2μ)] Re ∓ν + 12 + μ > 0 BU 84(3b)a
6.67–6.68 Combinations of Bessel and trigonometric functions 6.671
∞
1. 0
β sin ν arcsin α J ν (αx) sin βx dx = α2 − β 2 = ∞ or 0
[β = α]
αν cos νπ 2 ν = 2 2 β − α β + β 2 − α2
[β > α]
∞
2. 0
[β < α]
β cos ν arcsin α J ν (αx) cos βx dx = α2 − β 2
[Re ν > −2]
[β < α]
= ∞ or 0
[β = α]
−α sin ν = β 2 − α2 β + β 2 − α2 ν
0
[β > α] [Re ν > −1]
Y ν (ax) sin(bx) dx νπ − 1 b a2 − b2 2 sin ν arcsin = cot 2 a νπ − 1 1 b 2 − a2 2 = cosec 2 2 + + 1 ,ν 1 ,−ν 2 2 −ν 2 2 ν 2 2 × a cos(νπ) b − b − a −a b− b −a
0
4.
νπ 2
WA 444(5)
∞
3.
WA 444(4)
[0 < b < a, |Re ν| < 2]
[0 < a < b,
|Re ν| < 2] ET I 103(33)
∞
Y ν (ax) cos(bx) dx tan νπ b 2 = 1 cos ν arcsin a (a2 − b2 ) 2
+ νπ 1 1 ,ν − b2 − a2 2 a−ν b − b2 − a2 2 + cot(νπ) = − sin 2 + 1 ,−ν ν +a b − b2 − a2 2 cosec(νπ)
[0 < b < a,
|Re ν| < 1]
[0 < a < b,
|Re ν| < 1] ET I 47(29)
718
Bessel Functions
6.671
∞
5.
K ν (ax) sin(bx) dx
νπ ,ν + ,ν 1 + 2 1 1 1 −ν 2 2 −2 2 2 2 2 2 b +a a +b = πa cosec +b − b +a −b 4 2 [Re a > 0, b > 0, |Re ν| < 2, ν = 0] ET I 105(48)
0
∞
6.
K ν (ax) cos(bx) dx
+ + νπ 2 2 1 1 ,ν 1 ,−ν π 2 2 −2 −ν 2 2 ν 2 2 b+ b +a b +a = sec +a b+ b +a a 4 2 [Re a > 0, b > 0, |Re ν| < 1] ET I 49(40)
0
∞
7.
J 0 (ax) sin(bx) dx = 0 0
1 = √ 2 b − a2
0
∞
9. 0
1 J 0 (ax) cos(bx) dx = √ 2 a − b2
∞
10. 0
[a = b]
=0
[0 < a < b]
b 1 J 2n+1 (ax) sin(bx) dx = (−1)n √ T 2n+1 a a2 − b 2
∞
11.
b 1 J 2n (ax) cos(bx) dx = (−1)n √ T 2n 2 2 a a −b
12.
ET I 99(2)
[0 < b < a]
ET I 43(2)
b
2 arcsin a √ π a2 − b 2 % & 1 b2 b 2 − ln −1 = √ π b 2 − a2 a a2
[0 < b < a] [0 < a < b] ET I 103(31)
∞
Y 0 (ax) cos(bx) dx = 0 0
[0 < b < a]
[0 < a < b]
Y 0 (ax) sin(bx) dx = 0
ET I 43(1)
[0 < a < b]
=0
[0 < b < a]
=∞
=0
[0 < a < b] ET I 99(1)
∞
8.
[0 < b < a]
1 = −√ 2 b − a2
[0 < b < a] [0 < a < b] ET I 47(28)
6.672
Bessel and trigonometric functions
∞
13. 0
0
K 0 (βx) sin αx dx = ln α2 + β 2
. α2 +1 β2
α + β
[α > 0, ∞
14.8 6.672
1
719
π K 0 (βx) cos αx dx = 2 α2 + β 2
β > 0]
WA 425(11)a, MO 48
[α > 0]
WA 425(10)a, MO 48
∞
J ν (ax) J ν (bx) sin(cx) dx
1. 0
=0
b +a −c 1 = √ P ν− 12 2 ab 2ab2
b + a2 − c 2 cos(νπ) Q ν− 12 − =− √ 2ab π ab
2
2
∞
2.
J ν (x) J −ν (x) cos(bx) dx = 0
2
1 P 1 2 ν− 2
1 2 b −1 2
[Re ν > −1,
0 < c < b − a,
[Re ν > −1,
b − a < c < b + a,
[Re ν > −1,
b + a < c,
∞
0
∞
4. 0
π2 K ν (ax) K ν (bx) cos(cx) dx = √ sec(νπ) P ν− 12 a2 + b2 + c2 (2ab)−1 4 ab Re(a + b) > 0, c > 0, 1 K ν (ax) I ν (bx) cos(cx) dx = √ Q ν− 12 2 ab
2
0
|Re ν|
|Re b|,
c > 0,
Re ν > − 12
1 P ν− 12 1 − 2a2 2 1 = cos(νπ) Q ν− 12 2a2 − 1 π
Re ν > −1]
[0 < a < 1, [a > 1,
Re ν > −1] ET II 343(30)
∞
6.
2
1 Q ν− 12 1 − 2a2 π 1 = − sin(νπ) Q ν− 12 2a2 − 1 π
cos(2ax) [J ν (x)] dx = 0
7.
[0 < b < 2] [2 < b]
sin(2ax) [J ν (x)] dx =
ET I 102(27)
ET I 49(47) ∞
5.
0 < a < b]
ET I 46(21)
3.
0 < a < b]
=0
0 < a < b]
Re ν > − 12 Re ν > − 12
0 < a < 1, a > 1,
ET II 344(32) ∞
sin(2ax) J 0 (x) Y 0 (x) dx = 0 0
=−
+ 1 , K 1 − a−2 2 πa
[0 < a < 1] [a > 1] ET II 348(60)
720
Bessel Functions
∞
8. 0
1
K 0 (ax) I 0 (bx) cos(cx) dx = K c2 + (a + b)2
√ 2 ab c2 + (a + b)2
[Re a > |Re b|, ∞
9.
1 K(a) π
1 1 =− K πa a
cos(2ax) J 0 (x) Y 0 (x) dx = −
0
6.673
c > 0]
ET I 49(46)
[0 < a < 1] [a > 1] ET II 348(61)
∞
10.
2
1 K 1 − a2 π 2 1 = K 1− 2 πa a
cos(2ax) [Y 0 (x)] dx = 0
[0 < a < 1] [a > 1] ET II 348(62)
6.673 1.
∞
0
+ νπ νπ , J ν (ax) cos − Y ν (ax) sin sin(bx) dx 2 2 =0 =
2aν
1 √ b 2 − a2
+
b + b 2 − a2
1 ,ν
+
+ b − b 2 − a2
2
1 ,ν 2
[0 < b < a,
|Re ν| < 2]
[0 < a < b,
|Re ν| < 2] ET I 104(39)
∞
2. 0
+ νπ νπ , Y ν (ax) cos + J ν (ax) sin cos(bx) dx 2 2 =0 =−
2aν
1 √ b 2 − a2
+
b + b2 − a
+
,ν 2
+ b − b2 − a
1 2
,ν 2 1 2
[0 < b < a,
|Re ν| < 1]
[0 < a < b,
|Re ν| < 1] ET I 48(32)
3.∗
ea − 1 a
π/2
[cos x I 0 (a cos x) + I 1 (a cos x)] dx = 0
6.674
a
sin(a − x) J ν (x) dx = a J ν+1 (a) − 2ν
1. 0
(−1)n J ν+2n+2 (a)
n=0
[Re ν > −1] a
cos(a − x) J ν (x) dx = a J ν (a) − 2ν
2. 0
∞
∞
(−1)n J ν+2n+1 (a)
n=0
a
%
[Re ν > −1]
sin(a − x) J 2n (x) dx = a J 2n+1 (a) + (−1)n 2n cos a − J 0 (a) − 2
3. 0
ET II 334(12)
ET II 336(23) n
&
(−1)m J 2m (a)
m=1
[n = 0, 1, 2, . . .]
ET II 334(10)
6.676
Bessel and trigonometric functions
%
a
cos(a − x) J 2n (x) dx = a J 2n (a) − (−1)n 2n sin a − 2
4. 0
5.
n−1
721
& (−1)m J 2m+1 (a)
m=0
ET II 335(21) [n = 0, 1, 2, . . .] % & a n sin(a − x) J 2n+1 (x) dx = a J 2n+2 (a) + (−1)n (2n + 1) sin a − 2 (−1)m J 2m+1 (a) 0
m=0
[n = 0, 1, 2, . . .] %
a
ET II 334(11)
cos(a − x) J 2n+1 (x) dx = a J 2n+1 (a) + (−1) (2n + 1) cos a − J 0 (a) − 2 n
6. 0
n
&
m
(−1) J 2m (a)
m=1
[n = 0, 1, 2, . . .]
ET II 336(22)
z
7.
sin(z − x) J 0 (x) dx = z J 1 (z)
WA 415(2)
cos(z − x) J 0 (x) dx = z J 0 (z)
WA 415(1)
0
8.
z
0
6.675
∞
1. 0
2 2
2
2
√ √ a a a νπ νπ a a π − − J ν a x sin(bx) dx = cos J 12 ν− 12 − sin J 12 ν+ 12 3 8b 4 8b 8b 4 8b 2 4b [a > 0, b > 0, Re ν > −4] ET I 110(23)
∞
2.
√ J ν a x cos(bx) dx
0
=−
∞
3. 0
∞
4. 0
6.676
∞
1. 0
∞
2. 0
3. 0
∞
2
2
2
√ 2 a a a a π νπ νπ a 1 1 1 1 − − sin J + cos J 3 ν− 2 ν+ 2 2 2 8b 4 8b 8b 4 8b 4b 2 [a > 0, b > 0, Re ν > −2] ET I 53(22)a
√ 1 J 0 a x sin(bx) dx = cos b √ 1 J 0 a x cos(bx) dx = sin b
a2 4b a2 4b
[a > 0,
b > 0]
ET I 110(22)
[a > 0,
b > 0]
ET I 53(21)
√ √ 1 J ν a x J ν b x sin(cx) dx = J ν c
√ √ 1 J ν a x J ν b x cos(cx) dx = J ν c
√ √ 1 K0 J 0 a x K 0 a x sin(bx) dx = 2b
ab 2c
ab 2c
cos
sin
2
a 2b
a2 + b 2 νπ − 4c 2 [a > 0, b > 0,
c > 0,
ET I 111(29)a
a2 + b 2 νπ − 4c 2 [a > 0, b > 0,
Re ν > −2]
c > 0,
Re ν > −1] ET I 54(27)
[Re a > 0,
b > 0]
ET I 111(31)
722
Bessel Functions
∞
4.
J0 0
6.
a , √ √ π + a I0 − L0 ax K 0 ax cos(bx) dx = 4b 2b 2b
[Re a > 0, b > 0] √ √ √ a 1 Re a > 0, b > 0 K0 ax Y 0 ax cos(bx) dx = − K 0 2b 2b 0 ∞ √ a a , √ 1 1 π2 + H0 −Y0 K0 axe 4 πi K 0 axe− 4 πi cos(bx) dx = 8b 2b 2b 0
5.
6.677
ET I 54(29)
∞
ET I 54(30)
[Re a > 0, b > 0] 6.677
∞
1.
J 0 b x2 − a2 sin(cx) dx = 0
√ cos a c2 − b2 √ = c2 − b 2
a
√ ∞ exp −a b2 − c2 2 2 √ J 0 b x − a cos(cx) dx = 2 2 a b √− c − sin a c2 − b2 √ = c2 − b 2
∞
3.6
cos z J 0 α x2 + z 2 cos βx dx =
0
0
5. 0
ET I 113(47)
[0 < c < b] [0 < b < c]
α2 − β 2
α2 − β 2
[0 < β < α,
z > 0]
[0 < α < β,
z > 0] MO 47a
∞
4.
[0 < b < c]
ET I 57(48)a
=0
[0 < c < b]
2.
ET I 54(31)
1 Y 0 α x2 + z 2 cos βx dx = sin z α2 − β 2 α2 − β 2 1 = − exp −z β 2 − α2 β 2 − α2
[0 < β < α,
z > 0]
[0 < α < β,
z > 0] MO 47a
∞
+ , π K 0 α x2 + β 2 cos(γx) dx = exp −β α2 + γ 2 2 α2 + γ 2 [Re α > 0, Re β > 0,
√ a sin a b2 + c2 √ J 0 b a2 − x2 cos(cx) dx = b 2 + c2 0 √ ∞ cosh a b2 − c2 √ J 0 b x2 − a2 cos(cx) dx = b 2 − c2 0
γ > 0] ET I 56(43)
6. 7.
=0
[b > 0]
MO 48a, ET I 57(47)
[0 < c < b,
a > 0]
[0 < b < c,
a > 0] ET I 57(49)
6.681
Bessel and trigonometric functions
∞
8. 0
∞
9. 0
∞
6.678 0
6.679
i exp −iβ α2 + γ 2 (2) H 0 α β 2 − x2 cos(γx) dx = α2 + γ 2 + −π < arg β 2 − x2 ≤ 0,
α > 0,
γ>0
+ √ π √ , 1 π sin K 0 2 x + Y 0 2 x sin(bx) dx = 2 2b b
,
,
ET I 59(64) ∞
[a > 0, ∞
ET I 115(58)
[a > 0, + x , 2 2 cos(bx) dx = − cosh(πb) [K ib (a)] Y 0 2a sinh 2 π
b > 0]
ET I 59(62)
[a > 0, π 2 2 cos(bx) dx = [J ib (a)] + [Y ib (a)] 4
b > 0]
ET I 59(65)
J 0 2a sinh ∞
6. 0
∞
7.
x ,
+ K 0 2a sinh
0
2
cos(bx) dx = [I ib (a) + I −ib (a)] K ib (a)
x , 2
2
[Re a > 0,
π 2
cos(2μx) J 2ν (2a cos x) dx =
1. 0
2.
π 2
cos(2μx) Y 2ν (2a cos x) dx = 0
ET I 59(63)
b > 0]
+
0
6.681
ET I 111(34)
ET I 115(59)
5.
ET I 58(58)
[b > 0]
+ x , π cos(bx) dx = − [J ν+ib (a) Y ν−ib (a) + J ν−ib (a) Y ν+ib (a)] J 2ν 2a cosh 2 2 0 ∞ + x , 2 2 sin(bx) dx = sinh(πb) [K ib (a)] J 0 2a sinh 2 π 0
ET I 59(59)
+ x , cos(bx) dx = I ν−ib (a) K ν+ib (a) + I ν+ib (a) K ν−ib (a) J 2ν 2a sinh 2 a > 0, b > 0, Re ν > − 12
0
4.
γ>0
∞
2.
3.
α > 0,
+ x , sin(bx) dx = −i [I ν−ib (a) K ν+ib (a) − I ν+ib (a) K ν−ib (a)] J 2ν 2b sinh 2 [a > 0, b > 0, Re ν > −1]
0
exp iβ α2 + γ 2 (1) H 0 α β 2 − x2 cos(γx) dx = −i α2 + γ 2 + π > arg β 2 − x2 ≥ 0,
∞
1.
723
π J ν+μ (a) J ν−μ (a) 2
Re ν > − 12
b > 0]
ET I 59(66)
ET II 361(23)
π [cot(2νπ) J ν+μ (a) J ν−μ (a) − cosec(2νπ) J μ−ν (a) J −μ−ν (a)] 2 |Re ν| < 12 ET II 361(24)
724
Bessel Functions
π 2
3.
cos(2μx) I 2ν (2a cos x) dx = 0
π 2
4.
cos(νx) K ν (2a cos x) dx = 0
π I ν−μ (a) I ν+μ (a) 2
π I 0 (a) K ν (a) 2
6.682
Re ν > − 12
ET I 59(61)
[Re ν < 1]
WA 484(3)
π
J 0 (2z cos x) cos 2nx dx = (−1)n πJn2 (z).
5.
MO 45
0
π
J 0 (2z sin x) cos 2nx dx = πJn2 (z).
6.
WA 43(3), MO 45
0
π 2
7.
cos(2nπ) Y 0 (2a sin x) dx = 0
π J n (a) Y n (a) 2
[n = 0, 1, 2, . . .]
ET II 360(16)
π
8.
sin(2μx) J 2ν (2a sin x) dx = π sin(μπ) J ν−μ (a) J ν+μ (a) 0
[Re ν > −1]
9.
cos(2μx) J 2ν (2a sin x) dx = π cos(μπ) J ν−μ (a) J ν+μ (a) Re ν > − 12
0
π 2
10. π 2
11.
ET II 360(14)
J ν+μ (2z cos x) cos[(ν − μ)x] dx =
π J ν (z) J μ (z) 2
[Re(ν + μ) > −1]
cos[(μ − ν)x] I μ+ν (2a cos x) dx =
π I μ (a) I ν (a) 2
[Re(μ + ν) > −1]
0
ET II 360(13)
π
0
MO 42
WA 484(2), ET II 378(39)
π 2
12.
cos[(μ − ν)x] K μ+ν (2a cos x) dx =
0
π2 cosec[(μ + ν)π] [I −μ (a) I −ν (a) − I μ (a) I ν (a)] 4 [|Re(μ + ν)| < 1]
13.8
π 2
K ν−m (2a cos x) cos[(m + ν)x] dx = (−1)m
0
π I m (a) K ν (a) 2 [|Re(ν − m)| < 1]
6.682 1.
7
ET II 378(40)
WA 485(4)
π 2
π J ν (x) 2x 0 [ν may be zero, a natural number, one half, or a natural number plus one half; x > 0] J ν− 12 (x sin t) sin
2.
π 2
ν
ν+ 12
J ν (z sin x) sin x cos 0
t dt =
2ν
x dx = 2
ν−1
z √ 1 πΓ ν + z −ν Jν2 2 2 Re ν > − 12
MO 42a
MO 42a
6.683
Bessel and trigonometric functions
6.683 1. 0
π 2
2. 0
0
[Re ν > Re μ > −1] z1ν z2μ J ν+μ+1 z12 + z22 < J ν (z1 sin x) J μ (z2 cos x) sinν+1 x cosμ+1 x dx = ν+μ+1 (z12 + z22 )
π 2
4.
J μ (z sin θ) (sin θ)
1−μ
2ν+1
(cos θ)
dθ =
0
Re μ > −1]
WA 410(1)
k=0
Re μ > −1]
(see also 6.513 6)
J μ (z sin θ) (sin θ)
dθ =
μ+1
(cos θ)
0 π 2
6.
J μ (a sin θ) (sin θ)
WA 407(2)
Hμ− 12 (z) 2z π
1−μ
2+1
WA 407(3)
dθ = 2 Γ( + 1)a−−1 J +μ+1 (a)
0
[Re > −1,
Re μ > −1] WA 406(1),
π 2
7.
ν
J ν (2z sin θ) (sin θ) (cos θ) 0
2ν
WA 414(1)
s μ+ν,ν−μ+1 (z) 2μ−1 z ν+1 Γ(μ) [Re ν > −1]
π 2
5.
WA 407(4)
∞ 1 J ν z cos2 x J μ z sin2 x sin x cos x dx = (−1)k J ν+μ+2k+1 (z) z
[Re ν > −1,
Γ
[Re ν > −1, π 2
3.
μ−ν 2 2
J μ (z) J ν (z sin x) I μ (z cos x) tanν+1 x dx = μ+ν +1 Γ 2 z ν
π 2
725
EH II 46(5)
dθ
∞ 1 (−1)m z ν+2m Γ ν + m + 12 Γ ν + 12 = 2 m=0 m! Γ(ν + m + 1) Γ(2ν + m + 1) √ 1 2 = z −ν π Γ ν + 12 [J ν (z)] 2
Re ν > − 12 EH II 47(10)
π 2
8.
ν+1
J ν (z sin θ) (sin θ) 0
9. 0
π 2
−2ν
(cos θ)
z ν−1 dθ = 2−ν √ Γ π
2ν+1 2ν+1 J ν z sin2 θ J ν z cos2 θ (sin θ) (cos θ)
1 − ν sin z 2 −1 < Re ν < 12 Γ 12 + ν J 2ν+ 12 (z) dθ = 2ν+ 3 √ 2 Γ(ν + 1) 2 z 1 Re ν > − 2
EH II 68(39)
WA 409(1)
726
Bessel Functions
π 2
2
2
2μ+1
J μ z sin θ J ν z cos θ sin
10. 0
6.684
π
1.8
(sin x)
2ν
0
π
2.
(sin x)
2ν
0
θ cos
2ν+1
6.684
Γ μ + 12 Γ ν + 12 J μ+ν+ 12 (z) √ θ dθ = √ 2 π Γ(μ + ν + 1) 2z WA 417(1) Re μ > − 12 , Re ν > − 12
Jν α2 + β 2 − 2αβ cos x √ 1 J ν (α) J ν (β) ν dx = 2ν π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x Re ν > − 12
Yν α2 + β 2 − 2αβ cos x √ 1 J ν (α) Y ν (β) ν ν dx = 2 π Γ ν + 2 αν βν α2 + β 2 − 2αβ cos x
π 2
6.685
sec x cos(2λx) K 2μ (a sec x) dx = 0
6.686
∞
2
sin ax
1. 0
|α| < |β|,
π W λ,μ (a) W −λ,μ (a) 2a
Re ν > − 12
[Re a > 0]
ET II 362(27)
ET II 362(28) ET II 378(41)
2 2
√ b b ν+1 π √ 1 − π J 2ν J ν (bx) dx = − sin 8a 4 8a 2 a
[a > 0, b > 0, Re ν > −3] ET II 34(13)
√ ∞ b2 b2 ν+1 π − π J 12 ν cos ax2 J ν (bx) dx = √ cos 8a 4 8a 2 a 0 [a > 0, b > 0, Re ν > −1]
2.
ET II 38(38) ∞
3.
√ νπ π = − √ sec 2 4 a
2
2 2 b b b2 3ν + 1 b ν−1 − π J 12 ν + π Y 12 ν × cos − sin 8a 4 8a 8a 4 8a [a > 0, b > 0, −3 < Re ν < 3] ET II 107(7)
0
∞
4.
0
cos ax2 Y ν (bx) dx
√ νπ π = √ sec 4 a 2
2
2 2 b b b2 3ν + 1 b ν−1 − π J 12 ν + π Y 12 ν × sin + cos 8a 4 8a 8a 4 8a [a > 0, b > 0, −1 < Re ν < 1] ET II 107(8)
0
5.
sin ax2 Y ν (bx) dx
∞
b2 1 sin ax2 J 1 (bx) dx = sin b 4a
[a > 0,
b > 0]
ET II 19(16)
6.693
Bessel and trigonometric functions and powers
b2 6. cos ax 8a 0 2
∞ b 1 cos 7. sin2 ax2 J 1 (bx) dx = 2b 8a 0 2
∞ π π x 6.687 cos K 2ν xei 4 K 2ν xe−i 4 dx 2a 0
∞
2
2 J 1 (bx) dx = sin2 b
= 6.688
π 2
1. 0
π J ν (μz sin t) cos (μx cos t) dt = J ν2 2
Γ
1 4
[a > 0,
b > 0]
ET II 20(20)
[a > 0,
b > 0]
ET II 19(17)
√ π + ν Γ 14 − ν π √ W 14 ,ν aei 2 W 8 a a > 0, |Re ν| < 14
√
x2 + z 2 + x μ 2
√
J ν2
x2 + z 2 − x μ 2
1 4 ,ν
ae−i 2
π
ET II 372(1)
[Re ν > −1,
π 2
2. 0
727
π 2
3. 0
MO 46 Re z > 0] + , − 1 ν− 1 1 1√ ν+1 (sin x) cos (β cos x) J ν (α sin x) dx = 2− 2 παν α2 + β 2 2 4 J ν+ 12 α2 + β 2 2
[Re ν > −1] + , π cos [(z − ζ) cos θ] J 2ν 2 zζ sin θ dθ = J ν (z) J ν (ζ) 2 Re ν > − 12
ET II 361(19)
EH II 47(8)
6.69–6.74 Combinations of Bessel and trigonometric functions and powers
∞
6.691
x sin(bx) K 0 (ax) dx = 0
6.692
∞
1. 0
∞
0
1. 0
[Re a > 0,
b > 0]
ET I 105(47)
− 1 3 1 x K ν (ax) I ν (bx) sin(cx) dx = − (ab)− 2 c u2 − 1 2 Q 1ν− 1 (u), u = (2ab)−1 a2 + b2 + c2 2 2 Re a > |Re b|, c > 0, Re ν > − 32 ET I 106(54)
2.
6.693
− 3 πb 2 a + b2 2 2
∞
− 1 3 π x K ν (ax) K ν (bx) sin(cx) dx = (ab)− 2 c u2 − 1 2 Γ 32 + ν Γ 32 − ν P −1 (u) ν− 12 4 Re(a + b) > 0, c > 0, |Re ν| < 32 u = (2ab)−1 a2 + b2 + c2 ET I 107(61)
β dx 1 = sin ν arcsin J ν (αx) sin βx x ν α αν sin νπ 2 ν = 2 ν β + β − α2
[β ≤ α] [β ≥ α] [Re ν > −1]
WA 443(2)
728
Bessel Functions
2.
∞
8 0
β dx 1 = cos ν arcsin J ν (αx) cos βx x ν α αν cos νπ 2 ν = 2 ν β + β − α2
6.693
[β ≤ α] [β ≥ α]
[Re ν > 0] WA 443(3)
∞
3.
Y ν (ax) sin(bx) 0
dx x
νπ 1 b = − tan sin ν arcsin ν 2 a
[0 < b < a, |Re ν| < 1] + νπ + 1 ,ν 2 1 ,−ν 1 −ν 2 2 2 ν 2 2 sec −a b− b −a = a cos(νπ) b − b − a 2ν 2 [0 < a < b, |Re ν| < 1]
ET I 103(35) ∞
4.
dx 2 x√ b cos ν arcsin ab a2 − b2 sin ν arcsin ab − = ν 2− 1 √ ν (ν 2 − 1) νπ −aν cos 2 b + ν b2 − a2 = √ ν ν (ν 2 − 1) b + b2 − a2
J ν (ax) sin(bx) 0
0
dx J ν (ax) cos(bx) 2 x a cos (ν + 1) arcsin ab a cos (ν − 1) arcsin ab + = 2ν(ν − 1) 2ν(ν + 1) aν sin νπ aν+2 sin νπ 2 2 = √ √ ν−1 − ν+1 2ν(ν − 1) b + b2 − a2 2ν(ν + 1) b + b2 − a2
∞
J 0 (αx) sin x 0
7.
8. 9.
[0 < a < b,
Re ν > 0]
[0 < b < a,
Re ν > 1]
[0 < a < b,
Re ν > 1] ET I 44(6)
6.
Re ν > 0]
ET I 99(6) ∞
5.
[0 < b < a,
dx π = x 2 = arccosecα
[0 < α < 1] [α > 1] WH
∞
dx π = x 2 0 = arcsin β π =− 2 ∞ dx = ln 2α [J 0 (x) − cos αx] x 0 z ∞ 2 dx = J ν (x) sin(z − x) (−1)k J ν+2k+1 (z) x ν 0 J 0 (x) sin βx
k=0
[β > 1] 2 β 0]
WA 416(4)
6.697
Bessel and trigonometric functions and powers
z
J ν (x) cos(z − x)
10. 0
729
∞ 1 dx 2 = J ν (z) + (−1)k J ν+2k (z) x ν ν k=1
[Re ν > 0] 6.69410 0
6.695
∞
J 1 (ax) sin(bx) dx x
4a b b 1 b2 b2 = b− 1+ 2 E + 1− 2 K 2 3π% 4a 2a 4a 2a
2 −1 & 2a 2a 2b 4a 1 b2 = b− K 1+ 2 E − 1− 2 3π 4a b b2 b
∞
1. 0
∞
2. 0
[0 ≤ b ≤ 2a] ET I 102(22) [0 ≤ 2a ≤ b]
sin αx sinh αβ K 0 (βu) J 0 (ux) dx = 2 2 β +x β
[α > 0,
Re β > 0,
u > α]
cos αx π e−αβ I 0 (βu) J (ux) dx = 0 β 2 + x2 2 β
[α > 0,
Re β > 0,
−α < u < α]
MO 46
MO 46 ∞
3.
x2
0
WA 416(5)
2
x π sin(αx) J 0 (γx) dx = e−αβ I 0 (γβ) 2 +β 2
[α > 0,
Re β > 0,
0 < γ < α] ET II 10(36)
∞
4. 0
x cos(αx) J 0 (γx) dx = cosh(αβ) K 0 (βγ) x2 + β 2
[α > 0,
Re β > 0,
α < γ] ET II 11(45)
∞
6.696 0
[1 − cos(αx)] J 0 (βx)
dx = arccosh x
α β
=0
[0 < β < α] [0 < α < β] ET II 11(43)
6.697 1.
∞
sin[α(x + β)] J 0 (x) dx = 2 x+β −∞
α
0
cos βu √ du 1 − u2
∞
0
∞
3. 0
4. 5.
∞
WA 463(2)
[1 ≤ α < ∞]
WA 463(1), ET II 345(42)
sin(x + t) π J 0 (t) dt = J 0 (x) x+t 2
[x > 0]
WA 475(4)
cos(x + t) π J 0 (t) dt = − Y 0 (x) x+t 2
[x > 0]
WA 475(5)
= π J 0 (β) 2.
[0 ≤ α ≤ 1]
|x| sin[α(x + β)] J 0 (bx) dx = 0 x −∞ + β ∞ ,2 + ,2 sin[α(x + β)] + J n+ 12 (x) dx = π J n+ 12 (β) x+β −∞
[0 ≤ α < b] [2 ≤ α < ∞,
WA 464(5), ET II 345(43)a
n = 0, 1, . . .] ET II 346(45)
730
Bessel Functions
6.
7.
6.698
∞
sin[α(x + β)] J n+ 12 (x) J −n− 12 (x) dx = π J n+ 12 (β) J −n− 12 (β) x+β −∞ [2 ≤ α < ∞,
n = 0, 1, . . .]
√ 0]
6.698
1.
∞√
0
x J ν+ 14 (ax) J −ν+ 14 (ax) sin(bx) dx =
b 2 cos 2ν arccos 2a √ πb 4a2 − b2
=0 2.
∞√
0
x J ν− 14 (ax) J −ν− 14 (ax) cos(bx) dx =
ET II 346(46)
WA 463(3)
[0 < b < 2a] [0 < 2a < b]
b 2 cos 2ν arccos 2a √ πb 4a2 − b2
ET I 102(26)
[0 < b < 2a]
=0
[0 < 2a < b] ET I 46(24)
3.
∞√
0
x I 14 −ν
√ 2ν
1 1 π −2ν b + a2 + b2 √ ax K 14 +ν ax sin(bx) dx = a 2 2 2b a2 + b 2 Re a > 0, b > 0, Re ν < 54 ET I 106(56)
4.
∞√
0
x I − 14 −ν
√
2ν
1 1 π −2ν b + a2 + b2 √ ax K − 14 +ν ax cos(bx) dx = a 2 2 2b a2 + b 2 Re a > 0, b > 0, Re ν < 34 ET I 50(49)
6.699
1.
∞
λ
0
2+λ+ν
2 + λ + ν 2 + λ − ν 3 b2 , ; ; 2 2 2 2 a 0 < b < a, − Re ν − 1 < 1 + Re λ < 32 ν
1+λ+ν 1 Γ (ν + λ + 1) a b−(ν+λ+1) sin π = 2 Γ(ν + 1) 2
2+λ+ν 1+λ+ν a2 , ; ν + 1; 2 ×F 2 2 b 0 < a < b, − Re ν − 1 < 1 + Re λ < 32
x J ν (ax) sin(bx) dx = 2
1+λ −(2+λ)
a
b
Γ
2 F Γ ν−λ 2
ET I 100(11)
6.699
Bessel and trigonometric functions and powers
∞
2. 0
xλ J ν (ax) cos(bx) dx
2λ a−(1+λ) Γ 1+λ+ν 1 + λ + ν 1 + λ − ν 1 b2 2 ν−λ+1 F , ; ; 2 = 2 2 2 a Γ 2 0 < b < a, − Re ν < 1 + Re λ < 32 a ν −(ν+1+λ)
b Γ (1 + λ + ν) cos π2 (1 + λ + ν) 1+λ+ν 2+λ+ν a2 2 F , ; ν + 1; 2 = Γ(ν + 1) 2 2 b 0 < a < b, − Re ν < 1 + Re λ < 32 2+λ−μ Γ 2λ b Γ 2+μ+λ 2 2
∞
a2+λ
0
∞
4. 0
ET I 45(13)
xλ K μ (ax) sin(bx)dx =
3.
∞
5. 0
F
∞
6. 0
7. 0
2+μ+λ 2+λ−μ 3 b2 , ; ;− 2 2 2 2 a [Re (−λ ± μ) < 2, Re a > 0, b > 0] ET I 106(50)
μ+λ+1 1+λ−μ λ λ−1 −λ−1 x K μ (ax) cos(bx) dx = 2 a Γ Γ 2 2
μ+λ+1 1+λ−μ 1 b2 , ; ;− 2 ×F 2 2 2 a [Re (−λ ± μ) < 1, Re a > 0, −ν− 12 √ ν ν 2 π2 b a − b2 x sin(ax) J ν (bx) dx = Γ 12 − ν ν
=0
731
b > 0]
0 < b < a,
−1 < Re ν
0, xν cos(ax) J ν (ax) dx =
∞
12. 0
xμ K μ (ax) cos(bx) dx =
1√ 1 π(2a)μ Γ μ + 2 2
Re ν > − 32
ET I 105(49)
−μ− 12 b 2 + a2 Re a > 0,
b > 0,
Re μ > − 12
ET I 49(41) ∞
13.
xν Y ν−1 (ax) sin(bx) dx = 0
√ −ν− 12 2ν πaν−1 b 2 1 b − a2 = Γ 2 −ν
0
ET II 335(20)
0 < b < a,
|Re ν|
0,
ET I 47(26)
6.713
Bessel and trigonometric functions and powers
∞
5.
x 0
6.10 7.10 6.712 1.
2.
1−2ν
∞
733
Γ 32 − ν a 3 3 2 F − ν, − 2ν; 2 − ν; a sin(2ax) J ν (x) Y ν (x) dx = − 2 2 2 Γ 2ν − 12 Γ(2 − ν) 0 < Re ν < 32 , 0 < a < 1 ET II 348(63)
ρ2 a2 2z 2 Γ (ν) aμ ρ−ν − − μ−ν+3 2 Γ (μ + 1) ν − 1 μ + 1 3 0 2 ∞ μ −ν 2 ρ a Γ (ν) a ρ − − 2z 2 cos (zx)xν−μ−3 J μ (ax) J ν (ρx) dx = μ−ν+3 2 Γ (μ + 1) ν − 1 μ +1 0 √ −ν− 12 π(2a)ν 2 1 b + 2ab Γ 2 −ν 0 b > 0, −1 < Re ν < 12 √ ∞ −ν− 12 π(2a)ν 2 ν b + 2ab x [Y ν (ax) cos(ax) − J ν (ax) sin(ax)] cos(bx) dx = − 1 Γ 2 −ν 0
∞
xν [J ν (ax) cos(ax) + Y ν (ax) sin(ax)] sin(bx) dx =
ET I 104(40)
ET I 48(35) ∞
3.
xν [J ν (ax) cos(ax) − Y ν (ax) sin(ax)] sin(bx) dx
0
=0
√ −ν− 12 2ν πbν 2 b − 2ab = 1 Γ 2 −ν
arg sin (zx)xν−μ−4 J μ (ax) J ν (ρx) dx = z
−1 < Re ν < −1 < Re ν < 12
0 < b < 2a, 2a < b,
1 2
ET I 104(41) ∞
4.
xν [J ν (ax) sin(ax) + Y ν (ax) cos(ax)] cos(bx) dx
0
=0
√ −ν− 12 π(2a)ν 2 b − 2ab = − 1 Γ 2 −ν
0 < b < 2a, 0 < 2a < b,
|Re ν|
−2] α
+
cos(βx) π √ J ν (x) dx = J 0 (αβ) J 12 ν 12 α 2 2 2 α −x
, 2
+π
∞
(−1)n J 2n (αβ) J 12 ν+n
1.
∞√
0
2.
∞√
0
3.
∞√
0
4.
0
∞√
√ x J 14 a2 x2 sin(bx) dx = 2−3/2 a−2 πb J 14
b2 4a2
√ x J − 14 a2 x2 cos(bx) dx = 2−3/2 a−2 πb J − 14
xY
1 4
√ a2 x2 sin(bx) dx = −2−3/2 πba−2 H 14
2 2
x Y − 14 a x
cos(bx) dx = −2
−3/2
1 1 1 2 α J 2 ν−n 2 α
n=1
[Re ν > −1] 6.721
ET II 335(17)
ET II 336(27)
[b > 0]
2
ET I 108(1)
[b > 0]
2
ET I 51(1)
b 4a2 b 4a2
√ πba−2 H− 14
b2 4a2
ET I 108(7)
ET I 52(7)
736
Bessel Functions
2
2 √ b b −2 3 x K 14 a x sin(bx) dx = 2 π ba I 14 − L 14 2 4a 4a2 0 + , π |arg a| < , b > 0 4 2
2 ∞ √ 2 2 √ b b 1 x K − 14 a x cos(bx) dx = 2−5/2 π 3 ba−2 I − 14 − L −4 2 2 4a 4a 0
5.
6.
6.722
∞√
2 2
−5/2
[b > 0] 6.722
∞√
1. 0
ET I 109(11)
ET I 52(10)
2
2
√ Γ 58 − ν b b 5 W ν, 1 1 x K 18 +ν a2 x2 I 18 −ν a2 x2 sin(bx) dx = 2πb−3/2 M −ν, 8 2 2 8 8a 8a Γ 4 π 5 Re ν < , |arg a| < , b > 0 8 4 ET I 109(13)
2.10
∞√
x J − 18 −ν a2 x2 J − 18 +ν a2 x2 cos(bx) dx 4 √ Γ 14 b π 3 3 3 7 3 2F 3 = 3/4 3/2 3 5 − ν, + ν; , , ; − 8 8 8 4 8 4a 2 a Γ 4 Γ 8 − ν Γ 58 + ν 4 b 1 1 7 9 1 1 2b cos(πν) 2 F 3 − ν, + ν; , , ; − − 2 a π 2 2 2 8 8 4a 4 5/2 b 11 3 13 b ν 2 sin(πν) 2 F 3 1 − ν, 1 + ν; , , ; − − 4 15a π 8 2 8 4a 2 a > 0, Im b = 0 MC
∞√
x J 18 −ν a2 x2 J 18 +ν a2 x2 sin(bx) dx
0
3.
2 πi/2
2 πi/2
b e b e 2 −3/2 πi/8 = b W ν, 18 e W −ν, 18 2 π 8a 8a2 ⎤ 2 −πi/2
πi b e b 2 e− 2 ⎦ 1 + e−iπ/8 W ν, 18 W −ν, 8 8a2 8a2
0
[b > 0]
∞√
4. 0
x K 18 −ν a2 x2 I − 18 −ν a2 x2 cos(bx) dx =
6.723 0
∞
√
2πb
−3/2 Γ
ET I 108(6)
3 2
2
b b 8−ν 1 1 M W ν,− 8 −ν,− 8 3 2 2 8a 8a Γ 4 3 Re ν < 8 , b > 0 ET I 52(12)
1 x J ν x2 sin(νπ) J ν x2 − cos(νπ) Y ν x2 J 4ν (4ax) dx = J ν a2 J −ν a2 4 [a > 0, Re ν > −1] ET II 375(20)
6.726
Bessel and trigonometric functions and powers
6.724 1.
∞
x2λ J 2ν
0
∞
2. 0
x2λ J 2ν
a x
a x
737
sin(bx) dx
√ 2ν πa Γ(λ − ν + 1)b2ν−2λ−1 1 a2 b 2
0 F 3 2ν + 1, ν − λ, ν − λ + ; = 1 2 16 42ν−λ Γ(2ν + 1) Γ ν − λ + 2
3 a2 b 2 a2λ+2 Γ(ν − λ − 1)b , λ − ν + 2, λ + ν + 2; + 2λ+3 0F 3 2 Γ(ν + λ + 2) 2 16 − 54 < Re λ < Re ν, a > 0, b > 0 ET I 109(15) cos(bx) dx
√ 2ν 2ν−2λ−1 Γ λ − ν + 12 a2 b 2 1 =4 πa b 0 F 3 2ν + 1, ν − λ + , ν − λ; Γ(2ν + 1) Γ(ν − λ) 2 16
1 1 3 3 a2 b 2 −λ−1 2λ+1 Γ ν − λ − 2
,λ − ν + ,ν + λ + ; +4 a 0F 3 3 2 2 2 16 Γ ν+λ+ 2 3 − 4 < Re λ < Re ν − 12 , a > 0, b > 0 ET I 53(14) λ−2ν
6.725
∞
1. 0
∞
2. 0
∞
3.
2 2
√ a a νπ π sin(bx) π √ sin − − J ν a x dx = − J ν2 b 8b 4 4 8b x [Re ν > −3, √ cos(bx) √ J ν a x dx = x
π cos b
2
a νπ π − − 8b 4 4
√ 1 x 2 ν J ν a x sin(bx) dx = 2−ν aν b−ν−1 cos
0
∞
4.
√ 1 x 2 ν J ν a x cos(bx) dx = 2−ν b−ν−1 aν sin
0
2
a > 0,
b > 0] ET I 110(27)
a 8b [Re ν > −1,
J 12 ν
a > 0,
b > 0] ET I 54(25)
a2 νπ − 4b 2 −2 < Re ν < 12 , 2
a > 0,
b>0
ET I 110(28)
a νπ − 4b 2 −1 < Re ν < 12 ,
a > 0,
b>0
ET I 54(26)
6.726 1. 0
∞
− 1 ν x x2 + b2 2 J ν a x2 + b2 sin(cx) dx 1 ν− 34 π −ν −ν+ 3 2 2 c a − c2 2 a b = J ν− 32 b a2 − c2 2 =0
0 < c < a, 0 < a < c,
Re ν >
1 2
Re ν >
1 2
ET I 111(37)
738
Bessel Functions
∞
2. 0
2 − 1 ν x + b2 2 J ν a x2 + b2 cos(cx) dx 1 ν− 1 π −ν −ν+ 1 2 2 a b a − c2 2 4 J ν− 12 b a2 − c2 = 2 =0
Re ν > − 12
b > 0,
Re ν > − 12
ET I 55(37)
∞
2 ∓ 1 ν x + b2 2 K ν a x2 + b2 cos(cx) dx ± 1 ν− 1 π ∓ν 1 ∓ν 2 a b2 a + c2 2 4 K ±ν− 12 b a2 + c2 = 2 [Re a > 0, Re b > 0, c is real] ET I 56(45)
∞
2 − 1 ν x + a2 2 Y ν b x2 + a2 cos(cx) dx 1 ν− 1 aπ (ab)−ν b2 − c2 2 4 Y ν− 12 a b2 − c2 = 2 1 ν− 1 2a (ab)−ν c2 − b2 2 4 K ν− 12 a c2 − b2 =− π
11 0
b > 0,
1ν x x2 + b2 2 K ±ν a x2 + b2 sin(cx) dx − 1 ν− 3 π ν ν+ 3 2 a b 2 c a + c2 2 4 K −ν− 32 b a2 + c2 = 2 [Re a > 0, Re b > 0, c > 0] ET I 113(45)
0
4.
0 < c < a, 0 < a < c,
∞
3.
6.727
5. 0
0 < c < b,
a > 0,
Re ν > − 12
0 < b < c,
a > 0,
Re ν > − 12
ET I 56(41)
6.727
a
1.9 0
+ a + a , , π cos(cx) √ b 2 + c2 + c J ν b a2 − x2 dx = J 12 ν b2 + c2 − c J 12 ν 2 2 2 a2 − x2 [Re ν > −1, c > 0, a > 0] ET I 113(48)
∞
+a , +a , sin(cx) π √ J ν b x2 − a2 dx = J 12 ν c − c2 + b2 J − 12 ν c + c2 + b 2 2 2 2 x2 − a2 [0 < b < c, a > 0, Re ν > −1]
∞
+a , +a , cos(cx) π √ J ν b x2 − a2 dx = − J 12 ν c − c2 − b2 Y − 12 ν c + c2 − b 2 2 2 2 x2 − a2 [0 < b < c, a > 0, Re ν > −1]
2. a
ET I 113(49)
3. a
4.8 0
a
a2 − x2
1 2ν
a2 − x2 dx = cos x I ν
√ 2ν+1
ET I 58(54) 2ν+1
πa Γ ν + 32 Re ν > − 12
WA 409(2)
6.731
6.728
Bessel and trigonometric functions and powers
∞
1.
∞
2.
∞
3. 0
∞
4. 0
x cos ax2 J ν (bx) dx
2 2
2
2
√ b b b νπ b νπ πb = 3/2 cos − − J 12 ν+ 12 + sin J 12 ν− 12 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −2] ET II 38(39)
0
x sin ax2 J ν (bx) dx
2 2
2
2
√ b b b νπ b νπ πb = 3/2 cos − − J 12 ν− 12 − sin J 12 ν+ 12 8a 4 8a 8a 4 8a 8a [a > 0, b > 0, Re ν > −4] ET II 34(14)
0
∞
5.
β2 1 cos J 0 (βx) sin αx2 x dx = 2α 4α
[α > 0,
β > 0]
MO 47
β2 1 sin J 0 (βx) cos αx2 x dx = 2α 4α
[α > 0,
β > 0]
MO 47
b2 νπ − 4a 2 a > 0,
b > 0,
xν+1 sin ax2 J ν (bx) dx =
0
739
∞
6. 0
xν+1 cos ax2 J ν (bx) dx =
bν cos ν+1 2 aν+1
bν 2ν+1 aν+1
sin
−2 < Re ν
0,
b > 0,
−1 < Re ν
0, b > 0,
c > 0,
Re ν > −2] ET II 51(26)
b 2 + c2 bc νπ − Jν 4a 2 2a [a > 0, b > 0,
c > 0,
Re ν > −1] ET II 51(27)
6.731
1.11 0
∞
x sin ax2 J ν bx2 J 2ν (2cx) dx
ac2 bc2 1 sin 2 J = √ ν 2 2 2 2 b 2 − a2 b −2a b −2a
ac bc 1 cos = √ Jν 2 − b2 2 − b2 2 2 a a 2 a −b
[0 < a < b,
Re ν > −1]
[0 < b < a,
Re ν > −1] ET II 356(41)a
740
Bessel Functions
2.
∞
10
6.732
x cos ax2 J ν bx2 J 2ν (2cx) dx
ac2 bc2 1 = √ cos 2 Jν 2 2 2 2 b 2 − a2 b −2a b −2 a
ac bc 1 = √ sin Jν a2 − b 2 a2 − b 2 2 a2 − b 2
0
0 < a < b,
Re ν > − 12
0 < b < a,
Re ν > − 12
ET II 356(42)a
∞
6.732
x2 cos
0
6.733
∞
1.
sin 0
∞
2. 0
x2 Y 1 (x) K 1 (x) dx = −a3 K 0 (a) 2a
[a > 0]
ET II 371(52)
a √ √ dx = π J0 [sin x J 0 (x) + cos x Y 0 (x)] a Y0 a 2x x
[a > 0] a √ √ dx = π J0 [sin x Y 0 (x) − cos x J 0 (x)] cos a Y0 a 2x x
ET II 346(51)
ET II 347(52) [a > 0] a √ √ πa J1 3. x sin a K1 a [a > 0] ET II 368(34) K 0 (x) dx = 2x 2 0 ∞ a √ √ πa Y1 K 0 (x) dx = − 4. x cos a K1 a [a > 0] ET II 369(35) 2x 2 0 ∞ √ dx 6.734 cos a x K ν (bx) √ x 0
a a π a a = √ sec(νπ) D ν− 12 √ D −ν− 12 − √ + D ν− 12 − √ D −ν− 12 √ 2 b 2b 2b 2b 2b 1 Re b > 0, |Re ν| < 2 ET II 132(27) 6.735 ∞ √ √ 1. [a > 0] x1/4 sin 2a x J − 14 (x) dx = πa3/2 J 34 a2 ET II 341(10)
∞
0
∞
2. 0
∞
3. 0
∞
4. 0
6.736 1.11
∞
0
2. 0
∞
√ √ x1/4 cos 2a x J 14 (x) dx = πa3/2 J − 34 a2
[a > 0]
ET II 341(12)
√ √ x1/4 sin 2a x J 34 (x) dx = πa3/2 J − 14 a2
[a > 0]
ET II 341(11)
√ √ x1/4 cos 2a x J − 34 (x) dx = πa3/2 J 14 a2
[a > 0]
ET II 341(13)
, √ √ + π 2 π J 0 a − sin a2 − Y 0 a2 x−1/2 sin x cos 4a x J 0 (x) dx = −2−3/2 π cos a2 − 4 4 ET II 341(18) [a > 0] + , √ √ π π J 0 a2 + cos a2 − Y 0 a2 x−1/2 cos x cos 4a x J 0 (x) dx = −2−3/2 π sin a2 − 4 4
[a > 0]
ET II 342(22)
6.737
Bessel and trigonometric functions and powers
∞
3.
−1/2
x
√ sin x sin 4a x J 0 (x) dx =
0
∞
4.
√ x−1/2 cos x sin 4a x J 0 (x) dx =
0
741
π π 2 cos a2 + J0 a 2 4 [a > 0] π π cos a2 − J 0 a2 2 4
ET II 341(16)
[a > 0] ET II 342(20) ∞ , √ √ + π 2 π J 0 a − cos a2 − Y 0 a2 x−1/2 sin x cos 4a x Y 0 (x) dx = 2−3/2 π 3 sin a2 − 4 4 0
5.
[a > 0] ∞
6.
∞
1. 0
∞
2. 0
4.
5.
√ sin a x2 + b2 b b π 2 2 2 2 √ J ν (cx) dx = J 12 ν J − 12 ν a− a −c a+ a −c 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 35(19) √ cos a x2 + b2 b b π √ a − a2 − c2 Y − 12 ν a + a2 − c 2 J ν (cx) dx = − J 12 ν 2 2 2 x2 + b2 [a > 0, Re b > 0, c > 0, a > c, Re ν > −1] ET II 39(44)
√ , , + a + a cos b a2 − x2 π √ J ν (cx) dx = J 12 ν b2 + c2 − b J 12 ν b 2 + c2 + b 2 2 2 a2 − x2 0 [c > 0, Re ν > −1] ET II 39(47) √ √ a cos a2 − x2 πa2ν+1
xν+1 √ I ν (x) dx = ET II 365(9) [Re ν > −1] 3 a2 − x2 0 ν+1 2 Γ ν+ 2 √ ∞ 2 2 sin a b + x √ xν+1 J ν (cx) dx b2 + x2 0 π 1 +ν ν 2 2 − 14 − 12 ν 0 < c < a, Re b > 0, −1 < Re ν < 12 b 2 c a −c = J −ν− 12 b a2 −c2 2 =0 0 < a < c, Re b > 0, −1 < Re ν < 12
3.
√ x−1/2 cos x cos 4a x Y 0 (x) dx
, √ + π 2 π = −2−3/2 π 3 cos a2 − J 0 a + sin a2 − Y 0 a2 4 4 ET II 347(56) [a > 0]
0
6.737
ET II 347(55)
a
ET II 35(20)
742
Bessel Functions
∞
6.
ν+1 cos
x 0
6.738
√ − 1 − 1 ν a x2 + b2 π 1 +ν ν 2 √ b 2 c a − c2 4 2 Y −ν− 12 b a2 − c2 J ν (cx) dx = − 2 x2 + b2 =
0 < c < a,
−1 < Re ν
0,
− 1 − 1 ν 2 1 +ν ν 2 b 2 c c − a2 4 2 K ν+ 12 b c2 − a2 π 0 < a < c,
1 2
1 −1 < Re ν < 2
Re b > 0,
ET II 39(45)
6.738
a ν+1
x
1. 0
− 1 ν− 3 π ν+ 3 2 2 a 2 b 1 + b2 2 4 J ν+ 32 a 1 + b2 sin b a − x J ν (x) dx = 2
ET II 335(19) [Re ν > −1] xν+1 cos a x2 + b2 J ν (cx) dx , − 1 ν− 3 + π ν+ 3 ν 2 ab 2 c a − c2 2 4 cos(πν) J ν+ 32 b a2 − c2 − sin (πν) Y ν+ 32 b a2 − c2 = 2 0 < c < a, Re b > 0, −1 < Re ν < − 12
∞
2. 0
=0
0 < a < c,
−1 < Re ν < − 12
Re b > 0,
ET II 39(43)
t
6.739
x 0
6.741 1.
2.
−1/2 cos
√ √ √t √ b t−x t 2 2 2 2 √ J 2ν a x dx = π J ν a + b + b Jν a +b −b 2 2 t−x 1 Re ν > − 2 EH II 47(7)
a a cos (μ arccos x) π √ J 12 (ν−μ) J ν (ax) dx = J 12 (μ+ν) 2 2 2 1 − x2 0 [Re(μ + ν) > −1, 1 a a cos [(ν + 1) arccos x] π √ cos J ν+ 12 J ν (ax) dx = a 2 2 1 − x2 0
3. 0
1
1
cos [(ν − 1) arccos x] √ J ν (ax) dx = 1 − x2
a
π sin J ν− 12 a 2
[Re ν > −1, a 2 [Re ν > 0,
a > 0]
a > 0]
a > 0]
ET II 41(54)
ET II 40(53)
ET II 40(52)a
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers 6.751
Notation: 1 =
, , 1 + 1 + (b + c)2 + a2 − (b − c)2 + a2 , 2 = (b + c)2 + a2 + (b − c)2 + a2 2 2
6.752
Combinations of Bessel, trigonometric, and exponential functions and powers
< 1 1 1 ax dx = √ √ e sin(bx) I 0 b + b 2 + a2 2 2b b2 + a2 0 [Re a > 0,
∞ 1 1 a 1 ax dx = √ √ e− 2 ax cos(bx) I 0 √ 2 2 2 2b a + b b + a2 + b2 0
1.
2.
∞
− 12 ax
743
b > 0]
ET I 105(44)
[Re a > 0,
3.
< ∞
10
e−bx cos(ax) J 0 (cx) dx =
0
b > 0] 1/2 2 (b2 + c2 − a2 ) + 4a2 b2 + b2 + c2 − a2 √ < 2 2 (b2 + c2 − a2 ) + 4a2 b2
ET I 48(38)
[c > 0] alternatively, with a and b interchanged, ∞ 2 − b 2 −ax e cos(bx) J 0 (cx) dx = 2 2 2 2 − 1 0 6.752 1.
∞
10
e 0
2.10 0
∞
−ax
dx = arcsin J 0 (bx) sin(cx) x
ET II 11(46)
[c > 0]
2c
a2 + (c + b)2 + a2 + (c − b)2
= arcsin
c 2
[Re a > |Im b|, c > 0] ET I 101(17) b − b2 − 21 b dx = (1 − r) = , e−ax J 1 (cx) sin(bx) x c c c2 a2 − , c > 0 b2 = 1 − r2 r2 ET II 19(15)
Notation: For integrals 6.752 3–6.752 5 we define the auxiliary functions , 1 + (a + ρ)2 + z 2 − (a − ρ)2 + z 2 2 , 1 + 2 (a) ≡ 1 (a, ρ, z) = (a + ρ)2 + z 2 + (a − ρ)2 + z 2 2
1 (a) ≡ 1 (a, ρ, z) =
when a ≥ 0, ρ ≥ 0, and z ≥ 0. ∞ √ π 3.10 e−zx J ν+1/2 (ax) J ν+1 (ρx) x dx 2 0
a ρ2 − 21 2ν+2 1 =a ρ 2 ( 2 − 2 ) 2 1 ρ − 1 1 2 ν+1 2 2 − a ρ = aν+1/2 2ν+2 2 2 2 2 − 1 2 −ν−3/2 −ν−1
[Re z > |Im a| + |Im ρ|]
744
4.
10
5.10
6.753 1.
8
2.
Bessel Functions
6.753
∞ π dx e−zx J ν+1/2 (ax) J ν (ρx) √ 2 0 x
1/2 1 1 1 ν+1/2 ν ρ d =a 2ν 2 2 2 1 − a / 2 0 2 a/2 dx ν > − 12 , Re z > |Im a| + |Im ρ| = a−ν−1/2 ρν x2ν √ 2 1−x 0 2
2 − a2 a − ∞ 2 − 2 a 2 1 a ρ dx −zx + arcsin e sin(ax) J 1 (ρx) 2 = x 2aρ 2 2 0 [Re z > |Im a| + |Im ρ|]
∞
∞
ϕ ν sin (xa sin ψ) −xa cos ϕ cos ψ e J ν (xa sin ϕ) dx = ν −1 tan sin(νψ) x 2 0 + π π, Re ν > −1, a > 0, 0 < ϕ < , 0 < ψ < ET II 33(10) 2 2 ∞ ϕ ν cos (xa sin ψ) −xa cos ϕ cos ψ e J ν (xa sin ϕ) dx = ν −1 tan cos(νψ) x 0 + 2 π, Re ν > 0, a > 0, 0 < ϕ, ψ < 2 ET II 38(35)
3.8 0
2(2a) Γ(ν + 32 )R−2ν−3 b cos(ν + 32 )ϕ + s sin(ν + 32 )ϕ xν+1 e−sx sin(bx) J ν (ax) dx = − √ π ν
Re s > |Im a| + |Im b|, 2 2 2 ϕ = arg s + a − b − 2ibs
Re ν > − 32 , 2 R4 = s2 + a2 − b2 + 4b2 s2 ,
∞
4.8 0
xν+1 e−sx cos(bx) J ν (ax) dx =
2(2a)ν √ Γ(ν + 32 )R−2ν−3 s cos(ν + 32 )ϕ − b sin(ν + 32 )ϕ , π Re ν > −1,
2 R4 = s2 + a2 − b2 + 4b2 s2 , 5.10 0
∞
Re s > |Im a| + |Im b|, 2 2 2 ϕ = arg s + a − b − 2ibs
xν e−ax cos ϕ cos ψ sin (ax sin ψ) J ν (ax sin ϕ) dx 1 −ν− 12 ν νΓ ν + 2 √ a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ =2 sin ν + 12 β π + , π π β a > 0, 0 < ϕ < , 0 < ψ < , Re ν > −1 ET II 34(12) tan = tan ψ cos ϕ 2 2 2
6.755
Combinations of Bessel, trigonometric, and exponential functions and powers
∞
6. 0
6.754 1. 2.
xν e−ax cos ϕ cos ψ cos (ax sin ψ) J ν (ax sin ϕ) dx 1 −ν− 12 ν νΓ ν + 2 √ =2 a−ν−1 (sin ϕ) cos2 ψ + sin2 ψ cos2 ϕ cos ν + 12 β π π 1 β a > 0, 0 < ϕ, ψ < , Re ν > − ET II 38(37) tan = tan ψ cos ϕ 2 2 2
2
√ b π − b2 8 e e I0 ET I 108(9) [b > 0] 3/2 8 2 0 2
2
2
2
∞ 2 2 a a π π a a 1 π −ax − + e cos x J 0 x dx = J0 cos −Y0 cos 4 2 16 16 4 16 16 4 0
∞
∞
3. 0
−x2
sin(bx) I 0 x2 dx =
1 e−ax sin x2 J 0 x2 dx = 4
[a > 0] MI 42 2
2
2
2
a a π π a a π − + J0 sin −Y0 sin 2 16 16 4 16 16 4 [a > 0]
6.755
∞
1. 0
2.
3.
4.
6.
7.
ν−1 √ 2 x−ν e−x sin 4a x I ν (x) dx = 23/2 a e−a W
1 3 1 1 2 − 2 ν, 2 − 2 ν
[a > 0,
MI 42
2 2a Re ν > 0]
ET II 366(14)
∞
√ 2 1 3 x−ν− 2 e−x cos 4a x I ν (x) dx = 2 2 ν−1 aν−1 e−a W − 32 ν, 12 ν 2a2 0 a > 0, Re ν > − 12 ∞ ν−1 Γ 3 − 2ν 2 √ −ν x 3/2 ea W 3 ν− 1 , 1 − 1 ν 2a2 x e sin 4a x K ν (x) dx = 2 a π 21 2 2 2 2 Γ 2 +ν 0 a > 0, 0 < Re ν < 34 ∞ √ Γ 12 − 2ν a2 3 −ν− 12 x ν−1 ν−1 1 e W 3 ν,− 1 ν 2a2 x e cos 4a x K ν (x) dx = 2 2 πa 2 2 Γ 2 + ν 0 a > 0, − 12 < Re ν < 14
ET II 366(16)
ET II 369(38)
ET II 369(42)
√ ∞ √ 3 1 πa Γ( + ν) Γ( − ν) 3 2 , + ; −2a x− 2 e−x sin 4a x K ν (x) dx = + ν, − ν; 2F 2 2 2 2−2 Γ + 12 0 ET II 369(39) [Re > |Re ν|]
√ ∞ √ π Γ( + ν) Γ( − ν) 1 1 −1 −x 2 x e cos 4a x K ν (x) dx = 2 F 2 + ν, − ν; , + ; −2a 2 2 2 Γ + 12 0 ET II 370(43) [Re > |Re ν|] ∞ √ 2 1 x−1/2 e−x cos 4a x I 0 (x) dx = √ e−a K 0 a2 2π 0 [a > 0] ET II 366(15)
5.
745
746
Bessel Functions
∞
8.
√ e cos 4a x K 0 (x) dx =
−1/2 x
x
0
∞
9. 0
6.756 1.
∞
∞
2.
∞
3.
1
1
∞
0
6.757 1.
∞
√ cos a x J ν (bx) dx
a ia 1 1 ia Γ ν+ =√ D −ν− 12 √ D −ν− 12 √ + D −ν− 12 − √ 2 2πb b b b a > 0, b > 0, Re ν > − 12 ET II 39(42)
√ −1/2 −a x
e
√
x−1/2 e−a
x
√ 1 sin a x J 0 (bx) dx = a I 14 2b
√ a I 1 cos a x J 0 (bx) dx = 2b − 4
a2 4b
a2 4b
K 14 + π |arg a| < , 4 2
2
a a K 14 4b 4b + π |arg a| < , 4
=2
∞
, b>0
ET II 12(49)
ET I 193(26)
e−bx cos a 1 − e−x J ν ae−x dx
0
∞
=
J ν (a) Γ(ν − b + 2n) Γ(ν + b) + (ν + 2n) J ν+2n (a) 2(−1)n ν + b n=0 Γ(ν − b + 1) Γ(ν + b + 2n + 1) [Re b > − Re ν]
ET II 11(40)
∞ (−1)n Γ(ν − b + 2n + 1) Γ (ν + b) (ν + 2n − 1) J ν+2n+1 (a) Γ(ν − b + 1) Γ(ν + b + 2n + 2) n=0
[Re b > − Re ν]
2.
, b>0
e−bx sin a 1 − e−x J ν ae−x dx
0
ET II 369(41)
√ x
x− 2 e−a
x
ET II 369(40)
√ sin a x J ν (bx) dx
a ia 1 i ia Γ ν+ =√ D −ν− 12 √ − D −ν− 12 − √ D −ν− 12 √ 2 2πb b b b [a > 0, b > 0, Re ν > −1] ET II 34(17)
0
4.
[a > 0]
√ x
x− 2 e−a
0
π a2 e K 0 a2 2
√ 2 1 x−1/2 e−x cos 4a x K 0 (x) dx = √ π 3/2 e−a I 0 a2 2
0
6.756
π 2
6.758 −π 2
ei(μ−ν)θ (cos θ)
ν+μ
ET I 193(27)
(λz)−ν−μ J ν+μ (λz) dθ < = π(2az)−μ (2bz)−ν J μ (az) J ν (bz); λ = 2 cos θ (a2 eiθ + b2 e−iθ ) < λ = 2 cos θ (a2 eiθ + b2 e−iθ ) [Re(ν + μ) > −1] EH II 48(12)
6.775
Bessel functions and the logarithm, or arctangent
747
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions
∞
6.761 0
√ −x J 2ν 2 b2 − a2 √ dx = cosh x cos (2a sinh x) J ν (be ) J ν be 2 b 2 − a2 x
=0
∞
6.762
=0 =− ∞
6.763 0
Re ν > −1]
[0 < b < a,
Re ν > −1]
cosh x sin (2a sinh x) J ν (bex ) Y ν be−x − Y ν (bex ) J ν be−x dx
0
[0 < a < b,
+ −1/2 1/2 , 2 cos(νπ) a2 − b2 K 2ν 2 a2 − b2 π
cosh x cos (2a sinh x) Y ν (bex ) Y ν be−x dx + −1/2 1/2 , 1 = − b 2 − a2 J 2ν 2 b2 − a2 2 + −1/2 1/2 , 2 = cos(νπ) a2 − b2 K 2ν 2 a2 − b2 π
ET II 359(10)
0 < a < b,
|Re ν|
0, b > 0]
∞ √ 2 ab x + a2 + x dx 6.774 ln √ = K 20 J 0 (bx) √ [Re a > 0, b > 0] 2 + a2 − x 2 + a2 2 x x 0 ∞ + , 1 6.775 x ln 1 + a2 + x2 − ln x J 0 (bx) dx = 2 1 − e−ab b 0 [Re a > 0, b > 0] 1.
ET II 10(27) ET II 19(11) ET II 19(12) MO 46
ET II 10(28) ET II 10(29)
ET II 12(55)
748
Bessel Functions
2 1 − a K 1 (ab) 6.776 J 0 (bx) dx = b b 0 ∞ 2 6.777 J 1 (tx) arctan t2 dt = − kei x x 0
∞
a2 x ln 1 + 2 x
6.776
[Re a > 0,
b > 0]
ET II 10(30) MO 46
6.78 Combinations of Bessel and other special functions
∞
6.781 0
1 si(ax) J 0 (bx) dx = − arcsin b
b a
=0
[0 < b < a] [0 < a < b] ET II 13(6)
6.782 ∞ √ e−z − 1 1. Ei(−x) J 0 2 zx dx = z 0 ∞ √ sin z 2. si(x) J 0 2 zx dx = − z 0 ∞ √ cos z − 1 3. ci(x) J 0 2 zx dx = z 0 ∞ √ dx Ei(−z) − C − ln z √ 4. Ei(−x) J 1 2 zx √ = x z 0 ∞ π √ dx − si(z) 5. si(x) J 1 2 zx √ = − 2 √ x z 0 ∞ √ dx ci(z) − C − ln z √ 6. ci(z) J 1 2 zx √ = x z 0 ∞ √ C + ln z − e2 Ei(−z) 7. Ei(−x) Y 0 2 zx dx = πz 0 6.783 1. 2. 3.
NT 60(4) NT 60(6) NT 60(5) NT 60(7) NT 60(9) NT 60(8) NT 63(5)
2
b 2 x si a2 x2 J 0 (bx) dx = − 2 sin [a > 0] 2 b 4a 0 2 ∞ 2 2 b 2 x ci a x J 0 (bx) dx = 2 1 − cos [a > 0] 2 b 4a 0 2
2
∞ b b 1 ci a2 x2 J 0 (bx) dx = + ln + 2C ci b 4a2 4a2 0
∞
∞
4. 0
6.784 1. 0
∞
2
b 1 π 2 2 si a x J 1 (bx) dx = − − si b 4a2 2
xν+1 [1 − Φ(ax)] J ν (bx) dx = a−ν
ET II 13(7)a ET II 13(8)a
[a > 0]
ET II 13(8)a
[a > 0]
ET II 20(25)a
2
Γ ν + 32 b b2 1 1 1 1 exp − M 2 2 2 ν+ 2 , 2 ν+ 2 b2 Γ(ν + 2) 8a 4a + , π |arg a| < , b > 0, Re ν > −1 4 ET II 92(22)
6.792
Integration of Bessel functions
∞
2.
x [1 − Φ(ax)] J ν (bx) dx = ν
0
749
2
1 b 2 a 2 −ν Γ ν + 12 b2 1 1 1 1 M exp − 2 ν− 4 , 2 ν+ 4 π b3/2 Γ ν + 32 8a2 4a2 1 π |arg a| < , Re ν > − , b > 0 4 2 ET II 92(23)
∞ exp a2 − x 2x a π 5/2 2 2 sec(νπ) [J ν (a)] + [Y ν (a)] 6.785 1−Φ √ K ν (x) dx = x 4 2x 0 Re a > 0, |Re ν| < 12 ET II 370(46) ∞ 6.786 xν−2μ+2n+2 ex2 Γ μ, x2 Y ν (bx) dx 0 3 3 2
2
b b nΓ 2 −μ+ν +n Γ 2 −μ+n exp = (−1) W μ− 12 ν−n−1, 12 ν b Γ(1 − μ) 8 4 n is an integer, b > 0, Re(ν − μ + n) > − 32 , Re(−μ + n) > − 32 , Re ν < 12 − 2n
ET II 108(2) ∞
6.787 0
1
xν+2n− 2 J ν (bx) dx = 0 B(a + x, a − x)
π ≤ b < ∞,
−1 < Re ν < 2a − 2n −
7 2
ET II 92(21)
6.79 Integration of Bessel functions with respect to the order 6.791
∞
1. −∞ ∞
2. −∞ ∞
3. −∞
K ix+iy (a) K ix+iz (b) dx = π K iy−iz (a + b)
[|arg a| + |arg b| < π]
J ν−x (a) J μ+x (a) dx = J μ+ν (2a)
[Re(μ + ν) > 1]
ET II 382(21) ET II 379(1)
J κ+x (a) J λ−x (a) J μ+x (a) J ν−x (a) dx =
Γ(κ + λ + μ + ν + 1) Γ (κ +⎛ λ + 1) Γ(λ + μ + 1) Γ(μ + ν + 1) Γ (ν + κ + 1) × 4F 5 ⎝
κ+λ+μ+ν κ+λ+μ+ν+1 κ+λ+μ+ν+1 κ+λ+μ+ν , , + 1, + 1; 2 2 2 2 ⎞
κ + λ + μ + ν + 1, κ + λ + 1, λ + μ + 1, μ + ν + 1, ν + κ + 1; −4a2 ⎠ [Re(κ + λ + μ + ν) > −1] 6.792
∞
1. −∞
ET II 379(3)
eπx K ix+iy (a) K ix+iz (b) dx = πe−πz K i(y−z) (a − b) [a > b > 0]
ET II 382(22)
750
Bessel Functions
∞
2.
e −∞
ix
K ν+ix (α) K ν−ix (β) dx = π
αeρ + β α + βeρ
6.793
ν K 2ν
α2 + β 2 + 2αβ cosh [|arg α| + |arg β| + |Im | < π] ET II 382(23)
∞
3. −∞
e(π−γ)x K ix+iy (a) K ix+iz (b) dx = πe−βy−αz K iy−iz (c)
[0 < γ < π,
a > 0,
b > 0,
c > 0,
α, β, γ—the angles of the triangle with sides a, b, c] ET II 382(24), EH II 55(44)a
2ν ∞ h (2) (2) (2) e−cxi H ν−ix (a) H ν+ix (b) dx = 2i H 2ν (hk) k −∞ <
0,
c is real]
ET II 380(11)
a−μ−x b−ν+x ecxi J μ+x (a) J ν−x (b) dx % & 12 μ+ 12 ν
+ , +c c ,1/2 2 cos 2c 2 − 12 ci 2 12 ci = exp (ν − μ)i J μ+ν 2 cos +b e a e 1 1 2 2 a2 e− 2 ci + b2 e 2 ci [a > 0, b > 0, |c| < π, Re(μ + ν) > 1] =0 [a > 0,
b > 0,
|c| ≥ π,
Re(μ + ν) > 1]
EH II 54(41), ET II 379(2)
6.793 1.
∞
e−cxi [J ν−ix (a) Y ν+ix (b) + Y ν−ix (a) J ν+ix (b)] dx = −2 −∞ < < 1 1 1 1 k = ae− 2 c + be 2 c h = ae 2 c + be− 2 c ,
∞
2.
e
−cxi
−∞
∞
−∞
6.794
∞
1. 2.
α, β, γ ∈ R,
ae 2 c + be− 2 c , 1
1
k=
α, β > 0,
ae− 2 c + be 2 c 1
1
σ = α2 + β 2 − 2αβ cosh γ,
K ix (a) K ix (b) cosh[(π − ϕ)x] dx =
∞
cosh 0
Im c = 0]
ET II 380(9)
[a, b > 0,
Im c = 0]
ET II 380(10)
eiγx sech(πx) [J −ix (α) J ix (β) − J ix (α) J −ix (β)] dx = 2i H(σ) sign(β − α) J 0 σ 1/2
0
[a, b > 0,
2ν h [J ν−ix (a) J ν+ix (b) − Y ν−ix (a) Y ν+ix (b)] dx = 2 Y 2ν (hk) k < < h=
3.10
2ν h J 2ν (hk) k
π π x K ix (a) dx = 2 2
H(σ) the Heaviside step function
π K0 a2 + b2 − 2ab cos ϕ 2 [a > 0]
EH II 55(42) ET II 382(19)
6.794
Integration of Bessel functions
∞
3.
cosh(x) K ix+ν (a) K −ix+ν (a) dx = 0
5. 6.
+ , π K 2ν 2a cos 2 2
[2|arg a| + |Re | < π] π x J ix (a) dx = 2 sin a sech [a > 0] 2 −∞ ∞ π x J ix (a) dx = −2i cos a cosech [a > 0] 2 −∞ ∞ 2 2 sech(πx) [J ix (a)] + [Y ix (a)] dx = − Y 0 (2a) − E0 (2a)
4.
751
ET II 383(28)
∞
ET II 380(6) ET II 380(7)
0
∞
7. 0
∞
8.
x tanh(πx) K ix (β) K ix (α) dx = 0
9.
10.
π πa x K ix (a) dx = x sinh 2 2
[a > 0]
ET II 380(12)
[a > 0]
ET II 382(20)
π exp(−β − α) αβ 2 α+β
[|arg β| < π, |arg α| < π] ET II 175(4)
π α α2 x sinh(πx) K 2ix (α) K ix (β) dx = 5/2 √ exp −β − 8β 2 β 0 + π, β > 0, |arg α| < ET II 175(5) 4 ∞ x sinh(πx) π2 I n (β) K n (α) K ix (α) K ix (β) dx = [0 < β < α; n = 0, 1, 2, . . .] 2 2 x +n 2 0 2 π I n (α) K n (β) [0 < α < β; n = 0, 1, 2, . . .] = 2
∞
∞
11. 0
3/2
γ α β αβ π2 exp − + + 2 x sinh(πx) K ix (α) K ix (β) K ix (γ) dx = 4 2 +β α γ π |arg α| + |arg β| < , 2
, γ>0 ET II 176(9)
%
& ∞ π (α + β) γ 2 + 4αβ π2 γ √ x K 12 ix (α) K 12 ix (β) K ix (γ) dx = exp − x sinh 2 2 αβ 2 γ 2 + 4αβ 0 [|arg α| + |arg β| < π, γ > 0]
12.
ET II 176(8)
13. 0
ET II 176(10) ∞
x sinh(πx) K 12 ix+λ (α) K 12 ix−λ (α) K ix (γ) dx = 0 2
π γ 22λ+1 α2λ z z = γ 2 − 4α2
=
[0 < γ < 2α] , + 2λ (γ + z) + (γ − z)2λ [0 < 2α < γ]
ET II 176(11)
752
Bessel Functions
6.795 1.
∞
cos(bx) K ix (a) dx =
0
0
∞
3.
x sin(ax) K ix (b) dx = 0 −∞
4. −∞
x sin
0
1. 2. 3. 4.
∞
−1/2 1 1 − a2 4
πb sinh a exp (−b cosh a) 2
[|a| < 1] , b>0
ET II 175(1)
[0 < a < b;
n = 0, 1, . . .]
[0 < b < a;
n = 0, 1, . . .]
1 b2 π 3/2 b πx K 12 ix (a) K ix (b) dx = √ exp −a − 2 8a 2a + π |arg a| < , 2
ET II 382(25)
, b>0
ET II 175(6)
1
e 2 πx cos(bx) J ix (a) dx = −i exp (ia cosh b) [a > 0, −∞ sinh(πx)
∞ 1 π πx K ix (a) dx = cos (a sinh b) cos(bx) cosh 2 2 0
∞ 1 π πx K ix (a) dx = sin (a sinh b) sin(bx) sinh 2 2 0 ∞ b π2 2 cos(bx) cosh(πx) [K ix (a)] dx = − Y 0 2a sinh 4 2 0
b > 0]
ET II 380(8) EH II 55(47) EH II 55(48)
[a > 0,
∞ b π2 2 J 0 2a sinh sin(bx) sinh(πx) [K ix (a)] dx = 4 2 0
b > 0]
ET II 383(27)
[a > 0,
b > 0]
ET II 382(26)
5.
ET II 380(4)
+ π |Im a| < , 2
sin[(ν + ix)π] K ν+ix (a) K ν−ix (b) dx = π 2 I n (a) K n+2ν (b) n + ν + ix = π 2 K n+2ν (a) I n (b)
∞
5.
6.796
, a>0 EH II 55(46), ET II 175(2)
J x (ax) J −x (ax) cos(πx) dx =
+ π |Im b| < , 2
π −a cosh b e 2
∞
2.
6.795
6.797 1.
∞
0
(2)
(2)
xeπx sinh(πx) Γ(ν + ix) Γ(ν − ix) H ix (a) H ix (b) dx √ = i2ν π Γ 12 + ν (ab)ν (a + b)−ν K ν (a + b) [a > 0,
2. 0
∞
b > 0,
Re ν > 0]
ET II 381(14)
iπ 3/2 2ν (2) (2) (b−a)−ν H (2) xeπx sinh(πx) cosh(πx) Γ(ν +ix) Γ(ν −ix) H ix (a) H ix (b) dx = 1 ν (b−a) Γ − ν 2 0 < a < b, 0 < Re ν < 12 ET II 381(15)
6.812
Struve functions
∞
3.
πx
xe
sinh(πx) Γ
0
ν + ix 2
753
ν − ix (2) (2) Γ H ix (a) H ix (b) dx 2
− 1 ν a2 + b 2 = iπ22−ν (ab)ν a2 + b2 2 H (2) ν [a > 0,
4.
∞
11 0
b > 0,
Re ν > 0]
ET II 381(16)
x sinh(πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx = 2λ−1 π 3/2 (ab)λ (a + b)−λ Γ λ + 12 K λ (a + b) [|arg a| < π,
Re λ > 0,
b > 0] ET II 176(12)
∞
5.
x sinh(2πx) Γ(λ + ix) Γ(λ − ix) K ix (a) K ix (b) dx =
0
∞
6. 0
∞
7. 0
λ
5 2
ab 2λ π 1 K λ (|b − a|) |b − a| Γ 2 − λ a > 0, 0 < Re λ < 12 , b > 0
ET II 176(13)
ab √ x sinh(πx) Γ λ + 12 ix Γ λ − 12 ix K ix (a) K ix (b) dx = 2π 2 a2 + b 2 K 2λ 2 a2 + b 2 + , π |arg a| < , Re λ > 0, b > 0 2 x tanh(πx) K ix (a) K ix (b) 1 3 1 3 1 dx = 2 Γ 4 + 2 ix Γ 4 − 2 ix
ET II 177(14)
πab 2 + b2 exp − a a2 + b 2 + π |arg a| < , 2
, b>0 ,
ET II 177(15)
6.8 Functions Generated by Bessel Functions 6.81 Struve functions 6.811
∞
1.
Hν (bx) dx = −
0
∞
2.
Hν 0
Hν−1 0
a x
∞
3.
2
cot
νπ 2
b
Hν (bx) dx = −
√ J 2ν 2a b
b
a > 0,
b > 0,
b > 0]
ET II 158(1)
Re ν > − 32
ET II 170(37)
2
a x
[−2 < Re ν < 0,
Hν (bx)
√ 1 dx = − √ J 2ν−1 2a b x a b
a > 0,
b > 0,
Re ν > − 12
ET II 170(38)
6.812
1. 0
∞
H1 (bx) dx π [I 1 (ab) − L1 (ab)] = 2 2 x +a 2a
[Re a > 0,
b > 0]
ET II 158(6)
754
Functions Generated by Bessel Functions
∞
2. 0
6.813
b cot νπ Hν (bx) π 3 − ν 3 + ν a2 b 2 2 ; ; dx = − (ab) + 1; L F ν 2 1 νπ x2 + a2 1 − ν2 2 2 2 2a sin 2 [Re a > 0, b > 0, |Re ν| < 2] ET II 159(7)
6.813
∞
s−1
x
1. 0
∞
2.
2s−1 Γ s+ν s+ν 2 π Hν (ax) dx = s 1 tan 1 2
a Γ 2ν − 2s + 1 3 , 1 − Re ν a > 0, −1 − Re ν < Re s < min 2
x−ν−1 Hν (x) dx =
0
∞
3.
x−μ−ν Hμ (x) Hν (x) dx =
0
Re ν > − 32
1
√ 2−μ−ν π Γ(μ + ν) Γ μ + 12 Γ ν + 12 Γ μ + ν + 12
xν+1 Hν (ax) dx =
1 Hν+1 (a) a
x1−ν Hν (ax) dx =
aν−1 1 √ 1 − a Hν−1 (a) ν−1 πΓ ν + 2 2
0 1
5. 0
ET II 383(2)
[Re(μ + ν) > 0]
4.
2−ν−1 π Γ(ν + 1)
WA 429(2), ET I 335(52)
a > 0,
WA 435(2), ET II 384(8)
Re ν > − 32
[a > 0] 6.814 1.
∞
(x2 +
0
6.815
1.
1
1−μ a2 )
dx =
ET II 159(8)
√ 1 x 2 ν (1 − x)μ−1 Hν a x dx = 2μ a−μ Γ(μ) Hμ+ν (a)
0
2.
ET II 158(3)a
2μ−1 πaμ+ν b−μ [I −μ−ν (ab) − Lμ+ν (ab)] Γ(1 − μ) cos[(μ + ν)π] Re a > 0, b > 0, Re ν > − 32 , Re(μ + ν) < 12 , Re(2μ + ν) < 32
xν+1 Hν (bx)
ET II 158(2)a
Re ν > − 32 , Re μ > 0 ET II 199(88)a
1 √ 1 3 3 a2 B(λ, μ)aν+1 3 , ν + , λ + μ; − xλ− 2 ν− 2 (1 − x)μ−1 Hν a x dx = ν √ 1, λ; 2F 3 2 2 4 2 π Γ ν + 32 0 [Re λ > 0,
Re μ > 0]
ET II 199(89)a
6.82 Combinations of Struve functions, exponentials, and powers 6.821 1.6 0
∞
−n− 12 1 1 e−αx H−n− 12 (βx) dx = (−1)n β n+ 2 α + α2 + β 2 2 α + β2 [Re α > |Im β|]
ET I 206(6)
6.831
Struve and trigonometric functions
2.
∞
6 0
−n− 12 1 1 e−αx L−n− 12 (βx) dx = β n+ 2 α + α2 − β 2 2 α − β2 √
∞
755
2 e−αx H0 (βx) dx = π
α2 +β 2 +β α
ln
[Re α > |Re β|]
ET I 208(26)
[Re α > |Im β|] ET II 205(1) α2 + β 2 β ∞ arcsin α 2 −αx 4. e L0 (βx) dx = [Re α > |Re β|] ET II 207(18) π α2 + β 2 0 ∞ a a , + a a π cosec(νπ) sinh I ν+ 12 − cosh I −ν− 12 6.822 e(ν+1)x Hν (a sinh x) dx = a 2 2 2 2 0 [Re a > 0, −2 < Re ν < 0] 3.
0
ET II 385(11)
6.823
∞
λ −αx
x e
1. 0
λ+ν+3 3 3 b2 bν+1 Γ(λ + ν + 2) λ+ν
3 F 2 1, + 1, ; ,ν + ;− 2 Hν (bx) dx = √ 3 2 2 2 2 a 2ν aλ+ν+2 π Γ ν + 2 [Re a > 0, b > 0, Re(λ + ν) > −2] ET II 161(19)
ν
∞
2.
ν −αx
x e 0
β
Γ(2ν + 1) α (2β)ν Γ ν + 12 β −ν− 12 Lν (βx) dx = P −ν− 1 2ν+1 − √ 2 1 1 2 α π ν+ π α − β2 α β 2 − α2 2 4 2 Re α > |Re β|, Re ν > − 12 ET I 209(35)a
6.824 1.
∞
ν −at
t e
√ L2ν 2 t dt =
0
2. 6.825
∞
√ t dt =
1 a2ν+1
1 a
e Φ
1 √ a
MI 51
1 1 1 1 − 2ν, ea γ MI 51 2ν+1 2 a Γ 2 − 2ν a 0
∞ β ν+1 Γ 12 + 2s + ν2 2 2 3 β2 ν+s+1 3 ; , ν + ; − xs−1 e−α x Hν (βx) dx = ν+1 √ ν+s+1 1, 2F 2 2 2 2 4α2 2 πα Γ ν + 32+ 0 π, Re s > − Re ν − 1, |arg α| < 4
tν e−at L−2ν
1
ET I 335(51)a, ET II 162(20)
6.83 Combinations of Struve and trigonometric functions
6.831
∞
x−ν sin(ax) Hν (bx) dx = 0
0
=
√
π2
−ν −ν
b
ν− 12 b 2 − a2 Γ ν + 12
0 < b < a,
Re ν > − 12
0 < a < b,
Re ν > − 12
ET II 162(21)
756
Functions Generated by Bessel Functions
∞√
6.832 0
6.832
2
√ 2 2 √ a a −3/2 x sin(ax) H 14 b x dx = −2 π 2 Y 14 b 4b2 [a > 0]
ET I 109(14)
6.84–6.85 Combinations of Struve and Bessel functions
∞
6.841
Hν−1 (ax) Y ν (bx) dx = −aν−1 b−ν
0
=0
∞
6.842 0
6.843 1.
∞
0
∞
2. 0
4 K [H0 (ax) − Y 0 (ax)] J 0 (bx) dx = π(a + b)
√ 1 J 2ν a x Hν (bx) dx = − Y ν b
a2 4b
|Re ν|
0,
ET II 114(36)
b > 0] b > 0,
ET II 15(22)
−1 < Re ν
0,
b > 0,
Re ν > −1] ET II 168(27)
∞
6.844 0
6.845 1.
∞
0
√ √ √ μ−ν μ−ν π J μ a x − sin π Y μ a x K μ a x Hν (bx) dx cos 2 2 2
2
a a 1 = 2 W 12 ν, 12 μ W − 12 ν, 12 μ a 2b 2b + , π |arg a| < , b > 0, Re ν > |Re μ| − 2 ET II 169(35) 4 + a a , √ 4 cos(νπ) K 2ν 2 ab H−ν − Y −ν J ν (bx) dx = x x πb |arg a| < π,
b > 0,
|Re ν|
0, b > 0, − 32 < Re ν < 0 ∞
6.846 0
∞
6.847 0
ET II 170(39)
2
√ √ 2 a 1 K 2ν 2a x + Y 2ν 2a x Hν (bx) dx = J ν π b b a > 0,
b > 0,
|Re ν|
0, Re k > 0,
1 2
ET II 169(30)
− 12 < Re ν < 2
ET II 384(5)a, WA 467(8)
6.852
Combinations of Struve and Bessel functions
6.848 1.
∞
x [I ν (ax) − L−ν (ax)] J ν (bx) dx =
0
2 a ν−1 1 cos(νπ) 2 2 π b a + b Re a > 0,
757
b > 0,
−1 < Re ν < − 12
ET II 74(12) ∞
2.
x [H−ν (ax) − Y −ν (ax)] J ν (bx) dx = 2
0
cos(νπ) ν−1 1 b aν π a+ b |arg a| < π,
− 12 < Re ν,
b>0
ET II 73(5)
6.849 1.
∞
x K ν (ax) Hν (bx) dx = a−ν−1 bν+1
0
∞
2. 0
6.851 1. 0
a2
1 + b2
Re a > 0,
Re ν > − 32
2μ + (z − b)2μ 2 −μ−1 −2μ (z + b) sec(μπ), x [K μ (ax)] H0 (bx) dx = −2 πa bz z = 4a2 + b2 Re a > 0, b > 0, |Re μ| < 32
+ ,2 + x J 12 ν (ax) − Y
∞
,2 1 (ax) Hν (bx) dx 2ν =0 1 4 √ = 2 πb b − 4a2
b > 0,
ET II 164(12)
ET II 166(18)
0 < b < 2a,
− 32 < Re ν < 0
0 < 2a < b,
− 32 < Re ν < 0
ET II 164(7) ∞
2.
2 2 xν+1 [J ν (ax)] − [Y ν (ax)] Hν (bx) dx
0
=0 −ν− 12 23ν+2 a2ν b−ν−1 2 1 b − 4a2 = √ πΓ 2 −ν
0 < b < 2a,
− 34 < Re ν < 0
0 < 2a < b,
− 34 < Re ν < 0
ET II 163(6)
6.852 1.
∞
x1−μ−ν J ν (x) Hμ (x) dx =
0
2.
(2ν − 1)2−μ−ν (μ + ν − 1) Γ μ + 12 Γ ν + 12 Re ν > 12 ,
Re(μ + ν) > 1
ET II 383(4)
∞
xμ−ν+1 Y μ (ax) Hν (bx) dx
0
=0 =
ν−μ−1 21+μ−ν aμ b−ν 2 b − a2 Γ(ν − μ)
0 < b < a,
Re(ν − μ) > 0,
− 32 < Re μ
0,
− 32 < Re μ
0, b > 0, Re ν > − 32 , Re(μ + ν) > − 32 ET II 165(13)
x1−μ [sin (μπ) J μ+ν (ax) + cos(μπ) Y μ+ν (ax)] Hν (bx) dx
0
=0
μ−1 b ν b 2 − a2 = μ−1 μ+ν 2 a Γ(μ)
0 < b < a,
1 < Re μ < 32 ,
Re ν > − 32 ,
Re(ν − μ)
− 32 ,
Re(ν − μ)
− 32 , − Re ν − 52 < Re(λ − μ) < 1 ET II 76(21)
0
6.853
∞
3.
Re a > 0,
b > 0,
⎛
1
xλ+ 2 [Hμ (ax) − Y μ (ax)] J ν (bx) dx
0
! ⎞ !1−μ μ μ ! , 1 − , 1 + 2 2 1 cos(μπ) 3 ⎜b ! 2 ⎟ b−λ− 2 G 23 = 2λ+ 2 ⎠ 33 ⎝ 2 ! 3 π2 a ! + λ+ν , 1−μ , 3 + λ−ν ! 2 2 4 2 4 |arg a| < π, Re(λ + μ) < 1, Re(λ + ν) + 32 > |Re μ| ET II 73(6) ⎛
2
b > 0, 4.
∞√
0
0
∞
6.
μ−ν+1
∞
xH
1. 0
1 b2 a b 2 1 μ−ν+1 x [I μ (ax) − Lμ (ax)] J ν (bx) dx = √ F 1, ; ν − μ + ; − 2 2 2 a π Γ ν − μ + 12 1 −1 < 2 Re μ + 1 < Re ν + 2 , Re a > 0, b > 0 ET II 74(13) x
1 2
μ−ν+1 μ−1 ν−2μ−1
0
6.854
|Re ν|
0, b > 0, +
1 2ν
1 b2 2μ−ν+1 a−μ−1 bν−1 1 F 1, + μ; + ν; − 2 [I μ (ax) − L−μ (ax)] J ν (bx) dx = 1 2 2 a Γ 2 − μ Γ 12 + ν 1 Re a > 0, Re ν > − 2 , Re μ > −1, b > 0 ET II 75(18)
2
ax
K ν (bx) dx =
Γ
1
2ν + 1 1 1− 2 2 ν aπ
S
− 12 ν−1, 12 ν
b2 4a [a > 0,
Re b > 0,
Re ν > −2] ET II 150(75)
6.857
Combinations of Struve and Bessel functions
∞
2
x H 12 ν ax
2. 0
1 J ν (bx) dx = − Y 2a
1 2ν
b2 4a
a > 0,
759
b > 0,
−2 < Re ν
0, |Re ν| < 12
∞
2. 0
1 + a a , √ √ 3 aν+ 2 2 2ab K 2ν+1 2ab I ν+ 12 − Lν+ 12 J ν (bx) dx = 2 √ ν+1 J 2ν+1 x x πb Re a > 0, b > 0, −1 < Re ν < 12
ET II 74(8) ∞
3.
1
x2ν+ 2
+ a a , Hν+ 12 − Y ν+ 12 J ν (bx) dx x x
√ √ 1 1 1 = −25/2 π −3/2 aν+ 2 b−ν−1 sin(νπ) K 2ν+1 2abe 4 πi K 2ν+1 2abe− 4 πi |arg a| < π, b > 0, −1 < Re ν < − 16 ET II 74(9)
0
∞
6.856 0
2
√ √ 1 a x Y ν a x K ν a x Hν (bx) dx = 2 exp − 2b 2b+ b > 0,
|arg a|
− 32
,
ET II 169(32)
6.857
∞
1.
x exp 0
a2 x2 8
K 12 ν
k = 14 ν, 2. 0
∞
a2 x2 8
m=
1 2
Hν (bx) dx
2
2
νπ 1
b b 2 − ν −1 ν −1 2 2 Γ − ν exp =√ a b cos W k,m 2 2 2a2 a2 π + 14 ν |arg a| < 34 π, b > 0, − 32 < Re ν < 0 ET II 167(24)
1 2 2 1 a x Hν (bx) dx xσ−2 exp − a2 x2 K μ 2 2 √ + μ Γ ν+σ −μ π −ν−σ ν+1 Γ ν+σ 2 2 b = ν+2 a 2 Γ 32 Γ ν + 32 Γ ν+σ 2
ν+σ 3 3 ν+σ b2 ν+σ + μ, − μ; , ν + , ;− 2 × 3 F 3 1, 2 2 2 2 , 2 4a + π b > 0, |arg a| < , Re(σ + ν) > 2|Re μ| ET II 167(23) 4
760
Functions Generated by Bessel Functions
6.861
6.86 Lommel functions 6.861
∞
1.
λ−1
x
S μ,ν (x) dx =
Γ
1
2 (1
0
6.862
u
1.
+ λ + μ) Γ 12 (1 − λ − μ) Γ 12 (1 + μ + ν) Γ 12 (1 + μ − ν) 22−λ−μ Γ 12 (ν − λ) + 1 Γ 1 − 12 (λ + ν) ET II 385(17) − Re μ < Re λ + 1 < 52
√ 1 1 xλ− 2 μ− 2 (u − x)σ−1 s μ,ν a x dx
0
aμ+1 uλ+σ Γ(λ + 1) (μ − ν + 1) (μ + ν + 1) Γ(λ + σ + 1)
a2 u μ−ν+3 μ+ν+3 , , λ + σ + 1; − × 2 F 3 1, 1 + λ; 2 2 4 [Re λ > −1, Re σ > 0] ET II 199(92) 1 1 ∞ √ √ B μ, 12 (1 − λ − ν) − μ u 2 μ+ 2 ν 1 ν μ−1 2 x (x − u) s λ,ν a x dx = S λ+μ,μ+ν a u μ a u √ ! ! !arg a u ! < π, 0 < 2 Re μ < 1 − Re(λ + ν) ET II 211(71) = Γ(σ)
2.
6.863
∞√
0
6.864 0
6.865
xe−αx s μ, 14
x2 2
∞
2
√ α 3 dx = 2−2μ−1 α Γ 2μ + S −μ−1, 14 2 2 Re α > 0, Re μ > − 34
ET I 209(38)
exp[(μ + 1)x] s μ,ν (a sinh x) dx = 2μ−2 π cosec(μπ) Γ() Γ(σ) + a a a a , × I Iσ − I − I −σ 2 2 2 2 2 = μ + ν + 1, 2σ = μ − ν + 1 [a > 0, −2 < Re μ < 0] ET II 386(22) ∞√ μ−ν 1 B 14 − μ+ν 2 , 4 − 2 sinh x cosh(νx) S μ, 12 (a cosh x) dx = S μ+ 12 ,ν (a) √ μ+ 3 a2 2 0 |arg a| < π, Re μ + |Re ν| < 12 ET II 388(31)
6.866
∞
1.
x−μ−1 cos(ax) s μ,ν (x) dx
0
=0 =2 2. 0
∞
μ− 12
√ πΓ
μ+ν+1 2
1 μ+ 1 μ− 1 μ−ν+1 1 − a2 2 4 P ν− 12 (a) Γ 2 2
[a > 1] [0 < a < 1] ET II 386(18)
1 μ− 1 μ− 1 √ μ+ν μ−ν 2 −μ −μ− 12 a − 1 2 4 P ν− 12 (a) x sin(ax) S μ,ν (x) dx = 2 πΓ 1− Γ 1− 2 2 2 [a > 1, Re μ < 1 − |Re ν|]
ET II 387(23)
6.871
6.867
Thomson functions
761
π/2
1.
cos(2μx) S 2μ−1,2ν (a cos x) dx 0
a a a a , π22μ−3 a2μ cosec(2νπ) + J μ+ν Y μ−ν − J μ−ν Y μ+ν Γ(1 − μ − ν) Γ (1 − μ + ν) 2 2 2 2 [Re μ > −2, |Re ν| < 1] ET II 388(29) π/2 a a cos [(μ + 1) x] s μ,ν (a cos x) dx = 2μ−2 π Γ() Γ(σ) J Jσ 2 2 0 2 = μ + ν + 1, 2σ = μ − ν + 1 [Re μ > −2] ET II 386(21) =
2.
π/2
6.868 0
6.869
∞
1.
1−μ−ν
x 0
π π cos(2μx) π22μ−1 S 2μ,2ν (a sec x) dx = W μ,ν aei 2 W μ,ν ae−i 2 cos x a [|arg a| < π, Re μ < 1]
√ ν−1 12 (μ+ν−1) μ+ν−1 πa Γ(1 − μ − ν) 2 a J ν (ax) S μ,−μ−2ν (x) dx = − 1 P μ+ν (a) 1 2μ+2ν Γ ν + 2 a > 1, Re ν > − 12 , Re(μ + ν) < 1 ET II 388(28)
∞
2. 0
x−μ J ν (ax) s ν+μ,−ν+μ+1 (x) dx μ = 2ν−1 Γ(ν)a−ν 1 − a2 =0
ET II 388(30)
0 < a < 1, 1 < a,
Re μ > −1,
Re μ > −1,
−1e < Re ν < −1 < Re ν < 32
3 2
ET II 388(28)
∞
3. 0
2
2 b 1 1 1 Γ μ + ν + 1 Γ μ − ν + 1 S −μ−1, 12 ν x K ν (bx) s μ, 12 ν ax dx = 4a 2 2 4a Re μ > 12 |Re ν| − 2, a > 0, Re b > 0 ET II 151(78)
6.87 Thomson functions 6.871
1/2 β4 + 1 + β2 e−βx ber x dx = 2 (β 4 + 1) 0 1/2 ∞ β4 + 1 − β2 e−βx bei x dx = 2 (β 4 + 1) 0
1.
2.
∞
ME 40
ME 40
762
6.872
Functions Generated by Bessel Functions
∞
1. 0
∞
2. 0
3. 4. 5. 6. 7.
3. 4.
⎡
MI 49
1 3ν + π 2β 4 ⎤
1 3ν + 6 ⎦ 1 − J 12 (ν+1) + π sin 2β 2β 4
√ 1 e−βx beiν 2 x dx = 2β
π⎣ J 12 (ν−1) β
1 2β
sin
√ 1 1 e−βx ber 2 x dx = cos β β 0 ∞ √ 1 1 e−βx bei 2 x dx = sin β β 0 ∞ √ 1 1 1 1 1 e−βx ker 2 x dx = − cos ci + sin si 2β β β β β 0 ∞ √ 1 1 1 1 1 −βx e kei 2 x dx = − sin ci − cos si 2β β β β β 0
∞ √ √ 2 3νπ 2 1 Jν + e−βx berν 2 x beiν 2 x dx = sin 2β β β 2 0
0
2.
MI 49
6.873
1.
⎡
1 1 3νπ π⎣ J 12 (ν−1) + cos β 2β 2β 4 ⎤
1 1 3ν + 6 ⎦ + π − J 12 (ν+1) cos 2β 2β 4
√ 1 e−βx berν 2 x dx = 2β
∞
6.874
∞
2 √ √ 2 1 berν 2 x + bei2ν 2 x e−βx dx = I ν β β
ME 40 ME 40 MI 50 MI 50
[Re ν > −1]
MI 49
[Re ν > −1]
ME 40
√ 1 1 3π 3νπ e π √ ber2ν 2 2x dx = Jν − + cos β β β 4 2 x 0 Re ν > − 12
∞ −βx √ 1 1 3π 3νπ e π √ bei2ν 2 2x dx = Jν − + sin β β β 4 2 x 0 Re ν > − 12
∞ √ −βx ν 1 3νπ 2−ν 2 + x berν x e dx = 1+ν cos [Re ν > −1] β 4β 4 0
∞ √ −βx ν 1 3νπ 2−ν + x 2 beiν x e dx = 1+ν sin [Re ν > −1] β 4β 4 0
6.872
∞ −βx
MI 49
MI 49 ME 40 ME 40
6.921
6.875 1. 2. 6.876 1. 2.
Mathieu, hyperbolic, and trigonometric functions
763
√ √ 1 π 1 1 1 e ker 2 x − ln x ber 2 x dx = ln β cos + sin 2 β β 4 β 0 ∞ √ √ π 1 1 1 1 e−βx kei 2 x − ln x bei 2 x dx = ln β sin − cos 2 β β 4 β 0
∞
∞
−βx
1 arctan a2 2a 0 ∞ 1 ln (1 + a4 ) x ker x J 1 (ax) dx = 2a 0 x kei x J 1 (ax) dx = −
MI 50 MI 50
[a > 0]
ET II 21(32)
[a > 0]
ET II 21(33)
6.9 Mathieu Functions (m)
Notation: k 2 = q. For definition of the coefficients Ap
(m)
and Bp
, see section 8.6.
6.91 Mathieu functions 6.911
2π
1.
[m = p]
cem (z, q) cep (z, q) dz = 0
MA
0
2π
2.
∞ + ,2 ,2 + (2n) (2n) 2 A2r [ce2n (z, q)] dz = 2π A0 +π =π
0
2π
3.
2
[ce2n+1 (z, q)] dz = π 0
MA
r=1 ∞ + ,2 (2n+1) A2r+1 =π
MA
r=0 2π
4.
[m = p]
sem (z, q) sep (z, q) dz = 0
MA
0
2π
5.
2
[se2n+1 (z, q)] dz = π 0
MA
r=0 2π
6.
2
[se2n+2 (z, q)] dz = π 0
∞ + ,2 (2n+1) =π B2r+1 ∞ + ,2 (2n+2) B2r+2 =π
MA
r=0 2π
7.
sem (z, q) cep (z, q) dz = 0
[m = 1, 2, . . . ;
p = 1, 2, . . .]
MA
0
6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions 6.921
(2n)
π
cosh (2k cos u sinh z) ce2n (u, q) du =
1. 0
πA0 (−1)n Ce2n (z, −q) ce2n π2 , q [q > 0]
MA
764
Mathieu Functions
(2n)
π
2.
6.922
cosh (2k sin u cosh z) ce2n (u, q) du = 0
πA0 (−1)n Ce2n (z, −q) ce2n (0, q) [q > 0]
(2n+1)
π
3.
MA
sinh (2k sin u cosh z) se2n+1 (u, q) du = 0
πkB1 (−1)n Ce2n+1 (z, −q) se2n+1 (0, q) [q > 0]
(2n+1)
π
4.
MA
sinh (2k cos u sinh z) ce2n+1 (u, q) du = 0
πkA1 (−1)n+1 Se2n+1 (z, −q) ce2n+1 π2 , q [q > 0]
(2n+1)
π
5.
sinh (2k sin u sin z) se2n+1 (u, q) du = 0
MA
πkB1 se2n+1 (z, q) se2n+1 (0, q) [q > 0]
6.922
(2n+1)
π
πA1 Ce2n+1 (z, q) 2 ce2n+1 (0, q)
cos u cosh z cos (2k sin u sinh z) ce2n+1 (u, q) du =
1. 0
[q > 0] π
2.
sin u sinh z cos (2k cos u cosh z) se2n+1 (u, q) du = 0
MA
π
3.
sin u sinh z sin (2k cos u cosh z) se2n+2 (u, q) du = 0
MA
πB1 (2n+1) π Se2n+1 (z, q) ,q 2 se2n+1 2 [q > 0] (2n+2) πkB2 − 2 se2n+2 π2 , q
Se2n+2 (z, q)
[q > 0]
cos u cosh z sin (2k sin u sinh z) se2n+2 (u, q) du = 0
MA
(2n+2)
π
4.
πkB2 Se2n+2 (z, q) 2 se2n+2 (0, q) [q > 0]
π
5.
sin u cosh z cosh (2k cos u sinh z) se2n+1 (u, q) du =
πB1
MA
(2n+1)
2 se2n+1
0
π 2
,q
(−1)n Ce2n+1 (z, −q)
[q > 0]
π
6.
cos u sinh z cosh (2k sin u cosh z) ce2n+1 (u, q) du = 0
MA
(2n+1) πA1
2 ce2n+1 (0, q)
(−1)n Se2n+1 (z, −q)
[q > 0] 7.
π
sin u cosh z sinh (2k cos u sinh z) se2n+2 (u, q) du = 0
MA
MA
(2n+2) πkB2 π (−1)n+1 ,q 2 se2n+2
Se2n+2 (z, −q)
2
[q > 0]
MA
6.924
Mathieu, hyperbolic, and trigonometric functions
(2n+2)
π
8.
cos u sinh z sinh (2k sin u cosh z) se2n+2 (u, q) du = 0
πkB2 (−1)n Se2n+2 (z, −q) 2 se2n+2 (0, q) [q > 0]
6.923
∞
1.
[q > 0] ∞
2.
cos (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −
0
3.
[q > 0] ∞
4.
cos (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −
0
∞
5.
πA0 Ce2n (z, q) 2 ce2n 12 π, q
cos (2k cosh z cosh u) Ce2n (u, q) du =
(2n) πA0 1 − 2 ce2n 2 π, q
[q > 0] ∞
6. 0
∞
kπA1 Fey2n+1 (z, q) 2 ce2n+1 12 π, q
cos (2k cosh z cosh u) Ce2n+1 (u, q) du =
(2n+1) kπA1 2 ce2n+1 12 π, q
[q > 0] ∞
8. 0
cos (2k cos u cos z) ce2n (u, q) du = 0
πA0 1
ce2n
2 π, q
ce2n (z, q) [q > 0]
0
MA
(2n+1)
π
sin (2k cos u cos z) ce2n+1 (u, q) du = −
2.
MA
(2n)
π
1.
MA
Ce2n+1 (z, q)
[q > 0] 6.924
MA
(2n+1)
sin (2k cosh z cosh u) Ce2n+1 (u, q) du = 0
MA
Fey2n (z, q)
[q > 0] 7.
MA
(2n)
sin (2k cosh z cosh u) Ce2n (u, q) du = 0
MA
kπB2 (2n+2) Se2n+2 (z, q) 4 se2n+2 12 π, q
[q > 0]
MA
kπB2 (2n+2) Gey2n+2 (z, q) 4 se2n+2 12 π, q
sin (2k cosh z cosh u) sinh z sinh u Se2n+2 (u, q) du = −
0
MA
πB1 (2n+1) Gey2n+1 (z, q) 4 se2n+1 12 π, q
[q > 0] ∞
MA
πB1 (2n+1) Se2n+1 (z, q) 4 se2n+1 12 π, q
sin (2k cosh z cosh u) sinh z sinh u Se2n+1 (u, q) du = −
0
765
πkA1 ce2n+1 (z, q) ce2n+1 12 π, q [q > 0]
MA
766
Mathieu Functions
(2n)
π
3.
6.925
cos (2k cos u cosh z) ce2n (u, q) du = 0
πA0 1
ce2n
2 π, q
Ce2n (z, q) [q > 0]
(2n)
π
4.
cos (2k sin u sinh z) ce2n (u, q) du = 0
πA0 Ce2n (z, q) ce2n (0, q) [q > 0]
sin (2k cos u cosh z) ce2n+1 (u, q) du = − 0
πkA1 Ce2n+1 (z, q) ce2n+1 12 π, q [q > 0]
π
6.
sin (2k sin u sinh z) se2n+1 (u, q) du = 0
(2n+1) πkB1 se2n+1 (0, q)
Notation: z1 = 2k
MA
Se2n+1 (z, q) [q > 0]
6.925
MA
(2n+1)
π
5.
MA
MA
cosh2 ξ − sin2 η, and tan α = tanh ξ tan η
2π
sin [z1 cos(θ − α)] ce2n (θ, q) dθ = 0.
1.
MA
0
(2n)
2π
cos [z1 cos(θ − α)] ce2n (θ, q) dθ =
2. 0
2πA0 Ce2n (ξ, q) ce2n (η, q) ce2n (0, q) ce2n 12 π, q (2n+1)
2π
sin [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = −
3.
MA
0
2πkA1 Ce2n+1 (ξ, q) ce2n+1 (η, q) ce2n+1 (0, q) ce2n+1 12 π, q MA
2π
cos [z1 cos(θ − α)] ce2n+1 (θ, q) dθ = 0
4.
MA
0
(2n+1)
2π
sin [z1 cos(θ − α)] se2n+1 (θ, q) dθ =
5. 0
2πkB1 Se2n+1 (ξ, q) se2n+1 (η, q) se2n+1 (0, q) se2n+1 12 π, q MA
2π
6.
cos [z1 cos(θ − α)] se2n+1 (θ, q) dθ = 0
MA
sin [z1 cos(θ − α)] se2n+2 (θ, q) dθ = 0
MA
0
2π
7. 0
2π
cos [z1 cos(θ − α)] se2n+2 (θ, q) dθ =
8. 0
2πk 2 B2 (2n+2) Se2n+2 (ξ, q) se2n+2 (η, q) se2n+2 (0, q) se2n+2 12 π, q MA
(2n+2)
π
sin u sin z sin (2k cos u cos z) se2n+2 (u, q) du = −
6.926 0
πkB2 se2n+2 (z, q) 2 se2n+2 π2 , q [q > 0]
MA
6.941
Eigenfunctions of Helmholtz equation
767
6.93 Combinations of Mathieu and Bessel functions 6.931
,2 + (2n) π A0 π ce2n (z, q) 0 ,q ce2n (0, q) ce2n 2 + ,2 (2n) 2π 2π A0 1/2 π Fey2n (z, q) ce2n (u, q) du = Y 0 k [2 (cos 2u + cosh 2z)] 0 ,q ce2n (0, q) ce2n 2
1.
2.
π
1/2 ce2n (u, q) du = J 0 k [2 (cos 2u + cos 2z)]
MA
MA
6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems Notation: Particular solutions of the Helmholtz equation in three-dimensional infinite space ∇2 Ψ + k 2 Ψ = 0 in Cartesian (x, y, z), spherical (r, θ, φ), and cylindrical (ρ, z, φ) coordinates are Ψkx ky kz (x, y, z) ∝ ei(kx x+ky y+kz z) with k 2 = kx2 + ky2 + kz2 k imφ Z l+1/2 (kr) P m Ψlm (r, θ, φ) ∝ e l (cos θ) r Ψmkz (ρ, z, φ) ∝ ei(mφ+kz z) Z l+1/2 ρ k 2 − kz2 with P m l (cos θ) the associated Legendre function, Z is any Bessel function, m = 0, 1, . . . , l; l ∈ N, r2 = ρ2 + z 2 , ρ = r sin θ, z = r cos θ, φ = arccot(x/y), and kt2 = k 2 − kz2 . 6.941
k
1.
e −k
∞
2. −∞
3. 0
∞
iρz
z 2πk m p l−m 2 2 dp = i J l+1/2 (kr) P m J m ρ k − ρ Pl l k r r
e−iρz J l+1/2 (kr) P m l
z r
dz = im−l
[ρ > 0, l ≥ m ≥ 0] ρ 2πr J m ρ k 2 − ρ2 P m l k k
[ρ > 0, l ≥ m ≥ 0] x J m (ρkt ) cos kx x + m arcsin dx ρ < kx (−1)m 2 2 cos y kt − kx + m arccos = 2 2 kt kt − kx =0
kx2 < kt2 kx2 > kt2
768
Mathieu Functions
x Y m (ρkt ) cos kx x + m arcsin dx 0 ρ < m kx (−1) 2 2 sin y kt − kx + m arccos = 2 2 kt kt − kx <
|kx | (−1)m 2 2 = exp −y kx − kt − m sign (kx ) arccosh kt kx2 − kt2
∞ z kz 2πr (j) 2 (j) −ikz z m−l 2 Pm ρ e H H l+1/2 (kr) P m dx = i k − k l m l z r k k −∞
4.
5.
∞
7.
8. 9.
10.
[ρ > 0]
kx2 < kt2 kx2 > kt2
The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0.
∞ z kz 2πk (j) (j) m ikz z l−m 2 2 H l+1/2 (kr) P m H m ρ k − kz P l dkz = i e l k r r −∞ The result is true for j = 1 if π > arg k 2 − kz2 ≥ 0, for j = 2 if −π < arg k 2 − kz2 ≤ 0.
∞ 2 kz 2πr m z −ikz z m−l e J m ρ k 2 − kz2 P m kz < k2 J l+1/2 (kr) P l dz = i l r k k −∞ 2 =0 kz > k2
k z kz 2πk m ikz z l−m 2 2 J l+1/2 (kr) P m J m ρ k − kz P l dkz = i e l k r r −k
∞ 2 kz 2πr m z m −ikz z m−l 2 2 Y m ρ k − kz P l kz < k2 Y l+1/2 (kr) P l dz = i e r k k −∞
2 kz 2r K m ρ kz2 − k 2 P m kz > k2 = −2im−l l kπ k
k kz Y m ρ k 2 − kz2 P m il−m eikz z dkz l k −k
∞ m kz 4 1 − cos kz z + 2 π(m − l) P l K m ρ kz2 − k 2 eikz z dkz π k k z 2πk Y l+1/2 (kr) P m = l r r
6.
6.941
7.113
Associated Legendre functions
769
7.1–7.2 Associated Legendre Functions 7.11 Associated Legendre functions
1
P ν (x) dx = sin ϕ P −1 ν (cos ϕ)
7.111
MO 90
cos ϕ
7.112
1
1. −1
=
2 (n + m)! 2n + 1 (n − m)!
[n = k] SM III 185, WH
1
2. −1
[n = k]
m Pm n (x) P k (x) dx = 0
m m Qm n (x) P k (x) dx = (−1)
1 − (−1)n+k (n + m)! (k − n)(k + n + 1)(n − m)!
1
3. −1
P ν (x) P σ (x) dx 2π sin π(σ − ν) + 4 sin(πν) sin(πσ) [ψ(ν + 1) − ψ(σ + 1)] π 2 (σ − ν) (σ + ν + 1) 2 2 π − 2 (sin πν) ψ (ν + 1)
= 1 2 π ν+ 2
=
EH I 171(18)
1
4. −1
Q ν (x) Q σ (x) dx =
=
[σ + ν + 1 = 0] [σ = ν]
EH I 170(7)
EH I 170(9)a
[ψ(ν + 1) − ψ(σ + 1)] [1 + cos(πσ) cos(νπ)] − π2 sin π(ν − σ) (σ − ν) (σ + ν + 1) [σ + ν + 1 = 0; ν, σ = −1, −2, −3, . . .] 1 2 2π
+
− ψ (ν + 1) 1 + (cos νπ)
2
EH I 170(11)
,
2ν + 1 [ν = σ,
ν = −1, −2, −3, . . .] EH I 170(12)
1
5. −1
P ν (x) Q σ (x) dx =
1 − cos π(σ − ν) − 2π −1 sin(πν) cos(πσ) [ψ(ν + 1) − ψ(σ + 1)] (ν − σ) (ν + σ + 1) [Re ν > 0, Re σ > 0,
=−
sin(2νπ) ψ (ν + 1) π(2ν + 1)
σ = ν]
EH I 170(13)
[Re ν > 0,
σ = ν]
EH I 171(14)
7.113
Notation:
Γ 12 + ν2 Γ 1 + σ2 A = 1 σ Γ 2 + 2 Γ 1 + ν2
1
P ν (x) P σ (x) dx =
1. 0
πν πσ −1 sin πν A sin πσ 2 cos 2 − A 2 cos 2 1 2 π(σ − ν)(σ + ν + 1)
EH I 171(15)
770
Associated Legendre Functions
1
2.
Q ν (x) Q σ (x) dx =
ψ(ν + 1) − ψ(σ + 1) −
1
P ν (x) Q σ (x) dx = 0
7.114
−1 A−1 cos π(ν−σ) 2 (σ − ν)(σ + ν + 1)
∞
1.
+ , π(σ−ν) −1 A − A−1 sin π(σ+ν) A + A sin 2 2 (σ − ν)(σ + ν + 1) [Re ν > 0, Re σ > 0]
0
3.
π 2
7.114
P ν (x) Q σ (x) dx = 1
1 (σ − ν)(σ + ν + 1)
[Re ν > 0,
Re σ > 0]
[Re(σ − ν) > 0,
EH I 171(16) EH I 171(17)
Re(σ + ν) > −1] ET II 324(19)
∞
2.
Q ν (x) Q σ (x) dx = 1
∞
3.
ψ (ν + 1) 2ν + 1
2
[Q ν (x)] dx = 1
∞
7.115
Q ν (x) dx = 1
ψ(σ + 1) − ψ(ν + 1) (σ − ν)(σ + ν + 1) [Re(ν + σ) > −1;
1 ν(ν + 1)
σ, ν = −1, −2, −3, . . .]
Re ν > − 12
EH I 170(5)
EH I 170(6)
[Re ν > 0]
ET II 324(18)
7.12–7.13 Combinations of associated Legendre functions and powers
1
7.121
x P ν (x) dx = cos ϕ
7.122
1
1. 0
− sin ϕ sin ϕPν (cos ϕ) + cos ϕ P 1ν (cos ϕ) (ν − 1)(ν + 2)
MO 90
2
[P m 1 (n + m)! n (x)] dx = 1 − x2 2m (n − m)!
1
[P μν (x)]
2. 0
2
dx Γ(1 + μ + ν) =− 2 1−x 2μ Γ(1 − μ + ν)
[0 < m ≤ n] [Re μ < 0,
MO 74
ν + μ is a positive integer] EH I 172(26)
1
3.
2 P n−ν (x) ν
0
dx n! =− 2 1−x 2(n − ν) Γ(1 − n + 2ν)
[n = 0, 1, 2, . . . ;
Re ν > n] ET II 315(9)
1
7.123 −1
k Pm n (x) P n (x)
dx =0 1 − x2
[0 ≤ m ≤ n,
0 ≤ k ≤ n;
m = k] MO 74
1
7.124 −1
1m 2 1m m k z − 1 2 Qm xk (z − x)−1 1 − x2 2 P m n (x) dx = (−2) n (z) · z [m ≤ n; k = 0, 1, . . . , n − m; z is in the complex plane with a cut along the interval (−1, 1) on the real axis] ET II 279(26)
7.128
Associated Legendre functions and powers
1
7.125 −1
1 − x2
12 m
771
m m m −3/2 (k + m)!(l + m)! (n + m)!(s − m)! Pm k (x) P l (x) P n (x) dx = (−1) π (k − m)!(l − m)!(n − m)!(s − k)! Γ m + 12 Γ t − k + 12 Γ t − l + 12 Γ t − n + 12 × (s − l)!(s − n)! Γ s + 32 [2s = k + l + n + m and 2t = k + l − n − m are both even l ≥ m, m ≤ k − l − m ≤ n ≤ k + l + m]
ET II 280(32)
7.126 1.
√
1
P ν (x)xσ dx =
0
1
xσ P m ν (x) dx =
2. 0
> −1]
EH I 171(23)
ET II 313(2)
π 1/2 22μ−1 Γ 1+σ ν−μ+1 μ+ν μ 3+σ−μ 2 3+σ−μ 3 F 2 ,− , 1 − ; 1 − μ, ;1 x 2 2 2 2 Γ Γ 1−μ 0 2 2 [Re σ > −1, Re μ < 2] ET II 313(3) ∞ 1 μ xμ−1 Q ν (ax) dx = eμπi Γ(μ)a−μ a2 − 1 2 Q −μ ν (a)
4.
m+ν+1 m−ν m 3+σ+m , , + 1; m + 1, ;1 2 2 2 2 [Re σ > −1; m = 0, 1, 2, . . .]
× 3F 2
3.
π2−σ−1 Γ(1 + σ)
[Re σ Γ 1 + 12 σ − 12 ν Γ 12 σ + 12 ν + 32 (−1)m π 1/2 2−2m−1 Γ 1+σ Γ(1 + m + ν) 2 1 1 3 σ m Γ 2 + 2 m Γ 2 + 2 + 2 Γ(1 − m + ν)
1
σ
P μν (x) dx =
1
[|arg(a − 1)| < π,
−1
7.128 1.
1
Re(ν − μ) > −1]
ET II 325(26)
2
1
7.127
Re μ > 0,
(1 + x)σ P ν (x) dx = 1
2σ+1 [Γ(σ + 1)] Γ(σ + ν + 2) Γ(1 + σ − ν)
μ− 1
[Re σ > −1]
ET II 316(15)
μ− 3
(1 − x)− 2 μ (1 + x) 2 2 (z + x) 2 P μν (x) dx −1 1 −1/2 Γ μ − 12 (z − 1)μ− 2 (z + 1) = − 1/2 2μπi π e% Γ (μ + ν)&Γ(μ − ν % − 1) % %
1/2
1/2 &
1/2 &
1/2 & 1 + z 1+z 1+z 1+z μ−1 μ μ μ−1 × Qν Q −ν−1 + Qν Q −ν−1 2 2 2 2 1
[− 12 < Re μ < 1, z is in the complex plane with a cut along the interval (−1, 1) of the real axis]
ET II 317(20) 1
2. −1
1
(1 − x)− 2 μ (1 + x) 2 1
μ− 12
μ− 1
(z + x) 2 P μν (x) dx % %
1/2 &
1/2 & 2e−2μπi Γ 12 + μ 1+z 1+z μ μ μ = 1/2 Q −ν−1 (z − 1) Q ν 2 2 π Γ(μ − ν) Γ (μ + ν + 1) [− 12 < Re μ < 1, z is in the complex plane with a cut along the interval (−1, 1) of the real axis] ET II 316(18)
772
Associated Legendre Functions
4
1
7.129 −1
7.131 1.
∞
7.129
P ν (x) P λ (x)(1 + x)λ+ν dx =
1
(x − 1)− 2 μ (x + 1) 2 1
μ− 12
2λ+ν+1 [Γ(λ + ν + 1)] 2
[Γ(λ + 1) Γ(ν + 1)] Γ(2λ + 2ν + 2) [Re(ν + λ + 1) > 0]
(z + x)
μ− 12
EH I 172(30)
P μν (x) dx
%
1/2 &2 1+z − ν) Γ (1 − μ + ν) μ μ Pν =π (z − 1) 2 Γ 12 − μ [Re(μ + ν) < 0, Re(μ − ν) < 1, |arg(z + 1)| < π] ET II 321(6)
1
1/2 Γ(−μ
2.
∞
1
(x − 1)− 2 μ (x + 1) 2 1
μ− 12
(z + x)
μ− 32
P μν (x) dx
% %
1/2 &
1/2 & 1 −1/2 1+z 1+z π 1/2 Γ(1 − μ − ν) Γ (2 − μ + ν) (z − 1)μ− 2 (z + 1) μ μ−1 = Pν Pν 2 2 Γ 32 − μ [Re μ < 1, Re(μ + ν) < 1, Re(μ − ν) < 2, |arg(1 + z)| < π] ET II 321(7) 1
7.132 1.
2.
π2μ Γ λ + 12 μ Γ λ − 12 μ 1−x Γ λ + 12 ν + 12 Γ λ − 12 ν Γ − 12 μ + 12 ν + 1 Γ − 12 μ − 12 ν + 12 −1 [2 Re λ > |Re μ|] ET II 316(16) ∞ 2 λ−1 μ 2μ−1 Γ λ − 12 μ Γ 1 − λ + 12 ν Γ 12 − λ − 12 ν x −1 P n (x) dx = Γ 1 − 12 μ + 12 ν Γ 12 − 12 μ − 12 ν Γ 1 − λ − 12 μ 1 [Re λ > Re μ, Re(1 − 2λ − ν) > 0, Re(2 − 2λ + ν) > 0] ET II 320(2)
3.
1
∞
9
2 λ−1
P μν (x) dx =
2 λ−1 μ Γ x −1 Q ν (x) dx = eμπi
1
1
4.
− 1 μ xσ 1 − x2 2 P μν (x) dx =
0
1
5. 0
6. 0
1
1 2
+ 12 ν + 12 μ Γ 1 − λ + 12 ν Γ λ + 12 μ Γ λ − 12 μ 22−μ Γ 1 + 12 ν − 12 μ Γ 12 + λ + 12 ν [|Re μ| < 2 Re λ < Re ν + 2] μ−1
2 Γ 1 1 Γ 1 + 2σ − 2ν
1
ET II 324(23)
1 1 2 + 2 σ Γ 1 + 2 σ − 12 μ Γ 12 σ + 12 ν − 12 μ
+
3 2
EH I 172(24) [Re μ < 1, Re σ > −1] 1 1 1m (−1)m 2−m−1 Γ 2 + 2 σ Γ 1 + 2 σ Γ (1 + m + ν) xσ 1 − x2 2 P m ν (x) dx = Γ(1 − m + ν) Γ 1 + 12 σ + 12 m − 12 ν Γ 32 + 12 σ + 12 m + 12 ν [Re σ > −1, m is a positive integer] EH I 172(25), ET II 313(4)
1
2 η
1−x
P μν (x) dx =
2μ−1 Γ 1 + η − 12 μ Γ 12 + 12 σ Γ(1 − μ) Γ 32 + η + 12 σ − 12 μ
ν−μ+1 μ+ν μ 3+σ−μ ,− , 1 + η − ; 1 − μ, + η; 1 × 3F 2 2 2 2 2 Re η − 12 μ > −1, Re σ > −1 ET II 314(6)
7.135
Associated Legendre functions and powers
∞
7.
−ρ
x
x −1 2
− 12 μ
P μν (x) dx
=
[Re μ < 1,
∞
1.
[|arg(u − 1)| < π,
∞
2.
[|arg(u − 1)| < π,
∞
1. 1
∞
2. 1
7.135
1
1. −1
2.
0 < Re μ < 1 + Re ν]
MO 90a
2 1λ 1 λ+ 1 μ μ−1 x − 1 2 Q −λ dx = Γ(μ)eμπi u2 − 1 2 2 Q ν−λ−μ (u) ν (x)(x − u)
u
7.134
ET II 320(3)
1μ Q ν (x)(x − u)μ−1 dx = Γ(μ)eμπi u2 − 1 2 Q −μ ν (u)
u
ρ+μ+ν ρ+μ−ν−1 Γ 2 √2 π Γ(ρ) Re(ρ + μ + ν) > 0, Re(ρ + μ − ν) > 1]
2ρ+μ−2 Γ
1
7.133
773
0 < Re μ < 1 + Re(ν − λ)]
1μ 2λ+μ Γ(λ) Γ(−λ − μ − ν) Γ(1 − λ − μ + ν) (x − 1)λ−1 x2 − 1 2 P μν (x) dx = Γ(1 − μ + ν) Γ(−μ − ν) Γ(1 − λ − μ) [Re λ > 0, Re(λ + μ + ν) < 0, Re(λ + μ − ν) < 1]
ET II 204(30)
ET II 321(4)
− 1 μ 2λ−μ sin πν Γ(λ − μ) Γ(−λ + μ − ν) Γ(1 − λ + μ + ν) (x − 1)λ−1 x2 − 1 2 P μν (x) dx = − π Γ(1 − λ) [Re(λ − μ) > 0, Re(μ − λ − ν) > 0, Re(μ − λ + ν) > −1] ET II 321(5)
1 − x2
− 12 μ
− 1 μ (z − x)−1 P μμ+n (x) dx = 2e−iμπ z 2 − 1 2 Q μμ+n (z)
[n = 0, 1, 2, . . . , Re μ + n > −1, z is in the complex plane with a cut along the interval (−1, 1) ET II 316(17) of the real axis.] ∞ μ/2 −ρ (x − 1)λ−1 x2 − 1 (x + z) P μν (x) dx 1
=
2λ+μ−ρ Γ(λ − ρ) Γ(ρ − λ − μ − ν) Γ (ρ − λ − μ + ν + 1) Γ(1 − μ + ν) Γ(−μ − ν) Γ (1 + ρ − λ − μ)
1+z × 3 F 2 ρ, ρ − λ − μ − ν, ρ − λ − μ + ν + 1; ρ − λ + 1, ρ − λ − μ + 1; 2
Γ(ρ − λ) Γ(λ) μ 1+z λ−ρ 2 (z + 1) + λ, −μ − ν, 1 − μ + ν; 1 − μ, 1 − ρ + λ; 3F 2 Γ(ρ) Γ(1 − μ) 2 [Re λ > 0, Re(ρ − λ − μ − ν) > 0, Re(ρ − λ − μ + ν + 1) > 0, |arg(z + 1)| < π] ET II 322(9)
774
Associated Legendre Functions
∞
3.
7.136
−μ/2 −ρ (x − 1)λ−1 x2 − 1 (x + z) P μν (x) dx
1
=−
sin(νπ) Γ(λ − μ − ρ) Γ (ρ − λ + μ − ν) Γ(ρ − λ + μ + ν + 1) 2ρ−λ+μ π Γ (1 + ρ − λ)
1+z × 3 F 2 ρ, ρ − λ + μ − ν, ρ − λ + μ + ν + 1; 1 + ρ − λ, 1 + ρ − λ + μ; 2 Γ(λ − μ) Γ (ρ − λ + μ) λ−ρ−μ (z + 1) + Γ(ρ) Γ(1 − μ)
1+z × 3 F 2 λ − μ, −ν, ν + 1; 1 + λ − μ − ρ, 1 − μ; 2 [Re(λ − μ) > 0, Re (ρ − λ + μ − ν) > 0, Re(ρ − λ + μ + ν + 1) > 0, |arg(z + 1)| < π] ET II 322(10)
7.136 1.
1
1 − x2
λ−1
μ/2
P ν (ax) dx
π2μ Γ(λ) μ+ν 1−μ+ν 1 2 1 1 , ; + λ; a − 2F 1 2 2 2 Γ 2 + λ Γ 2 − 2 μ − 12 ν Γ 1 − 12 μ + 12 ν ET II 318(31) [Re λ > 0, −1 < a < 1] ∞ 2 λ−1 2 2 μ/2 μ x −1 a x −1 P ν (ax) dx 1 Γ(λ) Γ 1 − λ − 12 μ + 12 ν Γ 12 − λ − 12 μ − 12 ν = Γ 1 − 12 μ + 12 ν Γ 12 − 12 ν − 12 μ Γ(1 − λ − μ)
1−μ+ν μ−ν 1 ,1 − λ − ; 1 − λ − μ; 1 − 2 ×2μ−1 aμ−ν−1 2 F 1 2 2 a [Re a > 0, Re λ > 0, Re(ν − μ − 2λ) > −2, Re(2λ + μ + ν) < 1] ET II 325(25) −1
=
2.
1 − a2 x2
∞
3.
1
1
7.137 1.
∞
μ+ν+1
μ−2 μπi −μ−ν−1 Γ(λ) Γ 1 − λ + μ+ν 2 e a 2 3 Γ ν+2
μ+ν+1 μ+ν 3 −2 × 2F 1 ,1 − λ + ;ν + ;a 2 2 2 [|arg(a − 1)| < π, Re λ > 0, Re(2λ − μ − ν) < 2] ET II 325(27)
2 λ−1 2 2 − 1 μ Γ x −1 a x − 1 2 Q μν (ax) dx =
2
x− 2 μ− 2 (x − 1)−μ− 2 (1 + ax) 2 μ Q μν (1 + 2ax) dx 1
1
1
1
1
2. 1
∞
1
|arg a| < π,
x− 2 μ− 2 (x − 1)−μ− 2 (1 + ax) 2 Q μν (1 + 2ax) dx 1
1
3
,2 1μ μ + 1/2 2 − μ a Q (1 + a) ν 2 Re μ < 12 , Re(μ + ν) > −1 ET II 325(28)
= π −1/2 e−μπi Γ
1
μ
, + , 1 1 −1/2 μ+1 + 1/2 (1 + a) Q μν (1 + a)1/2 = −π −1/2 e−μπi Γ −μ − 12 a 2 μ+ 2 1 + a2 Qν |arg a| < π, Re μ < − 12 , Re(μ + ν + 2) > 0 ET II 326(29)
7.137
Associated Legendre functions and powers
1
3. 0
1
4.
,2 1 + 1 1 1 1 x− 2 μ− 2 (1 − x)−μ− 2 (1 + ax) 2 μ P μν (1 + 2ax) dx = π 1/2 Γ 12 − μ a 2 μ P μν (1 + a)1/2 Re μ < 12 , |arg a| < π ET II 319(32) 1
x− 2 μ− 2 (1 − x)−μ− 2 (1 + ax) 2 P μν (1 + 2ax) dx 1
1
3
μ
+ , 1 1 1/2 (1 + a) P μν (1 + a)2 = π 1/2 Γ − 12 − μ a 2 μ+ 2 P μ+1 ν ET II 319(33) Re μ < − 12 , |arg a| < π
0
1
5.
x 2 μ− 2 (1 − x)μ− 2 (1 + ax)− 2 μ P μν (1 + 2ax) dx 1
1
1
1
0
= π 1/2 Γ
1
1
1
− 12 μ
3
x 2 μ− 2 (1 − x)μ− 2 (1 + ax)
6. 0
=
1
7.
μ
1 2
+ , + , 1 1 1 1/2 −1/2 1/2 P 1−μ (1 + a) P μν (1 + a)1/2 π Γ μ − 12 a 2 − 2 μ (1 + a) ν 2 + , + , 1/2 +(μ + ν)(1 − μ + ν) P −μ (1 + a) P μν (1 + a)1/2 ν Re μ > 12 , |arg a| < π ET II 319(35)
x− 2 − 2 (1 − x)−μ− 2 (1 + ax) 2 μ Q μν (1 + 2ax) dx 1
1
1
= π 1/2 Γ 1
8.
μ
1
3
=
y
0
1
1 2
+ , + , 1 − μ a 2 μ P μν (1 + a)1/2 Q μν (1 + a)1/2 Re μ < 12 , |arg a| < π ET II 320(38)
x− 2 − 2 (1 − x)−μ− 2 (1 + ax) 2 Q μν (1 + 2ax) dx
0
9.
+ , + , 1 1/2 (1 + a) + μ a− 2 μ P μν (1 + a)1/2 P −μ ν Re μ > − 12 , |arg a| < π ET II 319(34)
P μν (1 + 2ax) dx
0
775
μ
1 1/2 −1/2 12 μ+ 12 π Γ −μ − 12 (1 + a) a 2 + , + , + , + , 1/2 1/2 (1 + a) Q μν (1 + a) + P μν (1 + a)1/2 Q μ+1 (1 + a)1/2 × P μ+1 ν ν Re μ < − 12 , |arg a| < π ET II 320(39)
− 12 λ λ (y − x)μ−1 x 1 + 12 γx P ν (1 + γx) dx
12 μ− 12 λ 12 μ 2 1 P λ−μ (1 + γy) y 1 + γy ν γ 2 [Re λ < 1, Re μ > 0, |arg γy| < π] ET II 193(52)
= Γ(μ) 10. 0
y
− 1 λ 1 (y − x)μ−1 xσ+ 2 λ−1 1 + 12 γx 2 P λν (1 + γx) dx γ − 12 λ
Γ(σ) Γ(μ)y σ+μ−1 1 γy = 2 −ν, 1 + ν, σ; 1 − λ, σ + μ; − 3F 2 Γ(1 − λ) Γ(σ + μ) 2 [Re σ > 0, Re μ > 0, |γy| < 1] ET II 193(53)
776
Associated Legendre Functions
y
11.
(y − x)μ−1 [x(1 − x)]− 2 λ P λν (1 − 2x) dx = Γ(μ)[y(1 − y)] 2 μ− 2 λ P λ−μ (1 − 2y) ν 1
1
0
12.
7.138
1
[Re λ < 1,
Re μ > 0,
0 < y < 1] ET II 193(54)
y
(y − x)μ−1 xσ+ 2 λ−1 (1 − x)− 2 λ P λν (1 − 2x) dx 1
1
0
=
∞
7.138
−μ−ν−2
(a + x)
Pμ
0
a−x a+x
Pν
Γ(μ) Γ(σ)y σ+μ−1 3 F 2 (−ν, 1 + ν, σ; 1 − λ, σ + μ; y) Γ (σ + μ) Γ(1 − λ) [Re σ > 0, Re μ > 0, 0 < y < 1] ET II 193(155)
a−x a+x
a−μ−ν−1 [Γ(μ + ν + 1)]
4
dx =
2
[Γ(μ + 1) Γ(ν + 1)] Γ(2μ + 2ν + 2) [|arg a| < π, Re(μ + ν) > −1] ET II 326(3)
7.14 Combinations of associated Legendre functions, exponentials, and powers 7.141
∞
1μ e−ax (x − 1)λ−1 x2 − 1 2 P μν (x) dx =
∞
e−ax (x − 1)λ−1 x2 − 1
1. 1
[Re a > 0,
2.
1 2μ
∞
3.
e
−ax
(x − 1)
λ−1
1
∞
4. 1
Re λ > 0]
ET II 323(13)
Q μν (x) dx
!
! 1 + μ, 1 Γ(ν + μ + 1)eμπi −λ−μ −a 22 ! a e G 23 2a ! = λ + μ, ν + 1, −ν 2 Γ(ν − μ + 1) [Re a > 0, Re λ > 0, Re(λ + μ) > 0] ET II 325(24)
1
!
! 1 + μ, 1 a−λ−μ e−a 31 G 23 2a !! λ + μ, −ν, 1 + ν Γ(1 − μ + ν) Γ(−μ − ν)
!
! 1, 1 − μ 2 − 12 μ μ −1 μ−λ −a 31 ! x −1 P ν (x) dx = −π sin(νπ)a e G 23 2a ! λ − μ, 1 + ν, −ν [Re a > 0, Re(λ − μ) > 0] ET II 323(15)
!
! 1 − μ, 1 2 − 12 μ μ 1 μπi μ−λ −a 22 −ax λ−1 ! x −1 e (x − 1) Q ν (x) dx = e a e G 23 2a ! λ − μ, ν + 1, −ν 2 [Re a > 0, Re λ > 0, Re(λ − μ) > 0]
ET II 323(14) ∞
5. 1
− 1 μ 1 e−ax x2 − 1 2 P μν (x) dx = 21/2 π −1/2 aμ− 2 K ν+ 12 (a) [Re a > 0,
Re μ < 1] ET II 323(11), MO 90
∞
7.142 1
e− 2 ax 1
x+1 x−1
12 μ
P μν− 1 (x) dx = 2
2 W μ,ν (a) a
Re μ < 1,
ν−
1 2
= 0, ±1, ±2, . . . BU 79(34), MO 118
7.146
7.143
Associated Legendre functions, exponentials, and powers
∞
1
[x(1 + x)]
1.
− 12 μ −βx
e
P μν (1
0
∞
2. 0
7.144 1.
∞
β μ− 2 1 + 2x) dx = √ e 2 β K ν+ 12 π
∞
1
∞
1. 0
∞
2. 0
Re β > 0]
ET I 179(1)
[Re μ < 1,
Re β > 0]
ET I 179(2)
1
sin(νπ) Γ(ν + μ + 1) E (−ν, ν + 1, λ + μ; μ + 1 : 2β) λ+μ Γ(ν − μ + 1) 2β sin(μπ) sin [(μ + ν) π] − 1−μ λ E (ν − μ + 1, −ν − μ, λ : 1 − μ : 2β) 2 β sin(μπ) [Re β > 0, Re λ > 0, Re (λ + μ) > 0] ET I 181(16) sin(νπ) E (−ν, ν + 1, λ − μ : 1 − μ : 2β) sin(μπ) sin[(μ − ν)π] − 1+μ λ E (μ + ν + 1, μ − ν, λ : 1 + μ : 2β) 2 β sin(μπ) [Re β > 0, Re λ > 0, Re(λ − μ)] > 0 ET I 181(17)
e−βx xλ− 2 μ−1 (x + 2) 2 μ Q μν (1 + x) dx = − 1
0
7.145
[Re μ < 1,
e−βx xλ+ 2 μ−1 (x + 2) 2 μ Q μν (1 + x) dx =
β 2
1μ 1 e2β 1 2 −βx μ W μ,ν+ 12 (β) e P ν (1 + 2x) dx = 1+ x β
0
2.
1
2β λ−μ
1 e−βx eβ Pν W ν+ 12 ,0 (β) W −ν− 12 ,0 (β) − 1 dx = 1+x (1 + x)2 β
x−1 e−βx Q − 12
[Re β > 0] 2 2 1 1 π2 −2 β β 1 + 2x dx = + Y0 J0 8 2 2 [Re β > 0]
∞
3. 0
∞
1. 0
2. 0
x− 2 μ e−βx P μν 1
√ β 1 5 1 + x dx = 2μ β 2 μ− 4 e 2 W
1 1 1 1 2 μ+ 4 , 2 ν+ 4
ET II 327(5)
√ β 1 1 3 e−βx x− 2 μ √ 1 + x dx = 2μ β 2 μ− 4 e 2 W P μν 1+x
Re ν > −1]
ET II 327(6)
(β)
[Re μ < 1, ∞
ET I 180(6)
1 2 x−1 e−ax Q ν 1 + 2x−2 dx = [Γ(ν + 1)] a−1 W −ν− 12 ,0 (ai) W −ν− 12 ,0 (−ai) 2 [Re a > 0,
7.146
777
1 1 1 1 2 μ+ 4 , 2 ν+ 4
Re β > 0]
ET I 180(7)
(β)
[Re μ < 1, Re β > 0]
ET I 180(8)a
778
Associated Legendre Functions
∞√
3.
−βx
xe
P 1/4 ν
1+
x2
P −1/4 ν
0
7.147
1 1 1 π (1) (2) 2 H 1 β H ν+ 1 β 1 + x dx = 2 2 2β ν+ 2 2 2 [Re β > 0]
∞
7.147
1ν xλ−1 x2 + a2 2 e−βx P μν
0
%
x (x2
ET I 180(9)
& 1/2
+ a2 )
dx
2−ν−2 aλ+ν 32 = G π Γ(−μ − ν) 24
! ! 1 − λ , 1−λ ! 2 2 ! 1 λ+μ+ν ! 0, 2 , − 2 , − λ−μ+ν 2 Re β > 0, Re λ > 0] ET II 327(7)
a2 β 2 4
[a > 0,
1 1 1 1 1 1 1−x y P μν (x) dx = 2ν y 2 μ+ν− 2 e 2 y W 7.148 (1 − x)− 2 μ (1 + x) 2 μ+ν−1 exp − 1+x −1 [Re y > 0] ∞ + 2 −1/2 1/2 , 2 2 2 α + β + 2αβx 7.149 exp − α + β + 2αβx P ν (x) dx
1 1 1 2 μ−ν− 2 , 2 μ
(y)
ET II 317(21)
1
= 2π −1 (αβ)−1/2 K ν+ 12 (α) K ν+ 12 (β) [Re α > 0,
Re β > 0]
ET II 323(16)
7.15 Combinations of associated Legendre and hyperbolic functions 7.151
∞
1.
(sinh x)
α−1
P −μ ν
(sinh x)
α−1
Q μν
0
∞
2. 0
2−1−μ Γ 12 α + 12 μ Γ 12 ν − 12 α + 1 Γ 12 − 12 α − 12 ν (cosh x) dx = Γ 12 μ + 12 ν + 1 Γ 12 + 12 μ − 12 ν Γ 1 + 12 μ − 12 α [Re(α + μ) > 0, Re(ν − α + 2) > 0, Re(1 − α − ν) > 0] EH I 172(28)
eiμπ 2μ−α Γ 12 + 12 ν + 12 μ Γ 1 + 12 ν − 12 α (cosh x) dx = Γ 1 + 12 ν − 12 μ Γ 12 + 12 ν + 12 α × Γ 12 α + 12 μ Γ 12 α − 12 μ [Re (α ± μ) > 0,
∞
7.152
e 0
−αx
sinh
2μ
Re(ν − α + 2) > 0]
EH I 172(29)
1 −2μ 1 Γ 2μ + 12 Γ(α − n − μ) Γ α + n − μ + 12 cosh 2 x dx = μ √ 2 x P 2n 1 4 π Γ(α + n + μ + 1) Γ α − n + μ +1 2 Re α > n + Re μ, Re μ > − 4 ET I 181(15)
7.162
Associated Legendre functions, powers, and trigonometric functions
779
7.16 Combinations of associated Legendre functions, powers, and trigonometric functions 7.161
1
1.
− 1 μ xλ−1 1 − x2 2 sin(ax) P μν (x) dx
0
=
2.
1
π 1/2 2μ−λ−1 Γ (λ + 1) a Γ 3+λ−μ+ν Γ 1 + λ−μ−ν 2 2
1+λ λ 3 λ−μ−ν 3+λ−μ+ν a2 ,1 + ; ,1 + , ;− × 2F 3 2 2 2 2 2 4 [Re λ > −1, Re μ < 1] ET II 314(7)
− 1 μ xλ−1 1 − x2 2 cos(ax) P μν (x) dx
0
=
∞
3.
2 1μ x − 1 2 sin(ax) P μν (x) dx =
0
π 1/2 2μ−λ Γ(λ)
1+λ−μ−ν λ−μ+ν Γ 1+ Γ 2 2
λ λ+1 1 1+λ−μ−ν λ−μ+ν a2 , ; , ,1 + ;− × 2F 3 2 2 2 2 2 4 [Re λ > 0, Re μ < 1] ET II 314(8) 2μ π 1/2 a−μ− 2 S 1 1 (a) 1 − 2 μ − 12 ν Γ 1 − 12 μ + 12 ν μ+ 2 ,ν+ 2 a > 0, Re μ < 32 , Re(μ + ν) < 1 1
Γ
1 2
ET II 320(1)
7.162
∞
2 −2
P ν 2x a
1. a
3. 0
πa − 1 sin(bx) dx = − 4 cos(νπ)
2 2 ab ab − J −ν− 12 J ν+ 12 2 2 [a > 0, b > 0, −1 < Re ν < 0]
P ν 2x2 a−2 − 1 cos(bx) dx
ab ab ab ab π = − a J ν+ 12 J −ν− 12 − Y ν+ 12 Y −ν− 12 4 2 2 2 2 [a > 0, b > 0, −1 < Re ν < 0] ET II 326(2)
a
ET II 326(1) ∞
2.
∞
+ ,2 2 2 −1/2 x +2 x + 1 dx = 2−1/2 π −1 a sin(νπ) K ν+ 12 2−1/2 a sin(ax) P −1 ν
[a > 0, −2 < Re ν < 1] ET I 98(22) ∞ −1/2 x2 + 2 sin(ax) Q 1ν x2 + 1 dx = −2−3/2 πa K ν+ 12 2−1/2 a I ν+ 12 2−1/2 a 0 ET 98(23) a > 0, Re ν > − 32
4.
780
Associated Legendre Functions
√
2 a 2 sin(νπ) K ν+ 12 √ cos(ax) P ν 1 + x2 dx = − π 2 0 [a > 0,
∞ a a π 2 I ν+ 12 √ cos(ax) Q ν 1 + x dx = √ K ν+ 12 √ 2 2 2 0 [a > 0, 1 a a π cos(ax) P ν 2x2 − 1 dx = J ν+ 12 J −ν− 12 2 2 2 0
5.
6.
7.
7.163
∞
−1 < Re ν < 0]
ET I 42(23)
Re ν > −1]
ET I 42(24)
[a > 0] 7.163
∞
1. a
2. 0
1
1 2 1 ν− 1 1 π νπ −ν + x − a2 2 4 sin(bx) P 02 ax−1 dx = b−ν− 2 cos ab − 2 4 a > 0, |Re ν| < 12
7.164 ∞
1.
,2 2 x2 x1/2 sin(bx) P −1/4 1 + a dx = ν
x1/2 sin(bx) P −1/4 ν
0
∞
3.
x1/2 sin(bx) P −1/4 ν
0
4. 0
Γ
−1 < Re ν < 0]
ET II 327(4)
2 b K ν+ 1 2 2a ν Re a > 0, b > 0, − 54 < Re ν < 14
2 −1 −1/2 b πa 1 Γ 4− 4 +ν
5
ET II 327(8) ∞
2.
0,
ET I 42(25)
∞
x1/2 sin(bx) P 1/4 ν
−1/4
1 + a2 x2 Q ν−1
1 + a2 x2 dx π − 1 πi
4 Γ ν + 54 b b 2e I ν+ 12 = K ν+ 12 1 3 2a 2a 2 ab Γ ν + 4 ET II 328(9) Re a > 0, b > 0, Re ν > − 54
dx −1/4 1 + a2 x2 P ν−1 1 + a2 x2 √ 1 + a2 x2
b b a−2 b1/2 5 5 K ν− 12 =√ K ν+ 12 2a 2a 2π Γ 4 + ν Γ 4 − ν 5 ET II 328(10) Re a > 0, b > 0, − 4 < Re ν < 54
dx 2 x2 √ 1 + a2 x2 P −3/4 1 + a ν 1 + a2 x2 2 b a−2 b1/2 7 3 K ν+ 1 =√ 2 2a 2π Γ 4 + ν Γ 4 − ν Re a > 0, b > 0, − 74 < Re ν < 34 ET II 328(11)
7.171
Associated Legendre function and probability integral
∞
5.
1/2
x
+ cos(bx)
P 1/4 ν
1+
a2 x2
,2
0
0
∞
7.
x1/2 cos(bx) P 1/4 1 + a2 x2 Q 1/4 1 + a2 x2 dx ν ν π 1 πi
4 Γ ν + 34 b b 2e = I ν+ 12 K ν+ 12 1/2 Γ ν + 5 2a 2a ab 4 ET II 328(13) Re a > 0, b > 0, Re ν > − 34 x1/2 cos(bx) P −1/4 ν
0
∞
8. 0
−1/2 2 a−1 πb b 2 1 K ν+ 1 dx = 3 2 2a Γ 4 + ν Γ − 4 − ν Re a > 0, b > 0, − 34 < Re ν < − 14 ET II 328(12)
∞
6.
781
x1/2 cos(bx) P 1/4 ν
dx 1 + a2 x2 P 3/4 1 + a2 x2 √ ν 1 + a2 x2 2 b a−2 b1/2 5 1 K ν+ 1 =√ 2 2a 2π Γ + ν Γ − ν 4 4 Re a > 0, b > 0, − 54 < Re ν < 14 ET II 328(14)
dx 1/4 1 + a2 x2 P ν−1 1 + a2 x2 √ 1 + a2 x2
b b a−2 b1/2 1 3 3 K ν− 1 =√ K ν+ 2 2 2a 2a 2π Γ 4 +ν Γ 4 −ν ET II 329(15) Re a > 0, b > 0, |Re ν| < 34
∞
7.165
cos(ax) P ν (cosh x) dx
1 + ν − iα ν + iα ν − iα Γ Γ − Γ − 2 2 2 [a > 0, −1 < Re ν < 0] ET II 329(18) π 2−μ π Γ 12 α + 12 μ Γ 12 α − 12 μ −μ α−1 7.166 P ν (cos ϕ) sin ϕ dϕ = 1 1 Γ 2 + 2 α + 12 ν Γ 12 α − 12 ν Γ 12 μ + 12 ν + 1 Γ 12 μ − 12 ν + 12 0 [Re (α ± μ) > 0] MO 90, EH I 172(27) η a η 1 η 2 Γ(μ − η) Γ η + sin(a − x) dx −η 2 (sin a) √ = P −μ 7.167 P −μ (cos x) P [cos(a − x)] ν ν ν (cos a) sin x sin x π Γ(η + μ + 1) 0 Re μ > Re η > − 12 ET II 329(16) 0
=−
sin(νπ) Γ 4π 2
1 + ν + iα 2
7.17 A combination of an associated Legendre function and the probability integral
7.171 1
∞
2 − 1 μ x − 1 2 exp a2 x2 [1 − Φ(ax)] P μν (x) dx
3 a2 1+μ+ν μ−ν −1 μ−1 =π 2 Γ Γ aμ− 2 e 2 W 2 2 [Re a > 0,
Re μ < 1,
Re (μ + ν) > −1,
1 1 1 1 4 − 2 μ, 4 + 2 ν
2 a
Re(μ − ν) > 0] ET II 324(17)
782
Associated Legendre Functions
7.181
7.18 Combinations of associated Legendre and Bessel functions 7.181 1.
∞
1
∞
2. 1
7.182
∞
P ν− 12 (x)x1/2 Y ν (ax) dx = 2−1/2 a−1 cos 12 a J ν 12 a − sin 12 a Y ν 12 a a > 0, Re ν < 12 1 cos 12 a Y ν 12 a + sin 12 a J ν 12 a P ν− 12 (x)x1/2 J ν (ax) dx = − √ 2a |Re ν| < 12
ν
x
1.
x −1 2
12 λ− 12
1
∞
3.
0
1
1
∞
6.
∞
7.
1
π μ− 1 a 2 J 12 −μ 12 a J ν+ 12 12 a 2 [Re μ < 1, Re(μ − ν) < 2]
1 −μ
x2 − 1
− 12 μ
P μν− 1 (x) K ν (ax) dx = (2π)−1/2 aμ− 2 1
2
− 1 μ 1 xμ+ 2 x2 − 1 2 P μν− 1 (x) K ν (ax) dx =
− 1 μ 3 xμ− 2 x2 − 1 2 P μν− 1 (x) K ν (ax) dx =
2
1
8.
2
1
+ a a a a , 1 = 2−3/2 π 1/2 aμ− 2 J ν J μ− 12 −Yν Y μ− 12 2 2 2 2 1 − 4 < Re μ < 1, a > 0, Re(2μ − ν) > − 12 ET II 349(67)a
ET II 337(33)a ∞
x2
+ a a a a , 1 = −2−3/2 π 1/2 aμ− 2 J μ− 12 Yν + Y μ− 12 Jν 2 2 2 1 2 1 − 4 < Re μ < 1, a > 0, |Re ν| < 2 + 2 Re μ ET II 344(37)a
− 1 μ 1 x 2 −μ 1 − x2 2 P μν (x) J ν+ 12 (ax) dx =
5.
3 2
− 1 μ 1 x 2 −μ x2 − 1 2 P μν− 1 (x) Y ν (ax) dx 2
1
Re(2λ + ν)
0, Re ν < 52 ,
ET II 344(36)a
ET II 345(38)a ∞
2.
P λ−1 (x) J ν (ax) dx λ
ET II 108(3)a
∞
− 1 μ 1 xμ− 2 x2 − 1 2 P μν− 3 (x) K ν (ax) dx = 2
K ν 12 a K μ− 12 12 a
[Re μ < 1,
Re a > 0]
ET II 135(5)a
π −3/2 − 1 a a e 2 W μ,ν (a) 2 [Re μ < 1,
Re a > 0]
ET II 135(3)a
π −1/2 − 1 a a e 2 W μ−1,ν (a) 2 [Re μ < 1,
Re a > 0]
ET II 135(4)a
π −1 − 1 a a e 2 W μ− 12 ,ν− 12 (a) 2 [Re μ < 1]
ET II 135(6)a
7.182
Associated Legendre and Bessel functions
∞
+ a ,2 1 ν− 1 1 −ν 2 2x − 1 K ν (ax) dx = π −1/2 a−ν 2ν−1 K μ+ 12 x1/2 x2 − 1 2 4 P μ2 2 1 Re ν > − 2 , Re a > 0 ET II 136(11)a
∞
1 ν− 1 1 −ν 2 2x − 1 Y ν (ax) dx x1/2 x2 − 1 2 4 P μ2 + a a a a , = π 1/2 2ν−2 a−ν J μ+ 12 J −μ− 12 − Y μ+ 12 Y −μ− 12 2 2 2 2 ET II 108(5)a Re ν > − 12 , a > 0, Re ν + |2 Re μ + 1| < 32
∞
1 ν− 1 1 −ν 2 2x − 1 J ν (ax) dx x1/2 x2 − 1 2 4 P μ2
9. 1
10. 1
11. 1
783
∞
12.
+ a ,2 + a ,2 J μ+ 12 − J −μ− 12 2 2 < 2 Re μ < 12 − Re ν ET II 345(39)a
= −2ν−2 a−ν π 1/2 sec(μπ) Re ν > − 12 ,
Re ν −
a > 0,
3 2
− 1 ν x x2 − 1 2 P νμ 2x2 − 1 K ν (ax) dx = 2−ν aν−1 K μ+1 (a)
1
[Re a > 0, ∞
13.
x x2 + a
1 2 2ν
P νμ 1 + 2x2 a
−2
ET II 136(10)a
K ν (xy) dx = 2−ν ay −ν−1 S 2ν,2μ+1 (ay)
0
Re ν < 1]
[Re a > 0,
Re y > 0,
Re ν < 1] ET II 135(7)
∞
14. 0
1ν x x2 + a2 2 (μ − ν) P νμ 1 + 2x2 a−2 + (μ + ν) P ν−μ 1 + 2x2 a−2 K ν (xy) dx = 21−ν μy −ν−2 S 2ν+1,2μ (ay) [Re a > 0,
∞
15. 0
Re y > 0,
Re ν < 1]
ET II 136(8)
1 ν−1 ν P μ 1 + 2x2 a−2 + P ν−μ 1 + 2x2 a−2 K ν (xy) dx = 21−ν y −ν S 2ν−1,2μ (ay) x x2 + a2 2 [Re a > 0,
Re y > 0,
Re ν < 1] ET II 136(9)
∞
16. 0
17. 0
− 1 ν− 1 −ν− 1 x1/2 x2 + 2 2 4 P μ 2
+ , 2 1 y −1/2 2 2 −ν π −1/2 K μ+ 12 2−1/2 y 2 x + 1 J ν (xy) dx = 3 1 Γ3 ν + μ + 2 Γ ν − μ + 2 1 − 2 − Re ν < Re μ < Re ν + 2 , y > 0 ET II 44(1)
∞
− 1 ν− 1 ν+ 1 x1/2 x2 + 2 2 4 Q μ 2 x2 + 1 J ν (xy) dx
1 1 = 2−ν− 2 π 1/2 e(ν+ 2 )πi y ν K μ+ 12 2−1/2 y I μ+ 12 2−1/2 y ET II 46(12) Re ν > −1, Re(2μ + ν) > − 52 , y > 0
784
Associated Legendre Functions
∞
7.183
− 1 μ− 1 μ+ 1 x1−μ 1 + a2 x2 2 4 Q ν− 12 (±iax) J ν (xy) dx 2
0
= i(2π)1/2 eiπ(μ∓ 2 ν∓ 4 ) a−1 y μ−1 I ν 1
3 −4 − 7.184
1 2
1
Re ν < Re μ < 1 + Re ν,
y > 0,
1
y K μ 12 a−1 y Re a > 0 ET II 46(11) 2a
−1
∞
1 μ− 1 − 1 −μ x1/2 x2 − 1 2 4 P − 21 +ν x−1 J ν (xa) dx = 21/2 a−1−μ π −1/2 cos a + 12 (ν − μ)π 2 |Re μ| < 12 , Re ν > −1, a > 0
∞
x−ν x2 − 1
1. 1
7.183
ET II 44(2)a
2.
1 1 4−2ν
ν− 12
Pμ
2x−2 − 1 K ν (ax) dx
1
= π 1/2 2−ν a−2+ν W μ+ 12 ,ν− 12 (a) W −μ− 12 ,ν− 12 (a) Re ν < 32 , a > 0 ET II 370(45)a
∞
3. 0
4. 0
1
1 ν+ 1 −ν− 1 xν 1 − x2 2 4 P μ 2 2x−2 − 1 J ν (xy) dx 3 1 ν+ 12 ν Γ 2 + μ + ν Γ 2 + ν − μ =2 y 1/2 Γ 3 + ν 2 (2π) 2
3 3 × 1 F 1 ν + μ + ; 2ν + 2; iy 1 F 1 ν + μ + ; 2ν + 2; −iy 2 2 y > 0, − 32 − Re ν < Re μ < Re ν + 12 ET II 45(3)
∞
1 − 1 ν 1 −ν 1 + 2a2 x−2 K ν (xy) dx x−ν x2 + a2 4 2 Q μ2 2 1 = ie−iπν π 1/2 2−ν−1 a−ν− 2 y ν−2 Γ 32 + μ − ν W −μ− 12 ,ν− 12 (iay) W −μ− 12 ,ν− 12 (−iay) Re a > 0, Re y > 0, Re μ > − 32 , Re(μ − ν) > − 32 ET II 137(13)
∞
1 − 1 ν 1 −ν 1 + 2x−2 J ν (ax) dx x−ν x2 + 1 4 2 Q μ2
5. 0
6.
2 1 + 2 J ν (ax) dx x 3 2 1 iπν − 12 ν −ν−2 Γ 2 +μ+ν Γ 2 +ν−μ = −ie π 2 a % & cos(μπ) sin(μπ) W μ+ 1 ,ν+ 1 (a) 1 (a) + M 1 × W −μ− 12 ,ν+ 12 (a) 2 2 Γ(2 + 2ν) μ+ 2 ,ν+ 2 Γ ν + μ + 32 3 1 a > 0, Re(μ + ν) > − 2 , Re(μ − ν) < 2 ET II 46(14)
1 + ν ν+ 1 xν 1 + x2 4 2 Q μ 2
0
=2
Γ 32 + μ − ν M μ+ 12 ,ν− 12 (a) W −μ− 12 ,ν− 12 (a) Γ(2ν) a > 0, 0 < Re ν < Re μ + 32 ET II 47(15)a
−iνπ 1/2 π −ν −ν−2 ie
a
7.187
Associated Legendre and Bessel functions
∞
7.
1 − 1 ν 12 −ν 1 + 2a2 x−2 K ν (xy) dx x−ν x2 + a2 4 2 Q − 1 2
0
−iπν 3/2 −ν−3
= ie
π
2
a
1 2 −ν
y
ν−1
2
[Γ(1 − ν)] ×
+
+ ay ,2 J ν− 12 + Y ν− 12 2 2 Re y > 0, Re ν < 1] ET II 136(12)
[Re a > 0,
785
ay ,2
+ 1/2 , 1 a2 + x2 x−1 J ν (xy) dx = 2−1/2 πy −1 exp − a2 − 14 y Jν 2y 0 ET II 46(10) Re ν > − 12 , y > 0
∞ 2 1 − x −ν−1 −2 x 1 + x2 Pν J 0 (xy) dx = y 2ν [2ν Γ(ν + 1)] K 0 (y) 1 + x2 0 ET II 13(10) [Re ν > 0]
7.185
7.186 7.187 1.
∞
∞
x1/2 Q ν− 12
x P νμ
0
∞
2.
+
[Re ν < 1, Re y > 0] ,2 y ,2 + 1 + a2 x2 J 0 (xy) dx = 2π −2 y −1 a−1 cos(λπ) K λ 2a Re a > 0, |Re λ| < 14 ,
0
− 12 ν
x Pμ
− 12 ν
1 + a2 x2 Q μ
0
∞
x P μσ− 1 2
∞
6.
x P μσ− 1 2
0
2
=y Re a > 0,
0
∞
y y +σ−μ W μ,σ M −μ,σ Γ(1 + 2σ) a a ET II 14(15) Re σ > − 14 , Re μ < 1
−2 μπi Γ
e
y > 0,
1 2
2 x2 J (xy) dx 1 + a2 x2 P −μ 1 + a 1 0 σ− 2
7.
ET II 137(15)
1 + a2 x2 J ν (xy) dx
1 + a2 x2 Q μσ− 1 1 + a2 x2 J 0 (xy) dx
Re y > 0]
1 y y y −1 e− 2 νπi Γ 1 + μ + 12 ν 1 1 = K I μ+ 2 μ+ 2 2a 2a a Γ 1 + μ − 12 ν 3 Re a > 0, y > 0, Re μ > − 4 , Re ν > −1 ET II 47(16)
0
5.
y>0
−1/2 ν x 1 + x2 Pμ 1 + x2 K ν (xy) dx = y −1/2 S ν− 12 ,μ+ 12 (y) [Re ν < 1,
∞
4.
ET II 137(14)
ET II 13(11) ∞
3.
1 + x2 K ν (xy) dx = y −3/2 S ν+ 12 ,μ+ 12 (y)
x P λ− 12
0
= 2π −1 y −2 cos(σπ) W μ,σ Re a > 0,
y > 0,
y
a |Re σ| < 14
W −μ,σ
y a
ET II 14(14)
2 y + y y , x P μσ− 1 1 + a2 x2 J 0 (xy) dx = −iπ −1 y −2 W μ,σ W μ,σ eπi − W μ,σ e−πi 2 a a a ET II 14(13) Re a > 0, y > 0, |Re σ| < 14 , Re μ < 1
786
Associated Legendre Functions
∞
8. 0
∞
9.
1 1 −1/2 − 12 − 12 ν − ν x 1 + a2 x2 Pμ 1 + a2 x2 P μ2 2 1 + a2 x2 J ν (xy) dx + y ,2 K μ+ 12 2a = πa2 Γ ν2 + μ + 32 Γ ν2 − μ + 12 ET II 46(9) Re a > 0, y > 0, − 54 < Re μ < 14
−1ν
x Pμ 2
1 + a2 x2
2
0
∞
10.
+ y ,2 −1 2 K μ+ 12 2a y J ν (xy) dx = 1 1 πa Γ 1 + μ + ν Γ ν − μ 2 2 Re a > 0, y > 0, − 34 < Re μ < − 14 , Re ν > −1
∞
1. 0
∞
2.
∞
3.
Re a > 0,
y y K μ+ 12 2a K μ+ 32 2a = πa2 Γ 2 + 12 ν + μ Γ 12 ν − μ ET II 45(8) y > 0, − 74 < Re μ < − 14
− 1 μ a y μ−2 e−ay √ x a2 + x2 2 P −ν J ν (xy) dx = μ−1 Γ(μ + ν) a2 + x2 Re a > 0, y > 0, Re ν > −1, 1ν xν+1 x2 + a2 2 P ν
0
ET II 45(7)
−1/2 − 12 ν − 12 ν x 1 + a2 x2 Pμ 1 + a2 x2 P μ+1 1 + a2 x2 J ν (xy) dx
0
7.188
7.188
x2 + 2a2 √ 2a x2 + a2
− 1 ν x1−ν x2 + a2 2 P ν−1
0
2
J ν (xy) dx =
2
x + 2a √ 2a x2 + a2
Re μ >
1 2
ya ,2 (2a)ν+1 y −ν−1 + K ν+ 12 π Γ(−ν) 2 [Re a > 0, −1 < Re ν < 0,
ET II 45(4)
y > 0] ET II 45(5)
ay y (2a) I ν− 12 K ν− 12 Γ(ν) 2 2 [Re a > 0, y > 0, 0 < Re ν < 1] 1−ν ν−1
J ν (xy) dx =
ay
ET II 45(6)
7.189
∞
μ −x
(a + x) e
1.
P −2μ ν
0
2.
∞
−μ −x
(x + a) 0
e
2x 1+ a
P −2μ ν
I μ (x) dx = 0 − 12 < Re μ < 0,
− 12 + Re μ < Re ν < − 12 − Re μ
ET II 366(18)
2x 1+ I μ (x) dx a 2μ−1 Γ μ + ν + 12 Γ μ − ν − 12 ea = W 12 −μ, 12 +ν (2a) π 1/2 Γ (2μ + ν + 1) Γ(2μ! − ν) ! |arg a| < π, Re μ > !Re ν + 12 ! ET II 367(19)
7.192
Associated Legendre functions and functions generated by Bessel functions
∞
3.
−μ x
x
e
P 2μ ν
0
2x 1+ K μ (x + a) dx a
= π −1/2 2μ−1 cos(μπ) Γ μ + ν + 12 Γ μ − ν + 12 W 12 −μ, 12 +ν (2a) ! ! |arg a| < π, Re μ > !Re ν + 12 ! ET II 373(11)
∞
4.
− 12 μ
x
−1/2 −x
(x + a)
e
0
787
P μν− 1 2
a−x a+x
K ν (a + x) dx =
π −1μ a 2 Γ(μ, 2a) 2
[a > 0, ∞
5.
(sinh x) 0
μ+1
(cosh x)
−2μ− 32
Re μ < 1]
ET II 374(12)
P −μ ν [cosh(2x)] I μ− 12 (a sech x) dx 1
=
2μ− 2 Γ(μ − ν) Γ (μ + ν + 1)
M ν+ 12 ,μ (a) M −ν− 12 ,μ (a) 3 2 π 1/2 aμ+ 2 [Γ (μ + 1)] [Re μ > Re ν, Re μ > − Re ν − 1] ET II 378(44)
7.19 Combinations of associated Legendre functions and functions generated by Bessel functions 7.191
∞
1.
2 1 2 = 2−ν−2 π 1/2 a cosec(μπ) cos(νπ) Y ν 12 a − Jν 2a −1 < Re μ < 0, Re ν < 12 ET II 384(6)
a
∞
2. 0
7.192
1. 0
− 1 − 1 ν ν+ 1 x1/2 x2 − a2 4 2 P μ 2 2x2 a−2 − 1 [Hν (x) − Y ν (x)] dx
1
−1/4−ν/2 ν+1/2 2 −2 2x a − 1 [I −ν (x) − Lν (x)] dx x1/2 x2 − a2 Pμ 2 2 = 2−ν−1 π 1/2 a cosec(2μπ) cos(νπ) I ν 12 a − I −ν 12 a −1 < Re μ < 0, Re ν < 12 ET II 385(15)
(ν−μ−2)/4 (μ−ν+2)/2 x(ν−μ−1)/2 1 − x2 P ν−1/2 (x) S μ,ν (ax) dx
μ+ν+3 μ − 3ν + 3 μ−ν μ−3/2 1/2 −(ν−μ−1)/2 π =2 π a Γ Γ cos 4 4 2 1 1 1 1 × J ν 2 a Y −(μ−ν+1)/2 2 a − Y ν 2 a J −(μ−ν+1)/2 2 a [Re(μ − ν) < 0,
a > 0,
|Re(μ + ν)| < 1,
Re(μ − 3ν) < 1]
ET II 387(24)a
788
Associated Legendre Functions
∞
2.
−β/2 β x1/2 x2 − 1 P ν (x) S μ,1/2 (ax) dx
1
1 Γ β−μ−ν − 2 4 S μ−β+1,ν+1/2 (a) −μ Re(μ + ν − β) < − 12 , Re(μ − ν − β) < 12 ET II 387(25)a
2−3/2+β−μ aβ−1 Γ
7.193 1.
∞
∞
β−μ+ν 2 π 1/2 Γ 12
= Re β < 1,
a > 0,
7.193
+
1 4
1/4−ν/2 ν−1/2 −2 x−ν x2 − 1 P μ/2−ν/2 2x − 1 S μ,ν (ax) dx 1 2μ−ν aν−2 π 1/2 Γ 3ν−μ−1 2 1+ν−μ = W ρ,σ aeiπ/2 W ρ,σ ae−iπ/2 Γ 2 ρ = 12 (μ + 1 − ν), σ = ν − 12 , Re(μ − ν) < 0, a > 0, Re ν < 32 , Re(3ν − μ) > 1 ET II 387(27)a
2.
−ν/2 ν 2 x x2 − 1 P λ 2x − 1 S μ,ν (ax) dx
1
+ λ Γ ν−μ−1 −λ 2 2 1−μ+ν 1−μ−ν S μ−ν+1,2λ+1 (a) Γ 2Γ 2 2 Re(μ − ν + λ) < −1, Re(μ − ν + λ) < 0] ET II 387(26)a =
[Re ν < 1,
a > 0,
aν−1 Γ
ν−μ+1
7.21 Integration of associated Legendre functions with respect to the order 7.211
1 1 cosec θ [0 < θ < π] 2 2 0
∞ 1 θ [0 < θ < π] 2. P x (cos θ) dx = cosec 2 −∞ ∞ 1 7.212 x−1 tanh(πx) P − 12 +ix (cosh a) dx = 2e− 2 a K e−a
∞
1.
P −x− 12 (cos θ) dx =
ET II 329(19) ET II 329(20)
0
[a > 0] x tanh(πx) P − 12 +ix (cosh b) dx = Q a− 12 (cosh b) [Re a > 0] 2 2 0 ∞ a + x 1 sinh(πx) cos(ax) P − 12 +ix (b) dx = 2 (b + cosh a) 0 [a > 0, |b| < 1]
7.213 7.214 7.215
0
∞
∞
cos(bx) P μ− 1 +ix (cosh a) dx = 0
π
2
= Γ
1 2
ET II 330(22) ET II 387(23)
ET I 42(27)
[0 < a < b] μ
2 (sinh a) μ+ 1 − μ (cosh a − cosh b) 2
[0 < b < a] ET II 330(21)
7.221
Integration of associated Legendre functions
∞
7.216
cos(bx) Γ(μ + ix) Γ(μ − ix) P
0
1 2 −μ − 12 +ix
789
π
μ− 1
Γ(μ) (sinh a) 2 (cosh a) dx = μ (cosh a + cosh b) [a > 0, b > 0, Re μ > 0] 2
ET II 330(24)
1 1 1 1 2 −ν − ix Γ 2ν − + ix P ν+ix−1 (cos θ) I ν− 12 +ix (a) K ν− 12 +ix (b) dx ν − + ix Γ 2 2 2 −∞ ν √ ab ν− 1 = 2π (sin θ) 2 K ν (ω) ω , + 1/2 ET II 383(29) ω = a2 + b2 + 2ab cos θ
7.217 1.
∞
∞
2. 0
∞
3.
2(ab)1/2 −ikR (2) (2) e xeπx tanh(πx) P − 12 +ix (− cos θ) H ix (ka) H ix (kb) dx = − ; πR 2 1/2 R = a + b2 − 2ab cos θ [a > 0, b > 0, 0 < θ < π, Im k ≤ 0] 1
−ν
(2)
(2)
2 xeπx sinh(πx) Γ (ν + ix) Γ(ν − ix) P − (− cos θ) H ix (a) H ix (b) dx 1 +ix 2
0
ν− 12
= i(2π)1/2 (sin θ) 1/2 R = a2 + b2 − 2ab cos θ
∞
4. 0
[a > 0,
b > 0,
ET II 381(17)
0 < θ < π,
Re ν > 0]
ab R
ν H (2) ν (R) ET II 381 (18)
λ 1 2 1 λ− 1 π 1/2 ab 2 −λ √ β − 1 2 4 K λ (z) x sinh(πx) Γ(λ+ix) Γ(λ−ix) K ix (a) K ix (b) P − (β) dx = 1 +ix 2 z 2 , + π 2 2 z = a + b + 2abβ |arg a| < , |arg(β − 1)| < π, Re λ > 0 ET II 177(16) 2
7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions 7.221 1.
1
−1
2.6
[m = n]
P n (x) P m (x) dx = 0 2 = 2n + 1
1
[m = n]
1 2n + 1 =0
[m = n]
P n (x) P m (x) dx = 0
= 3.
(−1)
2π
P 2n (cos ϕ) dϕ = 2π 0
[n − m is even,
2m+n−1 (m
WH, EH I 170(8, 10)
1 2 (m+n−1)
m!n! 2 ! − n) (n + m + 1) n2 ! m−1 2
2n −2n 2 n
m = n]
[n is even, m is odd] WH
2 .
MO 70, EH II 183(50)
790
7.222
Associated Legendre Functions
1
1. −1
[m < n] 4
−1
(1 + x)m+n P m (x) P n (x) dx =
2m+n+1 [(m + n)!]
ET II 277(15)
2
(m!n!) (2m + 2n + 1)!
1
3. −1
xm P n (x) dx = 0
1
2.
7.222
(1 + x)m−n−1 P m (x) P n (x) dx = 0
1
4. −1
1 − x2
n
P 2m (x) dx =
[m > n]
2n2 (n − m)(2m + 2n + 1)
1 −1
1 − x2
n−1
ET II 278(16)
P 2m (x) dx
[m < n]
1
x2 P n+1 (x) P n−1 (x) dx =
5. 0
7.223 7.224
1 2 {P n (x) P n−1 (x) − P n−1 (x) P n (z)} dx = − −x n [z belongs to the complex plane with a discontinuity along the interval from −1 to +1.] 1
−1
1
2. −1
1
3. −1
1
4. −1
1
5. −1
1
6. −1
1
7. −1
1
8. −1
2.
WH
1
1.
1.
n(n + 1) (2n − 1)(2n + 1)(2n + 3)
−1 z
7.225
WH
(z − x)−1 P n (x) dx = 2 Q n (z)
ET II 277(7)
x(z − x)−1 P 0 (x) dx = 2 Q 1 (z)
ET II 277(8)
xn+1 (z − x)−1 P n (x) dx = 2z n+1 Q n (z) −
2
2n+1 (n!) (2n + 1)!
ET II 277(9)
xm (z − x)−1 P n (x) dx = 2z m Q n (z)
[m ≤ n]
ET II 277(10)a
(z − x)−1 P m (x) P n (x) dx = 2 P m (z) Q n (z)
[m ≤ n]
ET II 278(18)a
(z − x)−1 P n (x) P n+1 (x) dx = 2 P n+1 (z) Q n (z) −
2 n+1
x(z − x)−1 P m (x) P n (x) dx = 2z P m (z) Q n (z) x(z − x)−1 [P n (x)] dx = 2z P n (z) Q n (z) − 2
[m < n]
2 2n + 1
−1 1 (x − t) P n (t) dt = n + (1 + x)−1/2 [T n (x) + T n+1 (x)] 2 −1
−1 1 1 −1/2 −1/2 (t − x) P P n (t) dt = n + (1 − x)−1/2 [T n (x) − T n+1 (x)] 2 x
x
WH
−1/2
ET II 278(19)
ET II 278(21)
ET II 278(20)
EH II 187(43)
EH II 187(44)
7.232
Legendre polynomials and powers
1
3. −1
1
4. −1
5.10
7.226 1.
2. 3.
1 2
(1 − x)−1/2 P n (x) dx =
−1/2
(cosh 2p − x)
23/2 2n + 1
EH II 183(49)
√ 2 2 exp[−(2n + 1)p] P n (x) dx = 2n + 1 [p > 0]
P (z) dz = P (x) Q (y) 2 (xy − z) − (x2 − 1) (y 2 − 1) −1 = P (y) Q (x)
1
7.227 0
(1 < x ≤ y) (1 < y ≤ x)
1
2 −1/2
ET II 276(4)
ET II 276(5)
[|p| < 1] 1/2 ,−2n−1
MO 71
+ a + a2 + 1 2 2 −1/2 2 x a +x P n 1 − 2x dx = 2n + 1
[Re a > 0] 1 1 −μ/2 Γ(1 + μ) P l (x)(z − x)−μ−1 dx = z 2 − 1 e−iπμ Q μl (z) 2 −1
7.2286
WH
1
% &2 Γ 12 + m 1−x P 2m (x) dx = m! −1 1 −1/2 Γ 12 + m Γ 32 + m x 1 − x2 P 2m+1 (x) dx = m!(m + 1)! −1 1 −m−3/2 2 (−p)m (1 + p)−m−1/2 1 + px2 P 2m (x) dx = 2m + 1 −1
791
[l = 0, 1, 2, . . . ,
ET II 278(23)
|arg(z − 1)| < π]
7.23 Combinations of Legendre polynomials and powers 7.231 1. 2.6
(−1)m Γ m − 12 λ Γ 12 + 12 λ x P 2m (x) dx = [Re λ > −1] 2 Γ − 12 λ Γ m + 32 + 12 λ 0 1 (−1)m Γ m + 12 − 12 λ Γ 1 + 12 λ λ x P 2m+1 (x) dx = 2 Γ 12 − 12 λ Γ m + 2 + 12 λ 0
1
λ
[Re λ > −2] 7.232
EH II 183(51)
EH II 183(52)
1
1. −1
(1 − x)a−1 P m (x) P n (x) dx =
2a Γ(a) Γ(n − a + 1) 4 F 3 (−m, m + 1, a, a; 1, a + n + 1, a − n; 1) Γ(1 − a) Γ(n + a + 1) [Re a > 0] ET II 278(17)
792
Associated Legendre Functions
1
2. −1
2a+b−1 Γ(a) Γ(b) 3 F 2 (−n, 1 + n, a; 1, a + b; 1) Γ(a + b)
(1 − x)a−1 (1 + x)b−1 P n (x) dx =
[Re a > 0,
1
(1 − x)μ−1 P n (1 − γx) dx =
3. 0
(1 − x)μ−1 xν−1 P n (1 − γx) dx = 0
[Re μ > 0]
Γ(μ) Γ(ν) 1 3 F 2 −n, n + 1, ν; 1, μ + ν; γ Γ(μ + ν) 2 [Re μ > 0,
1
7.233
x2μ−1 P n 1 − 2x2 dx =
0
Re b > 0]
ET II 276(6)
Γ(μ)n! P (μ,−μ) (1 − γ) Γ(μ + n + 1) n
1
4.
7.233
Re ν > 0]
ET II 190(37)a
ET II 190(38)
2
(−1)n [Γ(μ)] 2 Γ(μ + n + 1) Γ(μ − n) [Re μ > 0]
ET II 278(22)
7.24 Combinations of Legendre polynomials and other elementary functions
∞
7.241 0
∞
Pn e
7.242 0
7.243
n a
1 d e P n (1 − x)e−ax dx = e−a an a da a
n 1 1 d n =a 1+ 2 da an+1
−x
[Re a > 0] (a − 1)(a − 2) · · · (a − n + 1) e−ax dx = (a + n)(a + n − 2) · · · (a − n + 2) [n ≥ 2, Re a > 0]
P 2n (cosh x) e
−ax
0
[Re a > 2n] 2 2 2 2 2 a a −2 a − 4 · · · a − (2n)2 −ax P 2n+1 (cosh x) e dx = 2 (a − 1) (a2 − 32 ) · · · [a2 − (2n + 1)2 ]
ET I 171(6)
[Re a > 2n + 1] 2 2 2 2 2 a +1 a + 3 · · · a + (2n − 1)2 −ax P 2n (cos x) e dx = a (a2 + 22 ) (a2 + 42 ) · · · [a2 + (2n)2 ]
ET I 171(7)
∞
[Re a > 0] 2 2 2 2 2 a + 4 · · · a + (2n)2 a a +2 −ax P 2n+1 (cos x) e dx = 2 (a + 12 ) (a2 + 32 ) · · · [a2 + (2n + 1)2 ]
ET I 171(4)
∞
0
3. 0
4. 0
a2 − 12 a2 − 32 · · · a2 − (2n − 1)2 dx = a (a2 − 22 ) (a2 − 42 ) · · · [a2 − (2n)2 ]
∞
2.
ET I 171(3)
∞
1.
ET I 171(2)
1
5.11 −1
eixα P n (x) dx = in
[Re a > 0] 2π J 1 (α) α n+ 2
[n = 0, 1, 2, . . . ,
ET I 171(5)
a > 0] GH2 24 (171.10)
7.249
Legendre polynomials and elementary functions
7.244 1. 2.
793
a ,2 π+ J n+ 12 P n 1 − 2x2 sin ax dx = [a > 0] 2 2 0 1 a a π J −n− 12 P n 1 − 2x2 cos ax dx = (−1)n J n+ 12 2 2 2 0 1
ET I 94(2)
[a > 0] 7.245 1.
2π
P 2m+1 (cos θ) cos θ dθ =
0
π
2.
P m (cos θ) sin nθ dθ = 0
π 24m+1
2m m
2m + 2 m+1
ET I 38(1)
MO 70, EH II 183(5)
2(n − m + 1)(n − m + 3) · · · (n + m − 1) (n − m)(n − m + 2) · · · (n + m)
[n ≤ m or n + m is even]
=0
2π
3.10 0
√ 3 n+1 2 π Γ n + 2 P1 P 2n+1 (sin α sin φ) sin φ dφ = (−1) (cos α) (2n + 1) Γ (n + 2) 2n+1 α = 12 (2n + 1)π, n an integer
−1
cos(αx) P n (x) dx = 0
= (−1)v
7.246 7.247 7.248
[n is odd] 2π 1 (α) J α 2v+ 2
[n = 2v is even] GH2 24 (171.10a)
2 sin(2n + 1)θ P n 1 − 2 sin2 x sin2 θ sin x dx = 0 (2n + 1) sin θ 1 π dx 3 (a) J P 2n+1 (x) sin ax √ = (−1)n+1 2a 2n+ 2 x 0
π
1
1.
−1
MO 71
[a > 0]
1
−1
ET I 94(1)
+ −1/2 1/2 , P n (x) dx = π(ab)−1/2 J n+ 12 (aλ) J n+ 12 (bλ) a2 + b2 − 2abx sin λ a2 + b2 − 2abx [a > 0,
2.
b > 0]
ET II 277(11)
+ −1/2 1/2 , P n (x) dx = −π(ab)−1/2 J n+ 12 (aλ) Y n+ 12 (bλ) a2 + b2 − 2abx cos λ a2 + b2 − 2abx [0 ≤ a ≤ b]
7.249 1.
MO 71
1
4.
[n > m and n + m is odd]
ET II 277(12)
1
−1
P n (x) arcsin x dx = 0
=π
⎧ ⎪ ⎪ ⎨
⎫2 ⎪ ⎪ ⎬
(n − 2)!!
⎪ 1 n+1 ⎪ ⎪ ⎩ 2 2 (n+1) ⎭ !⎪ 2
[n is even]
[n is odd] WH
794
2.
Associated Legendre Functions
n t−1 1 2πr 2 P n (x) = x + x − 1 cos t t=0 t
7.251
[t > n]
7.25 Combinations of Legendre polynomials and Bessel functions 7.251
1
1.
x P n 1 − 2x2 Y ν (xy) dx = π −1 y −1 [S 2n+1 (y) + π Y 2n+1 (y)]
0
[n = 0, 1, . . . ;
y > 0,
ν > 0] ET II 108(1)
1 i 2 −1 n+1 x P n 1 − 2x K 0 (xy) dx = y K 2n+1 (y) + S 2n+1 (iy) (−1) 2 0
2.
1
3.
x P n 1 − 2x2 J 0 (xy) dx = y −1 J 2n+1 (y)
0
1
4.
2 x P n 1 − 2x2 [J 0 (ax)] dx =
0
1
5.
ET II 134(1)
[y > 0]
ET II 13(1)
1 2 2 [J n (a)] + [J n+1 (a)] 2(2n + 1)
x P n 1 − 2x2 J 0 (ax) Y 0 (ax) dx =
0
[y > 0]
ET II 338(39)a
1 [J n (a) Y n (a) + J n+1 (a) Y n+1 (a)] 2(2n + 1) ET II 339(48)a
6.
1
x2 P n 1 − 2x2 J 1 (xy) dx = y −1 (2n + 1)−1 [(n + 1) J 2n+2 (y) − n J 2n (y)]
0
[y > 0] ET II 20(23) 2 1 2−ν−1 aν Γ 12 μ + 12 ν 7. xμ−1 P n 2x2 − 1 J ν (ax) dx = Γ(ν + 1) Γ 12 μ + 12 ν + n + 1 Γ 12 + 12 ν − n 0
μ+ν μ+ν μ+ν μ+ν a2 , ; ν + 1, + n + 1, − n; − × 2F 3 2 2 2 2 4 [a > 0, Re(μ + ν) > 0] ET II 337(32)a 1 e−a [I n (a) + I n+1 (a)] 7.252 e−ax P n (1 − 2x) I 0 (ax) dx = 2n + 1 0 [a > 0] ET II 366(11)a π/2 7.253 sin(2x) P n (cos 2x) J 0 (a sin x) dx = a−1 J 2n+1 (a) ET II 361(20) 0 1 7.254 x P n 1 − 2x2 [I 0 (ax) − L0 (ax)] dx = (−1)n [I 2n+1 (a) − L2n+1 (a)]
0
[a > 0]
ET II 385(14)a
Gegenbauer polynomials Cnν (x) and powers
7.313
795
7.3–7.4 Orthogonal Polynomials 7.31 Combinations of Gegenbauer polynomials Cnν (x) and powers 7.311
1
1.
−1
3.
4.
ν− 12
C νn (x) dx = 0
Re ν > − 12
n > 0,
1 ν Γ(2ν + n) Γ(2ρ + n + 1) Γ ν + 12 Γ ρ + 12 2 ν− 2 1−x x C n (x) dx = 2n+1 Γ(2ν) Γ(2ρ + 1)n! Γ(n + ν + ρ + 1) 0 Re ρ > − 12 , Re ν > − 12 1 1 2β+ν+ 2 Γ(β + 1) Γ ν + 12 Γ(2ν + n) Γ β − ν + ν ν− 12 β (1 − x) (1 + x) C n (x) dx = n! Γ(2ν) Γ β − ν − n + 32 Γ β + ν + n + 32 −1 Re β > −1, Re ν > − 12 1 2α+β+1 Γ(α + 1) Γ(β + 1) Γ (n + 2ν) β (1 − x)α (1 + x) C νn (x) dx = −1
n! Γ(2ν) Γ(α + β + 2) 1 × 3 F 2 −n, n + 2ν, α + 1; ν + , α + β + 2; 1 2 [Re α > −1, Re β > −1]
2.
1 − x2
ET II 280(1)
1
n+2ρ
ET II 280(2) 3 2
ET II 280(3)
ET II 281(4)
7.312 In the following integrals, z belongs to the complex plane with a cut along the interval of the real axis from −1 to 1. 1 3 1 ν 1 ν− 1 ν− 12 π 1/2 2 2 −ν −(ν− 12 )πi m 2 m −1 2 ν− 2 e 1−x 1. x (z − x) C n (x) dx = z z − 1 2 4 Q n+ν− 1 (z) 2 Γ(ν) −1 1 m ≤ n, Re ν > − 2 ET II 281(5) 1 3 ν− 12 ν 1 ν− 1 ν− 12 π 1/2 2 2 −ν −(ν− 12 )πi n+1 2 e z − 1 2 4 Q n+ν− 2. xn+1 (z − x)−1 1 − x2 C n (x) dx = z 1 (z) 2 Γ(ν) −1 π21−2ν−n n! − Γ(ν) Γ(ν +n + 1) ET II 281(6) Re ν > − 12 1 3 ν− 12 ν 1 ν− 1 π 1/2 2 2 −ν −(ν− 12 )πi 2 ν− 12 e z − 1 2 4 C νm (z) Q n+ν− 3.6 (z−x)−1 1 − x2 C m (x) C νn (x) dx = 1 (z) 2 Γ(ν) −1 1 m ≤ n, Re ν > − 2 ET II 283(17) 7.313
1
1.
−1
1 − x2
ν− 12
C νm (x) C νn (x) dx = 0
m = n,
Re ν > − 12
ET II 282(12), MO 98a, EH I 177(16)
1
2. −1
1 − x2
ν− 12
2
[C νn (x)] dx =
π2
1−2ν
Γ(2ν + n)
n!(n + ν) [Γ(ν)]
2
Re ν > − 12
ET II 281(8), MO 98a, EH I 177(17)
796
7.314 1.
2.
Orthogonal Polynomials
7.314
π 1/2 Γ ν − 12 Γ(2ν + n) (1 − x) (1 + x) dx = n! Γ(ν) Γ(2ν) −1 Re ν > 12 1 1 2 23ν− 2 [Γ(2ν + n)] Γ 2n + ν + 12 1 2 (1 − x)ν− 2 (1 + x)2ν−1 [C νn (x)] dx = 2 (n!) Γ(2ν) Γ 3ν + 2n + 12 −1
1
ν− 32
ν− 12
2 [C νn (x)]
[Re ν > 0] 3.
4.
5.
6.
7.
1
3
1
ET II 281(9)
ET II 282(10)
2
(1 − x)3ν+2n− 2 (1 + x)ν− 2 [C νn (x)] dx −1 2 π 1/2 Γ ν + 12 Γ ν + 2n + 12 Γ (2ν + 2n) Γ 3ν + 2n − 12 = 2 1 1 n + Γ(2ν) 22ν+2n n! Γ ν + Γ 2ν + 2n + 2 2 ET II 282(11) Re ν > 16 1 1 3 (1 − x)ν− 2 (1 + x)ν+m−n− 2 C νm (x) C νn (x) dx −1 Γ ν − 12 + m − n Γ 12 − ν + m − n 22−2ν−m+n π 3/2 Γ(2ν + n) m = (−1) 2 Γ 12 − ν − n Γ 12 + m − n m!(n − m)! [Γ(ν)] Γ 12 + ν + m Re ν > − 12 ; n ≥ m ET II 282(13)a 1 ν− 1 (1 − x)2ν−1 (1 + x) 2 C νm (x) C νn (x) dx −1 1 23ν− 2 Γ ν + 12 Γ (2ν + m) Γ(2ν + n) Γ ν + 12 + m + n Γ 12 − ν + n − m = m!n! Γ(2ν) Γ 12 − ν Γ ν + 12 + n − m Γ 3ν + 12 + m + n ET II 282(14) [Re ν > 0] 1 1 3 (1 − x)ν− 2 (1 + x)3ν+m+n− 2 C νm (x) C νn (x) dx −1 2 24ν+m+n−1 Γ ν + 12 Γ(2ν + m + n) Γ ν + m + n + 12 Γ 3ν + m + n − 12 = Γ(2ν + n) Γ(4ν + 2m + 2n) Γ ν + m + 12 Γ ν + n + 12 Γ (2ν + m) Re ν > 16 ET II 282(15) 1 ν− 1 (1 − x)α (1 + x) 2 C μm (x) C νn (x) dx −1 1 2α+ν+ 2 Γ(α + 1) Γ ν + 12 Γ ν − α + n − 12 Γ(2μ + m) Γ (2ν + n) = 1 3 Γ(2μ) Γ(2ν) m!n!
Γ ν−α− 2 Γ ν−α+n+ 2 1 3 3 3 × 4 F 3 −m, m + 2μ, α + 1, α − ν + ; μ + , ν + α + n + , α − ν − n + ; 1 2 2 2 2 Re α > −1, Re ν > − 12 ET II 283(16)
1
7.315 −1
1−x
2
12 ν−1
C ν2n (ax) dx
1 π 1/2 Γ 12 ν ν 2 2a2 − 1 = 1 1 Cn Γ 2ν + 2 [Re ν > 0]
ET II 283(19)
Gegenbauer polynomials Cnν (x) and elementary functions
7.323
1
7.316 −1
7.317
1 − x2
1
2
C νn (cos α cos β + x sin α sin β) dx =
μ−1 λ− 12
(1 − x)
1.
ν−1
x
Γ (2λ + n) Γ λ + 12 Γ(μ) (α,β) P n (1 − γ) (1 − γx) dx = Γ(2λ) Γ λ + μ + n + 12 Re λ > −1, λ = 0, − 12 , Re μ > 0 β = λ − μ − 12
α = λ + μ − 12 ,
1
(1 − x)μ−1 xν−1 C λn (1 − γx) dx = 0
1 2ν
7.318
x
1−x
7.319
C νn
1
(1 − x)
μ−1 ν−1
1.
x
C λ2n
2
1/2
1
2. 0
ET II 191(40)a
ET II 283(21)
+ n) Γ(μ) Γ(ν) 1 2 3 F 2 −n, n + λ, ν; , μ + ν; γ n! Γ(λ) Γ(μ + ν) 2
n Γ(λ
dx = (−1)
0
γ 1 × 3 F 2 −n, n + 2λ, ν; λ + , μ + ν; 2 2 [2λ = 0, −1, −2, . . . , Re μ > 0, Re ν > 0]
γx
ET II 190(39)a
Γ(2λ + n) Γ(μ) Γ(ν) n! Γ(2λ) Γ(μ + ν)
Γ(2ν + n) Γ ν + 12 Γ(σ) (α,β) P n (1 − y), 1 − x y dx = 2 Γ(2ν) Γ n + ν + σ + 12 α = ν + σ − 12 , Re ν > − 12 , Re σ > 0 β = ν − σ − 12
2 σ−1
0
22ν−1 n! [Γ(ν)] C νn (cos α) C νn (cos β) Γ(2ν + n) [Re ν > 0] ET II 283(20)
C λn
0
2.
797
[Re μ > 0, Re ν > 0] ET II 191(41)a 1 n (−1) 2γ Γ(μ) Γ(λ + n + 1) Γ ν + 2 (1 − x)μ−1 xν−1 C λ2n+1 γx1/2 dx = 1 n! Γ(λ) Γ μ + ν + 2
1 2 1 3 × 3 F 2 −n, n + λ + 1, ν + 2 ; 2 , μ + ν + ; γ 2 Re μ > 0, Re ν > − 12 ET II 191(42)
7.32 Combinations of Gegenbauer polynomials Cnν (x) and elementary functions
1
7.321
−1
ν− 12
0
π21−ν in Γ(2ν + n) −ν a J ν+n (a) n! Γ(ν) Re ν > − 12 ET II 281(7), MO 99a x ν π Γ(2ν + n) a − 1 e−bx dx = (−1)n C νn e−ab I ν+n (ab) a n! Γ(ν) 2b Re ν > − 12 ET I 171(9)
eiax C νn (x) dx =
1
[x(2a − x)]ν− 2
7.322 7.323
2a
1 − x2
π
C νn (cos ϕ) (sin ϕ)
1. 0
2ν
dϕ = 0 = 2−2ν π Γ(2ν + 1) [Γ(1 + ν)]
[n = 1, 2, 3, . . .] −2
[n = 0] EH I 177(18)
798
Complete System of Orthogonal Step Functions
2.
π
11
C νn (cos ψ cos ψ + sin ψ sin ψ cos ϕ) (sin ϕ)
2ν−1
7.324
dϕ
0
= 22ν−1 n! [Γ(ν)] C νn (cos ψ) C νn (cos ψ ) [Γ(2ν + n)] 2
[Re ν > 0] 7.324
1
1.
1 − x2
ν− 12
C ν2n+1 (x) sin ax dx = (−1)n π
0
2.
1
1 − x2
ν− 12
C ν2n (x) cos ax dx =
0
EH I 177(20)
Γ(2n + 2ν + 1) J 2n+ν+1 (a) (2n + 1)! Γ(ν)(2a)ν Re ν > − 12 , a > 0
(−1)n π Γ(2n + 2ν) J ν+2n (a) (2n)! Γ(ν)(2a)ν Re ν > − 12 ,
−1
a>0
ET I 94(4)
ET I 38(3)a
7.325∗ Complete System of Orthogonal Step Functions Let sj (x) = (−1) 2jx for j ∈ N and cj (x) = (−1) 2jx+1/2 for j ∈ 0 + N where z denotes the integer part of z. Thus, cj (z) and sj (z) have minimal period j −1 and manifest even and odd symmetry about x = 1/2, respectively, and so are the discrete analogues of cos 2πjx and sin 2πjx. Furthermore, for j ∈ N let j denote its odd part: the quotient of j by its highest power-of-two factor. Then for all j and k ∈ N, if (j, k) denotes their highest common factor and [j, k] denotes their lowest common multiple: 1 (j,k) if j/j = k/k sj (x)sk (x) dx = [j,k] 1. 0 otherwise 0 1 if j/j = k/k (−1)(j+k)/2+1 (j,k) [j,k] 2. cj (x)ck (x) dx = 0 otherwise 0
7.33 Combinations of the polynomials C νn (x) and Bessel functions; Integration of Gegenbauer functions with respect to the index 7.331
∞
2n+1−ν
x
1. 1
x −1 2
ν−2n− 12
C ν−2n 2n
1 J ν (xy) dx x −1
−1
= (−1)n 22n−ν+1 y −ν+2n−1 [(2n)!] Γ(2ν − 2n) [Γ(ν − 2n)] cos y y > 0, 2n − 12 < Re ν < 2n + 12 ET II 44(10)a
7.334
7.332
Gegenbauer functions and Bessel functions
∞
1. 0
799
+ − 1 ν− 3 ν+ 12 + 2 −1/2 , 1/2 , x + β2 xν+1 x2 + β 2 2 4 C 2n+1 β J ν+ 32 +2n x2 + β 2 a J ν (xy) dx %
1/2 & + 2 , ν+ 12 y2 n 1/2 −1/2 12 −ν ν 2 −1/2 2 2 1/2 C 2n+1 a y a −y sin β a − y = (−1) 2 π 1− 2 a [0 < y < a] =0 [a < y < ∞]
[a > 0,
Re ν > −1]
Re β > 0,
ET II 59(23) ∞
2. 0
+ − 1 ν− 3 ν+ 1 + −1/2 , 1/2 , J ν+ 12 +2n x2 + β 2 xν+1 x2 + β 2 2 4 C 2n 2 β x2 + β 2 a J ν (xy) dx %
1/2 & + 2 , ν+ 12 y2 n 1/2 −1/2 12 −ν ν 2 −1/2 2 2 1/2 C 2n a y a −y cos β a − y = (−1) 2 π 1− 2 a [0 < y < a] =0 [a < y < ∞]
[a > 0,
Re ν > −1]
Re β > 0,
ET II 59(24)
7.333
π
(sin x)
1.
ν+1
ν+ 12
cos (a cos θ cos x) C n
0 n
= (−1) 2
(cos x) J ν (a sin θ sin x) dx
1/2 2π ν+ 1 ν (sin θ) C n 2 (cos θ) J ν+ 12 +n (a) a
=0
[n = 0, 2, 4, . . .] [n = 1, 3, 5, . . .]
[Re ν > −1] π
2.
(sin x)
ν+1
ν+ 12
sin (a cos θ cos x) C n
WA 414(2)a
(cos x) J ν (a sin θ sin x) dx
0
=0 = (−1)
n−1 2
2π a
[n = 0, 2, 4, . . .]
1/2 ν
ν+ 12
(sin θ) C n
(cos θ) J ν+ 12 +n (a)
[n = 1, 3, 5, . . .]
[Re ν > −1] 7.334
π
1.
(sin x)
2ν
0
2.
π
(sin x) 0
2ν
J ν (ω) π Γ(2ν + n) J ν+n (α) J ν+n (β) C νn (cos x) dx = ν−1 , ων 2 n! Γ(ν) αν βν 1/2 ω = α2 + β 2 − 2αβ cos x n = 0, 1, 2, . . . ; C νn (cos x)
Y ν (ω) π Γ(2ν + n) J ν+n (α) Y ν+n (β) dx = ν−1 , ων 2 n! Γ(ν) αν βν 2 1/2 ω = α + β 2 − 2αβ cos x |α| < |β|,
WA 414(3)a
Re ν > − 12
Re ν −
1 2
ET II 362(29)
ET II 362(30)
800
Complete System of Orthogonal Step Functions
7.335
Integration of Gegenbauer functions with respect to the index c+i∞ −ν −1 7.335 [sin(απ)] tα C να (z) dα = −2i 1 + 2tz + t2 c−i∞
[−2 < Re ν < c < 0,
EH I 178(25)
1 sech(πx) ν − + ix K ν− 12 +ix (a) I ν− 12 +ix (b) C ν− 1 +ix (− cos ϕ) dx 2 2 −∞ 2−ν+1 (ab)ν −ν ω K ν (ω) = Γ(ν) ω = a2 + b2 − 2ab cos ϕ EH II 55(45)
7.336
|arg (z ± 1)| < π]
∞
7.34 Combinations of Chebyshev polynomials and powers
1
7.341 −1
−1 2 [T n (x)] dx = 1 − 4n2 − 1 1
7.342 −1
7.343
1
1. −1
ET II 271(6)
, + 1/2 1/2 1 − z2 U n x 1 − y2 + yz dx =
T n (x) T m (x) √
2 U n (y) U n (z) n+1 [|y| < 1, |z| < 1]
dx =0 1 − x2 π = 2 =π
ET II 275(34)
[m = n] [m = n = 0] [m = n = 0] MO 104
1
2. −1
1 − x2 U n (x) U m (x) dx = 0 =
7.344
1
1. −1
1
2. −1
7.345
1
1. −1
1
2. −1
π 2
[m = n]
ET II 274(28)
[m = n]
ET II 274(27), MO 105a
−1/2 (y − x)−1 1 − y 2 T n (y) dy = π U n−1 (x)
[n = 1, 2, . . .]
EH II 187(47)
1/2 (y − x)−1 1 − y 2 U n−1 (y) dy = −π T n (x)
[n = 1, 2, . . .]
EH II 187(48)
(1 − x)−1/2 (1 + x)m−n− 2 T m (x) T n (x) dx = 0
[m > n]
ET II 272(10)
3
(1 − x)−1/2 (1 + x)m+n− 2 T m (x) T n (x) dx = 3
π(2m + 2n − 2)! − 1)!(2n − 1)!
2m+n (2m
[m + n = 0]
ET II 272(11)
7.349
Chebyshev polynomials and powers
1
3. −1
1
4. −1
5.
1
1
π(2m + 2n + 2)! + 1)!(2n + 1)!
3
(1 − x)1/2 (1 + x)m+n+ 2 U m (x) U n (x) dx =
2m+n+2 (2m
1
(1 − x)1/2 (1 + x)m−n− 2 U m (x) U n (x) dx = 0
[m > n]
801
ET II 274(31)
ET II 274(30)
25/2 (m + 1)(n + 1) (1 − x)(1 + x)1/2 U m (x) U n (x) dx = m + n + 32 m + n + 52 [1 − 4(m − n)2 ] −1 ET II 274(29)
6.
1
7.
−1/2
(1 − x)α−1 T m (x) T n (x) dx 1 π 1/2 2α− 2 Γ(α) Γ n − α + 12 1 1 1 1 = −m, m, α, α + ; , α + n + , α − n + F 4 3 1 2 2 2 Γ 2 −α Γ α+n+ 2 [Re α > 0] ET II
(1 + x) −1
(1 + x)
1/2
−1
1
xs−1 T n (x) √
0
7.347
−1
−1
dx π = s 1 1 2 s2 B 2 + 2 s + 12 n, 12 + 12 s − 12 n 1−x [Re s > 0] β
(1 − x)α (1 + x) T n (x) dx =
1 ;1 2
275(32)
ET II 324(2)
2α+β+2n+1 (n!) Γ(α + 1) Γ (β + 1)
(2n)! Γ(α + β + 2) 1 × 3 F 2 −n, n, α + 1; , α + β + 2; 1 2 [Re α > −1, Re β > −1]
ET II 271(2)
2
1
2.
272(12)
2
1
1.
(1 − x)α−1 U m (x) U n (x) dx
1 π 1/2 2α− 2 (m + 1) (n + 1) Γ(α) Γ n − α + 32 3 3 = Γ 2 −α Γ 2 +α+n 3 1 3 × 4 F 3 −m, m + 2, α, α − ; , α + n + , α − n − 2 2 2 [Re α > 0] ET II
7.346
1 ;1 2
β
(1 − x)α (1 + x) U n (x) dx =
2α+β+2n+2 [(n + 1)!] Γ(α + 1) Γ (β + 1)
(2n + 2)! Γ(α + β + 2) 3 × 3 F 2 −n, n + 1, α + 1; , α + β + 2; 1 2 ET II 273(22)
1
7.348 7.349
−1 1
−1
1 − x2
1 − x2
−1/2
−1/2
U 2n (xz) dx = π P n 2z 2 − 1
[|z| < 1]
1 T n 1 − x2 y dx = π [P n (1 − y) + P n−1 (1 − y)] 2
ET II 275(33) ET II 222(14)
802
Complete System of Orthogonal Step Functions
7.351
7.35 Combinations of Chebyshev polynomials and elementary functions
1
7.351 0
− 1 2a x−1/2 1 − x2 2 e− x T n (x) dx = π 1/2 D n− 12 2a1/2 D −n− 12 2a1/2 [Re a > 0]
7.352
∞
1. 0
∞
2.
+ −1/2 , x U n a a2 + x2
a−n a+1 −n−1 −2 dx = ζ n + 1, 1 n+1 2n 2 (a2 + x2 ) 2 (eπx + 1) [Re a > 0]
+ −1/2 , x U n a a2 + x2 (a2 + x2 )
0
1 2 n+1
(e2πx − 1)
dx =
2.
ET II 276(40)
+ , 1 a+1 a+3 2 2 −1/2 1−2n πx T n a a + x ζ n, dx = 2 − ζ n, 2 4 4 0
a + 1 = 21−n Φ −1, n, 2 [Re a > 0] ET II 273(19)
−2
∞ + , 2 1 2 1 a+1 2 −2n 2 −1/2 −1 1−n πx a +x Tn a a + x ζ n + 1, cosh dx = π n2 2 2 0 ∞
2 − 1 n a + x2 2 sech
[Re a > 0] 7.354
ET II 275(39)
a−n−1 a−n 1 ζ(n + 1, a) − − 2 4 2n [Re a > 0]
7.353 1.
ET II 272(13)
+
1
1.
sin(xyz) cos −1
1
sin(xyz) sin −1
1 − y2
1/2 , z T 2n+1 (x) dx = (−1)n π T 2n+1 (y) J 2n+1 (x)
+ 1/2 1/2 , 1/2 1 − y2 1 − x2 z U 2n+1 (x) dx = (−1)n π 1 − y 2 U 2n+1 (y) J 2n+2 (z) ET II 274(25)
1
3.
cos(xyz) cos −1
1/2
ET II 271(4)
2.
1 − x2
ET II 273(20)
1
4.
cos(xyz) sin −1
+ 1/2 1/2 , 1 − y2 1 − x2 z T 2n (x) dx = (−1)n π T 2n (y) J 2n (z) +
1 − x2
1/2
1 − y2
ET II 271(5)
1/2 , 1/2 z U 2n (x) dx = (−1)n π 1 − y 2 U 2n (y) J 2n+1 (z) ET II 274(24)
7.355
1
1. 0
2.
0
1
π dx T 2n+1 (x) sin ax √ = (−1)n J 2n+1 (a) 2 1 − x2 T 2n (x) cos ax √
π dx = (−1)n J 2n (a) 2 2 1−x
[a > 0]
ET I 94(3)a
[a > 0]
ET I 38(2)a
7.374
Hermite polynomials
803
7.36 Combinations of Chebyshev polynomials and Bessel functions
1
7.361
2 −1/2
1−x
0
∞
7.362
1 T n (x) J ν (xy) dx = π J 12 (ν+n) 2
2 − 1 x − 1 2 Tn
1
1 π W K 2μ (ax) dx = x 2a
1 2 n,μ
1 1 y J 12 (ν−n) y 2 2 [y > 0, Re ν > −n − 1]
ET II 42(1)
(a) W − 12 n,μ (a) [Re a > 0]
ET II 366(17)a
7.37–7.38 Hermite polynomials
x
7.371
0
H n (y) dy = [2(n + 1)]−1 [H n+1 (x) − H n+1 (0)] 1
7.372
−1
7.373
1.
x
1 − t2
α− 12
H 2n
EH II 194(27)
1
√ (−1)n π 1/2 (2n)! Γ α + 2 Lα n (x) xt dx = Γ(n + α + 1) Re a > − 12
e−y H n (y) dy = H n−1 (0) − e−x H n−1 (x) 2
2
EH II 195(34)
[see 8.956]
EH II 194(26)
0
∞
2. −∞
7.374 1.
∞
−∞
∞
2.11 −∞
e−x H 2m (xy) dx = 2
m √ (2m)! 2 y −1 π m!
e−x H n (x) H m (x) dx = 0 2
EH II 195(28)
[m = n]
√ = 2n · n! π
SM II 567
[m = n]
2 m+n−1 m n e−2x H m (x) H n (x) dx = (−1) 2 + 2 2 2 Γ
m+n+1 2
[m + n is even]
=0
[m + n is odd] ET II 289(10)a
∞
3. −∞ ∞
4. −∞ ∞
5.
e−x H m (ax) H n (x) dx = 0 2
e−x H 2m+n (ax) H n (x) dx = 2
e −∞
SM II 568
∞
6. −∞
−2α2 x2
[m < n]
ET II 290(20)a
m √ n (2m + n)! 2 a − 1 an π2 m!
ET II 291(21)a
m+n−1 2
−m−n−1
m+n 2
m+n+1 Γ
2
1 − 2α α α2 1−m−n ; 2 × 2 F 1 −m, n; 2 2α − 1 Re α2 > 0, α2 = 12 , m + n is even
H m (x) H n (x) dx = 2
e−(x−y) H n (x) dx = π 1/2 y n 2n 2
2
ET II 289(12)a
ET II 288(2)a, EH II 195(31)
804
Complete System of Orthogonal Step Functions
∞
7. −∞
∞
8. −∞
∞
7.375
2 −2y 2 e−(x−y) H m (x) H n (x) dx = 2n π 1/2 m!y n−m Ln−m m %
n 2 e−(x−y) H n (αx) dx = π 1/2 1 − α2 2 H n
αy
[m ≤ n] &
BU 148(15), ET II 289(13)a ET II 290(17)a
1/2
(1 − α2 )
e−(x−y) H m (αx) H n (αx) dx 2
9. −∞
min(m,n)
= π 1/2
k=0
m 2k k! k
% &
m+n n αy −k 1 − α2 2 H m+n−2k 1/2 k (1 − α2 ) ET II 291(26)a
∞
e−
10.
(x−y)2 2u
−∞
7.375 1.
∞
+ , n H n (x) dx = (2πu)1/2 (1 − 2u) 2 H n y(1 − 2u)−1/2 0 ≤ u < 12
e−2x H k (x) H m (x) H n (x) dx = π −1 2 2 (m+n+k−1) Γ(s − k) Γ(s − m) Γ(s − n) 2
−∞
1
2s = k + m + n + 1
∞
EH II 195(30)
[k + m + n is even]
ET II 290(14)a
m+n+k
e−x H k (x) H m (x) H n (x) dx = 2
2. −∞
2 2 π 1/2 k!m!n! , (s − k)!(s − m)!(s − n)! 2s = m + n + k
[k + m + n is even] ET II 290(15)a
7.376
∞
eixy e−
1. −∞
∞
2.
x2 2
H n (x) dx = (2π)1/2 e−
y2 2
H n (y)in
e−2αx xν H 2n (x) dx = (−1)n 22n− 2 − 2 ν 2
3
1
Γ
0
∞
3.
e−2αx xν H 2n+1 (x) dx = (−1)n 22n− 2 ν 2
1
Γ
0
MO 165a
ν+1
Γ n + 12 ν+1 1 1 2 ; ; F −n, √ 1 (ν+1) 2 2 2α πα 2
ν 2
[Re α > 0, Re ν > −1] BU 150(18a)
+ 1 Γ n + 32 3 1 ν F −n, + 1; ; √ 1 ν+1 2 2 2α 2 πα [Re α > 0,
∞
−∞
7.378 0
∞
BU 150(18b)
e−x H m (x + y) H n (x + z) dx = 2n π 1/2 m!z n−m Ln−m (−2yz) m 2
7.3778
Re ν > −2]
[m ≤ n]
xα−1 e−βx H n (x) dx = 2n
n
2 n! Γ(α + n − 2m)
m=0
m!(n − 2m)!
[Re α > 0, if n is even;
ET II 292(30)a
(−1)m 2−2m β 2m−α−n
Re α > −1, if n is odd;
Re β > 0]
ET I 172(11)a
7.385
7.379
∞
−∞ ∞
2. −∞
xe−x H 2m+1 (xy) dx = π 1/2 2
1.
ν
−∞
7.382
m (2m + 1)! 2 y y −1 m!
EH II 195(28)
2
∞
∞
805
xn e−x H n (xy) dx = π 1/2 n!Pn (y)
7.381
Hermite polynomials
(x ± ic) e
−x2
H n (x) dx = 2
EH II 195(29)
n−1−ν 1/2 Γ
π
n−ν 2
Γ(−ν)
exp ± 12 π(ν + n)i
ET II 288(3)a [c > 0] + √ , 2 2 1 −1 −x x−1 x2 + a2 e H 2n+1 (x) dx = (−2)n π 1/2 a−2 2ν n! − (2n + 1)!e 2 a D −2n−2 a 2
0
ET II 288(4)a
7.383
∞
1.
e−xp H 2n+1
√ 3 x dx = (−1)n 2n (2n + 1)!!π 1/2 (p − 1)n p−n− 2
0
∞
2. 0
∞
3. 0
∞
4.
[Re p > 0] EF 151(261)a, ET I 172(12)a √ (2n + 1)! (b − α)n e−(b−βx) H 2n+1 (α − β)x dx = (−1)n π α − β 3 n! (b − β)n+ 2 [Re(b − β) > 0] √ (2n)! (b − α)n 1 √ e−(b−β)x H 2n (α − β)x dx = (−1)n π n! (b − β)n+ 12 x
ET I 172(15)a
[Re(b − β) > 0]
ET I 172(16)a
xa− 2 n−1 e−bx H n 1
0
√ x dx = 2n Γ(a)b−a 2 F 1 − 12 n, 12 − 12 n; 1 − a; b Re a > 12 n, if n is even, Re a > 12 n − 12 , if n is odd, If a is even, only the first 1 +
:n; 2
terms are kept in the series for
2F 1
⎦
ET I 172(14)a
∞
5.
x−1/2 e−px H 2n
0
∞
7.384 0
√ 1 x dx = (−1)n 2n (2n − 1)!!π 1/2 (p − 1)n p−n− 2
1. 0
∞
MO 177a
⎛ ⎞ √
√ n α+ x a− x 1 2π α ⎠ √ e−bx H n 1 − λ−2 b−1 2 H n ⎝ < + Hn dx = λ λ b x 2 λ −1 b
[Re b > 0] 7.385
Re b > 0, ⎤
1 + , √ (2n)! Γ b + e−bx n 2n 2 √ x Ln (s) π s (1 − e−x ) dx = (−1) 2 H 2n Γ(n + b + 1) n e −1 Re b > − 12
ET I 173(17)a
ET I 174(23)a
806
Complete System of Orthogonal Step Functions
∞
2.
+√ √ , √ (2n + 1)! Γ(b) b Ln (s) s 1 − e−x dx = (−1)n 22n πs Γ n + b + 32
e−bx H 2n+1
0
∞
7.386
x−
n+1 2
7.386
q2
e− 4x H n
0
q √ 2 x
[Re b > 0] e−px dx = 2n π 1/2 p
n−1 2
e−q
ET I 174(24)a
√ p
EF 129(117)
7.387 ∞ √ 2 1 1 2 1. e−x sinh 2βx H 2n+1 (x) dx = 2n− 2 π 1/2 β 2n+1 e 2 β 0 ∞ √ 2 1 2 2. e−x cosh 2βx H 2n (x) dx = 2n−1 π 1/2 β 2n e 2 β
ET II 289(7)a ET II 289(8)a
0
7.388 1.
∞
e−x sin 2
√
1 1 2 2βx H 2n+1 (x) dx = (−1)n 2n− 2 π 1/2 β 2n+1 e− 2 β
√
n+ 12 − 1 β 2 2βx H 2n+1 (ax) dx = (−1)n 2−1 π 1/2 a2 − 1 e 2 H 2n+1
ET II 288(5)a
0
∞
2.
e
−x2
sin
ET II 290(18)a ∞
3.
√ 2 1 2 e−x cos 2βx H 2n (x) dx = (−1)n 2n−1 π 1/2 β 2n e− 2 β
0
∞
4.
e
−x2
0
ET II 289(6)a
% & √ aβ −1 1/2 2 n − 12 β 2 1−a cos 2βx H 2n (ax) dx = 2 π e H 2n √ 1/2 2 (a2 − 1) ET II 290(19)a
∞
5.
e−y [H n (y)] cos 2
2
√
2βy dy = π 1/2 2n−1 n!e−
β2 2
Ln β 2
EH II 195(33)
0
∞
6.11
√ 2 b2 e−x sin(bx) H n (x) H n+2m+1 (x) dx = 2n−1 (−1)m πn!b2m+1 e− 4 L2m+1 n
0
∞
7.
e−x cos(bx) H n (x) H n+2m (x) dx = 2n− 2 2
1
0
√ 1/2 2 (a2 − 1)
0
aβ
π
7.389 0
[b > 0] b2 π n!(−1)m b2m e− 4 L2m n 2
2
b 2
b2 2
[b > 0] + , (2n)! n 1/2 2 dx = 2−n (−1)n π (cos x) H 2n a (1 − sec x) 2 [H n (a)] (n!)
ET I 39(11)a
ET I 39(11)a ET II 292(31)
7.39 Jacobi polynomials 7.391 1.
1
−1
β
(1 − x)α (1 + x) P (α,β) (x) P (α,β) (x) dx n m =0
[m = n,
Re α > −1,
Re β > −1]
[m = n,
Re α > −1,
Re β > −1]
α+β+1
=
Γ(α + n + 1) Γ (β + n + 1) 2 n! (α + β + 1 + 2n) Γ(α + β + n + 1)
ET II 285(5, 9)
7.391
Jacobi polynomials
1
2. −1
σ
(1 − x)ρ (1 + x) P (α,β) (x) dx = n
807
2ρ+σ+1 Γ(ρ + 1) Γ(σ + 1) Γ(n + 1 + α) n! Γ(ρ + σ + 2) Γ(1 + α) × 3 F 2 (−n, α + β + n + 1, ρ + 1; α + 1, ρ + σ + 2; 1)
[Re ρ > −1, 1
3.6 −1
1
−1
5.
(1 − x)ρ (1 + x)β P (α,β) (x) dx = n
7.
Re σ > −1]
ET II 284(1)
2β+ρ+1 Γ(ρ + 1) Γ(β + n + 1) Γ(α − ρ + n) n! Γ(α − ρ) Γ(β + ρ + n + 2) [Re ρ > −1,
Re β > −1]
ET II 284(2)
,2 + 2α+β Γ(α + n + 1) Γ(β + n + 1) (1 − x)α−1 (1 + x)β P (α,β) (x) dx = n n!α Γ(α + β + n + 1) −1 1
ET II 285(6) [Re α > 0, Re β > −1] 2 1 + ,2 24α+β+1 Γ α + 12 [Γ(α + n + 1)] Γ(β + 2n + 1) (1 − x)2α (1 + x)β P (α,β) (x) dx = √ n 2 π (n!) Γ(α + 1) Γ(2α + β + 2n + 2) −1 Re α > − 12 , Re β > −1 ET II 285(7) 1 β (1 − x)ρ (1 + x) P (α,β) (x) P (ρ,β) (x) dx n n
6.
ET II 284(3)
2α+σ+1 Γ(σ + 1) Γ(α + 1) Γ(σ − β + 1) n! Γ(σ − β − n + 1) Γ(α + σ + n + 2) [Re α > −1,
4.
(1 − x)α (1 + x)σ P (α,β) (x) dx = n
Re σ > −1]
−1
=
1
8. −1
(1 − x)ρ−1 (1 + x)β P (α,β) (x) P (ρ,β) (x) dx = n n
2ρ+β Γ(α + n + 1) Γ(β + n + 1) Γ(ρ) n! Γ(α + 1) Γ(ρ + β + n + 1) [Re β > −1,
1
9.7 −1
−1
ET II 286(11)
σ
1
10.6
Re ρ > 0]
(1 − x)α (1 + x) P (α,β) (x) P (α,σ) (x) dx n m =
2ρ+β+1 Γ(ρ + n + 1) Γ(β + n + 1) Γ (α + β + 2n + 1) n! Γ(β + ρ + 2n + 2) Γ(α + β + n + 1) [Re ρ > −1, Re β > −1] ET II285(10)
2α+σ+1 Γ(α + n + 1) Γ (α + β + m + n + 1) Γ(σ + m + 1) Γ(σ − β + 1) m!(n − m)! Γ (α + β + n + 1) Γ(α + σ + m + n + 2) Γ (α − β + m − n + 1) [Re α > −1, Re σ > −1] ET II 286(12)
β
(1 − x)ρ (1 + x) P (α,β) (x) P (ρ,β) (x) dx n m
2β+ρ+1 Γ(α + β + m + n + 1) Γ (β + n + 1) Γ(ρ + m + 1) Γ (α − ρ − m + n) m!(n − m)! Γ(α + β + n + 1) Γ (β + ρ + m + n + 2) Γ(α − ρ) [Re β > −1, Re ρ > −1] ET II 287(16) x , 1 + (α+1,β+1) (α+1,β+1) P n−1 (1 − y)α (1 + y)β P (α,β) (y) dy = (0) − (1 − x)α+1 (1 + x)β+1 P n−1 (x) n 2n 0 =
11.
EH II 173(38)
808
7.392
Complete System of Orthogonal Step Functions
7.392
1
xλ−1 (1 − x)μ−1 P (α,β) (1 − γx) dx n
1. 0
=
Γ(α + n + 1) Γ(λ) Γ(μ) 1 γ −n, n + α + β + 1, λ; α + 1, λ + μ; 3F 2 n! Γ (α + 1) Γ(λ + μ) 2 [Re λ > 0, Re μ > 0] ET II 192(46)a
1
xλ−1 (1 − x)μ−1 P (α,β) (γx − 1) dx n
2. 0
+ n + 1) Γ(λ) Γ(μ) 1 3 F 2 −n, n + α + β + 1, λ; β + 1, λ + μ; γ a n! Γ (β + 1) Γ(λ + μ) 2 [Re λ > 0, Re μ > 0] ET II 192(47)a
n Γ(β
= (−1)
1
xα (1 − x)μ−1 P (α,β) (1 − γx) dx = n
3. 0
Γ(α + n + 1) Γ(μ) (α+μ,β−μ) P (1 − γ) Γ(α + μ + n + 1) n [Re a > −1,
1
xβ (1 − x)μ−1 P (α,β) (γx − 1) dx = n
4. 0
1
1.
2 ν
1−x
(ν,ν) sin bx P 2n+1 (x) dx
=
1
1 − x2
ν
(ν,ν)
cos bx P 2n (x) dx =
Re μ > 0]
ET II 191(44)a
√ (−1)n π Γ(2n + ν + 2) J 2n+ν+ 32 (b) 2 2 −ν (2n + 1)!bν+ 2 1
0
2.
ET II 191(43)a
Γ(β + n + 1) Γ(μ) (α−μ,β+μ) P (γ − 1) Γ(β + μ + n + 1) n [Re β > −1,
7.393
Re μ > 0]
(−1)n 2
ν− 12
1
[b > 0, Re ν > −1] √ π Γ(2n + ν + 1) J 2n+ν+ 12 (b)
ET I 94(5)
1
(2n)!bν+ 2
0
[b > 0,
Re ν > −1]
ET I 38(4)
7.41–7.42 Laguerre polynomials 7.411
t
Ln (x) dx = Ln (t) − Ln+1 (t)/(n + 1)
1. 0
t α α Lα n (x) dx = Ln (t) − Ln+1 (t) −
2. 0
t
Lα+1 n−1 (x) dx
3.
=
− Lα n (t)
+
0
n+α n
n+α n
MO 110
+
n+1+α n+1
EH II 189(16)a EH II 189(15)a
t
Lm (x) Ln (t − x) dx = Lm+n (t) − Lm+n+1 (t)
4.
EH II 191(31)
0
5.
2 ∞ t Lk (x) dx = et − 1 k! 0
k=0
[t ≥ 0]
MO 110
7.414
7.412
Laguerre polynomials
1
(1 − x)μ−1 xα Lα n (ax) dx =
1. 0
809
Γ(α + n + 1) Γ(μ) α+μ L (a) Γ(α + μ + n + 1) n [Re α > −1,
Re μ > 0] EH II 191(30)a, BU 129(14c)
1
(1 − x)μ−1 xλ−1 Lα n (βx) dx =
2. 0
Γ(α + n + 1) Γ(λ) Γ(μ) 2 F 2 (−n, λ; α + 1, λ + μ : β) n! Γ(α + 1) Γ(λ + μ) [Re λ > 0,
1
0
7.414 1.11
∞
α −y Ln (y) − Lα e−x Lα n (x) dx = e n−1 (y)
y
∞
2.
e
−bx
0
3.8
2 b − (λ + μ)b + 2λμ (b − λ − μ)n Ln (λx) Ln (μx) dx = Pn bn+1 b(b − λ − μ)
α e−x xα Lα n (x) Lm (x) dx = 0
0
=
EH II 191(29)
[Re b > 0] ∞
∞
Γ(α + n + 1) n!
ET I 175(34)
[m = n,
Re α > −1]
BU 115(8), ET II 293(3)
[m = n,
Re α > 0]
BU 115(8), ET II 292(2)
Γ(m + n + α + 1) (b − λ)n (b − μ)m bm+n+α+1 m!n! 0 b(b − λ − μ) × F −m, −n; −m − n − α, (b − λ)(b − μ) [Re α > −1, Re b > 0] ∞ Γ(α + n + 1)2 Γ(α + m + 1) Γ α + 32 Γ m − 12 α −x α+1/2 α 9 4(1) . e x Ln (x) Lm (x) dx = n!m! Γ(α + 1) Γ − 12 0 × 3 F 2 −n, α + 32 , 32 ; α + 1, 32 − m; 1 4.
α e−bx xα Lα n (λx) Lm (μx) dx =
∞
e
5. 0
−bx
Lan (x) dx
n a + m − 1 (b − 1)n−m
=
m=0 ∞
6.
m
bn−m+1
e−bx Ln (x) dx = (b − 1)n b−n−1
0
∞
7.
e−st tβ Lα n (t) dt =
Γ(β + 1) Γ(α + n + 1) −β−1 s F n! Γ(α + 1)
e−st tα Lα n (t) dt =
Γ(α + n + 1)(s − 1) n!sα+n+1
0
8. 0
ET II 192(50)a
(m + n)! Γ(α + m + 1) Γ(β + n + 1) α+β+1 Lm+n (y) m!n! Γ(α + β + m + n + 2) [Re α > −1, Re β > −1] ET II 293(7)
β xα (1 − x)β Lα m (xy) Ln [(1 − x)y] dx =
7.413
Re μ > 0]
ET I 175(35)
[Re b > 0]
ET I 174(27)
[Re b > 0]
ET I 174(25)
1 −n, β + 1; α + 1; s [Re β > −1, Re s > 0] BU 119(4b), EH II 191(133)
∞
n
[Re α > −1,
Re s > 0] EH II 191(32), MO 176a
810
Complete System of Orthogonal Step Functions
∞
9.
e
−x α+β
x
β Lα m (x) Ln (x) dx
m+n
= (−1)
0
∞
10.6
e−bx x2a [Lan (x)] dx = 2
0
∞
11.
e−x xγ−1 Lμn (x) dx =
0
∞
12.
e−x(s+
0
a1 +a2 2
α+m (α + β)! n
β+n m
[Re(α + β) > −1]
22a Γ a + 12 Γ n + 12
ET II 293(4)
2
2a+1 π (n!) b
2 2 1 1 Γ(a + n + 1) × F −n, a + ; − n; 1 − 2 2 b 1 Re a > − , Re b > 0 2
Γ(γ) Γ(1 + μ + n − γ) n! Γ(1 + μ − γ)
[Re γ > 0]
ET I 174(30) BU 120(4b)
) xμ+β Lμ (a x) Lμ (a x) dx 1 2 k k
⎤⎫ ⎧ ⎡ 1+μ+β μ+β A2 ⎬ ⎨ F , 1 + ; 1 + μ; k 2 2 2 B d ⎣ Γ(1 + μ + β) Γ(1 + μ + k) ⎦ = k 1+μ 1+μ+β ⎭ ⎩ dh k!k! Γ (1 + μ) (1 − h) B 4a1 a2 h A = ; (1 − h)2 2
13.
7.415
h=0
a1 + a 2 1 + h B =s+ 1−h 2
a1 + a2 BU 142(19) > 0, a1 > 0, a2 > 0, Re(μ + β) > −1 Re s + 2 2
∞ b1 a1 + a2 Γ(1 + μ + k) bk0 (μ,0) · P exp −x s + · xμ Lμk (a1 x) Lμk (a2 x) dx = k 1+μ+k 2 k! b 0 b2 b0 0 a1 + a2 a1 + a2 b0 = s + , b21 = b0 b2 + 2a1 a2 , b2 = s − 2 2
a1 + a 2 Re μ > −1, Re s + >0 2 BU 144(22)
1
Γ(α + n + 1) B(λ, μ) 2 F 2 (α + n + 1, λ; α + 1, λ + μ; −β) n! Γ(α + 1) 0 [Re λ > 0, Re μ > 0] ET II 193(51)a
∞ 1/2 n+m iy iy (2π) 1 2 n−m − 2 √ √ i x dx = 7.416 xm−n exp − (x − y)2 Lm−n 2 H H n m n 2 n! 2 2 −∞ 7.415
(1 − x)μ−1 xλ−1 e−βx Lα n (βx) dx =
BU 149(15b), ET II 293(8)a
7.417 1.
∞
0
2. 0
xν−2n−1 e−ax sin(bx) Lν−2n−1 (ax) dx = (−1)n i Γ(ν) 2n
b2n [(a − ib)−ν − (a + ib)−ν ] 2(2n)! [b > 0, Re a > 0, Re ν > 2n] ET I 95(12)
∞
−ν
−ν
[(a + ib) + (a − ib) ] 2(2n + 1)! [b > 0, Re a > 0, Re ν > 2n + 1]
xν−2n−2 e−ax sin(bx) Lν−2n−2 (ax) dx = (−1)n+1 Γ(ν) 2n+1
b
2n+1
ET I 95(13)
7.421
Laguerre polynomials
∞
3.
n+1 xν−2n e−ax cos(bx) L2n−1 Γ(ν) ν−2n (ax) dx = i(−1)
0
811
b2n−1 [(a − ib)−ν − (a + ib)−ν ] 2(2n − 1)! [b > 0, Re a > 0, Re ν > 2n − 1] ET I 39(12)
∞
4.
xν−2n−1 e−ax cos(bx) Lν−2n−1 (ax) dx = (−1)n Γ(ν) 2n
0
b2n [(a + ib)−ν + (a − ib)−ν ] 2(2n)! [b > 0, Re ν > 2n, Re a > 0] ET I 39(13)
7.418
∞
1. 0
1 2 1 i 2 2 [D −n−1 (ib)] − [D −n−1 (−ib)] e− 2 x sin(bx) Ln x2 dx = (−1)n n! √ 2 2π
[b > 0]
2 ∞ 2 b π −1 − 12 b2 −n − 12 x2 (n!) e e cos(bx) Ln x dx = 2 Hn √ 2 2 0 [b > 0]
∞ 1 2 π 2n+1 − 1 b2 n+ 12 b2 n+ 12 2n+1 − 12 x2 2 x b x e sin(bx) Ln e Ln dx = 2 2 2 0
2.
3.
∞
4.
n− 12
x2n e− 2 x cos(bx) Ln 2
1
0
∞
5.
xe− 2 x Lα n 2
1
0
∞
6.
e− 2 x Lα n 2
1
0
1 2 x Ln 2
1 2 −α
1 2 − 1 −α x Ln 2 2
1 2 x 2
1 2 x 2
ET I 95(14)
dx =
ET I 39(14)
[b > 0]
π 2n − 1 b2 n+ 12 1 2 2 b e b Ln 2 2
ET I 95(15)
[b > 0] ET I 39(16)
π 1/2 1 1 2 1 2 − 12 y 2 α 2 −α y Ln y ye Ln sin(xy) dx = 2 2 2
π 1/2 1 2 1 2 x cos(xy) dx = e− 2 y L α n 2 2
1 2 −α− 1 y Ln 2 2
ET II 294(11)
1 2 y 2
ET II 294(12)
∞
7.419 0
7.421
∞
1. 0
2. 0
1
xn+2ν− 2 exp[−(1 + a)x] L2ν n (ax) K ν (x) dx
π 1/2 Γ n + ν + 12 Γ n + 3ν + 12 1 1 1 F n + ν + , n + 3ν + ; 2ν + 1; − a = 1 2 2 2 2n+2ν+ 2 n! Γ (2ν + 1) 1 1 Re a > −2, Re(n + ν) > − 2 , Re(n + 3ν) > − 2 ET II 370(44)
xe− 2 αx Ln 1
2
1 2 βy 2 (α − β)n − 1 y2 2α βx J 0 (xy) dx = e L n 2 αn+1 2α(β − α) [y > 0,
∞
2 2−2n−1 2n − 1 y2 y e 4 xe−x Ln x2 J 0 (xy) dx = n!
Re α > 0]
ET II 13(4)a ET II 13(5)
812
Hypergeometric Functions
∞
3.
2n+ν+1 − 12 x2
x
e
Lν+n n
0
7.422
1 2 1 2 2n+ν − 12 y 2 ν+n x J ν (xy) dx = y y e Ln 2 2
Re ν > −1] ∞ y2 2 αy 2 xν+1 e−βx Lνn αx2 J ν (xy) dx = 2−ν−1 β −ν−n−1 (β − α)n y ν e− 4β Lνn 4β(α − β) 0 [y > 0,
4.
∞
5.
e− 2q x xν+1 Lνn 2
1
0
6.∗
x2 q n+ν+1 − qy2 ν ν e 2 y Ln J ν (xy) dx = 2q(1 − q) (q − 1)n
∞
∞
1.
xν+1 e−βx
2
y 2
y2 yν Γ n + 1 + 12 ν (2β)−ν−1 e− 4β πn!
2l n (−1)l Γ n − l + 12 Γ l + 12 αy 2 2α − β ν × L 2l β 2β(2α − β) Γ l + 1 + 12 ν (n − l)! l=0 [y > 0,
∞
9
MO 183
+ 1 ,2 ν Ln2 αx2 J ν (xy) dx =
2 σ 2 2 αx Ln αx J ν (xy) dx xν+1 e−αx Lν−σ m
∞
1.
e− 2 x Ln 1
2
0
∞
2.
e
− 12 x2
0
Ln
Re α > 0,
Re ν > −1,
y2 4α σ = 0,
y2
= (−1)m+n (2α)−ν−1 y ν e− 4α Lm−n−σ n [y > 0,
Re ν > −1]
Re β > 0,
0
7.423
ET II 43(5)
2
2 1 y 2n+ν − 1 y2 xν+1 e−x Lνn x2 J ν (xy) dx = e 4 2n! 2
0
2.
[ν > 0]
0
7.422
MO 183
n = 0,
ET II 43(7)
Ln−m+σ−ν m α = 1]
y2 4α
ET II 43(8)
π 1/2 1 2 1 2 x 1 2 y √ √ x H 2n+1 y H 2n+1 e− 2 y Ln sin(xy) dx = 2 2 2 2 2 2 2
1 2 x H 2n 2
x √ 2 2
cos(xy) dx =
π 1/2 2
e
− 12 y 2
Ln
1 2 y H 2n 2
ET II 294(13)a
y √ 2 2
ET II 294(14)a
7.5 Hypergeometric Functions 7.51 Combinations of hypergeometric functions and powers
7.511 0
∞
Γ(a + s) Γ(b + s) Γ(c) Γ(−s) Γ(a) Γ(b) Γ(c + s) [c = 0, −1, −2, . . . , Re s < 0, Re(a + s) > 0,
F (a, b; c; −z)z −s−1 dx =
Re(b + s) > 0]
EH I 79(4)
7.512
Hypergeometric functions and powers
7.512
1
1. 0
α α Γ 1+ Γ(γ) Γ (α − γ + 1) Γ γ − − β 2 2 xα−γ (1 − x)γ−β−1 F (α, β; γ; x) dx = α α Γ (1 + α) Γ 1 + − β Γ γ − 2 + 2 , α Re α + 1 > Re γ > Re β, Re γ − − β > 0 ET II 398(1) 2
1
Γ(γ) Γ(ρ) Γ(β − γ + 1) Γ(γ − ρ + n) Γ(γ + n) Γ(γ − ρ) Γ(β − γ + ρ + 1) [n = 0, 1, 2 . . . ; Re ρ > 0, Re(β − γ) > n − 1]
ET II 398(2)
Γ(γ) Γ(ρ) Γ(β − ρ) Γ(γ − α − ρ) Γ(β) Γ(γ − α) Γ(γ − ρ) [Re ρ > 0, Re(β − ρ) > 0, Re(γ − α − ρ) > 0]
ET II 399(3)
Γ(γ) Γ(ρ) Γ(γ + ρ − α − β) Γ(γ + ρ − α) Γ(γ + ρ − β) [Re γ > 0, Re ρ > 0, Re(γ + ρ − α − β) > 0]
ET II 399(4)
Γ(ρ) Γ(σ) 3 F 2 (α, β, ρ; γ, ρ + σ; 1) Γ(ρ + σ) [Re ρ > 0, Re σ > 0, Re(γ + σ − α − β) > 0]
ET II 399(5)
xρ−1 (1 − x)β−γ−n F (−n, β; γ; x) dx =
2. 0
1
xρ−1 (1 − x)β−ρ−1 F (α, β; γ; x) dx =
3. 0
1
xγ−1 (1 − x)ρ−1 F (α, β; γ; x) dx =
4. 0
1
xρ−1 (1 − x)σ−1 F (α, β; γ; x) dx =
5. 0
1
6.10 0
7.
813
zx xλ−1 (1 − x)β−λ−1 F α, β; λ; dx = B(λ, β − λ)(1 − z/b)−α b
BU 9
1
xγ−1 (1 − x)δ−γ−1 F (α, β; γ; xz) F (δ − α, δ − β; δ − γ; (1 − x)ζ) dx
11 0
= [0 < Re γ < Re δ,
1 γ−1
8.
x
(1 − x)
−1
(1 − xz)
0
9.
1
−δ
Γ(γ) Γ(δ − γ) (1 − ζ)α+β−δ F (α, β; δ; z + ζ − zζ) Γ(δ) |arg(1 − z)| < π, |arg(1 − ζ)| < π] ET II 400(11)
(1 − x)z F (α, β; γ; xz) F δ, β − γ; ; dx (1 − xz) Γ(γ) Γ() F (α + δ, β; γ + ; z) = Γ(γ + ) [Re γ > 0, Re > 0, |arg(z − 1)| < π] ET II 400(12), Eh I 78(3)
xγ−1 (1 − x)ρ−1 (1 − zx)−σ F (α, β; γ; x) dx
0
=
Γ(γ) Γ(ρ) Γ(γ + ρ − α − β) (1 − z)−σ Γ(γ + ρ − α) Γ(γ + ρ − β) × 3 F 2 ρ, σ, γ + ρ − α − β; γ + ρ − α, γ + ρ − β;
[Re γ > 0,
Re ρ > 0,
Re (γ + ρ − α − β) > 0,
|arg(1 − z)| < π]
z z−1
ET II 399(6)
814
Hypergeometric Functions
∞
10.
7.513
Γ(γ) Γ(α − γ + σ) Γ (β − γ + σ) Γ(σ) Γ(α + β − γ + σ)
xγ−1 (x + z)−σ F (α, β; γ; −x) dx =
0
× F (α − γ + σ, β − γ + σ; α + β − γ + σ; 1 − z) [Re γ > 0,
Re(α − γ + σ) > 0,
Re (β − γ + σ) > 0,
|arg z| < π]
ET II 400(10)
1
(1 − x)μ−1 xν−1 p F q (a1 , . . . , ap ; ν, b2 , . . . , bq ; ax) dx
11. 0
Γ(μ) Γ(ν) p F q (a1 , . . . , ap ; μ + ν, b2 , . . . , bq ; a) Γ(μ + ν) p ≤ q + 1; if p = q + 1, then |a| < 1] ET II 200(94) =
[Re μ > 0,
Re ν > 0,
1
(1 − x)μ−1 xν−1 p F q (a1 , . . . , ap ; b1 , . . . , bq ; ax) dx
12. 0
Γ(μ) Γ(ν) p+1 F q+1 (ν, a1 , . . . , ap ; μ + ν, b1 , . . . , bq ; a) Γ(μ + ν) Re ν > 0, p ≤ q + 1, if p = q + 1, then |a| < 1] ET II 200(95) =
[Re μ > 0,
1
7.513 0
ν s s 1 s xs−1 1 − x2 F −n, a; b; x2 dx = B ν + 1, 3 F 2 −n, a, ; b, ν + 1 + ; 1 2 2 2 2 [Re s > 0, Re ν > −1] ET I 336(4)
7.52 Combinations of hypergeometric functions and exponentials
∞
7.521
e−st p F q (a1 , . . . , ap ; b1 , . . . , bq , t) dt =
0
7.522 1.11
∞
∞
2.6 0
∞
3. 0
p+1 F q
1, a1 , . . . , ap ; b1 , . . . , bq , s−1
[p ≤ q] e−λx xγ−1 2 F 1 (α, β; δ; −x) dx =
0
1 s
e−bx xa−1 F
EH I 192
Γ(δ)λ−γ E (α, β, γ : δ : λ) Γ(α) Γ(β)
1 1 x + ν, − ν; a; − 2 2 2
[Re λ > 0,
Re γ > 0]
EH I 205(10)
1 1 dx = 2a eb √ Γ(a)(2b) 2 −a K ν (b) π
ET I 212(1) [Re a > 0, Re b > 0] α+β− 12
b b b e−bx xγ−1 F (2α, 2β; γ; −λx) dx = Γ(γ)b−γ e 2λ W 12 −α−β,α−β λ λ [Re b > 0, Re γ > 0, |arg λ| < π] BU 78(30), ET I 212(4)
∞
4.6
e−xt tb−1 F (a, a − c + 1; b; −t) dt = xa−b Γ(b)Ψ(a, c; x)
0
5.
[Re b > 0, ∞
e−x xs−1 p F q (a1 , . . . , ap , b1 , . . . , bq ; ax) dx = Γ(s)
p+1 F q
Re x > 0]
EH I 273(11)
(s, a1 , . . . , ap ; b1 , . . . , bq ; a)
0
[p < q,
Re s > 0]
ET I 337(11)
7.525
Hypergeometric functions and exponentials
∞
6.
β−1 −μx
x
e
2 F 2 (−n, n
−β
+ 1; 1, β; x) dx = Γ(β)μ
Pn
0
∞
7.
xβ−1 e−μx 2 F 2
0
2 1− μ
[Re μ > 0, Re β > 0]
1 1 −n, n; β, ; x dx = Γ(β)μ−β cos 2n arcsin √ 2 μ [Re μ > 0,
∞
8.
xρn −1 e−μx
mF n
815
Re β > 0]
ET I 218(6)
ET I 218(7)
(a1 , . . . , am ; ρ1 , . . . , ρn ; λx) dx
λ , . . . , a ; ρ , . . . , ρ ; a m F n−1 1 m 1 n−1 μ Re μ > 0, if m < n; Re μ > Re λ, if m = n] ET I 219(16)a
0
= Γ (ρn ) μ−ρn
[m ≤ n;
∞
9.
xσ−1 e−μx
mF n
Re ρn > 0,
(a1 , . . . , am ; ρ1 , . . . , ρn ; λx) dx
0
λ , . . . , a , σ; ρ , . . . , ρ ; a F m+1 n 1 m 1 n μ Re μ > 0, if m < n; Re μ > Re λ, if m = n] ET I 219(17) = Γ(σ)μ−σ
[m ≤ n,
∞
7.523 1
Re σ > 0,
(x − 1)μ−1 x−μ− 2 e− 2 ax W 2μ+ 12 ,λ (ax) dx = Γ(μ)e− 2 a W μ+ 12 ,λ (a) 1
1
1
[Re μ > 0, 7.524
∞
1. 0
∞
2.
e−λx F
1 α, β; ; −x2 dx = λα+β−1 S 1−α−β,α−β (λ) 2
e−st p F q a1 , . . . , ap ; b1 , . . . , bq ; t2 dx = s−1
0
∞
3.
e−st 0 F q
0
7.525
1. 0
∞
Re a > 0]
xσ−1 e−μx
1 2 q−1 t , ,..., , 1; q q q q q q
ET II 401(13) [Re λ > 0]
1 4 p+2 F q a1 , . . . , ap , 1, ; b1 , . . . , bq ; 2 2 s
[p < q] dt = s−1 exp s−q
MO 176 MO 176
a1 , . . . , am ; ρ1 , . . . , ρn ; (λx)k dx k kλ σ+k−1 σ σ+1 −σ = Γ(σ)μ m+k F n a1 , . . . , am , , ,..., ; ρ1 , . . . , ρn ; k k k μ m + k ≤ n + 1, Re σ > 0; Re μ > 0, if m + k ≤ n; 2πi Re μ + kλe k > 0; r = 0, 1, . . . , k − 1 for m + k = n + 1
mF n
ET I 220(19)
816
Hypergeometric Functions
∞
2. 0
7.526
xe−λx F α, β; 32 ; −x2 dx = λα+β−2 S 1−α−β,α−β (λ) [Re λ > 0]
7.526
γ+i∞
1.
st −b
e s γ−i∞
1 Γ(a + b − c + 1) b−1 t F a, b; a + b − c + 1; 1 − Ψ(a; c; t) dx = 2πi s Γ(b) Γ(b − c + 1) Re b > 0, Re(b − c) > −1,
γ>
1 2
EH I 273(12)
∞
2. 0
ET II 401(14)
∞
3.
t(x + y + t) −t γ−1 −α −a e t (x + t) (y + t) F a, a ; γ; dt = Γ(γ)Ψ(a, c; x)Ψ (a , c; y) , (x + t)(y + t) [Re γ > 0, xy = 0] EH I 287(21) γ = a + a − c + 1 γ−1
x 0
−α
(x + y)
−β −x
(x + z)
e
x(x + y + z) F α, β; γ; dx (x + y)(x + z) y+z
= Γ(γ)(zy)− 2 −μ e 2 W ν,μ (y) W λ,μ (z) 2ν = 1 − α + β − γ; 2λ = 1 + α − β − γ; 2μ = α + β − γ 1
[Re γ > 0,
|arg y| < π,
|arg z| < π] ET II 401(15)
7.527
∞
1.
λ−1 −μx 1 − e−x e F α, β; γ; δe−x dx = B(μ, λ) 3 F 2 (α, β, μ; γ, μ + λ; δ)
0
[Re λ > 0,
∞
2. 0
∞
0
∞
ET I 213(9)
μ B(α, μ + n + 1) B(α, β + n − α) 1 − e−x e−αx F −n, μ + β + n; β; e−x dx = B(α, β − α) [Re α > 0,
3.
4.
|arg(1 − δ)| < π]
Re μ > 0,
Re μ > −1]
γ−1 −μx Γ(μ) Γ(γ − α − β + μ) Γ(γ) 1 − e−x e F α, β; γ; 1 − e−x dx = Γ(γ − α + μ) Γ(γ − β + μ) [Re μ > 0, Re μ > Re(α + β − γ), Re γ > 0]
ET I 213(10)
ET I 213(11)
γ−1 −μx 1 − e−x e F α, β; γ; δ 1 − e−x dx = B(μ, γ) F (α, β; μ + γ; δ)
0
[Re μ > 0,
Re γ > 0,
|arg(1 − δ)| < π]
ET I 213(12)
7.542
Hypergeometric and Bessel functions
817
7.53 Hypergeometric and trigonometric functions 7.531
∞
1. 0
μ
3 2 2 −α−β+1 −α−β α+β−2 K α−β c x sin μx F α, β; ; −c x πc μ dx = 2 2 Γ(α) Γ(β) μ > 0, Re α > 12 ,
∞
2.
cos μx F 0
1 α, β; ; −c2 x2 2
μ
K α−β c Γ(α) Γ(β) [μ > 0, Re α > 0,
dx = 2−α−β+1 πc−α−β μα+β−1
Re β >
1 2
ET I 115(6)
Re β > 0,
c > 0] ET I 61(9)
7.54 Combinations of hypergeometric and Bessel functions
∞
7.541
xα+β−2ν−1 (x + 1)−ν exz K ν [(x + 1)z] F (α, β; α + β − 2ν; −x) dx
0
= π − 2 cos(νπ) Γ 1
γ = α + β − 2ν
Re(α + β − 2ν) > 0,
1
1 1 − α + ν Γ 12 − β + ν Γ(γ)(2z)− 2 − 2 γ W 12 γ, 12 (β−α) (2z) 1 1 Re 2 − α + ν > 0, Re 2 − β + ν > 0, |arg z| < 32 π 2
ET II 401(16)
7.542
∞
1.
xσ−1 p F p−1 a1 , . . . , ap ; b1 , . . . , bp−1 ; −λx2 Y ν (xy) dx
!
y 2 !! b∗0 , . . . , b∗p+1 , 1 4λ ! h, k, a∗1 , . . . , a∗p , l 2λ 2 σ Γ (a1 ) . . . Γ (ap ) σ σ σ a∗j = aj − , j = 1, . . . , p; b∗0 = 1 − ; b∗j = bj − , 2 2 2 ν 1+ν ν j = 1, . . . , p − 1; h = , k = − , l = − 2 2 2 |arg λ| < π, Re σ > |Re ν|, Re aj > 12 Re σ − 34 , y > 0
0
=
2.
Γ (b1 ) . . . Γ (bp−1 )
p+2,1
G p+2,p+3
ET II 118(53) ∞
xσ−1 p F p a1 , . . . , ap ; b1 , . . . , bp ; −λx2 Y ν (xy) dx
0
!
y 2 !! b∗0 , . . . , b∗p , l 1 4λ ! h, k, a∗1 , . . . , a∗p , l 2λ 2 σ Γ (a1 ) . . . Γ (ap ) ν ν 1+ν σ bj ∗ = bj − ; j = 1, . . . , p; h = , k = − , l = − 2 2 2 2 Re λ > 0, Re σ > |Re ν|, Re aj > 12 Re σ − 34 , y > 0 =
b∗0 = 1 −
σ ; 2
a∗j = aj −
σ , 2
Γ (b1 ) . . . Γ (bp )
p+2,1 G p+2,p+3
ET II 119(54)
818
Hypergeometric Functions
∞
3.
7.542
xσ−1 p F q a1 , . . . , ap ; b1 , . . . , bq ; −λx2 Y ν (xy) dx
, σ + ν σ − ν
= −π 2 y cos (σ − ν) Γ Γ 2 2 2
σ+ν σ−ν 4λ , ; b1 , . . . , bq ; − 2 × p+2 F q a1 , . . . , ap , 2 2 y [y > 0, p ≤ q − 1, Re σ > |Re ν|] ET II 119(55)
0
+π
−1 σ−1 −σ
∞
xσ−1 p F q a1 , . . . , ap ; b1 , . . . , bq ; −λx2 K ν (xy) dx
σ+ν σ−ν σ+ν σ−ν 4λ σ−2 −σ , ; b1 , . . . , bq ; 2 =2 y Γ Γ p+2 F q a1 , . . . , ap , 2 2 2 2 y [Re y > 0, p ≤ q − 1, Re σ > |Re ν|] ET II 153(88)
∞
x2ρ p F p a1 , . . . , ap ; b1 , . . . , bp ; −λx2 J ν (xy) dx
4. 0
5. 0
∞
6.
x2ρ
m+1 F m
0
∞
7.
y > 0,
y > 0,
2!
y !! 1, b1 , . . . , bp 22ρ Γ (b1 ) . . . Γ (bp ) p+1,1 = 2ρ+1 G y Γ (a1 ) . . . Γ (ap ) p+1,p+2 4λ ! h, a1 , . . . , ap , k h = 12 + ρ + 12 ν, k = 12 + ρ − 12 ν 1 Re λ > 0, −1 − Re ν < 2 Re ρ < 2 + 2 Re ar , r = 1, . . . , p ET II 91(18)
a1 , . . . , am+1 ; b1 , . . . , bm ; −λ2 x2 J ν (xy) dx
2 !
y !! 1, b1 , . . . , bm 22ρ Γ (b1 ) . . . Γ (bm ) y −2ρ−1 m+2,1 = G m+1,m+3 Γ (a1 ) . . . Γ (am+1 ) 4λ2 ! h, a1 , . . . , am+1 , k 1 h = 2 + ρ + 12 ν, k =12 + ρ − 12 ν, 1 Re λ > 0, Re(2ρ + ν) > −1, Re (ρ − ar ) < 4 ; r = 1, . . . , m + 1 ET II 91(19)
xδ F α, β; γ; −λ2 x2 J ν (xy) dx
∞
8. 0
9. 0
∞
! ⎞ ! ! 1 − α, 1 − β ! 1+δ−ν ⎠ !1+δ+ν , 0, 1 − γ, ! 2 2 −1 − Re ν − 2 min (Re α, Re β) < Re δ < − 12 ET II 82(9)
⎛ 2δ Γ(γ) −δ−1 22 ⎝ y 2 = y G 24 Γ(α) Γ(β) 4λ2
0
y > 0,
Re λ > 0,
! ⎞ ! 1, γ ! y Γ(γ) y 2 31 xδ F α, β; γ; −λ2 x2 J ν (xy) dx = G 24 ⎝ 2 !! 1 + δ + ν 1+δ−ν ⎠ Γ(α) Γ(β) 4λ ! , α, β, 2 2 y > 0, Re λ > 0, − Re ν − 1 < Re δ < 2 max (Re α, Re β) − 12 ET II 81(6) δ −δ−1
⎛
2
!
2ν+1 Γ(γ) −ν−2 30 y 2 !! γ y xν+1 F α, β; γ; −λ2 x2 J ν (xy) dx = G 13 Γ(α) Γ(β) 4λ2 ! ν + 1, α, β y > 0, Re λ > 0, −1 < Re ν < 2 max (Re α, Re β) − 32 ET II 81(5)
7.542
Hypergeometric and Bessel functions
∞
y 2ν−α−β+2 Γ(ν + 1) α+β−ν−2 y xν+1 F α, β; ν + 1; −λ2 x2 J ν (xy) dx = K α−β λα+β Γ(α) Γ(β) λ ET II 81(3) y > 0, Re λ > 0, −1 < Re ν < 2 max (Re α, Re β) − 32
∞
y xν+1 F α, β; ν + 1; −λ2 x2 K ν (xy) dx = 2ν+1 λ−α−β y α+β−ν−2 Γ(ν + 1) S 1−α−β,α−β λ [Re y > 0, Re λ > 0, Re ν > −1]
10. 0
11. 0
∞
12.
ν+1
x 0
∞
13. 0
∞
14.
ET II 152(86)
y 2 Γ β+ν+2 y β−1 λ−ν−β−1 2 β+ν 2 2 + 1; −λ x Jν (xy) dx = F α, β; K 12 (ν−β+1) 1 β−1 2 2λ π 2 Γ(α) Γ(β)2 y > 0, −1 < Re ν < 2 max (Re α, Re β) − 32 ET II 81(4)
!
1 1 λ−σ−1 y − 2 Γ(γ) 41 y 2 !! 1 − p, γ − p, l xσ+ 2 F α, β; γ; −λ2 x2 Y ν (xy) dx = √ G 35 4λ2 ! h, k, α − p, β − p, l 2 Γ(α) Γ(β) 1 1 1 1 h = 4 + 2 ν, k = 4 − 2 ν, l = − 14 − 12 ν, p = 12 + 12 σ y > 0, Re λ > 0, Re σ > |Re ν| − 32 , Re σ < 2 Re α, Re σ < 2 Re β ET II 118(52) ν+2
x
F
0
819
y y 1 1 3 2ν y −ν−1 2 2 , − ν; ; −λ x Y ν (xy) dx = 1 K ν+1 1 Kν 2 2 2 2λ 2λ π 2 λ2 Γ 2 − ν y > 0, Re λ > 0, − 32 < Re ν < − 12 ET II 117(49)
∞
15. 0
y ,2 1 Γ(ν + 2) + 3 K xν+2 F 1, 2ν + ; ν + 2; −λ2 x2 Y ν (xy) dx = π − 2 2−ν λ−2ν−3 ν 2 2λ Γ 2ν + 32 1 y > 0, Re λ > 0, − 2 < Re ν < 12
ET II 117(50)
1 ∞ y 3 3 π 2 2−μ−ν−1 λ−μ−2ν−3 y μ+ν ν+2 2 2 x F 1, μ + ν + ; ; −λ x Y ν (xy) dx = K μ 2 2 λ Γ μ + ν + 32 0 3 1 3 y > 0, Re λ > 0, − 2 < Re ν < 2 , Re(2μ + ν) > − 2 ET II 118(51)
16.
1 2 2 x F α − ν − , α; 2α; −λ x J ν (xy) dx 2 0 1 y + y , i Γ 2 + α Γ 12 + α + ν y W 12 −α,− 12 −ν e−iπ − W 12 −α,− 12 −ν eiπ W 12 −α,− 12 −ν = 1−ν−2α 2α−1 ν+2 π2 λ y λ λ λ ET II 80(1) y > 0, Re λ > 0, Re ν < − 12 , Re(α + ν) > − 12
17.
18. 0
∞
∞
2α+ν
x2α−ν F
1 ν + α − , α; 2α; −λ2 x2 J ν (xy) dx 2 y 22α−ν Γ 12 + α y ν−2 1 1 M = W α− ,ν− 2 2 λ2α−1 Γ(2ν) λ
y 1 1 2 −α,ν− 2
λ
ET II 80(2)
820
7.543
Confluent Hypergeometric Functions
∞
1.
−2α−1
x
F
0
∞
2.
x 0
1 4λ2 + α, 1 + α; 1 + 2α; − 2 J ν (xy) dx = λ−2α I 12 ν+α (λy) K 12 ν−α (λy) 2 x y > 0, Re λ > 0, Re ν > −1, Re α > − 12 ET II 81(7)
ν+1−4α
F
λ2 1 α, α + ; ν + 1; − 2 2 x
∞
7.544 0
7.543
y > 0,
J ν (xy) dx
1 1 Γ(ν) ν 1−2α 2α−ν−1 = 2 λ λy K 2α−ν−1 λy y Iν Γ(2α) 2 2 3 Re λ > 0, Re α − 1 < Re ν < 4 Re α − 2 ET II 81(8)
4x 1 xν+1 (1 + x)−2α F α, ν + ; 2ν + 1; J ν (xy) dx 2 (1 + x)2 Γ(ν + 1) Γ(ν − α + 1) 2ν−2α+1 2(α−ν−1) 2 y J ν (y) = Γ(α) y > 0, −1 < Re ν < 2 Re α − 32 ET II 82(10)
7.6 Confluent Hypergeometric Functions 7.61 Combinations of confluent hypergeometric functions and powers 7.611
∞
1. 0
∞
2.
3
x−1 W k,μ (x) dx =
Γ
3 4
π 2 2k sec(μπ) − 12 k + 12 μ Γ 34 − 12 k − 12 μ |Re μ| < 12
x−1 M k,μ (x) W λ,μ (x) dx =
0
3.
5.
Re μ > − 12 ,
Re(k − λ) > 0
BU 116(11), ET II 409(39) ∞
x−1 W k,μ (x) W λ,μ (x) dx
% & 1 1 1 − 1 = (k − λ) sin(2μπ) Γ 12 − k + μ Γ 12 − λ − μ Γ 2 − k − μ Γ 12 − λ + μ BU 116(12), ET II 409(40) |Re μ| < 12 1 1 ∞ ψ 2 +μ−κ −ψ 2 −μ−κ π 2 dz = {W κ,μ (z)} z sin 2πμ Γ 12 + μ − κ Γ 12 − μ − κ 0 BU 117(12a) |Re μ| < 12 ∞ 1 ψ −κ 1 2 [W κ,0 (z)] dx = 2 BU 117(12b) 2 0 z Γ 12 − κ 0
4.
Γ(2μ + 1) (k − λ) Γ 12 + μ − λ
ET II 406(22)
7.613
Confluent hypergeometric functions and powers
∞
6.
ρ−1
x 0
Γ(ρ + 1) Γ 12 ρ + 12 + μ Γ 12 ρ + 12 − μ W k,μ (x) W −k,μ (x) dx = 2 Γ 1 + 12 ρ + k Γ 1 + 12 ρ − k [Re ρ > 2|Re μ| − 1]
∞
7.11
=
∞
Γ(1 − μ + ν + ρ) Γ(1 + μ + ν + ρ) Γ(−2ν) Γ 12 − λ − ν Γ 32 − k + ν + ρ
3 1 × 3 F 2 1 − μ + ν + ρ, 1 + μ + ν + ρ, − λ + ν; 1 + 2ν, − k + ν + ρ; 1 2 2 Γ(1 + μ − ν + ρ) Γ(1 − μ − ν + ρ) Γ(2ν) + Γ 12 − λ + ν Γ 32 − k − ν + ρ
3 1 × 3 F 2 1 + μ − ν + ρ, 1 − μ − ν + ρ, − λ − ν; 1 − 2ν, − k − ν + ρ; 1 2 2 [|Re μ| + |Re ν| < Re ρ + 1] ET II 410(42)
tb−1 1 F 1 (a; c; −t) dt =
0
∞
2.
ET II 409(41)
xρ−1 W k,μ (x) W λ,ν (x) dx
0
7.612 1.
821
tb−1 Ψ(a, c; t) dt =
0
Γ(b) Γ(c) Γ(a − b) Γ(a) Γ(c − b)
[0 < Re b < Re a]
Γ(b) Γ(a − b) Γ(b − c + 1) Γ(a) Γ(a − c + 1)
EH I 285(10)
[0 < Re b < Re a
Re c < Re b + 1] EH I 285(11)
7.613
t
xγ−1 (t − x)c−γ−1 1 F 1 (a; γ; x) dx = tc−1
1. 0
BU 9(16)a, EH I 271(16) t
xβ−1 (t − x)γ−1 1 F 1 (t; β; x) dx =
2. 0
Γ(γ) Γ(c − γ) 1 F 1 (a; c; t) Γ(c) [Re c > Re γ > 0]
1
xλ−1 (1 − x)2μ−λ 1 F 1
3. 0
Γ(β) Γ(γ) β+γ−1 t 1 F 1 (t; β + γ; t) Γ(β + γ)
1 + μ − ν; λ; xz 2
[Re β > 0,
1
t
xβ−1 (t − x)δ−1 1 F 1 (t; β; x) 1 F 1 (γ; δ; t − x) dx = 0
0
Re(2μ − λ) > −1]
Γ(β) Γ(δ) β+δ−1 t 1 F 1 (t + γ; β + δ; t) Γ(β + δ) [Re β > 0, Re δ > 0] ET II 402(2), EH I 271(15)
t
1
1
xμ− 2 (t − x)ν− 2 M k,μ (x) M λ,ν (t − x) dx =
5.
1
BU 14(14)
4.
ET II 401(1)
dx = B(λ, 1 + 2μ − λ)e 2 z z − 2 −μ M ν,μ (z) [Re λ > 0,
Re γ > 0]
Γ(2μ + 1) Γ(2ν + 1) μ+ν t M k+λ,μ+ν+ 12 (t) Γ(2μ +2ν + 2) 1 1 Re μ > − , Re ν > − 2 2 BU 128(14), ET II 402(7)
822
Confluent Hypergeometric Functions
7.621
1
xβ−1 (1 − x)σ−β−1 1 F 1 (α; β; λx) 1 F 1 [σ − α; σ − β; μ(1 − x)] dx
6. 0
Γ(β) Γ(σ − β) λ e 1 F 1 (α; σ; μ − λ) Γ(σ) [0 < Re β < Re σ] ET II 402(3) =
7.62–7.63 Combinations of confluent hypergeometric functions and exponentials 7.621
∞
1.
e
Γ α + ν + 32 1 2 3 t M μ,ν (t) dt = α+ν+ 32 F α + ν + 2 , −μ + ν + 2 ; 2ν + 1; 2s + 1 1 2 +s Re α + μ + 32 > 0, Re s > 12
−st α
0
BU 118(1), MO 176a, EH I 270(12)a ∞
2. 0
e−st tμ− 2 M λ,μ (qt) dt = q μ+ 2 Γ(2μ + 1) s − 12 q 1
1
λ−μ− 12
s + 12 q
−λ−μ− 12
1 Re μ > − , 2
|Re q| Re s > 2
BU 119(4c), MO 176a, EH I 271(13)a
3.
4.
−α−μ− 32 1 ∞ Γ α + μ + 32 Γ α − μ + 32 q μ+ 2 1 −st α e t W λ,μ (qt) dt = s+ q Γ(α − λ + 2) 2 0
1 2s − q 3 × F α + μ + , μ − λ + ; α − λ + 2; 2 2 2s + q
3 q EH I 271(14)a, BU 121(6), MO 176 Re α ± μ + > 0, Re s > − , q > 0 2 2 ∞ [|s| > |k|] e−st tb−1 1 F 1 (a; c; kt) dt = Γ(b)s−b F a, b; c; ks−1
0 k = Γ(b)(s − k)−b F c − a, b; c; [|s − k > |k||] k−s [Re b > 0, Re s > max (0, Re k)] EH I 269(5)
∞
5.
−a tc−1 1 F 1 (a; c; t)e−st dt = Γ(c)s−c 1 − s−1
[Re c > 0,
Re s > 1]
EH I 270(6)
0
∞
6.
Γ(b) Γ(b − c + 1) F (b, b − c + 1; a + b − c + 1; 1 − s) Γ(a + b − c + 1) [Re b > 0, Re c < Re b + 1, Γ(b) Γ(b − c + 1) −b = s F a, b; a + b − c + 1; 1 − s−1 Γ(a + b − c + 1)
tb−1 Ψ (a, c; t) e−st dt =
0
7. 0
∞
e− 2 x xν−1 M κ,μ (bx) dx = b
1
|1 − s| < 1]
Re s >
1 2
EH I 270(7)
Γ(1 + 2μ) Γ (κ − ν) Γ 2 + μ + ν ν b Γ 12 + μ + κ Γ 12 + μ − ν Re ν + 12 + μ > 0,
Re (κ − ν) > 0
BU 119(3)a, ET I 215(11)a
7.622
Confluent hypergeometric functions and exponentials
∞
8.
e
−sx
0
2 Γ(1 + 2μ)e−iπκ dx = M κ,μ (x) x Γ 12 + μ + κ
s− s+
1 2 1 2
823
κ2 Q κμ− 1 (2s) 2 1 Re 2 + μ > 0,
Re s >
1 2
BU 119(4a)
∞
9.
e−sx W κ,μ (x)
0
π dx πμ = x cos 2
s− s+
κ 1 2
2 1 2
P κμ− 1 (2s) 2
Re
1 2
± μ > 0,
Re s > − 12
∞ Γ k + μ + 12 Γ 14 (2k + 6μ + 5) k+2μ−1 − 32 x x e W k,μ (x) dx = k + 3μ + 12 Γ 14 (2μ − 2k + 3) 0 Re(k + μ) > − 12 ,
∞
11. 0
∞
12. 0
Γ ν+ −μ Γ ν+ − 12 x ν−1 e x W κ,μ (x) dx = Γ (ν − κ + 1) 1 2
1 2
Re ν +
Re(k + 3μ) > − 12
BU 122(8a), ET II 406(23)
+μ
BU 121(7)
10.
1 2
±μ >0
Γ 12 + μ + ν Γ 12 − μ + ν 1 x ν−1 e2 x W κ,μ (x) dx = Γ (−κ − μ) 1 Γ 2 − μ − κ Γ 12 +μ − κ Re ν + 12 ± μ > 0,
BU 122(8b)
Re (κ + ν) < 0
BU 122(8c)a
7.622
∞
1.
e−st tc−1 1 F 1 (a; c; t) 1 F 1 (α; c; λt) dt
0
= Γ(c)(s − 1)−a (s − λ)−α sa+α−c F a, α; c; λ(s − 1)−1 (s − λ)−1 [Re c > 0,
∞
2.
Re s > Re λ + 1]
EH I 287(22)
e−t tρ 1 F 1 (a; c; t)Ψ (a ; c ; λt) dt
0
=C
Γ(c) Γ(β) σ λ F c − a, β; γ; 1 − λ−1 , Γ(γ) ρ = c − 1,
ρ = c + c − 2,
σ = −c,
β = c − c + 1,
σ = 1 − c − c ,
γ = c − a + a − c + 1,
β = c + c − 1,
γ = a − a + c,
C=
C=
Γ (a − a) , or Γ (a )
Γ (a − a − c + 1) Γ (a − c + 1) EH I 287(24)
824
Confluent Hypergeometric Functions
∞
3. 0
7.623
xν−1 e−bx M λ1 ,μ1 − 12 (a1 x) . . . M λn ,μn − 12 (an x) dx = aμ1 1 . . . aμnn (b + A)−ν−M Γ(ν + M )
an a1 ,..., × F A ν + M ; μ1 − λ1 , . . . , μn − λn ; 2μ1 , . . . , 2μn ; , b+A b+A A = 12 (a1 + · · · + an ) M = μ1 + · · · + μn , 1 1 Re(ν + M ) > 0, Re b ± 2 a1 ± · · · + 2 an > 0 ET I 216(14)
7.623
∞
1.
e−x xc+n−1 (x + y)−1 1 F 1 (a; c; x) dx = (−1)n Γ(c) Γ(1 − a)y c+n−1 Ψ(c − a, c; y)
0
3.
t
−1
k−1
|arg y| < π]
EH I 285(16)
1 2 (t−x)
ET II 402(6)
1
t Γ(λ) Γ 2 − k − λ + μ Γ 12 − k − λ − μ 1 1 1 x−k−λ−1 (t − x)λ−1 e 2 x W k,μ (x) dx = W k+λ,μ (t) tk+1 Γ 2 − k + μ Γ 2 − k − μ 0 Re λ > 0, Re(k + λ) < 12 − |Re μ|
4.
n = 0, 1, 2, . . . ,
1 Γ(k) Γ(2μ + 1) 1 k− 1 2 2 t x (t − x) e M k,μ (x) dx = lμ 1 π t 2 Γ k + μ + 0 2 Re k > 0, Re μ > − 12 ET II 402(5) k+λ−1 t 1 Γ(λ) Γ k + μ + 2 t 1 xk−1 (t − x)λ−1 e 2 (t−x) M k+λ,μ (x) dx = M k,μ (t) 1 Γ k+λ+ 0 μ+ 2 Re(k + μ) > − 12 , Re λ > 0
2.
[− Re c < n < 1 − Re a,
ET II 405(21)
∞
5. 1
∞
6.11 1
7.
Γ(μ) Γ 12 − k − λ − μ − 1 μ 1 a μ−1 λ− 12 12 ax 1 (x − 1) x e W k,λ (ax) dx = a 2 e 2 W k+ 12 μ,λ+ 12 μ (a) Γ − k − λ 2 |arg(a)| < 32 π, 0 < Re μ < 12 − Re(k + λ) ET II 211(72)a (x − 1)μ−1 xμ− 2 e− 2 ax W 2μ+ 12 ,λ (ax) dx = Γ(μ)e− 2 a W μ+ 12 ,λ (a) 1
1
1
[Re μ > 0, ∞
Re a > 0]
ET II 211(74)a
Re a > 0]
ET II 211(73)a
(x − 1)μ−1 xk−μ−1 e− 2 ax W k,λ (ax) dx = Γ(μ)e− 2 a W k−μ,λ (a) 1
1
1
[Re μ > 0,
7.624
Confluent hypergeometric functions and exponentials
8.
1
825
(1 − x)μ−1 xk−μ−1 e− 2 ax W k,λ (ax) dx 1
0
= Γ(μ)e− 2 a sec[(k − μ − λ)π] Γ k − μ + λ + 12 × sin(μπ) M k−μ,λ (a) + cos[(k − λ)π] W k−μ,λ (a) Γ(2λ + 1) 0 < Re μ < Re k − |Re λ| + 12 ET II 200(93)a 1
7.624
∞
1.
+ 1 , 1 2σ 1 xρ−1 x 2 + (a + x) 2 e− 2 x M k,μ (x) dx
! ! 1 , 1, 1 − k + ρ −σ Γ(2μ + 1)aσ !2 23 = 1 G 34 a ! 1 ! 2 + μ + ρ, −σ, σ, 12 − μ + ρ π 2 Γ 12 + k + μ 1 |arg a| < π, Re(μ + ρ) > − 2 , Re(k − ρ − σ) > 0 ET II 403(8)
0
∞
2.
ρ−1
x 0
∞
3.
! ! 1 , 1, 1 − k + ρ + 1 , 1 2σ 1 1 ! − x − σ 32 2 x 2 + (a + x) 2 e 2 W k,μ (x) dx = −π 2 σa G 34 a ! 1 ! 2 + μ + ρ, 12 − μ + ρ, −σ, σ |arg a| < π, Re ρ > |Re μ| − 12 ET II 406(24)
+ 1 , 1 2σ 1 xρ−1 x 2 + (a + x) 2 e− 2 x W k,μ (x) dx
! 1 ! 1 , 1, 1 + k + ρ σπ − 2 aσ ! 2 G 33 = − 1 34 a ! 1 ! 2 + μ + ρ, 12 − μ + ρ, −σ, σ Γ 2 − k + μ Γ 12 − k − μ |arg a| < π, Re ρ > |Re μ| − 12 , Re(k + ρ + σ) < 0 ET II 406(25)
0
∞
4.
+ 1 , 1 1 2σ 1 xρ−1 (a + x)− 2 x 2 + (a + x) 2 e− 2 x M k,μ (x) dx
0
! 1 1
! 0, , − k − ρ Γ(2μ + 1)aσ 23 2 2 ! a G 34 1 ! −σ, ρ + μ, ρ − μ, σ π 2 Γ 12 + k + μ Re(ρ + μ) > − 12 , Re(k − ρ − σ) > − 12 ET II 403(9) =
|arg a| < π,
∞
5.
+ 1 , 1 1 2σ 1 xρ−1 (a + x)− 2 x 2 + (a + x) 2 e− 2 x W k,μ (x) dx
! 1 1
1 ! 0, , + k + ρ π − 2 aσ 33 2 2 ! = G 34 a ! −σ, ρ + μ, ρ − μ, σ Γ 2 − k + μ Γ 12 − k − μ 1 ET II 406(26) |arg a| < π, Re ρ > |Re μ| − 2 , Re(k + ρ + σ) < 12
0
1
6.
∞
ρ−1
x 0
− 12
(a + x)
+
1 2
x + (a + x)
1 2
,2σ e
− 12 x
! 1 1
! 0, , − k + ρ 2 2 ! W k,μ (x) dx = π a a! −σ, ρ + μ, ρ − μ, σ |arg a| < π, Re ρ > |Re μ| − 12 ET II 406(27) − 12
σ
32 G 34
826
7.625
Confluent Hypergeometric Functions
∞
1. 0
xρ−1 exp − 12 (α + β)x M k,μ (αx) W λ,ν (βx) dx =
∞
2.
ρ−1
x 0
∞
3.
ρ−1
x 0
0
Γ(1 + μ + ν + ρ) Γ(1 + μ − ν + ρ) μ+ 1 −μ−ρ− 1 2 3 α 2β Γ − λ + μ + ρ 2
1 3 α + k + μ, 1 + μ + ν + ρ, 1 + μ − ν + ρ; 2μ + 1, − λ + μ + ρ; − × 3F 2 2 2 β [Re α > 0, Re β > 0, Re (ρ + μ) > |Re ν| − 1] ET II 410(43)
1 exp − (α + β)x W k,μ (αx) W λ,ν (βx) dx 2 −1 = β −ρ Γ 12 − k + μ Γ 12 − k − μ Γ 12 − λ + ν Γ 12 − λ − ν ! β !! 12 + μ, 12 − μ, 1 + λ + ρ 33 × G 33 ! α ! 12 + ν + ρ, 12 − ν + ρ, −k [|Re μ| + |Re ν| < Re ρ + 1, Re(k + λ + ρ) < 0] ET II 410(44)a ! 1 β !! 12 + μ, 12 − ν, 1 − λ + ρ −ρ 22 exp − (α + β)x W k,μ (αx) W λ,ν (βx) dx = β G 33 ! 2 α ! 12 + ν + ρ, 12 − ν + ρ, k ET II 411(46)
∞
4.
7.626
1
1 ρ−1 x exp − (α − β)x W k,μ (αx) W λ,ν (βx) dx 2 ! β !! 12 + μ, 12 − μ, 1 + λ + ρ −1 −ρ 23 1 1 Γ 2 −λ+ν Γ 2 −λ−ν =β ! G 33 α ! 12 + ν + ρ, 12 − ν + ρ, k [Re α > 0, |Re μ| + |Re ν| < Re ρ + 1] ET II 411(45)
1. 0
k 1 1 − (ξ + η) exp − (ξ + η)x xc 1 F 1 (a; c; ξx) 1 F 1 (a; c; ηx) dx x 4 2 =0 a 2 = e−ξ [ 1 F 1 (a + 1; c; ξ)] ξ [where ξ and η are any two zeros of the function
2. 1
7.625
∞
[ξ = η,
Re c > 0]
[ξ = η,
Re c > 0]
1F 1
(a; c; x)]
1 k 1 − (ξ + η) e− 2 (ξ+η)x xc Ψ (a, c; ξx) Ψ(a, c; ηx) dx = 0 x 4 = −ξ −1 e−ξ [Ψ(a − 1, c; ξ)]2 [where ξ and η are any two zeros of the function Ψ(a, c; x)]
EH I 285
[ξ = η] ; [ξ = η] EH I 286
7.627
7.627
Confluent hypergeometric functions and exponentials
∞
1.
2λ−1
x
−μ− 12
(a + x)
e
1 2x
0
∞
2.
Γ(2λ) Γ 12 − k + μ − 2λ λ−μ− 1 2 W
a W k,μ (a + x) dx = k+λ,μ−λ (a) 1 −k+μ Γ 2 1 ET II 411(50) |arg a| < π, 0 < 2 Re λ < − Re(k + μ) 2 −1x
x2λ−1 (a + x)−μ− 2 e− 2 x M k,μ2 (a + x) dx 1
1
Γ(2λ) Γ(2μ + 1) Γ k + μ − 2λ + 12 λ−μ− 1 2 M a = k−λ,μ−λ (a) 1 Γ(1 − 2λ + 2μ) Γ k+ μ + 2 Re λ > 0, Re(k + μ − 2λ) > − 12 ET II 405(20)
0
827
∞
3.
x2λ−1 (a + x)−μ− 2 e− 2 x W k,μ (a + x) dx = Γ(2λ)aλ−μ− 2 W k−λ,μ−λ (a) 1
1
1
0
[|arg a| < π, ∞
4.
Re λ > 0]
ET II 411(47)
xλ−1 (a + x)k−λ−1 e− 2 x W k,μ (a + x) dx = Γ(λ)ak−1 W k−λ,μ (a) 1
0
∞
5. 0
[|arg a| < π, Re λ > 0] ET II 411(48) !
! 0, 1 − k − σ 1 1 ! xρ−1 (a + x)−σ e− 2 x W k,μ (a + x) dx = Γ(ρ)aρ e 2 a G 30 a 23 ! −ρ, 1 + μ − σ, 1 − μ − σ 2 2 [|arg a| < π,
∞
6.
1
=
∞
7.
e
− 12 (a+x) (a
0
8. 0
∞
ET II 411(49)
xρ−1 (a + x)−σ e 2 x W k,μ (a + x) dx
0
Re ρ > 0]
2
!
1 ! k − σ + 1, 0 Γ(ρ)aρ e− 2 a ! 1 G 31 a 23 ! −ρ, 1 + μ − σ, 1 − μ − σ −k+μ Γ 2 −k−μ 2 2 [|arg a| < π, 0 < Re ρ < Re(σ − k)] ET II 412(51)
Γ + x)2κ−1 dx = W κ,μ (x) κ (ax) x
1 2
− μ − κ Γ 12 + μ − κ W κ,μ (a) a Γ (1 − 2κ) 1 Re 2 ± μ − κ > 0
BU 126(7a)
dx (x + a)α
Γ(1 + 2μ) Γ 12 + μ + γ Γ (κ − γ) 1 1 1 1 + μ − γ, − μ − γ; a = α, κ − γ; F 2 2 2 2 Γ 2 + μ − γ Γ 2 + μ + κ1 1 Γ α + γ + 2 + μ Γ −γ − 2 − μ γ+ 1 +μ a 2 + Γ(α)
1 3 1 × 2 F 2 α + γ + μ + , κ + μ + ; 1 + 2μ, + μ + γ; a 2 2 2 Re γ + α + 12 + μ > 0, Re (γ − κ) < 0 BU 126(8)a
e− 2 x xγ+α−1 M κ,μ (x) 1
Γ
1
828
Confluent Hypergeometric Functions
9. 0
7.628 1.
2.
7.628
1 dx n+1 n+μ+ 12 12 a = (−1) − μ + κ W −κ,μ (a) e x M κ,μ (x) a e Γ(1 + 2μ) Γ x+a 2
1 n > 0, Re κ − μ − BU 127(10a)a n = 0, 1, 2, . . . , Re μ + 1 + < n, |arg a| < π 2 2
∞
− 12 x n+μ+ 12
2 1 1 1 e−st e−t t2c−2 1 F 1 a; c; t2 dt = 21−2c Γ(2c − 1)Ψ c − , a + ; s2 2 2 4 0 1 Re c > 2 , Re s > 0 EH I 270(11) 2
2
∞ 2 1 2 t as 1 t2ν−1 e− 2a t e−st M −3ν,ν dt = √ Γ(4ν + 1)a−ν s−4ν eas /8 K 2ν a 8 2 π 0 Re a > 0, Re ν > − 14 , Re s > 0
∞
∞
3.
t2μ−1 e− 2a t e−st M λ,μ 1
2
0
2
t a
=2
7.629 1.8
∞
tk exp
0
∞
2. 0
7.631 1.
∞
a 2t
2.
∞
xρ−1 exp
−3μ−λ
a t
1 2 (λ+μ−1)
λ−μ−1
as2 8
as2 4
Γ(4μ + 1)a s e W − 12 (λ+3μ), 12 (λ−μ) Re a > 0, Re μ > − 14 , Re s > 0 ET I 215(13)
√ √ 1 dt = 21−2k as−k− 2 S 2k,2μ 2 as |arg a| < π, Re (k ± μ) > − 12 ,
Re s > 0
ET I 217(21)
Re s > 0]
1 −1 α x − βx−1 W k,μ α−1 x W λ,ν βx−1 dx 2 −1 = β ρ Γ 12 − k + μ Γ 12 − k − μ ! β !! 1 + k, 1 − λ − ρ 41 × G 24 α ! 12 + μ, 12 − μ, 12 + ν − ρ, 3 |arg α| < 2 π, Re β > 0, Re(k + ρ) < −|Re ν| − 12
ET I 217(22)
1 2
−ν−ρ
ET II 412(55)
ρ−1
x 0
dt
a a √ √ 1 t−k exp − e−st W k,μ dt = 2 ask− 2 K 2μ 2 as 2t t [Re a > 0,
0
e−st W k,μ
ET I 215(12)
1 −1 −1 α x − βx exp W k,μ α−1 x W λ,ν βx−1 dx 2 −1 = β ρ Γ 12 − k + μ Γ 12 − k − μ Γ 12 − λ + ν Γ 12 − λ − ν !
β !! 1 + k, 1 + λ − ρ × G 42 24 α ! 12 + μ, 12 − μ, 12 + ν − ρ, 12 − ν − ρ 3 3 |arg α| < 2 π, |arg β| < 2 π, Re(λ − ρ) < 12 − |Re μ|, Re(k + ρ) < 12 − |Re ν| ET II 412(57)
7.644
Confluent hypergeometric and trigonometric functions
829
1 −1 −1 α x + βx 3. x exp W k,μ α−1 x W λ,ν βx−1 dx 2 0 !
β !! 1 − k, 1 − λ − ρ ρ 40 = β G 24 α ! 12 + μ, 12 − μ, 12 + ν − ρ, 12 − ν − ρ [Re α > 0, Re β > 0] ET II 412(54)
∞ μ− 12 1 7.632 e−st et − 1 exp − λet M k,μ λet − λ dt 2 0 Γ(2μ + 1) Γ 12 + k − μ + s W −k− 12 s,μ− 12 s (λ) = Γ(s + 1) Re μ > − 12 , Re s > Re(μ − k) − 12 ET I 216(15)
∞
ρ−1
7.64 Combinations of confluent hypergeometric and trigonometric functions
∞
7.641
cos(ax) 1 F 1 (ν + 1; 1; ix) 1 F 1 (ν + 1; 1; −ix) dx
0
∞
7.64211 0
7.643
∞
4ν − 12 x2
sin(bx) 1 F 1
0
=0
[1 < a < ∞]
1 1 − 2ν; 2ν + 1; x2 2 2
dx =
π 4ν − 1 b2 b c 2 1F 1 2
2.
3.
4.
7.644
11
∞
∞
−μ− 12 − 12 x
x 0
e
b > 0,
ET II 402(4) EH I 285(12)
1 1 − 2ν; 1 + 2ν; b2 2 2 1
Re ν > − 4
1 2 1 2 1 2 π 2ν−1 − 1 b2 x b b x2ν−1 e− 4 x sin(bx) M 3ν,ν e 4 M 3ν,ν dx = 2 2 2 0 b > 0, Re ν > − 14
∞ 1 2 1 2 π −2ν−1 1 b2 −2ν−1 14 x2 x b b x e cos(bx) W 3ν,ν e 4 W 3ν,ν dx = 2 2 2 0 1 Re ν < 4 , b > 0
∞ 1 2 1 2 π −2ν 1 b2 −2ν 14 x2 x b b x e sin(bx) W 3ν−1,ν e 4 W 3ν−1,ν dx = 2 2 2 0 1 Re ν < 2 , b > 0
1.
[0 < a < 1] ;
[−1 < Re ν < 0] 2 1 Γ(c) 1 2α−1 |y| cos(2xy) 1 F 1 a; c; −x2 dx = π 2 e−y Ψ c − 12 , a + 12 ; y 2 2 Γ(a)
x e
1.
= −a−1 sin(νπ) P ν 2a−2 − 1
ET I 115(5)
ET I 116(10)
ET I 61(7)
ET I 116(9)
2
Γ(3 − 2μ) a 1 exp − sin 2ax M k,μ (x) dx = π a W ρ,σ a2 , 2 Γ 2 +k+μ 2ρ = k − 3μ + 1, 2σ = k + μ − 1 [a > 0, Re(k + μ) > 0] ET II 403(10)
1 2
1 2
k+μ−1
830
2.
3.
Confluent Hypergeometric Functions
∞
∞
7.651
1 1 c Γ(1 + μ + ρ) Γ (1 − μ + ρ) xρ−1 sin cx 2 e− 2 x W k,μ (x) dx = Γ 3 −k+ρ 0 2
c2 3 3 × 2 F 2 1 + μ + ρ, 1 − μ + ρ; , − k + ρ; − 2 2 4 [Re ρ > |Re μ| − 1] ET II 407(28) ∞ 1 1 xρ−1 sin cx 2 e 2 x W k,μ (x) dx 0 ! 1 π2 c2 !! 12 + μ − ρ, 12 − μ − ρ 22 G 23 = 1 ! 4 ! 12 , −k − ρ, 0 Γ 2 − k + μ Γ 12 − k − μ c > 0, Re ρ > |Re μ| − 1, Re(k + ρ) < 12 ET II 407(29)
4.
ρ−1
x
cos cx
1 2
e
0
∞
5.
− 12 x
1
+ μ + ρ Γ 12 − μ + ρ W k,μ (x) dx = Γ(1 − k + ρ)
1 1 1 c2 + μ + ρ, − μ + ρ; , 1 − k + ρ; − × 2F 2 2 2 4 2 Re ρ > |Re μ| − 12 ET II 407(30) Γ
2
1 1 xρ−1 cos cx 2 e 2 x W k,μ (x) dx
! 1 2 ! 1 + μ − ρ, 1 − μ − ρ π2 c ! 2 G 22 = 1 !2 23 4 ! 0, −k − ρ, 12 Γ 2 − k + μ Γ 12 − k − μ c > 0, Re ρ > |Re μ| − 12 , Re(k + ρ) < 12 ET II 407(31)
0
7.65 Combinations of confluent hypergeometric functions and Bessel functions 7.651
∞
J ν (xy) M − 12 μ, 12 ν (ax) W
1. 0
2. 0
1 1 2 μ, 2 ν
(ax) dx + 2 1 ,μ 2 − 1 Γ(ν + 1) 2 2 a + a a + y2 2 = ay −μ−1 1 1 + y 1 Γ 2 − 2μ + 2ν y > 0, Re ν > −1, Re μ < 12 , Re a > 0 ET II 85(19)
∞
M k, 12 ν (−iax) M −k, 12 ν (−iax) J ν (xy) dx ae− 2 (ν+1)πi [Γ(1 + ν)] 1 y −1−2k Γ 2 + k + 12 ν Γ 12 − k + 12 ν − 1 + 1 ,2k + 1 ,2k × a2 − y 2 2 + a − a2 − y 2 2 a + a2 − y 2 2 1
=
=0
2
a > 0,
Re ν > −1,
|Re k|
0, Re ν > −1, Re μ < 14 , Re a > 0, Re b > 0
7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers 7.661
∞
1.
x−1 W k,μ (ax) M −k,μ (ax) J 0 (xy) dx
% %
1 &
1 & 2 2 2 2 Γ(1 + 2μ) y y k k P μ− 1 = e−ikπ 1 1+ 2 Q μ− 1 1+ 2 2 2 a a Γ 2 +μ+k 1 3 y > 0, Re a > 0, Re μ > − 2 , Re k < 4 ET II 18(44)
0
∞
2.
−1
x 0
% %
1 &
1 & 1 y2 2 y2 2 k −k W k,μ (ax) W −k,μ (ax) J 0 (xy) dx = π cos(μπ) P μ− 1 1+ 2 P μ− 1 1+ 2 2 2 2 a a 1 y > 0, Re a > 0, |Re μ| < 2 ET II 18(45)
∞
3.
x2μ−ν W k,μ (ax) M −k,μ (ax) J ν (xy) dx
0
Γ(2μ + 1) = 22μ−ν+2k a2k y ν−2μ−2k−1 Γ ν − k − μ + 12
1 1 1 y2 − k, 1 − k, − k + μ; 1 − 2k, − k − μ + ν; − 2 × 3F 2 2 2 2 a y > 0, Re μ > − 12 , Re a > 0, Re(2μ + 2k − ν) < 12 ET II 85(20) 4. 0
5.
∞
x2ρ−ν W k,μ (iax) W k,μ (−iax) J ν (xy) dx !
−1 1 1 y 2 !! 12 , 0, 12 − μ, 12 + μ 2ρ−ν ν−2ρ−1 − 12 24 −k+μ Γ −k−μ =2 y π Γ ! G 44 2 2 a2 ! ρ + 12 , −k, k, ρ − ν + 12 y > 0, Re a > 0, Re ρ > |Re μ| − 1, Re(2ρ + 2k − ν) < 12 ET II 86(23)a
∞
x2ρ−ν W k,μ (ax) M −k,μ (ax) J ν (xy) dx
0
y > 0,
Re a > 0,
! 22ρ−ν Γ(2μ + 1) ν−2ρ−1 23 y 2 !! 12 , 0, 12 − μ, 12 + μ = 1 1 y ! G 44 a2 ! ρ + 12 , −k, k, ρ − ν + 12 π2 Γ 2 − k + μ Re ρ > −1, Re(ρ + μ) > −1, Re(2e + 2k + ν) < 12 ET II 86(21)a
832
Confluent Hypergeometric Functions
∞
6.
x2ρ−ν W k,μ (ax) W −k,μ (ax) J ν (xy) dx
0
=
7.662
∞
1.
−1
x 0
∞
2. 0
∞
3.
M −μ, 14 ν
Γ(ρ + 1 + μ) Γ(ρ + 1 − μ) Γ(2ρ + 2) ν −ν−1 −2ρ−1 y 2 a Γ 32 + k + ρ Γ 32 − k + ρ Γ(1 + ν)
3 3 y2 3 × 4 F 3 ρ + 1, ρ + , ρ + 1 + μ, ρ + 1 − μ; + k + ρ, − k + ρ, 1 + ν; − 2 2 2 2 a [y > 0, Re ρ > |Re μ| − 1, Re a > 0] ET II 86(22)a
Γ 1 + 12 ν 1 2 1 2 1 2 1 2 I 1 ν−μ 1 x W μ, 14 ν x J ν (xy) dx = 1 1 y y K 4 ν+μ 2 2 4 4 Γ 2 + 4ν − μ 4
x−1 M α−β, 14 ν−γ
x−1 M k,0
0
[y > 0,
4. 0
Re ν > −1]
ET II 86(24)
1 2 1 2 x W α+β, 14 ν+γ x J ν (xy) dx 2 2
Γ 1 + 12 ν − 2γ −2 1 2 1 2 y W α+γ, 14 ν+β y = y M α−γ, 14 ν−β 2 2 Γ 1 + 12 ν − 2β ET II 86(25) y > 0, Re β < 18 , Re ν > −1, Re(ν − 4γ) > −2
π iax2 M k,0 −iax2 K 0 (xy) dx = 16
2 2 2 y2 y Jk + Yk 8a 8a [a > 0]
∞
7.662
ET II 152(83)
iy iy 2 2 x−1 M k,μ iax2 M k,μ −iax2 K 0 (xy) dx = ay −2 [Γ(2μ + 1)] W −μ,k W −μ,k − 4a 4a a > 0, Re y > 0, Re μ > − 12
2
ET II 152(84)
7.663
∞
1.
2ρ
x 0
∞
2. 0
3. 0
∞
2!
y !! 1, b 22ρ Γ(b) 2 21 J ν (xy) dx = G 1 F 1 a; b; −λx Γ(a)y 2ρ+1 23 4λ ! 12 + ρ + 12 ν, a, 12 + ρ − 12ν y > 0, −1 − Re ν < 2 Re ρ < 12 + 2 Re a, Re λ > 0 ET II 88(6)
1 1 2 2ν−a+ 2 Γ(a + 1) 2a−ν−1 − 1 y2 1 4 1 y xν+1 1 F 1 2a − ν; a + 1; − x2 J ν (xy) dx = e K y 1 a−ν− 2 2 4 π 2 Γ(2a − ν) 1 y > 0, Re ν > −1, Re(4a − 3ν) > 2 ET II 87(1)
1 y2 1+a+ν 1+a+ν ; − x2 J ν (xy) dx = y a−1 1 F 1 a; ;− xa 1 F 1 a; 2 2 2 2 y > 0, Re a > − 12 , Re(a + ν) > −1 ET II 87(2)
7.664
Confluent hypergeometric functions, Bessel functions, and powers
∞
4.
ν+1−2a
x
1F 1
0
∞
5.
833
1 2 a; 1 + ν − a; − x J ν (xy) dx 2
1 1 2 π 2 Γ(1 + ν − a) −2a+ν+ 1 2a−ν−1 − 1 y2 2y 2 y = e 4 I a− 12 4 Γ(a) y > 0, Re a − 1 < Re ν < 4 Re a − 12 ET II 87(3)
−1 2λ−2 − 1 y2 x 1 F 1 λ; 1; −x2 J 0 (xy) dx = 22λ−1 Γ(λ) y e 4
0
[y > 0, ∞
6.
xν+1 1 F 1 a; b; −λx2 J ν (xy) dx
0
=
∞
7. 0
7.664
1
1
2
Γ(a)λ 2 a+ 2 ν
y
a−2 − y 8λ
e
xW
a
0
1 2 ν,μ
x
W − 12 ν,μ
a x
ET II 18(46)
y2 W k,μ , 2k = a − 2b + ν + 2, 2μ = a − ν − 1 4λ y > 0, −1 < Re ν < 2 Re a − 12 , Re λ > 0 ET II 88(4)
22b−2a−ν−1 Γ(b) −a 2a−2b+ν λ y x2b−ν−1 1 F 1 a; b; −λx2 J ν (xy) dx = Γ(a − b + ν + 1)
y2 × 1 F 1 a; 1 + a − b + ν; − 4λ y > 0, 0 < Re b < 34 + Re a + 12 ν , Re λ > 0
∞
1.
21−a Γ(b)
Re λ > 0]
ET II 88(5)
+ , + , 1 1 1 1 K ν (xy) dx = 2ay −1 K 2μ (2ay) 2 e 4 iπ K 2μ (2ay) 2 e− 4 iπ
ET II 152(85) [Re y > 0, Re a > 0]
∞ 2 2 x W 12 ν,μ W − 12 ν,μ J ν (xy) dx x x 0 1 1 1 K 2μ 2y 2 = −4y −1 sin μ − 12 ν π J 2μ 2y 2 + cos μ − 12 ν π Y 2μ 2y 2
2.
[y > 0,
4.
ET II 87(27)
2 2 W Y ν (xy) dx 1 1 ν,μ ν,μ − x x 2 0 2 1 1 1 cos μ − 12 ν π J 2μ 2y 2 − sin μ − 12 ν π Y 2μ 2y 2 K 2μ 2y 2 = 4y −1 y > 0, |Re μ| < 14 ET II 117(48)
∞ 1 1 2 2 4 Γ(1 + 2μ)y −1 J 2μ 2y 2 K 2μ 2y 2 x W − 12 ν,μ M 12 ν,μ J ν (xy) dx = 1 1 x x Γ 2 + 2 ν + μ 0 y > 0, Re ν > −1, Re μ > − 14
3.
Re (ν ± 2μ) > −1]
∞
xW
ET II 86(26)
834
Confluent Hypergeometric Functions
5.
x W − 12 ν,μ
0
7.665
∞
1. 0
∞
∞
2.
ia x
ia W − 12 ν,μ − J ν (xy) dx x + , + , −1 1 1 K μ (2iay) 2 K μ (−2iay) 2 = 4ay −1 Γ 12 + μ + 12 ν Γ 12 − μ + 12 ν y > 0, Re a > 0, |Re μ| < 12 , Re ν > −1 ET II 87(28)
1 1 1 x M k,μ (x) dx x− 2 J ν ax 2 K 12 ν−μ 2 2
2
a a Γ(2μ + 1) = M 12 (k+μ), 12 k+ 14 ν W 12 (k−μ), 12 k− 14 ν 1 2 2 a Γ k + ν + 1 2 1 1 ET II 405(18) a > 0, Re k > − 4 , Re μ > − 2 , Re ν > −1 + , 1 1 1 x 2 c+ 2 c −1 Ψ(a, c; x) 1 F 1 (a ; c ; −x) J c+c −2 2(xy) 2 dx
0
=
∞
7.666
7.665
1 1 Γ (c ) y 2 c+ 2 c −1 Ψ (c − a , c + c − a − a ; y) 1 F 1 (a ; a + a ; −y) Γ (a + a ) Re c > 0, 1 < Re (c + c ) < 2 Re (a + a ) + 12 EH I 287(23)
+ , 1 1 1 1 1 x 2 c− 2 1 F 1 a; c; −2x 2 Ψ a, c; 2x 2 J c−1 2(xy) 2 dx
0
= 2−c
, 1 c−2a 1 Γ(c) a− 1 c− 1 + y 2 2 1 + (1 + y) 2 (1 + y)− 2 Γ(a) Re c > 2, Re(c − 2a) < 12 EH I 285(13)
7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers 7.671
∞
k− 32
x
1. 0
2. 0
∞
1 1 ax M k,ν (x) dx exp − (a + 1)x K ν 2 2 1 π 2 Γ(k) Γ(k + 2ν) −1 = k+ν 1 2 F 1 k, k + 2ν; 2ν + 1; −a a Γ k+ν+ 2 [Re a > 0, Re k > 0, Re(k + 2ν) > 0] ET II 405(17)
3 1 1 ax W k,μ (x) dx x−k− 2 exp − (a − 1)x K μ 2 2 π Γ(−k) Γ(2μ − k) Γ(−2μ − k) 22k+1 ak−ν 2 F 1 −k, 2μ − k; −2k; 1 − a−1 = 1 Γ 2 − k Γ 12 + μ − k Γ 12 − μ − k [Re a > 0, Re k < 2 Re μ < − Re k] ET II 408(36)
7.672
7.672
Confluent hypergeometric functions, Bessel functions, exponentials, and powers
∞
1.
2 1 x2ρ e− 2 ax M k,μ ax2 J ν (xy) dx
! Γ(2μ + 1) 2ρ −2ρ−1 21 y 2 !! 12 − μ, 12 + μ 2 y = ! G 23 4a ! 12 + ρ + 12 ν, k, 12 + ρ − 12 ν Γ μ + k + 12 y > 0, −1 − Re 12 ν + μ < Re ρ < Re k − 14 , Re a > 0 ET II 83(10)
0
∞
2.
2 1 x2ρ e− 2 ax W k,μ ax2 J ν (xy) dx
Γ 1 + μ + 12 ν + ρ Γ 1 − μ + 12 ν + ρ 2−ν−1 − 1 ν−ρ− 1 ν 2y = a 2 Γ(ν + 1) Γ 32 − k + 12 ν + ρ
y2 1 × 2 F 2 λ + μ, λ − μ; ν + 1, − k + λ; − , 2 4a y > 0, Re a > 0, Re ρ ± μ + 12 ν > −1 ET II 85(16) λ = 1 + 12 ν + ρ
0
∞
3.
2 1 x2ρ e 2 ax W k,μ ax2 J ν (xy) dx =
0
∞
4.
∞
5. 0
7.
e
y > 0,
M k,μ
22ρ y −2ρ−1 Γ 2 +μ − k! Γ 12 − μ − k y 2 !! 12 − μ, 12 + μ 22 × G 23 ! 4a ! 12 + ρ + 12 ν, −k, 12 + ρ − 12 ν |arg a| < π, −1 − Re 12 ν ± μ < Re ρ < − 14 − Re k ET II 85(17) 1
!
1 2 2λ y −1/2 Γ(2μ + 1) 31 y 2 !! −μ − λ, μ − λ, 1 G 34 x Y ν (xy) dx = 2 2 ! h, κ, −λ − 12 , l Γ 2 +k+μ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = − 14 − 12 ν y > 0, Re(k − λ) > 0, Re (2λ + 2μ ± ν) > − 52 ET II 116(45)
1 1 2 1 2 x Y ν (xy) dx x2λ+ 2 e 4 x W k,μ 2
−1 2!
1 1 y !! −μ − λ, μ − λ, l 32 −k+μ Γ −k−μ = 2λ Γ y −1/2 , G 34 2 2 2 ! h, κ, − 12 − k − λ, l h = 14 + 12 ν, κ = 14 − 12 ν, l = − 14 − 12 ν y > 0, Re(k + λ) < 0, Re (2λ ± 2μ ± ν) > − 52 ET II 117(47)
1 2 1 y x−1/2 e− 2 x M 12 ν− 14 , 12 ν+ 14 x2 J ν (xy) dx = (2ν + 1)2−ν y ν−1 1 − Φ 2 0 y > 0, Re ν > − 12 ∞ 1 2 1 Γ(ν + 2)y ν y x−1 e− 2 x M 12 ν+ 12 , 12 ν+ 12 x2 J ν (xy)dx = 1 − Φ 3 ν 2 Γ ν+2 2 0
6.
2λ+ 12 − 14 x2
x 0
835
∞
[y > 0, Re ν > −1]
ET II 82(1)
ET II 82(2)
836
Confluent Hypergeometric Functions
∞
8.
e
− 14 x2
0
1 2−k Γ(ν + 1) 2k−1 − 1 y2 y M k, 12 ν e 2 x2 J ν (xy) dx = 2 Γ k + 12 ν + 12 y > 0, Re ν > −1,
0
∞
10.
ν−2μ
x
e
1 2 4x
∞
11.
W k,±μ
xν−2μ e− 4 x W k,±μ 1
2
0
y > 0, ∞
12.
2μ−ν − 14 x2
x
e
13.
∞
2μ−ν − 14 x2
x
e
M k,μ
Re(ν − 2μ) > −1] ET II 84(14)
1 2 x J ν (xy) dx 2
1 2 Γ(1 + ν − 2μ) 12 ( 12 +k−3μ+ν ) μ−k− 3 1 y2 2 4 = y , y e W α,β 2 2 Γ 12 + μ − k 1 2α = k + 3μ −ν − 12 , 2β = k1 − μ + ν + 2 1 Re ν > −1, Re(ν − 2μ) > −1, Re k − μ + 2 ν < − 4 ET II 84(15)
1 2 x J ν (xy) dx 2
1 1 3 1 2 1 2 Γ(2μ + 1) 2 2 ( 2 −k+3μ−ν ) y k−μ− 2 e− 4 y M α,β = 1 y 2 Γ 2 +k−μ+ν 1 1 + k + 3μ − ν, 2β = − + k − μ +ν 2α = 2 2 y > 0, − 12 < Re μ < Re k + 12 ν − 14 ET II 83(8)
1 2 x Y ν (xy) dx 2
M k,μ
0
Re ν > −1,
0
1 2 1 2 Γ(1 + ν − 2μ) β−μ k+μ− 3 − 1 y2 2 4 x J ν (xy) dx = 2 y y e M α,β 2 Γ(1 + 2β) 2 1 1 2α = 2 + k + ν − 3μ, 2β = 2 − k + ν − μ [y > 0,
1 2
1 ν−2μ − 14 x2 x e M k,μ x2 J ν (xy) dx 2
1 1 1 2 Γ(2μ + 1) k+μ− 32 − 14 y 2 = 2 2 ( 2 −k−3μ+ν ) y e W , y α,β 2 Γ μ + k + 12 1 2β = k + μ − ν − 12 2α = k − 3μ + ν + 2 , y > 0, −1 < Re ν < 2 Re(k + μ) − 12 ET II 83(9)
0
Re k
0,
1
2
2α = 3μ − ν + k + 12 , 2β = μ − ν − k + 12 1 −1 < 2 Re μ < Re(2k + ν) + 2 , Re(2μ − ν) > −1 ET II 116(44)
7.673
Confluent hypergeometric functions, Bessel functions, exponentials, and powers
∞
14.
2μ+ν − 14 x2
x
e
M k,μ
0
837
1 2 x Y ν (xy) dx 2 = π −1 2μ+β y k−μ− 2 Γ(2μ + 1) 1 2 − 1 y2 1 Γ(2μ + ν + 1) ×Γ 2 − μ − k e 4 cos(2μπ) 3 M α,β 2 y Γ μ+ν−k+ 2 1 2 + sin[(μ − k)π] W α,β 2 y 3
∞
15.
y > 0,
2α = 3μ + ν + k + 12 , 2β = μ + ν − k + 12 1 −1 < 2 Re μ < Re(2k − ν) + 2 , Re(2μ + ν) > −1 ET II 116(43)
2 1 1 1 1 3 x2μ+ν e− 2 ax M k,μ ax2 K ν (xy) dx = 2μ−k− 2 a 4 − 2 (μ+ν+k) y k−μ− 2
0
2
y2 y W κ,m , 8a 4a 2m = μ + ν − k + 12 2κ = −3μ − ν − k − 12 , 1 Re a > 0, Re μ > − 2 , Re(2μ + ν) > −1 ET II 152(82) × Γ(2μ + 1) Γ(2μ + ν + 1) exp
7.673 1.
∞
10 0
Re y > 0,
√ 1 1 e− 2 ax x 2 (μ−ν−1) M κ, 12 μ (ax) J ν 2 bx dx 1+μ κ−1 2 − 4 1 b b a− 2 (μ+1−ν) Γ(1 + μ)e− 2a = a
b × M 1 (κ−ν−1)+ 3 (1+μ), κ+ν − 1+μ 2 4 2 4 a
ν−μ 3 Re(1 + μ) > 0, Re κ + >− , 2 4
2. 0
∞
1
1+μ κ+ν − Γ 1+ 2 4 Im b = 0
BU 128(12)a
b √ 1 1 1 Γ (ν + 1 ∓ μ) e 2a a 12 (κ+1)+ 14 (1∓ν) 1±μ e 2 ax x 2 (ν−1∓μ) W κ, 12 μ (ax) J ν 2 bx dx = a− 2 (ν+1∓μ) b Γ 2 −κ
b × W 12 (κ+1−ν)− 34 (1∓μ), 12 (κ+ν)+ 14 (1∓μ)
a ν∓μ 3 + κ < , Re ν > −1 BU 128(13) Re 2 4
838
Confluent Hypergeometric Functions
7.674 1.
∞
0
2.
7.674
1 xρ−1 e− 2 κ J λ+ν ax1/2 J λ−ν ax1/2 W k,μ (x) dx 1 2λ 1 Γ 2 + λ + μ + ρ Γ 12 + λ − μ + ρ 2a = Γ (1 + λ + ν) Γ(1 + λ − ν) Γ(1 + λ − k + ρ)
× 4 F 4 1 + λ, 12 + λ, 12 + λ + μ + ρ, 12 + λ − μ + ρ; 1 + λ + ν,
1 + λ − ν, 1 + 2λ, 1 + λ − k + ρ; −a2 |Re μ| < Re(λ + ρ) + 12 ET II 409(37) ∞ 1 xρ−1 e− 2 κ I λ+ν ax1/2 K λ−ν ax1/2 W k,μ (x) dx 0 !
π −1/2 24 2 !! 0, 12 , 12 + μ − ρ, 12 − μ − ρ = G 45 a ! 2 λ, ν, −λ, −ν, k − ρ |Re μ| < Re(λ + ρ) + 12 , |Re μ| < Re(ν + ρ) + 12 ET II 409(38)
Combinations of Struve functions and confluent hypergeometric functions 7.675 1.
∞
1
1
2
0
∞
2.
1
1
∞
3.
2λ+ 12
x 0
y > 0, 4. 0
∞
2
ET II 171(42)
1 2 x Hν (xy) dx 2 Γ 74 + 12 ν + λ + μ Γ 74 + 12 ν + λ − μ 1 1 −λ− ν −1/2 ν+1 2 π y = 24 Γ ν + 32 Γ 94 + λ − k − 12 ν
7 ν 3 3 9 ν y2 7 ν × 3 F 3 1, + + λ + μ, + + λ − μ; , ν + , + λ − k + ; − 4 2 2 2 4 2 2 4 2 Re(2λ + ν) > 2|Re μ| − 74 , y > 0 ET II 171(43)
x2λ+ 2 e− 4 x W k,μ
0
2!
1 2 y !! l, −μ − λ, mu − λ 2−λ Γ(2μ + 1) G 22 x Hν (xy) dx = 1/2 1 34 2 2 ! l, k − λ − 12 , h, κ y Γ 2 +k+μ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = 34 + 12 ν Re(2λ + 2μ + ν) > − 72 , Re(k − λ) > 0, y > 0, Re(2λ − 2k + ν) < − 12
x2λ+ 2 e− 4 x M k,μ
1 2 x Hν (xy) dx e W k,μ 2
−1 2!
1 1 y !! l, −μ − λ, μ − λ −k+μ Γ −k−μ = 2λ Γ y −1/2 G 23 34 2 2 2 ! l, −k − λ − 12 , h, κ 1 1 h = 4 + 2 ν, κ = 14 − 12 ν, l = 34 + 12 ν 7 Re(2λ + ν) > 2|Re μ| − 2 , Re (2k + 2λ + ν) < − 12 , Re(k + λ) < 0 ET II 172(46)a 1 2 4x
+ y , 1 2 1 2 e 2 x W − 12 ν− 12 , 12 ν x2 Hν (xy) dx = 2−ν−1 y ν πe 4 y 1 − Φ 2 [y > 0,
Re ν > −1]
ET II 171(44)
7.681
Confluent hypergeometric functions and other special functions
839
7.68 Combinations of confluent hypergeometric functions and other special functions Combinations of confluent hypergeometric functions and associated Legendre functions 7.681
∞
1. 0
1 x x−1/2 (a + x)μ e− 2 x P −2μ 1+2 M k,μ (x) dx ν a 1 sin(νπ) Γ(2μ + 1) Γ k − μ + ν + 12 Γ k − μ − ν − 12 e 2 a W ρ,σ (a), =− π Γ(k) ρ = 12 !− k + μ, ! σ = 12 + ν 1 |arg a| < π, Re μ > − , Re(k − μ) > !Re ν + 1 ! ET II 403(11) 2
∞
1 x 1+2 M k,μ (x) dx x−1/2 (a + x)−μ e− 2 x P −2μ ν a 1 Γ(2μ + 1) Γ k + μ + ν + 12 Γ k + μ − ν − 12 e 2 a W 12 −k−μ, 12 +ν (a) = Γ k + μ + 12 Γ(2μ + ν + 1) Γ(2μ − ν) ! ! |arg a| < π, Re μ > − 12 , Re(k + μ) > !Re ν + 12 ! ET II 403(12)
∞
1 1 1 1 x W k,ν (x) dx x− 2 − 2 μ−ν (a + x) 2 μ e− 2 x P μk+ν− 3 1 + 2 2 a 1 1 1 1 Γ(1 − μ − 2ν) a− 4 + 2 k− 2 ν e 2 a W ρ,σ (a) = 3 Γ 2 −k−μ−ν 2ρ = 12 + 2μ + ν − k, 2σ = k + 3ν − 32
2. 0
2
3. 0
[|arg a| < π,
x− 2 − 2 μ−ν (a + x)− 2 μ e− 2 x P μk+μ+ν− 3 1 + 2 1
1
1
1
2
0
x a =
W k,ν (x) dx Γ
1 1 1 1 Γ(1 − μ − 2ν) 3 a− 2 + 2 k− 2 ν e 2 a W ρ,σ (a) − k − μ − ν 2 2ρ = 12 − k + ν, 2σ = k + 2μ + 3ν − 32
[|arg a| < π,
5. 0
Re(μ + 2ν) < 1] ET II 407(32)
∞
4.
Re μ < 1,
Re μ < 1,
Re(μ + 2ν) < 1] ET II 408(33)
∞
xμ− 4 k− 2 ν− 2 (a + x) 2 ν e− 2 x Q νμ−k+ 3 1 + 2 1
1
1
1
1
x
M k,ν (x) dx a eνπi Γ(1 + 2μ − ν) Γ(1 + 2μ) Γ 52 − k + μ + ν 1 (κ+2μ−2ν+5) 1 a = e 2 W ρ,σ (a) a4 2 Γ 12 + k + μ 2ρ = 12 − k − μ + 2ν, 2σ = k − 3μ − 32 |arg a| < π, Re μ > − 12 , Re(2μ − ν) > −1 2
ET II 404(14)
840
7.682
Confluent Hypergeometric Functions
∞
1.
x−1/2 e− 2 x P −2μ ν 1
1+
0
∞
2. 0
∞
3. 0
∞
4.
x 1/2 M k,μ (x) dx a 1 Γ(2μ + 1) Γ k + 12 ν Γ k − 12 ν − 12 e 2 a W 3 −k, 1 + 1 ν (a) = 2μ 1/4 1 1 1 1 4 4 2 2 a Γ k + μ + 2 1Γ μ + 21ν + 2 Γ μ − 12 ν |arg a| < π, Re k > 2 Re ν − 2 , Re k > − 2 Re ν ET II 404(13)
x 1/2 1+ M k,μ (x) dx 1a Γ 2 − ν Γ(1 + 2μ) Γ(k + μ + ν) 1 a 1 = e(1−k+μ−ν)πi 2μ−k−ν a 2 (k+μ−1) e 2 W ρ,σ (a), Γ k + μ + 12 ρ = 12 − k − 12 ν, σ = μ + 12 ν 1 |arg a| < π, Re μ > − 2 , Re(k + μ + ν) > 0 ET II 404(15)
1−k+μ−ν x 2 (k+μ+ν)−1 (a + x)−1/2 e− 2 x Q k−μ−ν−1 1
1
x 1/2 M k,μ (x) dx a 1 2(μ−ν)πi 2μ−2ν−1 12 (k+μ−1) 12 a Γ(2μ + 1) Γ(ν + 1) Γ k + μ − 2ν − 2 2 a e =e W ρ,σ (a), Γ k + μ + 12 2ρ = 1 − k + μ − 2ν, 2σ = k − μ − 2ν − 2 |arg a| < π, Re μ > − 12 , Re ν > −1, Re(k + μ − 2ν) > 12 ET II 404(16)
xν− 2 e− 2 x Q 2μ−2ν 2k−2ν−3 1
1
− 12 − 12 μ−ν − 12 x
x 0
e
1+
P μ2k+μ+2ν−3
x 12 1+ W k,ν (x) dx a 2μ Γ(1 − μ − 2ν) − 1 + 1 k− 1 ν 1 a a 2 2 2 e 2 W ρ,σ (a), = 3 Γ 2 −k−μ−ν 2ρ = 1 − k + μ + ν, 2σ = k + μ + 3ν − 2 [|arg a| < π,
5.8 0
∞
7.682
Re μ < 1,
x 1/2 − 12 − 12 μ−ν −1/2 − 12 x μ x (a + x) e P 2k+μ+2ν−2 1 + W k,ν (x) dx a μ 2 Γ(1 − μ − 2ν) − 1 + 1 k− 1 ν 1 a a 2 2 2 e 2 W ρ,σ (a), 2ρ = μ + ν − k, = 3 Γ 2 −k−μ−ν [|arg a| < π, Re μ > 0,
Re(μ + 2ν) < 1] ET II 408(34)
2σ = k + μ + 3ν − 1 Re ν > 0]
ET II 408(35)
A combination of confluent hypergeometric functions and orthogonal polynomials 1 μ−α 1 8 e− 2 ax xα (1 − x) 2 −1 Lα 7.683 n (ax) M α− 1+α , μ−α−1 [a(1 − x)] dx 0
2
1
Γ(μ − α) Γ(1 + n + α) − 1+α a 2 M α+n, μ2 (a) Γ(1 + μ) n! Re(μ − α) > 0, n = 0, 1, 2, . . .] BU 129(14b) =
[Re a > −1,
7.711
Parabolic cylinder functions
841
A combination of hypergeometric and confluent hypergeometric functions
∞ λ ρ−1 − 12 x 7.684 x e M γ+ρ,β+ρ+ 12 (x) 2 F 1 α, β; γ; − dx x 0 Γ(α + β + 2ρ) Γ(2β + 2ρ) Γ(γ) 1 β+ρ− 1 1 λ 2 e2 λ2 W k,μ (λ); = Γ(β) Γ(β + γ + 2ρ) 1 1 k = 2 − α − 2 β − ρ, μ = 12 β + ρ [|arg λ| < π, Re(β + ρ) > 0, Re (α + β + 2ρ) > 0, Re γ > 0] ET II 405(19)
7.69 Integration of confluent hypergeometric functions with respect to the index 1 (aβ)1/2 exp − (α + β) 7.691 sech(πx) W ix,0 (α) W −ix,0 (β) dx = 2 α+β 2 −∞ i∞ 7.692 Γ(−a) Γ(c − a)Ψ(a, c; x)Ψ(c − a, c; y) da = 2πi Γ(c)Ψ(c, 2c; x + y)
∞
ET II 414(61) EH I 285(15)
−i∞
7.693
∞
Γ(ix) Γ(2k + ix) W k+ix,k− 12 (α) W −k−ix,k− 12 (β) dx
1. −∞
= 2π
1/2
k
Γ(2k)(aβ) (α + β)
1 2 −2k
K
2k− 12
a+β 2
ET II 414(62)
i∞
2.
Γ −i∞
1 2
+ ν + μ + x Γ 12 + ν + μ − x Γ 12 + ν − μ + x Γ 12 + ν − μ − x × M μ+ix,ν (α) M μ−ix,ν (β) dx 1
2
2π(aβ)ν+ 2 [Γ(2ν + 1)] Γ(2ν + 2μ + 1) Γ(2ν − 2μ + 1) M 2μ,2ν+ 12 (α + β) = (α + β)2ν+1 Γ(4ν+ 2) 1 Re ν > |Re μ| − 2 ET II 413(59) ∞
7.69411 −∞
e−2ρxi Γ
1 2
+ ν + ix Γ 12 + ν − ix M ix,ν (α) M ix,ν (β) dx 1 2 = π αβ [Γ(2ν + 1)] sech ρ exp − (α + β) tanh ρ J 2ν αβ sech ρ 2 |Im ρ| < 12 π, Re ν > − 12
7.7 Parabolic Cylinder Functions 7.71 Parabolic cylinder functions 7.711
∞
1. −∞
D n (x) D m (x) dx = 0 = n!(2π)1/2
[m = n] [m = n] WH
842
Parabolic Cylinder Functions
%
& 1 1 ∓ 1 1 1 Γ 12 − 12 μ Γ − 12 ν Γ 2 − 2ν Γ −2μ [when the lower sign is taken, Re μ > Re ν] BU 11 117(13a), EH II 122(21)
∞
1
π2 2 (μ+ν+1) D μ (±t) D ν (t) dt = μ−ν
2. 0
7.721
∞
2
3.
[D ν (t)] dt = π
1/2 −3/2 ψ
2
0
1 2
− 12 ν − ψ − 12 ν Γ(−ν)
BU 117(13b)a, EH II 122(22)a
7.72 Combinations of parabolic cylinder functions, powers, and exponentials 7.721
∞
e− 4 x (x − z)−1 D n (x) dx = ±ie∓nπi (2π)1/2 n!e− 4 z D −n−1 (∓iz) 1
1. −∞
2
1
2
[The upper or lower sign is taken accordingly as the imaginary part of z is positive or negative.]
∞
2.
xν (x − 1)
1
1 1 2 μ− 2 ν−1
μ ν μ−ν (x − 1)2 a2 exp − D μ (ax) dx = 2μ−ν−2 a 2 − 2 −1 Γ D ν (a) 4 2 [Re(μ − ν) > 0]
7.722
∞
1. 0
2. 3.11 7.723
3 2 1 1 1 e− 4 x xν D ν+1 (x) dx = 2− 2 − 2 ν Γ(ν + 1) sin (1 − ν)π 4
1/2 − 12 μ− 12 ν
Γ(μ) π 2 1 1 Γ 2 μ + 2 ν + 12 0
∞ 1 − 34 x2 ν − 12 ν πν e x D ν−1 (x) dx = 2 Γ(ν) sin 4 0
∞
1. 0
ET II 395(4)a
[Re ν > −1] ∞
WH
e− 4 x xμ−1 D −ν (x) dx = 1
2
WH
[Re μ > 0]
EH II 122(20)
[Re ν > −1]
ET II 395(2)
π 1/2 −1 1 2 1 2 e− 4 x xν x2 + y 2 D ν (x) dx = Γ(ν + 1)y ν−1 e 4 y D −ν−1 (y) 2 [Re y > 0, Re ν > −1] EH II 121(18)a, ET II 396(6)a
∞
2.
−1/2 1 2 1 2 e− 4 x xν−1 x2 + y 2 D ν (x) dx = y ν−1 Γ(ν)e 4 y D −ν (y)
0
3. 0
[Re y > 0, 1
ET II 396(7)
λ−1 a2 x2 Γ(λ) Γ(2ν) λ−1 a2 2 x2ν−1 1 − x2 e 4 D −2λ−2ν (ax) dx = e 4 D −2ν (a) Γ(2λ + 2ν) [Re λ > 0,
Re ν > 0]
∞
7.724 −∞
e−
(x−y)2 2μ
Re ν > 0]
+ , y2 1 2 1 e 4 x D ν (x) dx = (2πμ)1/2 (1 − μ) 2 ν e 4−4μ D ν y(1 − μ)−1/2
ET II 395(3)a
[0 < Re μ < 1] EH II 121(15)
7.731
7.725
Parabolic cylinder and hyperbolic functions
∞
1.
e
−pt
(2t)
ν−1 2
e
− 2t
0
∞
2.
e−pt (2t)
ν−1 2
0
4.
5.
6.
√ π 1/2 2t dt = 2
[Re ν > −1] ν p+1−1 √ pν p + 1
√
MO 175
MO 175 [Re ν > −1] √ 3 n −n− 2 b + 12 e−bx D 2n+1 2x dx = (−2)n Γ n + 32 b − 12 0 Re b > − 12 ET I 210(3) 1
n
−n− 2 ∞ √ √ −1 −bx 1 1 1 x e D 2n 2x dx = (−2)n Γ n + b+ b− 2 2 2 0 1 Re b > − 2 ET I 210(5)
ν ∞ < √ √ 1 1 < x− 2 (ν+1) e−sx D ν x dx = π 1 + 12 + 2s 1 0 4 +s Re s > − 14 , Re ν < 1 ET I 210(7)
∞ ν + , 1−β− 1/2 2π β ν β ν+β+1 z−k 2 Γ(β) −zt −1+ β 1/2 − 2 2 (z + k) dt = 1 , ; ; e t D −ν 2(kt) F 2 2 2 z+k Γ 2 ν + 12 β + 12 0 , + z Re(z + k) > 0, Re > 0 k
3.
π 1/2 √p + 1 − 1ν+1 √ D −ν−2 2t dt = 2 (ν + 1)pν+1
e− 2 D −ν t
843
∞
EH II 121(11)
∞
eixy−
7.726
+ , + 1/2 , (1+λ)y 2 1 D ν x(1 − λ)1/2 dx = (2π)1/2 λ 2 ν e− 4λ D ν i λ−1 − 1 y
(1+λ)x2 4
−∞
∞
7.727 0
∞
7.728
e 2 x e−bx 1
μ+ 12
(ex − 1)
− ν2
(2t) 0
e−pt e
exp −
2 − q8t
a 1 − e−x
D ν−1
q √ 2t
D 2μ
dt =
EH II 121(16)
√ √ 2 a √ dx = e−a 2b+μ Γ(b + μ) D −2b 2 a −x 1−e [Re a > 0, Re b > − Re μ]
π 12 2
[Re λ > 0]
ET I 211(13)
p
1 2 ν−1
e
√ −q p
MO 175
7.73 Combinations of parabolic cylinder and hyperbolic functions 7.731
1. 0
∞
, + 3 2 cosh(2μx) exp − (a sinh x) D 2k (2a cosh x) dx = 2k− 2 π 1/2 a−1 W k,μ 2a2 2 Re a > 0
ET II 398(20)
844
Parabolic Cylinder Functions
∞
2. 0
7.741
+ , Γ(μ − k) Γ(−μ − k) 2 cosh(2μx) exp (a sinh x) D 2k (2a cosh x) dx = W k+ 12 ,μ 2a2 5 k+ 2 a Γ(−2k) 2 3π , Re k + |Re μ| < 0 |arg a| < 4 ET II 398(21)
7.74 Combinations of parabolic cylinder and trigonometric functions 7.741
∞
1. 0
∞
2.
1 2 i √ 2 2 sin(bx) [D −n−1 (ix)] − [D −n−1 (−ix)] dx = (−1)n+1 π 2πe− 2 b Ln b2 n! e− 4 x sin(bx) D 2n+1 (x) dx = (−1)n 1
2
0
3. 4.
5.
7.742
∞
[b > 0]
ET I 115(3)
[b > 0]
ET I 115(1)
π 2n+1 − 1 b2 b e 2 2
π 2n − 1 b2 b e 2 [b > 0] 2 0 ∞ + , √ 1 2 1 1 2 e− 4 x sin(bx) D 2ν− 12 (x) − D 2ν− 12 (−x) dx = 2π sin ν − 14 π b2ν− 2 e− 2 b 0 Re ν > 14 , b > 0 √ ∞ 1 1 1 2 + , 2 4 −2ν πb2ν− 2 e− 4 b − 12 x2 e cos(bx) D 2ν− 12 (x) + D 2ν− 12 (−x) dx = cosec ν + 14 π 0 Re ν > 14 , b > 0
∞
1.
e− 4 x cos(bx) D 2n (x) dx = (−1)n 1
2
x2ρ−1 sin(ax)e−
∞
2.
x2ρ−1 sin(ax)e
x2 4
0
ET I 115(2)
ET I 61(4)
x2 4
0
1 Γ (2ρ + 1) D 2ν (x) dx = 2ν−ρ− 2 π 1/2 a Γ(ρ − ν + 1)
3 a2 1 × 2 F 2 ρ + , ρ + 1; , ρ − ν + 1; − 2 2 2 Re ρ > − 12 ! 2ρ−ν−2 22 a2 !! 12 − ρ, 1 − ρ D 2ν (x) dx = ! G Γ(−2ν) 23 2 ! −ρ − ν, 12 , 0 a > 0, Re ρ > − 12 ,
ET I 60(2)
ET II 396(8)
Re(ρ + ν)
0]
ET II 396(10)a
7.752
Parabolic cylinder and Bessel functions
∞
4.
2ρ−1
x
cos(ax)e
0
x2 4
2ρ−ν−2 22 D 2ν (x) dx = G Γ(−2ν) 23
a2 2
845
! ! 1 − ρ, 1 − ρ !2 ! ! −ρ − ν, 0, 12 a > 0, Re ρ > 0,
Re(ρ + ν)
0] ET I 115(4) √ , √ 2x dx − 2x D 2ν− 12 √ 2ν √ π sin ν − 14 π 1 + 1 + b2 √ =− 1 1 + b2 b2ν+ 2 [b > 0] ET I 60(3)
7.75 Combinations of parabolic cylinder and Bessel functions 7.751 1. 2.
3.
+ y ,2 2 [D n (ax)] J 1 (xy) dx = (−1)n−1 y −1 D n [y > 0] a 0 ∞ y y D n+1 J 0 (xy) D n (ax) D n+1 (ax) dx = (−1)n y −1 D n a a 0 y > 0, |arg a| < 14 π ∞ J 0 (xy) D ν (x) D ν+1 (x) dx = 2−1 y −1 [D ν (−y) D ν+1 (y) − D ν+1 (−y) D ν (y)] ∞
ET II 20(24)
ET II 17(42) ET II 397(17)a
0
7.752
∞
1. 0
ET II 76(1), MO 183 ∞
2. 0
3. 0
1 2 1 2 1 xν e− 4 x D 2ν−1 (x) J ν (xy) dx = − sec(νπ)y ν−1 e− 4 y [D 2ν−1 (y) − D 2ν−1 (−y)] 2 y > 0, Re ν > − 12
∞
1 2 1 1 2 xν e 4 x D 2ν−1 (x) J ν (xy) dx = 2 2 −ν π sin(νπ)y −ν Γ(2ν)e 4 y K ν 14 y 2 y > 0, − 12 < Re ν < 12 xν+1 e− 4 x D 2ν (x) J ν (xy) dx = 1
2
ET II 77(4)
1 2 1 sec(νπ)y ν−1 e− 4 y [D 2ν+1 (y) − D 2ν+1 (−y)] 2
[y > 0,
Re ν > −1]
ET II 78(13)
846
Parabolic Cylinder Functions
∞
4.
xν e− 4 x D 2ν+1 (x) J ν (xy) dx = 2
1
0
∞
5. 0
0
0
∞
8.
xν+1 e− 4 x D 2ν+2 (x) J ν (xy) dx = − 12 sec(νπ)y ν e− 4 y [D 2ν+2 (y) + D 2ν+2 (−y)] 2
1
1
9.
10.
2
y > 0]
ET II 78(16)
1 2 1 2 xν+1 e 4 x D 2ν+2 (x) J ν (xy) dx = π −1 sin(νπ) Γ(2ν + 3)y −ν−2 e 4 y K ν+1 14 y 2 y > 0, −1 < Re ν < − 56 xν e− 4 x D −2ν (x) J ν (xy) dx = 2−1/2 π 1/2 y −ν e− 4 y I ν 2
1
2
1
2
1
1
2
xν e 4 x D −2ν (x) J ν (xy) dx = y ν−1 e 4 y D −2ν (y)
0
ET II 77(5)
ET II 78(19) ∞
7.
1 2 1 sec(νπ)e− 4 y y ν [D 2ν (y) + D 2ν (−y)] 2 y > 0, Re ν > − 12
[Re ν > −1, ∞
6.
7.752
1
4y
2
y > 0,
Re ν > − 12
Re ν > − 12 ,
y>0
ET II 77(8)
ET II 77(9), EH II 121(17) ∞
xν e 4 x D −2ν−2 (x) J ν (xy) dx = (2ν + 1)−1 y ν e 4 y D −2ν−1 (y) 0 y > 0, Re ν > − 12 ET II 77(10)
1 ∞ 2μ− 2 Γ ν + 12 y ν 1 2 2 y2 1 ; ν − μ + 1; − xν e− 4 a x D 2μ (ax) J ν (xy) dx = ν + F 1 1 Γ(ν − μ + 1)a1+2ν 2 2a2 0 y > 0, |arg a| < 14 π, Re ν > − 12
∞
11.
2
1
1
2
xν e 4 a
1
x2
D 2μ (ax) J ν (xy) dx =
Γ
0
∞
12.
1
0
∞
1
2
x2
1
2
xν+1 e 4 a
x2
+ ν a2k 2m+μ y22 y2 1 μ+ 3 e 4a W k,m 4a2 Γ 2 −μ y 2 2k = 12 + μ − ν, 2m = 12 + μ + ν y > 0, |arg a| < 14 π, − 12 < Re ν < Re 12 − 2μ
D 2μ (ax) J ν (xy) dx =
D 2μ (ax) J ν (xy) dx =
2
0
13.
ET II 77(11)
xν+1 e− 4 a
2
Γ
ET II 78(12)
3 y2 3 ν + ;ν − μ + ;− 2 2 2 2a y > 0, |arg a| < 14 π, Re ν > −1
2μ Γ ν + 32 y ν 1F 1 Γ ν − μ + 32 a2ν+2 3 2
ET II 79(23) 1
+ ν 2 2 +m+μ a2k+1 y22 y2 4a W e k,m Γ(−μ)y μ+2 2a2 2k = μ − ν − 1, 2m = μ + ν + 1 y > 0, |arg a| < 34 π, −1 < Re ν < − 12 − 2 Re μ ET II 79(24)
7.754
Parabolic cylinder and Bessel functions
∞
14.
x
e
1 2 2 4a x
0
∞
15. 0
∞
16.
2λ− 2 μ π − 2 1
λ+ 12
1
22 G 23
D μ (ax) J ν (xy) dx = 3 Γ(−μ)y λ+ 2 y > 0, |arg a| < 34 π,
2
1
847
! ! 1, 1 !2 ! 3 λ+ν ! 4 + 2 , − μ2 , 34 + λ−ν 2 Re μ < − Re λ < Re ν + 32 ET II 80(26) y2 2a2
2
1
xν+1 e 4 x D −2ν−1 (x) J ν (xy) dx = (2ν + 1)y ν−1 e 4 y D −2ν−2 (y) y > 0, Re ν > − 12 xν+1 e− 4 x D −2ν−3 (x) J ν (xy) dx = 1
2
1 2 2−1/2 π 1/2 y −ν−2 e− 4 y
I ν+1
0
[y > 0, ∞
17.
2
1
1
1
4y
2
ET II 79(20)
Re ν > −1]
ET II 79(21)
2
xν+1 e 4 x D −2ν−3 (x) J ν (xy) dx = y ν e 4 y D −2ν−3 (y)
0
[y > 0, ∞
18.
ν
x e
1 2 2 4a x
0
7.753
∞
1. 0
−1
3 3 4 ν+ 4
D 12 ν− 12 (ax) Y ν (xy) dx = −π 2 y > 0,
xν− 2 e−(x+a) I ν− 12 (2ax) D ν (2x) dx = 2
1
a
−ν −1
y
Re ν > −1]
Γ(ν + 1)e
|arg a| < 34 π,
2. 0
xν− 2 e−(x+a) I ν− 32 (2ax) D ν (2x) dx = 2
3
∞
1.
y2 2a2
1 1 −1/2 π Γ(ν)aν− 2 D −ν (2a) 2
Re ν > 0]
ET II 397(12)
3 1 −1/2 π Γ(ν)aν− 2 D −ν (2a) 2
[Re a > 0, 7.754
ET II 79(22)
W − 12 ν− 12 , 12 ν − 12 < Re ν < 23 ET II 115(39)
[Re a > 0, ∞
y2 4a2
Re ν > 1]
ET II 397(13)
xν e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν−1 (x) − D 2ν−1 (−x)} J ν (xy) dx 1
2
0
= ±y ν−1 e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν−1 (y) − D 2ν−1 (−y)} y > 0, Re ν > − 12 ET II 76(2, 3) 1
∞
2.
2
xν e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν+1 (x) − D 2ν+1 (−x)} J ν (xy) dx 1
2
0
= ∓y ν e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν (y) + D 2ν (−y)} y > 0, Re ν > − 12 ET II 77(6, 7) 1
3.
∞
2
xν+1 e− 4 x {[1 ± 2 cos(νπ)] D 2ν (x) + D 2ν (−x)} J ν (xy) dx 1
2
0
= ±y ν−1 e− 4 y {[1 ± 2 cos(νπ)] D 2ν+1 (y) − D 2ν+1 (−y)} 1
2
[y > 0,
Re ν > −1]
ET II 78(14, 15)
848
Parabolic Cylinder Functions
∞
4.
7.755
xν+1 e− 4 x {[1 ∓ 2 cos(νπ)] D 2ν+2 (x) + D 2ν+2 (−x)} J ν (xy) dx 1
2
0
= ±y ν e− 4 y {[1 ∓ 2 cos(νπ)] D 2ν+2 (y) + D 2ν+2 (−y)} 1
2
[y > 0, 7.755
∞
1.
x−1/2 D ν
√ √ ax D −ν−1 ax J 0 (xy) dx
0
=2
∞
0
∞
4. 0
0
%
4y 2 1+ 2 a
1/2 & P
1 1 2 ν− 4 1 4
%
4y 2 1+ 2 a
1/2 &
ET II 17(43)
1 1 x1/2 D − 12 −ν ae 4 πi x1/2 D − 12 −ν ae− 4 πi x1/2 J ν (xy) dx
∞
a
5.
P
1 1 2 ν+ 4 − 14
ET II 78(17, 18)
,2ν −1 + 2 −1/2 1/2 a + 2y = 2−ν π 1/2 y −ν−1 a2 + 2y Γ ν + 12 −a y > 0, Re a > 0, Re ν > − 12 ET II 80(27)
0
πa
−1/2
[y > 0, Re a > 0]
2.
3.
−3/2
Re ν > −1]
∞
1 1 D − 12 −ν ae 4 πi x−1/2 D − 12 −ν ae− 4 πi x−1/2 J ν (xy) dx
, + −1 = 21/2 π 1/2 y −1 Γ ν + 12 exp −a(2y)1/2 y > 0, Re a > 0, e Re ν > − 12 ET II 80(28)a
x1/2 D ν− 12 ax−1/2 D −ν− 12 ax−1/2 Y ν (xy) dx
x1/2 D ν− 12 ax−1/2 D −ν− 12
+ , = y −3/2 exp −ay 1/2 sin ay 1/2 − 12 ν − 12 π y > 0, |arg a| < 14 π ET II 115(40) , + ax−1/2 K ν (xy) dx = 2−1 y −3/2 π exp −a(2y)1/2 Re y > 0, |arg a| < 14 π ET II 151(81)
Combinations of parabolic cylinder and Struve functions ∞ 1 2 7.756 x−ν e− 4 x [D μ (x) − D μ (−x)] Hν (xy) dx 0
23/2 Γ 1 μ + 12 μ+ν 1 1 1 1 1 y μπ 1 F 1 μ + ; μ + ν + 1; − y 2 sin = 1 2 2 2 2 2 2 Γ 2 μ + ν + 1 3 y > 0, Re(μ + ν) > − 2 , Re μ > −1 ET II 171(41)
7.773
Integration of a parabolic cylinder function
849
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric functions 7.761
∞
1. 0
∞
2. 0
1 2 1 e 4 t t2c−1 D −ν (t) 1 F 1 a; c; − pt2 dt 2
1 1 π 1/2 Γ(2c) Γ 12 ν − c + a 1 F a, c + ; a + + ν; 1 − p = c+ 1 ν 1 2 2 2 2 2 Γ 2 ν Γ a + 12 + 12 ν [|1 − p| < 1, Re c > 0, Re ν > 2 Re(c − a)] EH II 121(12)
1 2 1 e 4 t t2c−2 D −ν (t) 1 F 1 a; c; − pt2 dt 2
1 π 1/2 Γ(2c − 1) Γ 12 ν + 12 − c + a 1 = c+ 1 ν− 1 F a, c − ; a + ν; 1 − p 2 2 Γ 12 + 12 ν Γ a + 12 ν 2 2 2 1 |1 − p| < 1, Re c > 2 , Re ν > 2 Re(c − a) − 1 EH II 121(13)
7.77 Integration of a parabolic cylinder function with respect to the index
7.771 0
∞
1 π 1/2 β 2 cos a cos(ax) D x− 12 (β) D −x− 12 (β) dx = exp − 2 cos a 2 =0
|a| < 12 π |a| > 12 π ET II 298(22)
7.772
− 12 +i∞
1. − 12 −i∞
⎡ ν 1 1 tan 12 ϕ ⎣ 4 iπ ξ −e e 4 iπ η D D ν −ν−1 cos 12 ϕ ⎤ ν 1 1 cot 12 ϕ dν D −ν−1 e 4 iπ ξ D ν −e 4 iπ η ⎦ + 1 sin νπ sin 2 ϕ = −2i(2π)1/2 exp − 14 i ξ 2 − η 2 cos ϕ − 12 iξη sin ϕ
− 12 +i∞ 2. − 12 −i∞
ν
tan 12 ϕ cos 12 ϕ
EH II 125(7)
1 1 dν D ν −e 4 iπ ζ D −ν−1 e 4 iπ η sin νπ + 1 + 1 , , iπ 1 ζ cos 2 ϕ + η sin 12 ϕ D −1 e 4 iπ η cos 12 ϕ − ζ sin 12 ϕ = −2i D 0 e 4 EH II 125(8)
7.773
c+i∞
1. c−i∞
D ν (z)tν Γ(−ν) dν = 2πie− 4 z 1
2
−zt− 12 t2
+ c < 0,
|arg t|
0; j = 1, . . . , m; Re σ > 0, ⎡ ⎤
p q 1 ⎦ > −1, Re ⎣ aj − bj + (p − q) ρ − 2 2 j=1 j=1
or p 0;
j = 1, . . . , m;
Re σ > 0 ET II 417(1)
!
!
! a1 , . . . , ap ! a1 , . . . , ap , ρ m+1,n ! ! x−ρ (x − 1)σ−1 G mn αx dx = Γ(σ) α G p+1,q+1 pq ! b1 , . . . , bq ! ρ − σ, b1 , . . . , bq 1 where
3.
(or p ≤ q for |α| < 1) ,
∞
• • • • •
p + q < 2(m + n) 1 |arg α| < m + n − 2 p − 12 q π Re (ρ − σ − aj ) > −1; j = 1, . . . , n Re σ > 0 either p + q ≤ 2(m + n),
|arg α| ≤ m + n − 12 p − 12 q π,
Re (ρ − σ − aj ) > −1; j = 1, . . . , n; Re σ > 0, ⎡ ⎤
p q 1 ⎦ > −1, aj − bj + (q − p) ρ − σ + Re ⎣ 2 2 j=1 j=1 or q 1) ,
Re (ρ − σ − aj ) > −1;
Re σ > 0 ET II 417(2)
$n $m !
∞ ! a1 , . . . , ap Γ (bj + ρ) j=1 Γ (1 − aj − ρ) j=1 ρ−1 mn $p α−ρ x dx = $q G pq αx !! b , . . . , b Γ (1 − b − ρ) Γ (a + ρ) 1 q j j 0 j=m+1 j=n+1 p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, − min Re bj < Re ρ < 1 − max Re aj
4.
j = 1, . . . , n;
1≤j≤m
1≤j≤n
ET II 418(3)a, ET I 337(14)
852
Meijer’s and MacRobert’s Functions (G and E)
∞
5.
ρ−1
x
−σ
(x + β)
mn G pq
0
7.812
!
!
! a1 , . . . , ap ! 1 − ρ, a1 , . . . , ap β ρ−σ m+1,n+1 ! ! αx ! dx = αβ ! G b1 , . . . , bq Γ(σ) p+1,q+1 σ − ρ, b1 , . . . , bq
where • • • • • •
p + q < 2(m + n) |arg α| < m + n − 12 p − 12 q π |arg β| < π Re (ρ + bj ) > 0, j = 1, . . . , m Re (ρ − σ + aj ) < 1, j = 1, . . . , n either p ≤ q,
p + q ≤ 2(m + n),
|arg α| ≤ m + n − 12 p − 12 q π,
|arg β| < π
Re (ρ = bj ) > 0, j = 1, . . . , m, Re (ρ − σ + aj ) < 1, j = 1, . . . , n, ⎤ ⎡
p q 1 ⎦ > 1, Re ⎣ aj − bj − (q − p) ρ − σ − 2 j=1 j=1 or p ≥ q,
p + q ≤ 2(m + n),
1 1 |arg α| ≤ m + n − p − q π, 2 2
|arg β| < π,
j = 1, . . . , m, Re (ρ − σ + aj ) < 1, j = 1, . . . , n, ⎡ ⎤
p q 1 ⎦>1 Re ⎣ aj − bj + (p − q) ρ − 2 j=1 j=1
Re (ρ + bj ) > 0,
ET II 418(4)
7.812
1. 0
1
z xβ−1 (1 − x)γ−β−1 E a1 , . . . , ap : ρ1 , . . . , ρq ; m dx x = Γ(γ − β)mβ−γ E (a1 , . . . , ap+m : ρ1 , . . . , ρq+m : z) β+k−1 γ+k−1 , ρq+k = , k = 1, . . . , m ap+k = m m [Re γ > Re β > 0, m = 1, 2, . . .] ET II 414(2)
2.
∞
xρ−1 (1 + x)−σ E [a1 , . . . , ap : ρ1 , . . . , ρq : (1 + x)z] dx
0
= Γ(ρ) E (a1 , . . . , ap , σ − ρ; ρ1 , . . . , ρq , σ; z) [Re σ > Re ρ > 0]
ET II 415(3)
7.815
G and E and the elementary functions
∞
3.
−β s−1
(1 + x)
x
mn G pq
0
853
!
!
! 1 − s, a1 , . . . , ap ax !! a1 , . . . , ap m,n+1 ! dx = Γ(β − s) G p+1,q+1 a ! 1 + x ! b1 , . . . , bq b1 , . . . , bq , 1 − β − min Re bk < Re s < Re β,
1 ≤ k ≤ m;
(p + q) < 2(m + n), 1 1 |arg a| < m + n − 2 p − 2 q π ET I 338(19)
7.813 1.
∞
−ρ −βx
x
e
mn G pq
0
!
!
! a1 , . . . , ap α !! ρ, a1 , . . . , ap m,n+1 ρ−1 ! αx ! dx = β G p+1,q b1 , . . . , bq β ! b1 , . . . , bq p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, |arg β| < 12 π, Re (bj − ρ) > −1, j = 1, . . . , m ET II 419(5)
∞
2. 0
! !
! 4α !! 0, 12 , a1 , . . . , ap m,n+2 2 ! a1 , . . . , ap −1/2 −1 e−βx G mn β αx dx = π G p+2,q pq ! b1 , . . . , bq β 2 ! b1 , . . . , bq p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, 1 1 |arg β| < 2 π, Re bj > − 2 ; j = 1, . . . , m
ET II 419(6)
7.814 1.
∞
0
xβ−1 e−x E (a1 , . . . , ap : ρ1 , . . . , ρq : xz) dx = π cosec(βπ) E a1 , . . . , ap : 1 − β, ρ1 , . . . , ρq : e±iπ z −z
2.
−β
E a1 + β, . . . , ap + β : 1 + β, ρ1 + β, . . . , ρl + β : e±iπ z
[p ≥ q + 1, Re (ar + β) > 0, r = 1, . . . , p, |arg z| < π. The formula holds also for p < q + 1, ET II 415(4) provided the integral converges.] ∞ xβ−1 e−x E a1 , . . . , ap : ρ1 , . . . , ρq : x−m z dx 0
Re β > 0, 7.815 1. 0
∞
ap+k
1 1 1 = (2π) 2 − 2 m mβ− 2 E a1 , . . . , ap+m : ρ1 , . . . , ρq : m−m z β+k−1 , k = 1, . . . , m; m = 1, 2, . . . = ET II 415(5) m
! !
! √ −1 m,n+1 4α ! 0, a1 , . . . , ap , 12 2 ! a1 , . . . , ap ! sin(cx) G mn πc αx dx = G p+2,q pq ! b1 , . . . , bq c2 ! b1 , . . . , bq p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, c > 0, Re bj > −1, j = 1, 2, . . . , m, Re aj < 12 , j = 1, . . . , n ET II 420(7)
854
Meijer’s and MacRobert’s Functions (G and E)
∞
2.
cos(cx) G mn pq
0
7.821
! !
! 4α !! 12 , a1 , . . . , ap , 0 m,n+1 2 ! a1 , . . . , ap 1/2 −1 αx ! dx = π c G p+2,q b1 , . . . , bq c2 ! b1 , . . . , bq p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, c > 0, Re bj > − 12 , j = 1, . . . , m, Re aj < 12 , j = 1, . . . , n ET II 420(8)
7.82 Combinations of the functions G and E and Bessel functions 7.821 1.
∞
0
!
!
! a1 , . . . , ap ! ρ − 1 ν, a1 , . . . , ap , ρ + 1 ν √ m,n+1 2 2 ! ! x−ρ J ν 2 x G mn αx dx = α G p+2,q pq ! b1 , . . . , bq ! b1 , . . . , bq p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π 3 1 − 4 + max Re aj < Re ρ < 1 + 2 Re ν + min Re bj 1≤j≤n
∞
2.
√ x−ρ Y ν 2 x G mn pq
1≤j≤m
ET II 420(9)
!
! a1 , . . . , ap αx !! dx b1 , . . . , bq
! ! ρ − 1 ν, ρ + 1 ν, a , . . . , a , ρ + 1 + 1 ν 1 p ! 2 2 2 2 = α! ! b1 , . . . , bq , ρ + 12 + 12 ν p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, − 34 + max Re aj < Re ρ < min Re bj + 12 |Re ν| + 1
0
m,n+2 G p+3,q+1
1≤j≤n
∞
3.
√ x−ρ K ν 2 x G mn pq
0
1≤j≤m
ET II 420(10)
!
!
! a1 , . . . , ap ! ρ − 1 ν, ρ + 1 ν, a1 , . . . , ap 1 m,n+2 2 2 ! ! αx ! dx = G p+2,q α ! b1 , . . . , bq 2 b1 , . . . , bq p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, Re ρ < 1 − 12 |Re ν| + min Re bj 1≤j≤m
ET II 421(11)
7.822 1. 0
∞
!
!
! a1 , . . . , ap 4λ !! h, a1 , . . . , ap , k 22ρ m,n+1 ! λx ! dx = 2ρ+1 G p+2,q b1 , . . . , bq y y 2 ! b1 , . . . , bq h = 12 − ρ − 12 ν, k = 12 − ρ + 12 ν p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, Re bj + ρ + 12 ν > − 12 , j = 1, 2, . . . , m, Re (aj + ρ) < 34 , j = 1, . . . , n, y > 0
2ρ
x
J ν (xy) G mn pq
2
ET II 91(20)
7.823
G and E and Bessel functions
∞
2.
1/2
x 0
Y ν (xy) G mn pq
855
!
! 2 ! a1 , . . . , ap λx ! dx b1 , . . . , bq
= (2λ)−1/2 y −1/2 G n+2,m q+1,p+3
! y 2 !! 12 − b1 , . . . , 12 − bq , l ! 4λ ! h, k, 12 − a1 , . . . , 12 − ap , l
+ 12 ν, k = 14 − 12 ν, l = − 14 − 12 ν p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, y > 0, 3 Re aj < 1, j = 1, . . . , n, Re bj ± 12 ν > − , j = 1, . . . , m 4 h=
∞
3.
1 4
ET II 119(56)
!
! 1/2 mn 2 ! a1 , . . . , ap x K ν (xy) G pq λx ! dx b1 , . . . , bq
! y 2 !! 12 − b1 , . . . , 12 − bq =2 λ y ! 4λ ! h, k, 12 − a1 , . . . , 12 − ap h = 14 + 12 ν, k = 14 − 12 ν Re y > 0, p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π , Re bj > 12 |Re ν| − 34 , j = 1, . . . , m
0
−3/2 −1/2 −1/2
n+2,m G q,p+2
ET II 153(90)
7.823
∞
1. 0
xβ−1 J ν (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−2m z dx = (2π)−m (2m)β−1 exp 12 π (β − ν − 1) i E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m ze−mπi + exp − 12 π(β − ν − 1)i E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m zemπi , β + ν + 2k − 2 β − ν + 2k − 2 ap+k = , ap+m+k = , m = 1, 2, . . . , ; k = 1, . . . , m 2m 2m Re(β + ν) > 0, Re (2ar m − β) > − 32 , r = 1, . . . , p ET II 415(7)
2.
∞
xβ−1 K ν (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−2m z dx
0
ap+k
= (2π)1−m 2β−2 mβ−1 × E a1 , . . . , ap+2m : ρ1 , . . . , ρq : (2m)−2m z , β + ν + 2k − 2 β − ν + 2k − 2 , ap+m+k = , k = 1, 2, . . . , m = 2m 2m [Re β > |Re ν|, m = 1, 2, . . .] ET II 416(8)
856
Meijer’s and MacRobert’s Functions (G and E)
7.824
∞
1.
!
! 2 ! a1 , . . . , ap x1/2 Hν (xy) G mn λx dx pq ! b1 , . . . , bq
! y 2 !! l, 12 − b1 , . . . , 12 − bq = (2λy) ! 4λ ! l, 12 − a1 , . . . , 12 − ap , h, k 1 ν 3 ν 1 ν h= + , k= − , l= + 4 2 4 2 4 2 p + q < 2(m + n), |arg λ| < m + n − 12 p − 12 q π, y > 0, 3 1 5 Re aj < min 1, 4 − 2 ν , j = 1, . . . , n, Re (2bj + ν) > − 2 , j = 1, . . . , m
0
−1/2
7.824
∞
2.
n+1,m+1 G q+1,p+3
ET II 172(47)
!
! a1 , . . . , ap √ mn −ρ ! x Hν 2 x G pq αx ! dx b1 , . . . , bq
! !ρ − ! α! !ρ −
− 12 ν, a1 , . . . , ap , ρ + 12 ν, ρ − 12 ν = − 12 ν, b1 , . . . , bq p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π ,
3 ν−1 1 3 max − , Re + max Re aj < Re ρ < min Re bj + 2 Re ν + 2 1≤j≤n 1≤j≤m 4 2
0
m+1,n+1 G p+3,q+1
1 2 1 2
ET II 421(12)
7.83 Combinations of the functions G and E and other special functions
7.831
∞
−ρ
x 1
(x − 1)
σ−1
!
! a1 , . . . , ap ! F (k + σ − ρ, λ + σ − ρ; σ; 1 − αx ! dx b1!, . . . , bq
! a1 , . . . , ap , k + λ + σ − ρ, ρ m+2,n ! = Γ(σ) G p+2,q+2 α ! k, λ, b1 , . . . , bq x) G mn pq
where
ET II 421(13)
⎡ ⎤
p q 1 ⎦ 1 • Re ⎣ aj − bj + (q − p) k + >− 2 2 j=1 j=1 ⎡ ⎤
p q 1 ⎦ 1 ⎣ • Re aj − bj + (q − p) λ + >− 2 2 j=1 j=1 • either p + q < 2(m + n), |arg α| < m + n − 12 p − 12 q π, Re σ > 0, Re k ≥ Re λ > Re aj − 1, j = 1, . . . , n,
7.832
G and E and other special functions
857
or p + q ≤ 2(m + n), |arg α| ≤ m + n − 12 p − 12 q π, Re σ > 0, Re k ≥ Re λ > Re aj − 1, j = 1, . . . , n,
∞
7.832
1 xβ−1 e− 2 x W κ,μ (x) E a1 , . . . , ap : ρ1 , . . . , ρq : x−m z dx
0
1 1 1 = (2π) 2 − 2 m mβ+κ− 2 E a1 , . . . , ap+2m : ρ1 , . . . , ρq+m : m−m z ,
ap+k =
β + k + μ − 12 , m
ap+m+k =
β − μ + k − 12 β−κ+k , ρq+k = , k = 1, . . . , m m m 1 Re β > |Re μ| − 2 , m = 1, 2, . . . ET II 416(10)
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8–9 Special Functions 8.1 Elliptic Integrals and Functions 8.11 Elliptic integrals 8.110 1.
Every integral of the form
R x, P (x) dx, where P (x) is a third- or fourth-degree polyno-
mial, can be reduced to a linear combination of integrals leading to elementary functions and the following three integrals: √ 1 − k 2 x2 dx dx √ , , dx, 2 2 2 2 2 1−x (1 − x ) (1 − k x ) (1 − nx ) (1 − x2 ) (1 − k 2 x2 ) which are called respectively elliptic integrals of the first, second, and third kind in the Legendre normal form. The results of this reduction for the more frequently encountered integrals are given√in formulas 3.13–3.17. The number k is called the modulus ∗ of these integrals; the number k = 1 − k 2 is called the complementary modulus, and the number n is called the parameter of BY (110.04) the integral of the third kind. 2.
3.11 4.∗
By means of the substitution x = sin ϕ, elliptic integrals can be reduced to the normal trigonometric forms < dϕ dϕ 2 2 1 − k sin ϕ dϕ, , . BY (110.04) 2 2 2 1 − n sin ϕ 1 − k sin ϕ 1 − k 2 sin2 ϕ The results of reducing integrals of trigonometric functions to normal form are given in 2.58– 2.62. π in the 8.110 2 formuElliptic integrals from 0 to 1 in the 8.110 1 formulation (or from 0 to 2 lation) are called complete elliptic integrals. Take note that in mathematical software, and elsewhere, the notation for elliptic integrals is often modified by replacing the parameter k2 that is used here with k.
8.111 Notations: 1. ∗ The
Δϕ =
0, all roots e1 , e2 , and e3 of the equation 4z 3 − g2 z − g3 = 0 (where g2 and g3 are real numbers) are real. In this case, the roots e1 , e2 , and e3 are numbered in such a way that e1 > e2 > e3 . 1.
If Δ > 0, then
2.
dz , − g2 z − g3 g3 + g2 z − 4z 3 e1 −∞ where ω1 is real and ω2 is a purely imaginary number. Here, the values of the radical in the ω2 integrand are chosen in such a way that ω1 and will be positive. i If Δ < 0, the root e2 of the equation 4z 3 − g2 z − g3 = 0 is real, and the remaining two roots (e1 and e3 ) are complex conjugates. Suppose that e1 = α + iβ, and e3 = α − iβ. In this case, it is convenient to take ∞ ∞ dz dz and ω = ω = 3 3 4z − g2 z − g3 4z − g2 z − g3 e1 e3 as basic semiperiods.
ω1 =
∞
dz
4z 3
,
e3
ω2 = i
In the first integral, the integration is taken over a path lying entirely in the upper half-plane and in the second over a path lying entirely in the lower half-plane. SI 151(21, 22)
8.169
8.165 1.
875
1 g3 u4 g 2 u6 g2 u2 3g2 g3 u8 + + 4 2 + ... + + 4 2 2 u 4·5 4·7 2 ·3·5 2 · 5 · 7 · 11
WH
Series representation: ℘(u) =
8.166
The Weierstrass function ℘(u)
Functional relations
℘ (u) = − ℘ (−u) 2 1 ℘ (u) − ℘ (v) 2. ℘(u + v) = − ℘(u) − ℘(v) + 4 ℘(u) − ℘(v)
g2 g3 2 8.167 ℘ (u; g2 , g3 ) = μ ℘ μu; 4 , μ μ6 1.
℘(u) = ℘(−u),
SI 163(32)
(the formula for homogeneity) SI 149(13)
The special case: μ = i. 1.
℘ (u; g2 , g3 ) = − ℘ (iu; g2 , −g3 )
8.168 An arbitrary elliptic function can be expressed in terms of the elliptic function ℘(u) having the same periods as the original function and its derivative ℘ (u). This expression is rational with respect to ℘(u) and linear with respect to ℘ (u). 8.169 A connection with the Jacobian elliptic functions. For Δ > 0 (see 8.164 1).
u cn 2 (u; k) 1. ℘ √ = e1 + (e1 − e3 ) 2 sn (u; k) e1 − e2 dn 2 (u; k) = e2 + (e1 − e3 ) 2 sn (u; k) 1 = e3 + (e1 − e3 ) 2 sn (u; k) SI 145(5), ZH 120(197–199)a
2.
3.
4. 5.
6.11
K iK ω1 = √ , ω2 = √ , e 1 − e3 e1 − e3 where e2 − e 3 e1 − e2 , k = k= e1 − e 3 e1 − e3 For Δ < 0 (see 8.164 2) 1 + cn(2u; k) u ; = e2 + 9α2 + β 2 ℘ 4 1 − cn(2u; k) 9α2 + β 2 K − iK K + iK ω = , ω = , 4 2 9α2 + β 2 9α2 + β 2 where . . 1 1 3e2 3e2 − + k= ; k = 2 2 2 4 9α + β 2 4 9α2 + β 2
SI 154(29)
SI 145(7)
SI 147(12)
SI 153(28)
SI 147
For Δ = 0, all the roots e1 , e2 , and e3 are real, and if g2 g3 = 0, two of them are equal to each other. If e1 = e2 = e3 , then
876
7.
8.
Elliptic Integrals and Functions
3g3 9g3 9g3 2 ℘(u) = − coth u − g2 2g2 2g2 If e1 = e2 = e3 , then 1 9g3 3g3 < + ℘(u) = − 2g2 2g2 sin2 u 9g3
8.171
SI 148
SI 149
2g2
9.
If g2 = g3 = 0, then e1 = e2 = e3 = 0, and 1 ℘(u) = 2 u
SI 149
8.17 The functions ζ(u) and σ(u) 8.171 1. 2. 8.172 1.
2.
Definitions:
1 ℘(z) − 2 dz z 0
u
1 σ(u) = u exp ℘(z) − 2 dz z 0 1 ζ(u) = − u
u
Series and infinite-product representation 1 1 u 1 ζ(u) = + + + 2 u u − 2mω1 − 2nω2 2mω1 + 2nω2 (2mω1 − 2nω2 ) m,n
- u2 u u σ(u) = u + 1− exp 2 2mω1 + 2nω2 2mω1 + 2nω2 2 (2mω1 + 2nω2 ) mn,
SI 181(45) SI 181(46)
SI 307(8)
SI 308(9)
8.173 g2 u 3 g3 u5 g 2 u7 3g2 g3 u9 − 2 − 4 2 2 − 4 − ··· · 3 · 5 2 · 5 · 7 2 · 3 · 5 · 7 2 · 5 · 7 · 9 · 11 g3 u7 g 2 u9 3g2 g3 u11 g2 u 5 − 3 − 9 22 − 7 2 2 − ··· 2. σ(u) = u − 4 2 · 3 · 5 2 · 3 · 5 · 7 2 · 3 · 5 · 7 2 · 3 · 5 · 7 · 11
∞ πu ζ (ω1 ) π πu π ω2 8.174 ζ(u) = u+ cot + + nπ cot ω1 2ω1 2ω1 2ω1 n=1 2ω1 ω1
πu ω2 + cot − nπ 2ω1 ω1 ∞ ζ (ω1 ) π πu 2π q 2n πnu = u+ cot + sin ω1 2ω1 2ω1 ω1 n=1 1 − q 2n ω1 1.
ζ(u) = u −
22
SI 181(49) SI 181(49)
MO 154 MO 155
Functional relations and properties 8.175 ζ(u) = − ζ(−u), 8.176 1.
σ(u) = − σ(−u)
ζ (u + 2ω1 ) = ζ(u) + 2ζ (ω1 )
SI 181
SI 184(57)
8.181
Theta functions
877
2.
ζ (u + 2ω2 ) = ζ(u) + 2ζ (ω2 )
SI 184(57)
3.
σ (u + 2ω1 ) = − σ(u) exp {2 (u + ω1 ) ζ (ω1 )} .
SI 185(60)
4.
σ (u + 2ω2 ) = − σ(u) exp {2 (u + ω2 ) ζ (ω2 )} . π ω2 ζ (ω1 ) − ω1 ζ (ω2 ) = i 2
SI 185(60)
5.
SI 186(62)
8.177 1 ℘ (u) − ℘ (v) 2 ℘(u) − ℘(v)
1.
ζ(u + v) − ζ(u) − ζ(v) =
2.
℘(u) − ℘(v) = −
3.
ζ(u − v) + ζ(u + v) − 2ζ(u) =
SI 182(53)
σ(u − v) σ (u + v) σ 2 (u) σ 2 (v)
SI 183(54)
℘ (u) ℘(u) − ℘(v)
SI 182(51)
8.178 1.
ζ (u; ω1 , ω2 ) = t ζ (tu; tω1 , tω2 )
MO 154
2.8
σ (u; ω1 , ω2 ) = t−1 σ (tu; tω1 , tω2 )
MO 156
For the indefinite integrals of Weierstrass elliptic functions, see 5.14.
8.18–8.19 Theta functions 8.180 1.
Theta functions are defined as the sums (for |q| < 1) of the following series: ϑ4 (u) =
∞
2
(−1)n q n e2nui = 1 + 2
n=−∞
2.
3.
ϑ1 (u) = 11
ϑ2 (u) =
ϑ3 (u) =
2
(−1)n q n cos 2nu
∞ ∞ 1 2 1 2 1 (−1)n q (n+ 2 ) e(2n+1)ui = 2 (−1)n+1 q (n− 2 ) sin(2n − 1)u i n=−∞ n=1 ∞
∞
2
1 q (n+ 2 ) e(2n+1)ui = 2
∞
2
q (n− 2 ) cos(2n − 1)u 1
WH
WH
n=1 2
q n e2nui = 1 + 2
n=−∞
WH
n=1
n=−∞
4.
∞
∞
2
q n cos 2nu
WH
n=1
The notations ϑ(u, q) and ϑ (u | τ ), where τ and q are related by q = eiπτ , are also used. Here, q is called the nome of the theta function and τ its parameter. 8.181 Representation of theta functions in terms of infinite products 1.
ϑ4 (u) =
∞ -
1 − 2q 2n−1 cos 2u + q 2(2n−1)
1 − q 2n
SI 200(9), ZH 90(9)
n=1
2.
ϑ3 (u) =
∞ n=1
1 + 2q 2n−1 cos 2u + q 2(2n−1)
1 − q 2n
SI 200(9), ZH 90(9)
878
Elliptic Integrals and Functions
8.182
∞
3. 4.8
- √ 1 − 2q 2n cos 2u + q 4n 1 − q 2n ϑ1 (u) = 2 4 q sin u √ ϑ2 (u) = 2 4 q cos u
n=1 ∞ -
1 + 2q 2n cos 2u + q 4n
1 − q 2n
SI 200(9), ZH 90(9)
SI 200(0), ZH 90(9)
n=1
Functional relations and properties 8.182 Quasiperiodicity. Suppose that q = eπτ i (Im τ > 0). Then, theta functions that are periodic functions of u are called quasiperiodic functions of τ and u. This property follows from the equations 1.
ϑ4 (u + π) = ϑ4 (u)
SI 200(10)
2.
1 ϑ4 (u + τ π) = − e−2iu ϑ4 (u) q
SI 200(10)
3.
ϑ1 (u + π) = − ϑ1 (u)
SI 200(10)
4.
1 ϑ1 (u + τ π) = − e−2iu ϑ1 (u) q
SI 200(10)
5.
ϑ2 (u + π) = − ϑ2 (u)
SI 200(10)
6.
ϑ2 (u + τ π) =
7.
ϑ3 (u + π) = ϑ3 (u)
8.
ϑ3 (u + τ π) =
8.183 1. ϑ4 u + 2. ϑ1 u + 3. ϑ2 u + 4. ϑ3 u + 5. ϑ4 u + 6. ϑ1 u + 7. ϑ2 u + 8. ϑ3 u + 8.184
1 2π 1 2π 1 2π 1 2π
1 −2iu e ϑ2 (u) q
SI 200(10) SI 200(10)
1 −2iu e ϑ3 (u) q
SI 200(10)
= ϑ3 (u)
WH
= ϑ2 (u)
WH
= − ϑ1 (u)
WH
= ϑ4 (u) −1/4 −iu 1 e ϑ1 (u) 2 πτ = iq −1/4 −iu 1 e ϑ4 (u) 2 πτ = iq −1/4 −iu 1 e ϑ3 (u) 2 πτ = q −1/4 −iu 1 e ϑ2 (u) 2 πτ = q
WH WH WH WH WH
Even and odd theta functions
1.
ϑ1 (−u) = − ϑ1 (u)
WH
2.
ϑ2 (−u) = ϑ2 (u)
WH
3.
ϑ3 (−u) = ϑ3 (u)
WH
4.
ϑ4 (−u) = ϑ4 (u)
8.185 8.1867
ϑ44 (u)
ϑ42 (u)
WH
ϑ41 (u)
ϑ43 (u)
+ = + WH Considering the theta functions as functions of two independent variables u and τ , we have
8.192
Theta functions
879
∂ 2 ϑk (u | τ ) ∂ ϑk (u | τ ) =0 [k = 1, 2, 3, 4] +4 WH 2 ∂u ∂τ 8.187 We denote the partial derivatives of the theta functions with respect to u by a prime and consider them as functions of the single argument u. Then, πi
1.
ϑ1 (0) = ϑ2 (0) ϑ3 (0) ϑ4 (0)
2.
ϑ 1 (0) ϑ1 (0)
=
ϑ2 (0) ϑ2 (0)
ϑ3 (0)
+
ϑ3 (0)
WH
+
ϑ4 (0)
WH
ϑ4 (0)
8.188 ϑ1 (u) ϑ2 (u) ϑ3 (u) ϑ4 (0) = 12 ϑ1 (2u) ϑ2 (0) ϑ3 (0) ϑ4 (0) 8.189 The zeros of the theta functions: πτ π 1.8 ϑ4 (u) = 0 for u = 2m + (2n − 1) 2 2 πτ π 10 2. ϑ1 (u) = 0 for u = 2m + 2n 2 2 πτ π 3. ϑ2 (u) = 0 for u = (2m − 1) + 2n 2 2 πτ π 4. ϑ3 (u) = 0 for u = (2m − 1) + (2n − 1) 2 2
WH
SI 201 SI 201 SI 201
[m and n are integers or zero]
SI 201
For integrals of theta functions, see 6.16. 8.191 Connections with the Jacobian elliptic functions: K K For τ = i K , i.e. for q = exp −π K ,
1.
2.
3.
πu ϑ 1 1 1 H (u) 2K =√ sn u = √ πu k ϑ4 k Θ (u) 2K πu k ϑ2 2K k H 1 (u) πu = cn u = k ϑ k Θ(u) 4 2K πu √ ϑ3 √ Θ1 (u) dn u = k 2K πu = k Θ(u) ϑ4 2K
SI 206(22), SI 209(35)
SI 207(23), SI 209(35)
SI 207(24), SI 209(35)
8.192 Series representation of H , H 1 , Θ, Θ1 . the functions In these formulas, q = exp −π KK . 1.
2.
3.
Θ(u) = ϑ4
H (u) = ϑ1
πu 2K πu
Θ1 (u) = ϑ3
2K
=1+2
2
(−1)n q n cos
n=1
=2
πu 2K
∞
∞
0] 3.7
Ei(x) =
1 2
{Ei(x + i0) + Ei(x − i0)}
8.212 1.8
2.7 3. 4. 5. 6. 7.8
e
[x > 0]
NT 11(1)
[x > 0]
NT 11(10)
0
∞ −t 1 e dt + 2 x 0 (x − t) ∞ −t e dt 1 −x Ei(−x) = e − + 2 x 0 (x + t) 1 dt Ei (±x) = ±e±x x ± ln t 0 ∞ −xt e Ei (±xy) = ±e±xy dt 0 y∓t ∞ −it e ±x dt Ei (±x) = −e t ± ix 0 1 y−1 t dt Ei(xy) = exy x + ln t 0 Ei(x) = ex
ET I 386
x −t
−1 dt t 0 x = C + e−x ln x + e−t ln t dt
Ei(−x) = C + ln x +
[x > 0]
[x > 0]
(cf. 8.211 1)
[x > 0]
(cf. 8.211 1)
[x > 0]
(cf. 8.211 1)
[Re y > 0, [x > 0]
x > 0]
LA 281(28)
NT 19(11) NT 23(2, 3)
LA 282(44)a
884
8.
The Exponential Integral Function and Functions Generated by It
= x−1 e
9. 10. 11. 12. 13. 14. 15. 16.
1
ty−1 dt 0 %x − ln t 1 tx−1 −xy
Ei(−xy) = −e−xy
0
LA 282(45)a
& 2
(y − ln t)
dt − y −1
[x > 0,
y > 0]
LA 283(47)a
∞
dt 1 2 x − ln t t 1 ∞ dt 1 Ei(−x) = −e−x 2 x + ln t t 1 ∞ t cos t + x sin t Ei(−x) = −e−x dt t2 + x2 0 ∞ t cos t − x sin t Ei(−x) = −e−x dt t2 + x2 0 t 2 ∞ cos t arctan dt Ei(−x) = π 0 t x 2e−x ∞ x cos t − t sin t Ei(−x) = ln t dt π 0 t2 + x2 2ex ∞ x cos t + t sin t Ei(x) = 2 ln x − ln t dt π 0 t2 + x2 ∞ Ei(−x) = −x e−tx ln t dt Ei(x) = ex
8.213
[x > 0]
LA 283(48)
[x > 0]
LA 283(48)
[x > 0]
NT 23(6)
[x < 0]
NT 23(6)
[Re x > 0]
NT 25(13)
[x > 0]
NT 26(7)
[x > 0]
NT 27(8)
[x > 0]
NT 32(12)
1
See also 3.327, 3.881 8, 3.916 2 and 3, 4.326 1, 4.326 2, 4.331 2, 4.351 3, 4.425 3, 4.581. For integrals of the exponential integral function, see 6.22–6.23, 6.78. Series and asymptotic representations 8.213 1.
li(x) = C + ln (− ln x) +
∞ k (ln x) k=1
2.
li(x) = C + ln ln x +
k · k!
∞ k (ln x) k=1
k · k!
[0 < x < 1]
[x > 1]
NT 3(9)
NT 3(10)
8.214 1.
Ei(x) = C + ln(−x) +
∞ xk k · k!
[x < 0]
k=1
2.
Ei(x) = C + ln x +
∞ xk k · k!
[x > 0]
k=1
3.
Ei(x) − Ei(−x) = 2x
∞ k=0
x2k (2k + 1)(2k + 1)!
[x > 0]
NT 39(13)
8.219
The exponential integral function Ei(x)
8.2157
& % n ez k! Ei(z) = + Rn (z) z zk
885
−n−1 |Rn (z)| = O |z|
k=0
−n−1
[Re z ≤ 0] |arg(−z)| ≤ π − δ; δ > 0 small] |Rn (z)| ≤ (n + 1)!|z|
, 1 1 kn nx + Ei(nx) − Ei(−nx) = e + 3 3 , nx n2 x2 n x where x = x sign Re(x), kn = O(1), and n → ∞ NT 39(15) [z → ∞,
8.2167
8.217
Functional relations: ∞ x sin t ex Ei (−x ) − e−x Ei (x ) = −2 NT 24(11) dt 2 2 0∞ t + x x cos t 4 [x = x sign Re x] NT 27(9) = ln t dt − 2e−x ln x 2 2 π 0 t +x ∞ t cos t 4 ∞ t sin t −x ex Ei (−x ) + e−x Ei (x ) = −2 dt = 2e ln x − ln t dt 2 2 π 0 t2 + x2 0 t +x [x = x sign Re x] NT 24(10), NT 27(10)
∞ 1 t x− x cos t 1 2 arctan Ei(−x) − Ei − dt = x π 0 t 1 + t2
1.
2.
3.
Ei(−αx) Ei(−βx) − ln(αβ) Ei[−(α + β)x] = e−(α+β)x
4.
0
[Re x > 0] ∞ −tx
e
ln[(α + t)(β + t)] dt t+α+β
NT 25(14) NT 32(9)
See also 3.723 1 and 5, 3.742 2 and 4, 3.824 4, 4.573 2. • For a connection with a confluent hypergeometric function, see 9.237. • For integrals of the exponential integral function, see 5.21, 5.22, 5.23, 6.22, and 6.23. 8.218
Two numerical values:
1.
Ei(−1) = −0.219 383 934 395 520 273 665 . . .
NT 89
2.
Ei(1) = 1.895 117 816 355 936 755 478 . . .
NT 89
8.219∗ 1.∗ 2.∗ 3.∗
Definite integrals of exponential functions ∞ π2 Ei2 (x)e−2x dx = 4 0 ∞ π2 Ei2 (−x)e2x dx = 4 0 ∞ Ei(x) Ei(−x) dx = 0 0
886
The Exponential Integral Function and Functions Generated by It
8.221
8.22 The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x 8.221 1.
+π , sinh t dt = −i + si(ix) t 2 0 x cosh t − 1 dt chi x = C + ln x + t 0 x
shi x =
2.11
(see 8.230 1)
EH II 146(17) EH II 146(18)
8.23 The sine integral and the cosine integral: si x and ci x 8.230 1.
10
2.10 8.231 1.
si(x) = −
∞
π sin t dt = − + Si(x), where Si(x) = t 2 x x ∞ cos t cos t − 1 dt = C + ln x + dt ci(x) = − t t x 0 si(xy) = −
∞ x
ci(xy) = −
2.
x
si(x) = −
3.
∞
π/2
0
x
sin t dt t
NT 11(3)
[ci(x) is also written Ci(x)]
NT 11(2)
sin ty dt t
NT 18(7)
cos ty dt t
NT 18(6)
e−x cos t cos (x sin t) dt
NT 13(26)
0
8.232
∞
si(x) = −
1.
π (−1)k+1 x2k−1 + 2 (2k − 1)(2k − 1)!
NT 7(4)
k=1
2.7
∞
ci(x) = C + ln(x) +
(−1)k
k=1
x2k 2k(2k)!
NT 7(3)
8.233 1. ci(x) ± i si(x) = Ei (±ix) 2. ci(x) − ci xe±πi = ∓πi
8.234 7
2
NT 7(7)
π/2
Ei(−x) − ci(x) = 0
2.
NT 7(5)
si(x) + si(−x) = −π
3.
1.
NT 6a
e−x cos ϕ sin (s sin ϕ) dϕ
π/2
2
[ci(x)] + [si(x)] = −2 0
NT 13(27)
exp (−x tan ϕ) ln cos ϕ dϕ sin ϕ cos ϕ [Re x > 0]
(see also 4.366) NT 32(11)
See also 3.341, 3.351 1 and 2, 3.354 1 and 2, 3.721 2 and 3, 3.722 1, 3, 5 and 7, 3.723 8 and 11, 4.338 1, 4.366 1.
8.250
The probability integral, Fresnel integrals and error functions
887
8.235 lim (x si(x)) = 0,
1.
x→+∞
lim si(x) = −π,
2.
x→−∞
lim (x ci(x)) = 0
x→+∞
[ < 1]
lim ci(x) = ±πi
NT 38(5) NT 38(6)
x→−∞
• For integrals of the sine integral and cosine integral, see 6.24–6.26, 6.781, 6.782, and 6.783. • For indefinite integrals of the sine integral and cosine integral, see 5.3.
8.24 The logarithm integral li(x) 8.240
x
dt = Ei (ln x) [x < 1] 0 ln t 1−ε x dt dt + li(x) = lim = Ei (ln x) [x > 1] ε→0 ln t 0 1+ε ln t li exp −xe±πi = Ei −xe±iπ = Ei (x ∓ i0) = Ei(x) ± iπ = li (ex ) ± iπ
1.
li(x) =
2. 3.
JA JA
[x > 0]
JA, NT 2(6)
[x < 1]
LA 281(33)
Integral representations 8.241 1.
ln x t
li(x) = −∞
li(x) = x
e 1 dt = x ln ln − t x
3.
∞
e−t ln t dt
− ln x
1
dt ln x + ln t 0 1 dt x +x = 2 ln x (ln x + ln t) 0 ∞ dt 1 =x 2 ln x − ln t t 1 x t a 1 dt li (ax ) = ln a −∞ t
2.
LA 280(22) LA 280(29)
[x < 1]
LA 280(30)
[x > 0]
For integrals of the logarithm integral, see 6.21
8.25 The probability integral Φ(x), the Fresnel integrals S (x) and C (x), the error function erf (x), and the complementary error function erfc(x) 8.250 1. 2.
11
Definition:
x 2 2 Φ(x) = erf(x) = √ e−t dt π 0 x 2 S (x) = √ sin t2 dt 2π 0
(called the error function)
888
3. 4.11 5.∗
6.∗
7.∗ 8.∗
9.∗
The Exponential Integral Function and Functions Generated by It
2 C (x) = √ 2π
x
cos t2 dt 0
erfc(x) = 1 − erf(x) (called the complementary error function) ∞ −(p+x)y √ e sin a x dx 0 π(p + x)
a a 1 −a√p 1 a√p √ √ √ = − sinh (a p) + e Φ Φ √ − py + e √ + py 2 2 y 2 2 y 2
√
∞ −(p+x)y √ p −a√p a a e 1 √ cos a x dx = √ exp − − py − e Φ √ − py πy a y 0 π(p + x) 4y
2 √ p √p a √ √ √ + e Φ √ + py − p cosh (a p) 2 2 y [Re p > 0, a, b are real]
2 √ p p p π exp −x2 Φ(p − x) dx = exp −x2 erf(p − x) dx = Φ √ 2 2 0 0 p p x2 exp −x2 Φ(p − x) dx = x2 exp −x2 erf(p − x) dx 0
2 2
√0 π p p x p − √ Φ − = Φ √ erf √ 4 2 2 2 2 2 √ √ (a+b)/ 2 (b+a)/ 2 2 √ 2 √ √ Φ b Φ a exp −x 2 − x dx + exp −x 2 − x dx = π Φ(a) Φ(b) √ √ (b−a)/ 2
(a−b)/ 2
Integral representations 8.251 1. 2. 3.
1 Φ(x) = √ π
1 S (x) = √ 2π
2. 3.
2y Φ(xy) = √ π
x2 −t
e √ dt t
0
1 C (x) = √ 2π
8.252 1.
8.251
x2 0
x2 0
x 0
(see also 3.361 1)
sin t √ dt t cos t √ dt t
e−t
2 2
y
dt
x 2y S (xy) = √ sin t2 y 2 dt 2π 0 x 2y C (xy) = √ cos t2 y 2 dt 2π 0
Re y 2 > 0
8.255
4.
5.7 6.8
The probability integral, Fresnel integrals and error functions
∞ −t2 y2 2 2 2 e ty dt Re y 2 > 0 Φ(xy) = 1 − √ e−x y √ π t2 + x2 0 2 2 2x −x2 y2 ∞ e−t y dt Re y 2 > 0 e =1− 2 2 π t +x 0
y 4xie y2 ∞ 2 2 −y 2 = √ 4x Φ e−t y sin(ty) dt Re x2 > 0 −Φ 2xi 2xi π 0 ∞ y y2 2 2 2 Φ e−t x −ty dt Re x2 > 0 = 1 − √ xe− 4 2x π 0
See also 3.322, 3.362 2, 3.363, 3.468, 3.897, 6.511 4 and 5. 8.2538 Series representations:
∞ x2k−1 2 −x2 3 2 2 11 erf(x) = √ e x F 1 1; ; x = √ (−1)k+1 1. 2 (2k − 1)(k − 1)! π π k=1 ∞ k 2k+1 2 2 x 2 = √ e−x (2k + 1)!! π k=0
2 5 3 1 2 7 5 1 2 2 3 2 2 x sin x F 1; , ; − x − x cos x F 1; , ; − x 2. S (x) = √ 4 4 4 3 4 4 4 2π ∞ k 4k+3 (−1) x 2 =√ 2π k=0 (2k + 1)!(4k + 3) ∞ ∞ (−1)k 22k x4k+1 (−1)k 22k+1 x4k+3 2 2 2 − cos x sin x =√ (4k + 1)!! (4k + 3)!! 2π k=0 k=0
2 3 2 7 5 1 2 5 3 1 2 2 2 x sin x F 1; , ; − x − x cos x F 1; , ; − x 3. C (x) = √ 4 4 4 4 4 4 2π 3 ∞ k 4k+1 (−1) x 2 =√ 2π k=0 (2k)!(4k + 1) ∞ ∞ (−1)k 22k+1 x4k+3 (−1)k 22k x4k+1 2 2 2 + cos x sin x =√ (4k + 3)!! (4k + 1)!! 2π k=0
889
NT 19(11)a NT 19(13)a
NT 28(3)a NT 27(1)a
NT 7(9)a NT 10(11)a
NT 8(14)a
NT 10(13)a
NT 8(13)a
NT 10(12)a
k=0
For the expansions in Bessel functions, see 8.515 2, 8.515 3. Asymptotic representations % n & 2 e−z −2n−z k (2k − 1)!! 8 Φ(z) = 1 − √ (−1) + O |z| , 8.254 k πz (2z 2 ) k=0
[z → ∞, where |Rn |
0 small] NT 37(10)a
[x → ∞]
MO 127a
890
The Exponential Integral Function and Functions Generated by It
1 1 sin x2 + O C (x) = + √ 2 2πx
2. 8.256 1. 2.
4.
5.11
1 x2
Functional relations:
z 2 z i 2 Φ √ =√ C (z) + i S (z) = eit dt 2 2π 0 i z √ 2 1 2 C (z) − i S (z) = √ Φ z i = √ e−it dt 2π 0 2i
3.
1 2 cos u C (u) + sin u S (u) = cos u + sin u2 + 2 2
2
8.256
[x → ∞]
MO 127a
∞ 2 e−2ut sin t2 dt π 0
[Re u ≥ 0] ∞ 2 1 2 2 cos u − sin u2 − cos u S (u) − sin u2 C (u) = e−2ut cos t2 dt 2 π 0
NT 28(6)a
[Re u ≥ 0] 2 2 2 √ π/2 exp −x tan ϕ sin ϕ2 cos ϕ 1 2 1 dϕ + S (x) − = C (x) − 2 2 π 0 sin 2ϕ
NT 28(5)a
(see also 6.322)
NT 33(18)a
• For a connection with a confluent hypergeometric function, see 9.236. • For a connection with a parabolic cylinder function, see 9.254. 8.257 1. 2.
x S (x) − 12 = 0 lim x C (x) − 12 = 0
lim
x→+∞ x→+∞
[ < 1]
NT 38(11)
[ < 1]
NT 38(11)
1 2 1 lim C (x) = x→+∞ 2
3.
lim S (x) =
NT 38(12)a
x→+∞
4.
NT 38(12)a
• For integrals of the probability integral, see 6.28–6.31. • For integrals of Fresnel’s sine integral and cosine integral, see 6.32. 8.25810
Integrals involving the complementary error function
∞
2 1 1 − arccos erfc2 (x)e−βx dx = √ β + 2 arctan 1+β βπ 0
1.
2. 0
∞
x erfc2 (x)e−βx dx = 2
1 2β
√ 4 arctan 1 + β √ 1− π 1+β
[β > 0]
[β > 0]
Lobachevskiy’s function Lfunction]Lobachevskiyfunction(ZdddZLZdddZ)(x)
8.262
√ 4 arctan 1 + β √ x erfc (x)e 1− π 1+β 0 √ arctan 1 + β 1 1 − + 3 βπ (1 + β) (β 2 + 2β + 2) (1 + β) 2 [β > 0] ∞ √ 1 + 32 β 1 x erfc x e−βx dx = 2 1 − [β > 0] 3 β (1 + β) 2 0 √ ∞ √ −βx √ 1 arctan β 1 1 x erfc x e dx = √ − 3 2β(1 + β) π 2 β2 0
3.
4. 5.11
891
∞
3
−βx2
2
1 dx = 2β 2
[β > 0] 8.259∗ 1.
2.
Integrals involving the error function and an exponential function ∞ √ a p π −px2 Φ [Re p > 0] , a, b real e Φ(a + bx) dx = p b2 + p −∞
∞ √ a p 2 π ab2 1 a2 p 2 −px Φ − x e Φ(a + bx) dx = exp − 2 3/2 2p p b +p b2 + p −∞ p (b2 + p)
∞
3.
x2n e−px Φ(a + bx) dx = (−1) 2
n
−∞
∂n ∂pn
%
π Φ p
[Re p > 0, √ & a p b2 + p
a, b are real]
[n = 0, 1, . . . ,
Re p > 0,
a, b are real]
8.26 Lobachevskiy’s function L(x) 8.260
Definition:
L(x) = −
x
ln cos t dt
LO III 184(10)
0
For integral representations of the function L(x), see also 3.531 8, 3.532 2, 3.533, and 4.224. 8.261 Representation in the form of a series: ∞ sin 2kx 1 L(x) = x ln 2 − (−1)k−1 LO III 185(11) 2 k2 k=1
8.262
Functional relationships:
1.
L(−x) = − L(x)
2.
L(π − x) = π ln 2 − L(x)
3.
L(π + x) = π ln 2 + L(x) π π 1 π −x = x− ln 2 − L − 2x L(x) − L 2 4 2 2
4.
+ π π, − ≤x≤ 2 2
LO III 185(13) LO III 286
+ π, 0≤x< 4
LO III 286 LO III 186(14)
892
Euler’s Integrals of the First and Second Kinds
8.310
8.3 Euler’s Integrals of the First and Second Kinds and Functions Generated by Them 8.31 The gamma function (Euler’s integral of the second kind): Γ(z) 8.310 1.
Definition: ∞ Γ(z) = e−t tz−1 dt
[Re z > 0]
(Euler)
FI II 777(6)
0
2.
Generalization: 1 Γ(z) = − (−t)z−1 e−t dt 2i sin πz C for z not an integer. The contour C is shown in the drawing:
WH
Γ(z) is an analytic function z with simple poles at the points z = −l (for l = 0, 1, 2,. . . ) to which (−1)l . Γ(z) satisfies the relation Γ(1) = 1. correspond to residues WH, MO 1 l! Integral representations 8.311 Γ(z) = 8.312 1.
∞
z−1
1
Γ(z) =
1 ln t
∞
Γ(z) = xz 0
3.
4.
5.
2az ea Γ(z) = sin πz
Γ(z) =
(0+)
e2πiz − 1
0
2.
1
e−t tz−1 dt
MO 2
dt
[Re z > 0]
e−xt tz−1 dt
[Re z > 0,
∞
e−at
2
1 + t2
z− 12
1 2 sin πz
Γ(y) = xy e−iβy
bz 2 sin πz
FI II 779(8)
cos [2at + (2z − 1) arctan t] dt
[a > 0] ∞
WH
z 2 e−t tz−1 1 + t2 2 {3 sin [t + z arccot(−t)] + sin [t + (z − 2) arccot(−t)]} dt
0
[arccot denotes an obtuse angle]
∞
ty−1 exp −xte−iβ dt
+ x, y, β real,
Γ(z) =
Re x > 0]
0
0
6.
FI II 778
∞
−∞
ebti (it)z−1 dt
[b > 0,
x > 0,
y > 0,
0 < Re z < 1]
|β|
0,
b ≥ 0,
[b > 0,
0 < Re z < 1]
Re z > 0]
∞
b cos(bt)tz−1 dt cos πz 0 2 ∞ bz = sin(bt)tz−1 dt πz sin 2 0
Γ(z) =
9.
NH 152(1)a
∞
Γ(z) =
NH 152(5)
e−t (t − z)tz−1 ln t dt
0
10.
∞
Γ(z) = −∞
NH 173(7)
[Re z > 0]
NH 145(14)
exp zt − et dt
∞
11.11 Γ(x) cos αx = λx
tx−1 e−λt cos α cos (λt sin α) dt
0
12.
[Re z > 0]
∞
Γ(x) sin αx = λx
tx−1 e−λt cos α sin (λt sin α) dt
0
x > 0,
−
π, π 0,
x > 0,
−
π π, 0,
8.313
8.314∗
Γ
z+1 v
1.11 2.8
= vu
z+1 v
∞
exp (−utv ) tz dt
[Re u > 0,
MO 2
Re v > 0,
Re z > −1]
0
JA, MO 7a
∞
Γ(z) = 1
8.315
[n = Re z ]
e−t tz−1 dt +
∞
(−1)k k!(z + k) n=0
i 1 = (−t)−z e−t dt Γ(z) 2π C ∞ ebti 2πe−ab bz−1 dt = 2 Γ(z) −∞ (a + it) ∞ −bti e dt = 0 Re a > 0, z (a + it) −∞
[z → 0, in |arg z| < π]
[for the contour C, see 8.310 2]
b > 0,
Re z > 0,
|arg(a + it)| < 12 π
894
3.
Euler’s Integrals of the First and Second Kinds
ea 1 = a1−z Γ(z) π
8.321
π/2
cos (a tan θ − zθ) cosz−2 θ dθ
[Re z > 1]
NH 157(14)
0
See also 3.324 2, 3.326, 3.328, 3.381 4, 3.382 2, 3.389 2, 3.433, 3.434, 3.478 1, 3.551 1, 2, 3.827 1, 4.267 7, 4.272, 4.353 1, 4.369 1, 6.214, 6.223, 6.246, 6.281.
8.32 Representation of the gamma function as series and products 8.321 1.6
Representation in the form of a series: Γ(z + 1) =
∞
ck z k
k=0 c0 = 1,
#n cn+1 =
k+1 sk+1 cn−k k=0 (−1)
n+1
;
s1 = C,
sn = ζ(n) for n ≥ 2,
|z| < 1 NH 40(1, 3)
2.11
∞
1 = dk z k Γ(z + 1) #n k=0 (−1)k sk+1 dn−k ; d0 = 1, dn+1 = k=0 n+1
s1 = C,
sn = ζ(n) for n ≥ 2
NH 41(4, 6)
Infinite-product representation ∞
1 - ez/k [Re z > 0] z 1 + kz k=1 z ∞ 1 - 1 + k1 = [Re z > 0] z 1 + kz k=1 n nz - k = lim [Re z > 0] n→∞ z z+k k=1 ∞ < 2k 8.3237 Γ(z) = 2z z e−z B 2k−1 z, 12 k=1
1 z + Γ ∞ 2 2k √ 8.3247 Γ(1 + z) = 4z π k=1 8.32211 Γ(z) = e−Cz
8.325 1. 2.11 3.7
∞ Γ(α) Γ(β) γ γ = 1+ 1− Γ(α + γ) Γ(β − γ) α+k β+k k=0
∞ eCx Γ(z + 1) x = 1− [z = 0, −1, −2, . . . ; ex/k Γ(z − x + 1) z+k k=1
√ ∞ π z z 1 − = 1 + 2k − 1 2k Γ 1 + z2 Γ 12 − z2 k=1
SM 269 WH SM 267(130)
NH 98(12)
MO 3
NH 62(2)
Re z > 0,
Re(z − x) > 0]
MO 2
8.331
Functional relations involving the gamma function
895
8.326 2
1.
[Γ(x)] ! !
∞ ! Γ(x) !2 y2 Γ(2x) ! = = !! 1 + B(x + iy, x − iy) Γ(x − iy) ! (x + k)2 k=0
[x, y are real,
2.11
∞ xe−iCy - exp Γ(x + iy) = Γ(x) x + iy n=1 1 +
x = 0, −1, −2, . . .] LO V, NH 63(4)
iy n iy x+n
[x, y are real,
x = 0, −1, −2, . . .] MO 2
8.327 1.∗
Asymptotic representation for large arguments:
√ −5 1 1 139 571 z− 12 −z + Γ(z) ∼ z e 2π 1 + − − +O z 12z 288z 2 51840z 3 2488320z 4 [|arg z| < π]
WH
2.∗
For z real and positive, the remainder of the series is less than the last term that is retained. n n n n √ √ n! ∼ 2πn or equivalently Γ(n + 1) ∼ 2πn e e [Stirling’s asymptotic formula for n 0] AS 6.1.38
3.∗
ln Γ(z) ∼
1 1 1 1 1 1 − z− + − + ... ln z − z + ln(2π) + 2 2 12z 360z 3 1260z 5 1680z 7 [z → ∞,
8.328 1. 2.
1
lim |Γ(x + iy)|e 2 |y| |y| 2 π
−x
|y|→∞
lim
|z|→∞
=
√ 2π
[x and y are real]
Γ(z + a) −a ln z e =1 Γ(z)
8.33 Functional relations involving the gamma function 8.331 1.
Γ(x + 1) = x Γ(x)
2.∗
Γ(x + a) = (x + a − 1) Γ(x + a − 1) =
3.∗
Γ(x + a + 1) (x + a)
Γ(x − a) = (x − a − 1) Γ(x − a − 1) =
Γ(x − a + 1) (x − a)
|arg z| < π]
AS 6.1.38
MO 6 MO 6
896
Euler’s Integrals of the First and Second Kinds
8.332
8.332 π y sinh πy ! 1 ! π !Γ + iy !2 = 2 cosh πy 2
|Γ(iy)| =
1. 2.
Γ(1 + ix) Γ(1 − ix) =
[y is real]
MO 3
[y is real]
πx sinh xπ
[x is real] 2π 2 x2 + y 2 Γ(1 + x + iy) Γ(1 − x + iy) Γ (1 + x − iy) Γ(1 − x − iy) = cosh 2yπ − cos 2xπ
3. 4.
LO V
[x and y are real] n
8.333 [Γ(n + 1)] = G(n + 1)
n -
LO V
kk ,
k=1
where n is a natural number and ∞
z z n z(z + 1) C 2 - z2 2 1+ − z G(z + 1) = (2π) exp − exp −z + 2 2 n 2n n=1 8.334 1. 2. 3.
z n , -+ 1 n 1 − = −z k Γ −z exp 2πki n k=1 k=1 1 1 π Γ 2 +x Γ 2 −x = cos πx π Γ(1 − x) Γ(x) = sin πx n -
WH
∞
[n = 2, 3, 3 . . .]
MO 2
FI II 430
Special cases 8.3357
Γ(nx) = (2π)
1−n 2
1
nnx− 2
n−1 k=0
1. 2. 3. 4.10
k Γ x+ n
22x−1 Γ(2x) = √ Γ(x) Γ x + 12 π
[product theorem]
FI II 782a, WH
[doubling formula]
1 33x− 2 Γ(x) Γ x + 13 Γ x + 23 2π
n−1 - k k (2π)n−1 Γ Γ 1− = n n n k=1 ∞ Γ2 n − 12 1 1 1 25 1 1 2 2 1 = 4 + 16 + 256 + 1024 + 65536 + · · · = π − n=0 4 (n!) Γ
Γ(3x) =
WH
2
8.336
∞ yz + xi yz − xi z+1 Γ − e−tx sinz (ty) dt Γ(1 − z) = (2i) y Γ 1 + 2y 2y 0 [Re(yi) > 0, Re(x − yzi) > 0] NH 133(10)
8.339
Functional relations involving the gamma function
897
• For a connection with the psi function, see 8.361 1. • For a connection with the beta function, see 8.384 1. • For integrals of the gamma function, see 8.412 4, 8.414, 9.223, 9.242 3, 9.242 4. 8.337 1. 2.
2 Γ (x) < Γ(x) Γ (x)
[x > 0]
For x > 0, min Γ(1 + x) = 0.88560 . . . is attained when x = 0.46163 . . .
MO 1 JA
Particular values 8.338 1. 2. 3. 4.
Γ(1) = Γ(2) = 1 √ Γ 12 = π √ Γ − 12 = −2 π 4 ∞ (4k − 1)2 (4k + 1)2 − 1 1 2 Γ = 16π 4 [(4k − 1)2 − 1] (4k + 1)2
MO 1a
k=1
5.
8 k=1
8.339 1. 2. 3. 4. 5.∗
3 k π 640 Γ = 6 √ 3 3 3
For n a natural number Γ(n) = (n − 1)! √ π Γ n + 12 = n (2n − 1)!! 2 √ n 1 π n 2 Γ 2 − n = (−1) (2n − 1)!! 2 1 Γ p+n+ 2 4p − 12 4p2 − 32 . . . 4p2 − (2n − 1)2 = 22n Γ p − n + 12 Γ(n + k) = (n + k − 1)! =
6.∗
WH
Γ(n + k + 1) (n + k)
[n + k ≥ 0, 1, . . .]
Γ(n − k) = (n − k − 1)! =
Γ(n − k + 1) (n − k)
[n − k ≥ 0, 1, . . .]
WA 221
898
Euler’s Integrals of the First and Second Kinds
8.341
8.34 The logarithm of the gamma function 8.341 1.
Integral representation:
−tz ∞ 1 1 1 e 1 1 − + t dt ln Γ(z) = z − ln z − z + ln 2π + 2 2 2 t e −1 t 0
2.11
3. 4.
5.
7.
WH
√ arctan 1 dt ln Γ(z) = z ln z − z − ln z + ln 2π + 2 2 e2πt − 1 0 w du Re z > 0 and arctan w = is taken over a rectangular path in the w-plane 2 0 1+u ∞ −zt e dt − e−t −t [Re z > 0] ln Γ(z) = + (z − 1)e −t 1−e t 0 ∞ (1 + t)−z − (1 + t)−1 dt −t ln Γ(z) = (z − 1)e + ln(1 + t) t 0
ln Γ(x) =
ln π − ln sin πx 1 + 2 2
∞ 0
t z
[Re z > 0] sinh 2 − x t dt − (1 − 2x)e−t t t sinh 2 1
dt tz − t − t(z − 1) ln Γ(z) = t − 1 t ln t 0 ∞ −tz −t dt −e e −t ln Γ(z) = (z − 1)e + −t 1 − e t 0
6.
[Re z > 0] ∞
1
WH
WH
WH
[0 < x < 1]
WH
[Re z > 0]
WH
[Re z > 0]
NH 187(7)
See also 3.427 9, 3.554 5. 8.342 Series representations: 1.11
ln Γ(z + 1) ∞ 1 πz 1 − ζ(2k + 1) 2k+1 1+z = z ln − ln + (1 − C) z + 2 sin πz 1−z 2k + 1 k=1 ∞ zk = −Cz + (−1)k ζ(k) k
[|z| < 1] NH 38(16, 12)
k=2
2.
ln Γ(1 + x) =
∞ x2n+1 πx 1 ln − Cx − ζ(2n + 1) 2 sin πx 2n + 1 n=1
[|x| < 1] 8.343 1.
∞ √ ln Γ(x) = ln 2π + n=1
NH 38(14)
1 1 cos 2nπx + (C + ln 2nπ) sin 2nπx 2n nπ [0 < x < 1]
FI III 558
8.352
The incomplete gamma function
ln Γ(z) = z ln z − z −
2.
899
∞ ∞ √ m 1 1 1 ln z + ln 2π + 2 2 m=1 (m + 1)(m + 2) n=1 (z + n)m+1
[|arg z| < π]
MO 9
Asymptotic expansion for large values of |z|:
8.3447
√ B2k 1 + Rn (z), ln Γ(z) = z ln z − z − ln z + ln 2π + 2 2k(2k − 1)z 2k−1 n−1
k=1
where |Rn (z)|
0]
EH II 133(1), NH 1(1)
0
2.11
∞
Γ(α, x) =
e−t tα−1 dt
EH II 133(2), NH 2(2), LE 339
x
3.∗
Γ(z, 0) = Γ(z)
4.∗
Γ(a, ∞) = 0
5.
∗
γ(a, 0) = 0
8.351
x−α γ(α, x) is an analytic function with respect to α and x Γ(α)
1.
γ ∗ (α, x) =
2.
Another definition of Γ(α, x) that is also suitable for the case Re α ≤ 0: γ(α, x) =
xα xα −x e Φ (1, 1 + α; x) = Φ(a, 1 + a; −x) α α
EH II 133(5)
EH II 133(3)
3.
For fixed x, Γ(α, x) is an entire function of α. For non-integral α, Γ(α, x) is a multiple-valued function of x with a branch point at x = 0.
4.
A second definition of Γ(α, x): Γ(α, x) = xα e−x Ψ(1, 1 + α; x) = e−x Ψ(1 − α, 1 − α; x)
8.352
Special cases: %
1.
EH II 133(4)
γ(1 + n, x) = n! 1 − e
−x
n xm m! m=0
& [n = 0, 1, . . .] EH II 136(17, 16), NH 6(11)
2.
Γ(1 + n, x) = n!e−x
n
xm m! m=0
[n = 0, 1, . . .]
EH II 136(16, 18)
900
Euler’s Integrals of the First and Second Kinds
3.11
Γ(−n, x) =
8.353
n 1 1 z k−n−1 (−1)n 1 Ei(−z) − ln(−z) + ln − − ln z − e−z n! 2 2 z (−n)k k=1
[n = 1, 2, . . .] n−1
xm m! m=0 %
4.∗
Γ(n, x) = (n − 1)!e−x
5.∗
n−2 m! (−1)n+1 Γ(−n + 1, x) = (−1)m m+1 Γ(0, x) − e−z (n − 1)! x m=0
% 6.∗
γ(n, x) = (n − 1)! 1 − e−x −x
n−1
xm m! m=0
&
[n = 2, 3, . . .]
&
[n = 1, 2, . . .]
n−1
xm m! m=0 %
7.
∗
Γ(n, x) = (n − 1)!e
8.
∗
n−k−1 m! (−1)n−k Γ(0, x) − e−x Γ(−n + k, x) = (−1)m m+1 (n − k)! x m=0
[n = 1, 2, . . .] &
[n − k ≥ 1, 8.353 1.
Integral representations: π γ(α, x) = xα cosec πα ex cos θ cos (αθ + x sin θ) dθ
[x = 0,
k = 0, 1, . . .]
Re α > 0,
α = 1, 2, . . .]
0
1
EH II 137(2)
∞
γ(α, x) = x 2 α
2.
√ 1 e−t t 2 α−1 J α 2 xt dt
0
3.
Γ(α, x) =
ρ−x xα Γ(1 − α)
[Re α > 0]
EH II 138(4)
∞ −t −α
e t dt x+t
0
[Re α < 1,
x > 0] EH II 137(3), NH 19(12)
1 2α
−x
2x e Γ(1 − α) α −xy Γ(α, xy) = y e
4.
Γ(α, x) =
5.
∞
0 ∞
+ √ , 1 e−t t− 2 α K α 2 xt dt
[Re α < 1]
EH II 138(5)
e−ty (t + x)α−1 dt
0
[Re y > 0,
x > 0,
Re α > 1]
(See also 3.936 5, 3.944 1–4)
NH 19(10)
For integrals of the gamma function, see 6.45. 8.354 1.
Series representations: γ(α, x) =
∞ (−1)n xα+n n!(α + n) n=0
EH II 135(4)
8.356
2.
The incomplete gamma function
Γ(α, x) = Γ(α) −
∞ (−1)n xα+n n!(α + n) n=0
901
[α = 0, −1, −2, . . .] EH II 135(5), LE 340(2)
3.
Γ(α, x) − Γ (α, x + y) = γ(α, x + y) − γ(α, x) = e−x xα−1
∞ (−1)k [1 − e−y ek (y)] Γ(1 − α + k) k=0
xk Γ(1 − α) ek (x) =
4.
γ(α, x) = Γ(α)e−x x 2 α 1
∞
n √ 1 (−1)m x 2 n I n+α 2 x m! n=0 m=0
k xm m! m=0
[x = 0,
[|y| < |x|] α = 0,
EH II 139(2)
−1, −2, . . .] EH II 139(3)
5.
8.355
Γ(α, x) = e−x xα
∞
Lα n (x) n +1 n=0
[x > 0]
Γ(α, x) γ(α, y) = e−x−y (xy)α
EH II 140(5)
∞
n! Γ(α) α Lα n (x) Ln (y) (n + 1) Γ(α + n + 1) n=0 [y > 0, x ≥ y,
α = 0, −1, . . .] EH II 139(4)
8.356 1.11
Functional relations: γ(α + 1, x) = α γ(α, x) − xα e−x α −x
EH II 134(2)
2.
Γ(α + 1, x) = α Γ(α, x) + x e
EH II 134(3)
3.
Γ(α, x) + γ(α, x) = Γ(α)
EH II 134(1)
4. 5.
d Γ(α, x) d γ(α, x) =− = xα−1 e−x dx dx n−1 Γ(α, x) xα+s Γ(α + n, x) = + e−x Γ(α + n) Γ(α) Γ(α + s + 1) s=0
6.11
Γ(α) Γ(α + n, x) − Γ(α + n) Γ(α, x) = Γ(α + n) γ(α, x) − Γ(α) γ(α + n, x)
7.∗
Γ(a + k, x) = (a + k − 1) Γ(a + k − 1, x) + xa+k−1 e−x 1 Γ(a + k + 1, x) − xa+k e−x = a+k
8.∗
Γ(a − k, x) = (a − k − 1) Γ(a − k − 1, x) + xa−k−1 e−x 1 Γ(a − k + 1, x) − xa−k e−x = a−k
9.∗
γ(a + k, x) = (a + k − 1) γ(a + k − 1, x) − xa+k−1 e−x 1 Γ(a + k + 1, x) + xa+k e−x = a+k
EH II 135(8) NH 4(3) NH 5
902
10.∗
8.357 1.
8.358
8.359 1. 2. 3. 4.
11
Euler’s Integrals of the First and Second Kinds
8.357
γ(a − k, x) = (a − k − 1) γ(a − k − 1, x) − xa−k−1 e−x 1 γ(a − k + 1, x) + xa−k e−x = a−k Asymptotic representation for large values of |x|: %M −1 & (−1)m Γ(1 − α + m) −M α−1 −x + O |x| Γ(α, x) = x e xm Γ(1 − α) m=0 3π 3π < arg x < , M = 1, 2, . . . |x| → ∞, − 2 2
EH II 135(6), NH 37(7), LE 340(3)
Representation as a continued fraction: e−x xα Γ(α, x) = 1−α x+ 1 1+ 2−α x+ 2 1+ 3−α x+ 1 + ... Relationships with other functions:
EH II 136(13), NH 42(9)
Γ(0, x) = − Ei(−x)
1 Γ 0, ln = − li(x) x √ √ Γ 12 , x2 = π − π Φ(x) √ γ 12 , x2 = π Φ(x)
EH II 143(1) EH II 143(2) EH II 147(2) EH II 147(1)
8.36 The psi function ψ(x) 8.360 1. 8.361 1.8 2. 3. 4.
Definition: ψ(x) =
d ln Γ(x) dx
Integral representations:
∞ −t e e−zt d ln Γ(z) = − ψ(z) = dt dz t 1 − e−t 0
∞ dt 1 ψ(z) = e−t − z (1 + t) t 0 ∞ t dt 1 −2 ψ(z) = ln z − 2 2 2πt − 1) 2z 0 (t + z ) (e
1 1 tz−1 − ψ(z) = dt − ln t 1 − t 0
[Re z > 0]
NH 183(1), WH
[Re z > 0]
NH 184(7), WH
[Re z > 0]
WH
[Re z > 0]
WH
8.363
The psi function ψ(x)
5.
∞ −t
− e−zt dt − C, 1 − e−t
e
ψ(z) = 0
6.
∞
ψ(z) =
(1 + t)−1 − (1 + t)−z
0
7. 8.
903
WH
dt − C, t
−1 dt − C t − 1 0 ∞ 1 1 − ψ(z) = ln z + e−tz dt t 1 − e−t 0
[Re z > 0]
WH
1 z−1
ψ(z) =
t
FI II 796, WH
[Re z > 0]
MO 4
See also 3.244 3, 3.311 6, 3.317 1, 3.457, 3.458 2, 3.471 14, 4.253 1 and 6, 4.275 2, 4.281 4, 4.482 5. For integrals of the psi function, see 6.46, 6.47. Series representation 8.362 1.
2.
∞
1 1 − x+k k+1 k=0 ∞ 1 1 = −C − + x x k(x + k) k=1
∞ 1 1 − ln 1 + ψ(x) = ln x − x+k x+k ψ(x) = −C −
FI II 799(26), KU 26(1) FI II 495
MO 4
k=0
∞
3.
π2 (x − 1) − (x − 1) ψ(x) = −C + 6
k=1
1 1 − k+1 x+k
k−1
1 x+n n=0
NH 54(12)
8.363 1.
ψ(x + 1) = −C +
∞
(−1)k ζ(k)xk−1
NH 37(5)
k=2
2.
3.
∞ π x2 1 − cot πx − − C + [1 − ζ(2k + 1)] x2k 2x 2 1 − x2 k=1
∞ 1 1 − ψ(x) − ψ(y) = y+k x+k
ψ(x + 1) =
NH 38(10)
k=0
(see also 3.219, 3.231 5, 3.311 7, 3.688 20, 4.253 1, 4.295 37)
4.
5.
∞
2yi y 2 + (x + k)2 k=0
∞ p 1 q − ψ = −C + q k + 1 p + kq ψ(x + iy) − ψ(x − iy) =
k=0
NH 99(3)
(see also 3.244 3)
NH 29(1)
904
Euler’s Integrals of the First and Second Kinds
8.364
6.8
[ q+1
2 ]−1 p pπ kπ 2kpπ π +2 ln sin ψ cos = −C − ln(2q) − cot q 2 q q q k=1 [q = 2, 3, . . . , p = 1, 2, . . . , q − 1]
7.
∞ ∞ p p−1 1 ψ −ψ =q n−1 q q (p + kq) n=2
MO 4, EH I 19(29) NH 59(3)
k=0
ψ (n) (x) = (−1)n+1 n!
8.
∞ k=0
1 = (−1)n+1 n! ζ(n + 1, x) (x + k)n+1
NH 37(1)
Infinite-product representation 8.364 ψ(x)
∞ =x 1+
1.
e
2.
ey ψ(x)
1 1 e− x+k x+k k=0
∞ y Γ(x + y) y = 1+ e− x+k Γ(x) x+k
NH 65(12)
NH 65(11)
k=0
See also 8.37. • • • •
For a connection with Riemann’s zeta function, see 9.533 2. For a connection with the gamma function, see 4.325 12 and 4.352 1. For a connection with the beta function, see 4.253 1. For series of psi functions, see 8.403 2, 8.446, and 8.447 3 (Bessel functions), 8.761 (derivatives of associated Legendre functions with respect to the degree), 9.153, 9.154 (hypergeometric function), 9.237 (confluent hypergeometric function). • For integrals containing psi functions, see 6.46–6.47. 8.365 1. 2. 3.
Functional relations: 1 ψ(x + 1) = ψ(x) + x
x x+1 = 2 β(x) ψ −ψ 2 2 ψ(x + n) = ψ(x) +
n−1 k=0
4.
JA
(cf. 8.37 0)
1 x+k
GA 154(64)a
n 1 ψ(n + 1) = −C + k
MO 4
k=1
5. 6.
lim [ψ(z + n) − ln n] = 0
MO 3
n→∞
n−1 1 k ψ(nz) = ψ z+ + ln n n n k=0
[n = 2, 3, 4, . . .]
MO 3
8.367
7.
The psi function ψ(x)
ψ(x − n) = ψ(x) −
n k=1
8. 9. 10. 8.366
1 x−k
ψ(1 − z) = ψ(z) + π cot πz ψ 12 + z = ψ 12 − z + π tan πz ψ 34 − n = ψ 14 + n + π
10. 11.
ψ (n) =
2. 3. 4. 5. 6. 7. 8. 9.
GA 155(68)a JA
[n = 0,
±1,
±2, . . .]
Particular values ψ(1) = −C ψ 12 = −C − 2 ln 2 = −1.963 510 026 . . . & % n 1 1 − ln 2 ψ 2 ± n = −C + 2 2k − 1 k=1 π ψ 14 = −C − − 3 ln 2 2 3 π ψ 4 = −C + − 3 ln 2 2 < 1 π 1 3 ψ 3 = −C − − ln 3 2 3 2 < π 1 3 ψ 23 = −C + − ln 3 2 3 2 π2 = 1.644 934 066 848 . . . ψ (1) = 6 π2 = 4.934 802 200 5 . . . ψ 12 = 2 ψ (−n) = ∞
1.
905
(cf. 8.367 1) GA 155a JA GA 157a GA 157a GA 157a GA 157a JA JA
[n is a natural number]
JA
[n is a natural number]
JA
n π2 1 −4 +n = 2 (2k − 1)2
[n is a natural number]
JA
n π2 1 +4 −n = 2 (2k − 1)2
[n is a natural number]
JA
π2 1 − 6 k2 n−1
k=1
12.
ψ
1 2
k=1
13.
ψ
1 2
k=1
8.367 1. 2. 3.
Euler’s constant (also denoted by γ): C = − ψ(1) = 0.577 215 664 90 . . . & %n−1 1 − ln n C = lim n→∞ k k=1 1 C = lim ζ(x) − x→1+0 x−1
FI II 319, 795 FI II 801a
FI II 804
906
Euler’s Integrals of the First and Second Kinds
8.370
Integral representations: ∞ 4. C=− e−t ln t dt
FI II 807
0
5. 6. 7. 8. 9. 10. 11. 12.
13.
1 ln ln FI II 807 dt t 0 1 1 1 + C= DW dt ln t 1 − t 0 ∞ dt 1 C=− MO 10 cos t − 1+t t 0 ∞ sin t 1 dt − C=1− MO 10 t 1 + t t 0 ∞ dt 1 −t C=− FI II 795, 802 e − 1 + t t 0 ∞ dt 1 C=− DW, MO 10 e−t − 2 1+t t 0 ∞ 1 1 − C= DW dt et − 1 tet 0 ∞ −t 1 e −t dt − dt 1−e C= FI II 802 t t 0 1 See also 8.361 5–8.361 7, 3.311 6, 3.435 3 and 4, 3.476 2, 3.481 1 and 2, 3.951 10, 4.283 9, 4.331 1, 4.421 1, 4.424 1, 4.553, 4.572, 6.234, 6.264 1, 6.468. C=−
1
Asymptotic expansions C =
n−1 k=1
1 1 1 1 1 1 − ln n + + − + − + ... k 2n 12n2 120n4 252n6 240n8
[0 < θ < 1]
FI II 827
θ B2r+2 B2r 1 + ··· + 2r 2r+2 2r n 2 (r + 1) n
8.37 The function β(x) Definition:
x x+1 1 β(x) = ψ −ψ 2 2 2 8.371 Integral representations: 1 x−1 t 3 dt β(x) = 1. 1 +t 0 ∞ −xt e 2. β(x) = dt −t 0 1+e
∞ −xt x+1 e dt 3. β = 2 0 cosh t 8.370
NH 16(13)
[Re x > 0]
WH
[Re x > 0]
MO 4
[Re x > −1]
See also 3.241 1, 3.251 7, 3.522 2 and 4, 3.623 2 and 3, 4.282 2, 4.389 3, 4.532 1 and 3.
8.377
The function β(x)
907
Series representation 8.372 1.7
β(x) =
∞ (−1)k k=0
2.7
β(x) =
∞ k=0
x+k 1 (x + 2k)(x + 2k + 1)
[−x ∈ N]
NH 37, 101(1)
[−x ∈ N]
NH 101(2)
∞
3.8
β(x) =
1 k! 1 2 x(x + 1) . . . (x + k) 2k k=0
[−x ∈ N]
[β has simple poles at x = −n with residue (−1)n ]
NH 246(7)
8.373 1.6
β(x + 1) = ln 2 +
∞
(−1)k 1 − 2−k ζ(k + 1)xk
[|x| < 1]
NH 37(5)
k=1
2.
6
∞ π 1 1 − + 1 − 1 − 2−2k ζ(2k + 1) x2k β(x + 1) = ln 2 − 1 + − 2x 2 sin πx 1 − x2 k=1
[0 < |x| < 2;
x = ±1]
NH 38(11)
∞
8.374
(−1)k dn β(x) = (−1)n n! n dx (x + k)n+1
[−x ∈ N]
NH 37(2)
k=0
8.375
1.
6
Representation in the form of a finite sum: q−1
2 p (2k + 1)π p(2k + 1)π π ln sin β − cos = q 2 sin pπ q 2q q
k=0
2.
β(n) = (−1)n+1 ln 2 +
n−1 k=1
[q = 2, 3, . . . , p = 1, 2, 3, . . . , q − 1]
NH 23(9)
(−1)k+n+1 k
Functional relations
2n x+k 8.376 (−1)k β = (2n + 1) β(x) 2n + 1 k=0 n 8.377 β 2k x = ψ (2n x) − ψ(x) − n ln 2 k=1
(see also 8.362 5–7)
NH 19 NH 20(10)
908
Euler’s Integrals of the First and Second Kinds
8.380
8.38 The beta function (Euler’s integral of the first kind): B(x, y) Integral representation 8.380 1.
1
B (x, y) = 0
tx−1 (1 − t)y−1 dt∗ 1
=2 2.
4. 5. 6.
sin2x−1 ϕ cos2y−1 ϕ dϕ
[Re x > 0, t
B(x, y) = z (1 + z)
π/2
x 0
10. 11.
Re y > 0]
KU 10
Re y > 0]
FI II 775
Re y > 0]
MO 7
Re y > 0]
BI (1)(15)
(1 − t)y−1 dt (t + z)x+y [Re x > 0, Re y > 0,
dt Re y > 0]
BI (1)(15)
1 x−1
B (x, y) = z y (1 + z)x
9.
[Re x > 0,
∞ tx−1 t2x−1 B(x, y) = dt = 2 dt [Re x > 0, x+y 2 x+y 0 (1 + t) 0 (1 + t ) 1 (1 + t)2x−1 (1 − t)2y−1 B(x, y) = 22−y−x dt [Re x > 0, x+y (1 + t2 ) −1 1 x−1 ∞ x−1 t + ty−1 t + ty−1 B(x, y) = dt = dt [Re x > 0, x+y (1 + t)x+y 0 (1 + t) 1 1+ , 1 x−1 B(x, y) = x+y−1 (1 − t)y−1 + (1 + t)y−1 (1 − t)x−1 (1 + t) 2 0
y
FI II 774(1)
0 ∞
0
8.
Re y > 0]
π/2
7.
[Re x > 0,
0
B(x, y) = 2
3.
y−1 t2x−1 1 − t2 dt
cos2x−1 ϕ sin2y−1 ϕ x+y
(z + cos2 ϕ) [Re x > 0,
0 > z > −1,
Re(x + y) < 1]
NH 163(8)
0 > z > −1,
Re(x + y) < 1]
NH 163(8)
dϕ
Re y > 0,
See also 3.196 3, 3.198, 3.199, 3.215, 3.238 3, 3.251 1–3, 11, 3.253, 3.312 1, 3.512 1 and 2, 3.541 1, 3.542 1, 3.621 5, 3.623 1, 3.631 1, 8, 9, 3.632 2, 3.633 1, 4, 3.634 1, 2, 3.637, 3.642 1, 3.667 8, 3.681 2. 1 1 (1 − t)x−1 1 1 2 x−1 √ 1−t dt = 2x−1 B(x, x) = 2x−2 dt 2 2 t 0 0 See 8.384 4, 8.382 3, and also 3.621 1, 3.642 2, 3.665 1, 3.821 6, 3.839 6. ∞ cosh 2yt B(x + y, x − y) = 41−x MO 9 dt [Re x > |Re y|, Re x > 0] cosh2x t 0 1 , y + y x−1 y−1 =z B x, (1 − tz ) t dt Re z > 0, Re > 0, Re x > 0 z z 0 FI II 787a
∗ This
equation is used as the definition of the function B(x, y).
8.384
8.381 1.
The beta function (Euler’s integral of the first kind): B(x, y)
1−x−y
dt 2π (a + b) = x y (x + y − 1) B(x, y) −∞ (a + it) (b − it) [a > 0,
2.
∞
b > 0;
x and y are real,
x + y > 1]
MO 7
b > 0;
x and y are real,
x + y > 1]
MO 7
∞
dt =0 x (b − it)y (a − it) −∞ [a > 0,
3.
909
B(x + iy, x − iy) = 2
1−2x
−2iγy
αe
∞
e2iαyt dt 2x −∞ cosh (αt − γ) [y, α, γ are real,
α > 0;
Re x > 0] MI 8a
4.
For an integral representation of ln B(x, y), see 3.428 7. 2x+y−1 (x + y − 1) π/2 1 = cos[(x − y)t] cosx+y−2 t dt B (x, y) π 0 2x+y−2 (x + y − 1) π = cos[(x − y)t] sinx+y−2 t dt π cos (x − y) π2 0 2x+y−2 (x + y − 1) π = sin[(x − y)t] sinx+y−2 t dt π sin (x − y) π2 0
NH 158(5)a NH 159(8)a NH 159(9)a
Series representation 8.382 1.
2.
∞ 1 (y − 1) . . . (y − n) [y > 0] (−1)n y WH y n=0 n!(x + n)
∞ √ 1 − 1 − 2−2k ζ(2k + 1) 2k+1 tan πx 1+x 1 1+x 1 2 , x ln B ln 2π + ln − ln + 2 2 2 x 1−x 2k + 1
B(x, y) =
k=0
[|x| < 2] 3. 8.383
B z,
1 2
=
∞ (2k − 1)!! k=1
2k k!
1 1 + z+k z
NH 39(17)
(see also 8.384 and 8.380 9)
WH
Infinite-product representation: (x + y + 1) B(x + 1, y + 1) =
∞ k(x + y + k) (x + k)(y + k)
[x,
y = −1,
−2, . . .]
MO 2
k=1
8.384
Functional relations involving the beta function: Γ(x) Γ(y) = B(y, x) Γ(x + y)
1.
B(x, y) =
2.
B(x, y) B(x + y, z) = B(y, z) B(y + z, x)
FI II 779 MO 6
910
3.
Bessel Functions and Functions Associated with Them ∞
B(x, y + k) = B(x − 1, y)
k=0
4.
8.391
B(x, x) = 21−2x B
1
WH
2, x
(see also 8.380 9 and 8.382 3) FI II 784
5. 6.
π B(x, x) B x + 12 , x + 12 = 4x−1 2 x
n+m−1 n+m−1 1 =m =n n−1 m−1 B(n, m)
WH
[m and n are natural numbers]
For a connection with the psi function, see 4.253 1.
8.39 The incomplete beta function Bx (p, q)
8.392 I x (p, q) =
x
tp−1 (1 − t)q−1 dt =
8.3917 Bx (p, q) =
0 Bx (p, q)
xp 2 F 1 (p, 1 − q; p + 1; x) p
ET I 373 ET II 429
B(p, q)
8.4–8.5 Bessel Functions and Functions Associated with Them 8.40 Definitions 8.401
Bessel functions Z ν (z) are solutions of the differential equation
ν2 d2 Z ν 1 dZν + 1 − 2 Zν = 0 + KU 37(1) dz 2 z dz z Special types of Bessel functions are what are called Bessel functions of the first kind J ν (z), Bessel functions of the second kind Y ν (z) (also called Neumann functions and often written N ν (z)), and Bessel (2) functions of the third kind H (1) ν (z) and H ν (z) (also called Hankel’s functions). ∞ z 2k zν [|arg z| < π] (−1)k 2k KU 55(1) 8.402 J ν (z) = ν 2 2 k! Γ(ν + k + 1) k=0
8.403 1.
Y ν (z) =
1 [cos νπ J ν (z) − J −ν (z)] sin νπ
[for non-integer ν,
|arg z| < π] KU 41(3)
8.407
Definitions
911
z (n − k − 1)! z 2k−n − 2 k! 2 n−1
2.
π Y n (z) = 2 J n (z) ln
k=0
−
∞
(−1)k
k=0
= 2 J n (z) ln
z n+2k 1 [ψ(k + 1) + ψ(k + n + 1)] k! (k + n)! 2
z +C − 2
n−1 k=0
KU 43(10)
(n − k − 1)! z 2k−n k!
2
& n+2k % n+k n ∞ k z n 1 1 1 (−1)k z2 1 − + − 2 n! k k! (k + n)! m m=1 m m=1 k=1 k=1 [n + 1 a natural number, |arg z| < π] KU 44, WA 75(3)a
8.404 1.
Y −n (z) = (−1)n Y n (z)
[n is a natural number]
KU 41(2)
2.
J −n (z) = (−1)n J n (z)
[n is a natural number]
KU 41(2)
8.4057 1.
H (1) ν (z) = J ν (z) + i Y ν (z)
KU 44(1)
2.
H (2) ν (z) = J ν (z) − i Y ν (z)
KU 44(1)
In all relationships that hold for an arbitrary Bessel function Z ν (z), that is, for the functions J ν (z), (2) Y ν (z), and linear combinations of them, for example, H (1) ν (z) and H ν (z), we shall write simply the letter Z instead of the letters J, Y , H (1) , and H (2) . Modified Bessel functions of imaginary argument I ν (z) and K ν (z) 8.406 1. 2.
π π I ν (z) = e− 2 νi J ν e 2 i z 3 3 I ν (z) = e 2 πνi J ν e− 2 πi z
+ π, −π < arg z ≤ 2 , +π < arg z ≤ π 2
WA 92 WA 92
For integer ν, I n (z) = i−n J n (iz)
3. 8.407 1.8 2.8
πi π νi (1) 1 πi e 2 H ν ze 2 2 −πi − π νi (2) − 1 πi e 2 H −ν ze 2 K ν (z) = 2 K ν (z) =
KU 46(1)
−π < arg z ≤ 12 π − 12 π < arg z ≤ π
For the differential equation defining these functions, see 8.494.
WA 92(8)
912
Bessel Functions and Functions Associated with Them
8.411
8.41 Integral representations of the functions Jν (z) and Nν (z) 8.411 1.
11
2.
1 π −niθ+iz sin θ e dθ J n (z) = 2π −π π 1 [n = 0, 1, 2, . . .] cos (nθ − z sin θ) dθ = π 0 1 π 2 π/2 J 2n (z) = cos 2nθ cos (z sin θ) dθ = cos 2nθ cos (z sin θ) dθ π 0 π 0
WH
[n an integer]
WA 30(7)
3.11
4.
5.
6.
7. 8.
9.
10. 11.
12.
π
1 sin(2n + 1)θ sin (z sin θ) dθ π 0 π/2 2 WA 30(6) [n an integer] = sin(2n + 1)θ sin (z sin θ) dθ π 0 z ν π/2 2 J ν (z) = 2 sin2ν θ cos (z cos θ) dθ Γ ν + 12 Γ 12 0 Re ν > − 12 WH z ν π 2
J ν (z) = sin2ν θ cos (z cos θ) dθ Re ν > − 12 1 0 Γ ν + 12 Γ 2 z ν π/2 2
cos (z sin θ) cos2ν θ dθ J ν (z) = 1 −π/2 1 Γ ν+2 Γ 2 KU 65(5), WA 35(4)a Re ν > − 12 z ν π 2 J ν (z) = e±iz cos ϕ sin2ν ϕ dϕ Re ν + 12 > 0 WH 1 1 Γ ν+2 Γ 2 0 z ν 1 ν− 12 2 1 − t2 J ν (z) = cos zt dt Re ν > − 12 KU 65(6), WH 1 1 Γ ν + 2 Γ 2 −1 x −ν ∞ 1 sin xt J ν (x) = 2 1 2 1 − 2 < Re ν < 12 , x > 0 MO 37 1 dt ν+ Γ 2 − ν Γ 2 1 (t2 − 1) 2 z ν 1 ν− 12 2 1 J ν (z) = eizt 1 − t2 dt Re ν > − 12 WA 34(3) 1 Γ ν + 2 Γ 2 −1 2 ∞ νπ J ν (x) = sin x cosh t − WA 199(12) cosh νt dt π 0 2 π/2 cosν− 12 θ sin z − νθ + 1 θ 2 2ν+1 z ν 1 J ν (z) = e−2z cot θ dθ 2ν+1 1 Γ ν+2 Γ 2 0 sin θ , + π |arg z| < , Re ν + 12 > 0 WH 2 J 2n+1 (z) =
8.414
13.
10
14.
Integral representations of the functions Jν (z) and Nν (z)
1 J ν (z) = π
J ν (z) =
e
π
0
±νπi
π
sin νπ cos (νθ − z sin θ) dθ − π
π
0
∞
913
e−νθ−z sinh θ dθ
0
[Re z > 0] ∞
WA 195(4)
cos (νθ + z sin θ) dθ − sin νπ e−νθ+z sinh θ dθ 0 π π for < |arg z| < π, with the upper sign taken for |arg z| > 2 2 π and the lower sign taken for |arg z| < − 2 WH
8.412 1. 2. 3.8
4.
5.7
+ z 1 π, t exp t− dt |arg z| < 2 t 2 −∞
(0+) zν z2 J ν (z) = ν+1 t−ν−1 exp t − dt 2 πi −∞ 4t ∞ z ν (−1)k z 2k (0+) t −ν−k−1 J ν (z) = ν+1 et dt 2 πi 22k k! −∞ k=1 i∞ x ν+2t Γ(−t) 1 J ν (x) = dt [Re ν > 0, x > 0] 2πi −i∞ Γ(ν + t + 1) 2 z ν (1+,−1−) Γ 12 − ν 2 ν− 12 1 2 t −1 J ν (z) = cos(zt) dt 2πi Γ 2 A 1 J ν (z) = 2πi
(0+)
−ν−1
WH, WA 195(2)
WA 195(1)
WA 195(1)
WA 214(7)
ν = 12 , 32 , . . . ; The pointA falls to the right of the point t = 1, and arg(t − 1) = arg(t + 1) = 0 at the point A
6.8
J ν (z) =
1 2π
WH π+∞i
e−iz sin θ+iνθ dθ
[Re z > 0]
−π+∞i
The path of integration being taken around the semi-infinite strip y ≥ 0, −π ≤ x ≤ π.
∞ Jν z2 + ζ 2 1 8 8.413 = eζ cos t cos (z sin t − νt) dt ν π(z + ζ)ν (z 2 − ζ 2 ) 2 0 ∞
− sin νπ
exp (−z sinh t − ζ cosh t − νt) dt
0
1 1 − 2 +i∞ Γ(−t) 2t J 0 (t) dt = x dt 8.414 t 4π − 12 −i∞ t Γ(1 + t) 2x See 3.715 2, 9, 10, 13, 14, 19–21, 3.865 1, 2, 4, 3.996 4.
[Re(z + ζ) > 0]
MO 40
[x > 0]
MO 41
∞
914
Bessel Functions and Functions Associated with Them
8.415
• For an integral representation of J 0 (z), see 3.714 2, 3.753 2, 3, and 4.124. • For an integral representation of J 1 (z), see 3.697, 3.711, 3.752 2, and 3.753 5. 8.415 1.
Y 0 (x) =
1 0
arcsin t 4 √ sin(xt) dt − 2 π 1 − t2
∞ 1
√ ln t + t2 − 1 √ sin(xt) dt t2 − 1 [x > 0]
x −ν ∞ cos xt Y ν (x) = −2 1 2 1 1 dt Γ 2 − ν Γ 2 1 (t2 − 1)ν+ 2
2.
Y ν (x) = −
3. 4.
4 π2
8
5.
6.
1 Y ν (z) = π
2 π
− 12 < Re ν < 12 ,
x>0
KU 89(28)a, MO 38
0
MO 37
∞
cos x cosh t −
0 π
νπ 2
cosh νt dt
1 sin (z sin θ − νθ) dθ − π
∞
[−1 < Re ν < 1,
x > 0]
WA 199(13)
eνt + e−νt cos νπ e−z sinh t dt
0
[Re z > 0] WA 197(1) & % z ν π/2 ∞ 2 2 2ν Y ν (z) = sin (z sin θ) cos θ dθ − e−z sinh θ cosh2ν θ dθ Γ ν + 12 Γ 12 0 0 Re ν > − 12 , Re z > 0 WA 181(5)a 1 π2 cosν− 2 θ cos z − νθ + 12 θ −2z cot θ 2ν+1 z ν Y ν (z) = − dθ e Γ ν + 12 Γ 12 0 sin2ν+1 θ , + π |arg z| < , Re ν + 12 > 0 2 WA 186(8)
For an integral representation of Y 0 (z), see 3.714 3, 3.753 4, 3.864. See also 3.865 3.
8.42 Integral representations of the functions Hν(1) (z) and Hν(2) (z) 8.421 1.
2.
νπi ∞ e− 2 eix cosh t−νt dt πi −∞ νπi ∞ 2e− 2 = eix cosh t cosh νt dt πi 0
H (1) ν (x) =
[−1 < Re ν < 1,
x > 0]
WA 199(10)
[−1 < Re ν < 1,
x > 0]
WA 199(11)
∞ e e−ix cosh t−νt dt πi −∞ νπi ∞ 2e 2 =− e−ix cosh t cosh νt dt πi 0
H (2) ν (x) = −
νπi 2
(1)
8.422
3.
4.
5.
6. 7.
8.
9.
10.
11. 8.422
H (1) ν (z)
2ν+1 iz ν =− Γ ν + 12 Γ 12
H (2) ν (z) =
2ν+1 iz ν Γ ν + 12 Γ 12
2.
π/2 0
π/2
915
t 1 cosν− 2 t ei(z−νt+ 2 ) exp (−2z cot t) dt sin2ν+1 t Re ν > − 12 , Re z > 0
cos
te−i( sin2ν+1 t
ν− 12
0
z−νt+ 2t
WA 186(5)
) exp (−2z cot t) dt
Re ν > − 12 ,
Re z > 0
WA 186(6)
−ν ∞ 1 2i x2 eixt (1) 1 1 H ν (x) = − √ − < Re ν < , x > 0 WA 87(1) 1 dt 2 2 π Γ 2 − ν 1 (t2 − 1)ν+ 2 −ν ∞ 1 2i x2 e−ixt (2) 1 1 H ν (x) = √ − < Re ν < , x > 0 WA 187(2) 1 dt 2 2 π Γ 2 − ν 1 (t2 − 1)ν+ 2
1 1 i − 1 iνπ ∞ 2 e iz t + H (1) (z) = − exp t−ν−1 dt ν π 2 t 0 [0 < arg z < π; or arg z = 0 and − 1 < Re ν < 1] MO 38
1 z2 i − 1 iνπ ν ∞ 2 e ix t + H (1) (xz) = − z exp t−ν−1 dt ν π 2 t 0 , + π π 0 < arg z < , x > 0, Re ν > −1; or arg z = , x > 0 and − 1 < Re ν < 1 2 2 , + xν exp i xz − π ν − π ∞
ν− 12 1 2 it (1) 2 4 H ν (xz) = tν− 2 e−xt dt 1+ 1 πz 2z Γ ν+2 0 Re ν > − 12 , − 12 π < arg z < 32 π, x > 0 z ν ∞ −2ie−iνπ 2 H (1) (z) = eiz cosh t sinh2ν t dt √ ν π Γ ν + 12 0 0 < arg z < π, Re ν > − 12 or arg z = 0 and − (1) H 0 (x)
i =− π
√ exp i x2 + t2 √ dt x2 + t2 −∞
1 2
< Re ν
0]
z ν (1+) − ν ν− 12 2 (1) izt 2 2 H ν (z) = t e − 1 dt [−π < arg z < 2π] πi Γ 12 1+∞i z ν (−1−) Γ 12 − ν ν− 12 (2) 2 H ν (z) = eizt t2 − 1 dt 1 πi Γ 2 −1+∞i [−2π < arg z < π] Γ
1.
(2)
Integral representations of the functions Hν (z) and Hν (z)
MO 38
1
The paths of integration are shown in the drawing.
WA 183(4)
916
Bessel Functions and Functions Associated with Them
8.423 1. 2.
H (1) ν (z)
1 =− π
H (2) ν (z)
1 =− π
−π+∞i
e−iz sin θ+iνθ dθ
8.423
[Re z > 0]
WA 197(2)a
[Re z > 0]
WA 197(3)a
−∞i −∞i
e−iz sin θ+iνθ dθ
π+∞i
The path of integration for 8.423 1 is shown in the left-hand drawing and for 8.423 2 in the right-hand drawing.
8.424 1.
H (1) ν (z) J ν (ζ) =
1 πi
γ+i∞
exp 0
1 zζ dt z2 + ζ 2 t− Iν 2 t t t
[γ > 0,
1 zζ dt i γ−i∞ z2 + ζ 2 H (2) (z) J (ζ) = exp t − I ν ν ν π 0 2 t t t
Re ν > −1,
|ζ| < |z|]
MO 45
[γ > 0,
Re ν > −1,
|ζ| < |z|]
MO 45
2.
8.43 Integral representations of the functions Iν (z) and Kν (z) The function I ν (z) 8.431 1.
I ν (z) =
2.
I ν (z) =
3.
I ν (z) =
4.
I ν (z) =
z ν 1 ν− 12 ±zt 2 1 − t2 e dt 1 1 Γ ν + 2 Γ 2 −1 z ν 1 ν− 12 2 1 − t2 cosh zt dt 1 1 Γ ν + 2 Γ 2 −1 z ν π 2 e±z cos θ sin2ν θ dθ Γ ν + 12 Γ 12 0 z ν π 2 cosh (z cos θ) sin2ν θ dθ Γ ν + 12 Γ 12 0
Re ν + 12 > 0
WA 94(9)
Re ν + 12 > 0
WA 94(9)
Re ν + 12 > 0
WA 94(9)
Re ν + 12 > 0
WA 94(9)
8.432
5.
Integral representations of the functions Iν (z) and Kν (z)
1 I ν (z) = π
π
e
z cos θ
0
sin νπ cos νθ dθ − π
∞
0
e−z cosh t−νt dt + π |arg z| ≤ , 2
917
, Re ν > 0
WA 201(4)
See also 3.383 2, 3.387 1, 3.471 6, 3.714 5. For an integral representation of I 0 (z) and I 1 (z), see 3.366 1, 3.534 3.856 6. The function K ν (z) 8.432 1.
∞
K ν (z) =
, + π |arg z| < or Re z = 0 and ν = 0 2
e−z cosh t cosh νt dt
0
2.
3.
4. 5.
z ν 1 ∞ Γ 2 K ν (z) = 2 e−z cosh t sinh2ν t dt Γ ν + 12 0 Re ν > − 12 , Re z > 0; or Re z = 0 and − z ν 1 ∞ ν− 12 Γ 2 K ν (z) = 2 e−zt t2 − 1 dt 1 Γ ν+2 1 + Re ν + 12 > 0,
|arg z|
0,
|arg z|
− 12 , x > 0
MO 39
√ x ν ∞ exp −x√t2 + z 2 π √ K ν (xz) = t2ν dt 2 + z2 2z Γ ν + 12 t 0 + Re ν > − 12 ,
Re z 2 > 0
Re
t2 + z 2 > 0,
, x>0
MO 39
918
Bessel Functions and Functions Associated with Them
8.433
See also 3.383 3, 3.387 3, 6, 3.388 2, 3.389 4, 3.391, 3.395 1, 3.471 9, 3.483, 3.547 2, 3.856, 3.871 3, 4, 7.141 5. √
∞ 2x x 3 √ 8.433 K 13 cos t3 + xt dt KU 98(31), WA 211(2) =√ x 0 3 3 For an integral representation of K 0 (z), see 3.754 2, 3.864, 4.343, 4.356, 4.367.
8.44 Series representation The function J ν (z) 8.440 8.441 1.
J ν (z) =
∞ z ν
2
k=0
[|arg z| < π]
WH 358 a
Special cases: J 0 (z) =
∞
(−1)k
k=0
2.
z 2k (−1)k k! Γ(ν + k + 1) 2
z 2k 22k (k!)
J 1 (z) = − J 0 (z) =
2
∞
z (−1)k z 2k 2 22k k!(k + 1)! k=0
3.
4.
J 13 (z) =
Γ
1 4 3
3
z 2
∞ k=0
√ 2k z 3 (−1) 2k 2 k! · 1 · 4 · 7 · · · · · (3k + 1) k
√ 2k ∞ z 3 1 3 2 1+ J − 13 (z) = 2 (−1)k 2k z 2 k! · 2 · 5 · 8 · · · · · (3k − 1) Γ 3 k=1
For the expansion of J ν (z) in Laguerre polynomials, see 8.975 3. 8.442 μ+ν+2m ∞ (−1)m 12 z (μ + ν + m + 1)m 7 J ν (z) J μ (z) = 1. m! Γ(μ + m + 1) Γ(ν + m + 1) m=0
2k az ν bz μ b2 k az (−1) F −k, −ν − k; μ − 1; ∞ 2 2 2 a2 2.8 J ν (az) J μ (bz) = Γ(μ + 1) k! Γ(ν + k + 1) k=0
The function Y ν (z)
∞ z ν z 2k 1 (−1)k 2k 8.44311 Y ν (z) = cos νπ sin νπ 2 2 k! Γ (ν + k + 1) k=0 ∞ z −ν z 2k k − (−1) 2k 2 2 k! Γ(k − ν + 1) k=0 [ν = an integer] (cf. 8.403 1) For ν + 1 a natural number, see 8.403 2.; for ν a negative integer, see 8.404 1
MO 28
8.447
8.444 1.
2.11
Series representation
919
Special cases, ∞ k z (−1)k z 2k 1 π Y 0 (z) = 2 J 0 (z) ln + C − 2 2 2 2 m (k!) m=1 k=1 z 2k−1 ∞ (−1)k+1 k−1 2 z z 1 1 2 +2 π Y 1 (z) = 2 J 1 (z) ln + C − − − 2 z 2 k!(k − 1)! k m m=1
KU 44
k=2
The functions I ν (z) and K n (z) 8.445 I ν (z) =
∞
z ν+2k 1 k! Γ(ν + k + 1) 2
k=0 n−1
8.4468 K n (z) =
1 2
(−1)k
k=0
n+1
+(−1)
WH 372a
(n − k − 1)! n−2k k! z2
∞ k=0
z n+2k
1 1 z ln − ψ(k + 1) − ψ(n + k + 1) k!(n + k)! 2 2 2 2
l
n+l ∞ z n+2l 1 1 1 1 n+1 n 2 + = (−1) I n (z) ln z + C + (−1) 2 2 l!(n + l)! k k
l=0
+
k=1
WA 95(15)
k=1
n−1 1 (−1)l (n − l − 1)! z 2l−n 2 l! 2 l=0
[n + 1 is a natural number] MO 29
8.447 1.
Special cases: I 0 (z) =
∞
I 1 (z) =
2 2
k=0
2.
z 2k (k!)
I 0 (z)
=
∞ k=0
z 2k+1 2
k!(k + 1)! ∞
3.
K 0 (z) = − ln
z 2k z I 0 (z) + 2 ψ(k + 1) 2 22k (k!) k=0
WA 95(14)
920
Bessel Functions and Functions Associated with Them
8.451
8.45 Asymptotic expansions of Bessel functions 8.451 1.
For large values of |z| ∗ & %n−1 Γ ν + 2k + 12 π (−1)k 2 π + R1 J ±ν (z) = cos z ∓ ν − πz 2 4 (2z)2k (2k)! Γ ν − 2k + 12 k=0 & %n−1 Γ ν + 2k + 32 π (−1)k π + R2 − sin z ∓ ν − 2 4 (2z)2k+1 (2k + 1)! Γ ν − 2k − 12 k=0
[|arg z| < π]
2.
WA 222(1, 3)
& %n−1 Γ ν + 2k + 12 π (−1)k π + R1 sin z ∓ ν − Y ±ν (z) = 2 4 (2z)2k (2k)! Γ ν − 2k + 12 k=0 & %n−1 Γ ν + 2k + 32 π (−1)k π + R2 + cos z ∓ ν − 2 4 (2z)2k+1 (2k + 1)! Γ ν − 2k − 12
11
(see 8.339 4)
2 πz
k=0
[|arg z| < π]
(see 8.339 4)
WA 222(2, 4, 5)
%n−1 & 2 i(z− π2 ν− π4 ) (−1)k Γ ν + k + 12 (−1)n Γ ν + n + 12 + θ1 e πz (2iz)k k! Γ ν − k + 12 (2iz)n k! Γ ν − n + 12 k=0 (see 8.339 4) WA 221(5) Re ν > − 12 , |arg z| < π
%n−1 & Γ ν + k + 12 Γ ν + n + 12 2 −i(z− π2 ν− π4 ) 1 1 + θ2 e πz (2iz)k k! Γ ν − k + 12 (2iz)n k! Γ ν − n + 12 k=0 (see 8.339 4) WA 221(6) Re ν > − 12 , |arg z| < π
11
H (1) ν (z)
4.11
H (2) ν (z)
5.
For indices of the form ν = 2n−1 (where n is a natural number), the series 8.451 terminate. In 2 this case, the closed formulas 8.46 are valid for all values. ∞ ez (−1)k Γ ν + k + 12 I ν (z) ∼ √ 2πz k=0 (2z)k k! Γ ν − k + 12 ∞ exp −z ± ν + 12 πi 1 Γ ν + k + 12 √ + (2z)k k! Γ ν − k + 12 2πz k=0
3.
=
=
[The + sign is taken for − 12 π < arg z < 32 π, the − sign for − 32 π < arg z < 12 π]∗ 6.11
K ν (z) =
π −z e 2z
%n−1 k=0
& Γ ν+k+ 2 Γ ν + n + 12 1 + θ3 (2z)k k! Γ ν − k + 12 (2z)n n! Γ ν − n + 12
1
(see 8.339 4)
(see 8.339 4)] WA 226(2,3)
WA 231, 245(9)
An estimate of the remainders of the asymptotic series in formulas 8.451: ∗ An
estimate of the remainders in formulas 8.451 is given in 8.451 7 and 8.451 8. contradiction that this condition contains at first glance is explained by the so-called Stokes phenomenon (see Watson, G.N., A Treatise on the Theory of Bessel Functions, 2nd Edition, Cambridge Univ. Press, 1944, page 201). ∗ The
8.452
7.
8.
Asymptotic expansions of Bessel functions
! ! ! ! Γ ν + 2n + 12 ! !! |R1 | < ! 1 ! (2z)2n (2n)! Γ ν − 2n + 2 ! ! ! ! ! Γ ν + 2n + 32 ! !! |R2 | < ! 1 ! (2z)2n+1 (2n + 1)! Γ ν − 2n − 2 ! For −
3 π < arg z < π, ν real, and n + 2 2
1 2
1 2
ν 3 n≥ − 2 4
> |ν|
1, |θ1 | < |sec (arg z)|, 3 π For − π < arg z < , ν real, and n + 2 2
ν 1 n> − 2 4
WA 231
WA 231
WA 245
if Im z ≥ 0 if Im z ≤ 0
> |ν|
1, |θ2 | < |sec (arg z)|,
921
WA 246
if Im z ≤ 0 if Im z ≥ 0
For ν real,
WA 245
1 if Re z ≥ 0 |θ3 | < |cosec (arg z)|, if Re z < 0 Re θ3 ≥ 0,
if Re z ≥ 0
For ν and z real and n ≥ ν − 12 ,
WA 231
0 ≤ |θ3 | ≤ 1 In particular, it follows from 8.451 7 and 8.451 8 that for real positive values of z and ν, the errors |R1 | and |R2 | are less than the absolute value of the first discarded term. For values of |arg z| close to π, the series 8.451 1 and 8.451 2 may not be suitable for calculations. In particular, the error for |arg z| > π can be greater in absolute value than the first discarded term. “Approximation by tangents” 8.45211 For large values of the index (where the argument is less than the index). Suppose that x > 0 and ν > 0. Let us set ν/x = cosh α. Then, for large values of ν, the following expansions are valid:
ν exp (ν tanh α − να) 5 1 1 √ ∼ coth α − coth3 α 1. Jν 1+ cosh α ν 8 24 2νπ tanh α
9 1 231 1155 2 4 + 2 coth α − coth α + coth6 α + . . . ν 128 576 3456 WA 269(3)
922
Bessel Functions and Functions Associated with Them
2.
Yν
8.453
ν 5 1 1 exp (να − ν tanh α) 3 π coth α − coth α 1− ∼− cosh α ν 8 24 ν tanh α
2 9 1 231 1155 coth2 α − coth4 α + coth6 α + . . . + 2 ν 128 576 3456 WA 270(5)
8.453 For large values of the index (where the argument is greater than the index). Suppose that x > 0 and ν > 0. Let us set ν/x = cos β. Then, for large values of ν, the following expansions are valid: ⎧⎡ ⎛ ⎨ 2 1 ⎝ 9 cot2 β + 231 cot4 β ⎣1 − 1. J ν (ν sec β) ∼ νπ tan β ⎩ ν 2 128 576 ⎞ ⎤ 1155 π + cot6 β ⎠ + . . .⎦ cos ν tan β − νβ − 3456 4 ⎤
1 1 5 π ⎦ cot β + cot3 β − . . . sin ν tan β − νβ − + ν 8 24 4
2.
3.
4.
⎧⎡ ⎛ ⎨ 2 1 ⎝ 9 cot2 β + 231 cot4 β ⎣1 − Y ν (ν sec β) ∼ νπ tan β ⎩ ν 2 128 576 ⎞ ⎤ 1155 π + cot6 β ⎠ + . . .⎦ sin ν tan β − νβ − 3456 4 ⎤ ⎤
5 π ⎦ 1 1 cot β + cot3 β − . . .⎦ cos ν tan β − νβ − − ν 8 24 4
exp νi (tan β − β) − π4 i 5 i 1 (1) 3 π cot β + cot β H ν (ν sec β) ∼ 1− ν 8 24 ν tan β
2 9 1 231 1155 cot2 β + cot4 β + cot6 β + . . . − 2 ν 128 576 3456
exp −νi (tan β − β) + π4 i 5 i 1 (2) 3 cot β + cot β H ν (ν sec β) ∼ 1+ π ν 8 24 2 ν tan β
9 1 231 1155 cot2 β + cot4 β + cot6 β + . . . − 2 ν 128 576 3456
WA 271(4)
WA 271(5)
WA 271(1)
WA 271(2) 1
Formulas 8.453 are not valid when |x − ν| is of a size comparable to x 3 . For arbitrary small (and also large) values of |x − ν|, we may use the following formulas:
8.457
8.454
Asymptotic expansions of Bessel functions
923
Suppose that x > 0 and ν > 0, we set w=
x2 − 1; ν2
Then,
π 1 w3 (1) ν 3 +ν w− − arctan w i H 1 w +O 3 6 3 3 |ν|
1 w3 π w (2) ν 3 (2) − arctan w i H 1 w +O 2. H ν (x) = √ exp − − ν w − MO 34 3 6 3 3 |ν| 3 ! !
√ !1! 1 The absolute value of the error O is then less than 24 2!! !!. |ν| ν 8.455 For x real and ν a natural number (ν = n), if n 1, the following approximations are valid: 3 2 2(n − x) 1 [2(n − x)] √ 1.7 K 13 J n (x) ≈ π 3x 3 x [n > x] (see also 8.433) 1.
w H (1) ν (x) = √ exp 3
1 2 ≈ e 3 πi 2
2 (n − x) (1) H1 3 3x
3
i [2(n − x)] 2 √ 3 x
WA 276(1)
[n > x] 1 ≈√ 3
2(x − n) 3x
%
J 13
3
{2(x − n)} 2 √ 3 x
&
% + J − 13
3
{2(x − n)} 2 √ 3 x
&
MO 34
(see also 8.441 3, 8.441 4) WA 276(2)
2.
Y n (x) ≈
2(x − n) 3x
J − 13
%
3
{2(x − n)} 2 √ 3 x
&
% − J 13
3
{2(x − n)} 2 √ 3 x
&
[x > n] An estimate of the error in formulas 8.455has not yet been achieved. ∞ 1 Γ ν + k + (2k − 1)!! 2 2 8.45611 J 2ν (z) + Y 2ν (z) ≈ [|arg z| < π] πz 2k z 2k k! Γ ν − k + 12
WA 276(3)
(see also 8.479 1)
k=0
WA 250(5)
8.457
2 J 2ν (x) + J 2ν+1 (x) ≈ πx
[x |ν|]
WA 223
924
Bessel Functions and Functions Associated with Them
8.461
8.46 Bessel functions of order equal to an integer plus one-half The function J ν (z) 8.461 1.11
J n+ 12 (z) =
⎧ ⎪
n2 π (−1)k (n + 2k)! 2 ⎨ sin z − n (2z)−2k πz ⎪ 2 (2k)!(n − 2k)! ⎩ k=0
⎫ n−1 ⎪
2 ⎬ k (−1) (n + 2k + 1)! π −(2k+1) (2z) + cos z − n ⎪ 2 (2k + 1)!(n − 2k − 1)! ⎭ k=0
[n + 1 is a natural number] 2.
J −n− 12 (z) =
(cf. 8.451 1)
KU 59(6), WA 66(2)
⎧ ⎪
n2 π (−1)k (n + 2k)! 2 ⎨ cos z + n πz ⎪ 2 (2k)!(n − 2k)!(2z)2k ⎩ k=0
⎫ n−1 ⎪
2 ⎬ k (−1) (n + 2k + 1)! π − sin z + n 2 (2k + 1)!(n − 2k − 1)!(2z)2k+1 ⎪ ⎭
k=0
[n + 1 is a natural number] 8.462 1.
2.
1 J n+ 12 (z) = √ 2πz
eiz
n i−n+k−1 (n + k)! k=0
1 J −n− 12 (z) = √ 2πz
k!(n − k)!(2z)k
+ e−iz
(cf. 8.451 1)
n (−i)−n+k−1 (n + k)! k=0
k!(n − k)!(2z)k [n + 1 is a natural number]
eiz
KU 58(7), WA 67(5)
KU 59(6), WA 66(1)
n n in+k (n + k)! (−i)n+k (n + k)! −iz + e k!(n − k)!(2z)k k!(n − k)!(2z)k k=0 k=0 [n + 1 is a natural number]
KU 59(7), WA 67(4)
8.463 1.
n n+ 12
J n+ 12 (z) = (−1) z
2. 8.464 1. 2.
J −n− 12 (z) = z
n+ 12
sin z 2 dn π (z dz)n z n cos z d
2 π (z dz)n
Special cases: 2 sin z J 12 (z) = πz 2 cos z J − 12 (z) = πz
z
KU 58(4)
KU 58(5)
DW DW
8.469
Bessel functions of order equal to an integer plus one-half
925
3. 4. 5.8 6.
2 sin z − cos z πz z 2 cos z − sin z − J − 32 (z) = πz z
3 2 3 J 52 (z) = − 1 sin z − cos z πz z2 z
3 2 3 sin z + J − 52 (z) = − 1 cos z πz z z2 J 32 (z) =
DW
DW DW
DW
The function Y n+ 12 (z) 8.465 1.
Y n+ 12 (z) = (−1)n−1 J −n− 12 (z)
JA
2.
Y −n− 12 (z) = (−1)n J n+ 12 (z)
JA
(1,2)
The functions H n+ 1 (z), I n+ 12 (z), K n+ 12 (z) 2
8.466 1.
(1) H n− 1 (z) 2
=
1 2 −n iz (n + k − 1)! i e (−1)k πz k!(n − k − 1)! (2iz)k n−1
k=0
(cf. 8.451 3)
2.
n−1
1 2 n −iz (n + k − 1)! i e (cf. 8.451 4) πz k!(n − k − 1)! (2iz)k k=0 % & n n (−1)k (n + k)! (n + k)! 1 z n+1 −z e ± (−1) e I ±(n+ 1 ) (z) = √ 2 k!(n − k)!(2z)k k!(n − k)!(2z)k 2πz (2)
H n− 1 (z) = 2
8.467
k=0
8.468 K n+ 12 (z) = 8.469 1. 2. 3. 4.
k=0
(cf. 8.451 5) π −z e 2z
n k=0
(n + k)! k!(n − k)!(2z)k
(cf. 8.451 6)
KU 60a KU 60
Special cases:
2 Y 12 (z) = − cos z πz 2 Y − 12 (z) = sin z πz π −z e K ± 12 (z) = 2z 2 eiz (1) H 1 (z) = 2 πz i
WA 95(13)
MO 27
926
Bessel Functions and Functions Associated with Them
8.471
5. 6. 7.
2 e−iz 2 πz −i 2 iz (1) e H − 1 (z) = 2 πz 2 −iz (2) e H − 1 (z) = 2 πz (2)
H 1 (z) =
MO 27 MO 27 MO 27
8.47–8.48 Functional relations 8.4718 1.
Recursion formulas: z Z ν−1 (z) + z Z ν+1 (z) = 2ν Z ν (z)
KU 56(13), WA 56(1), WA 79(1), WA 88(3)
d Z ν (z) KU 56(12), WA 56(2), WA 79(2), We 88(4) dz Sonin and Nielsen, in their construction of the theory of Bessel functions, defined Bessel functions as analytic functions of z that satisfy the recursion relations 8.471. Z denotes J, N , H (1) , H (2) or any linear combination of these functions, the coefficients of which are independent of z and ν. 8.472 Consequences of the recursion formulas for Z defined as above: d 1. z Z ν (z) + ν Z ν (z) = z Z ν−1 (z) KU 56(11), WA 56(3), WA 79(3), WA 88(5) dz d 2. z Z ν (z) − ν Z ν (z) = −z Z ν+1 (z) KU 56(10), WA 56(4), WA 79(4), WA 88(6) dz
m d 3. (z ν Z ν (z)) = z ν−m Z ν−m (z) KU 56(8), WA 57(5), WA 89(9) z dz
m −ν d z Z ν (z) = (−1)m z −ν−m Z ν+m (z) 4. WA 89(10), Ku 55(5), WA 57(6) z dz 2.
Z ν−1 (z) − Z ν+1 (z) = 2
5.
Z −n (z) = (−1)n Z n (z)
8.473 1. 2. 3. 4. 5. 6.
[n is a natural number]
(cf. 8.404)
Special cases: 2 J 2 (z) = J 1 (z) − J 0 (z) z 2 Y 2 (z) = Y 1 (z) − Y 0 (z) z 2 (1,2) (1,2) (1,2) H 2 (z) = H 1 (z) − H 0 (z) z d J 0 (z) = − J 1 (z) dz d Y 0 (z) = − Y 1 (z) dz d (1,2) (1,2) H (z) = − H 1 (z) dz 0
8.4748 Each of the pairs of functions J ν (z) and J −ν (z) (for ν = 0, ±1, ±2,. . . ), J ν (z) and Y ν (z), and (2) H (1) ν (z) and H ν (z), which are solutions of equation 8.401, and also the pair I ν (z) and K ν (z) is a pair of linearly independent functions. The Wronskians of these pairs are, respectively,
8.478
Functional relations
927
2 4i 1 2 sin νπ, , − , − KU 52(10, 11, 12), WA 90(1, 4) πz πz πz z 8.4756 The functions J ν (z), and Y ν (z), H (1,2) (z), I ν (z), K ν (z), with the exception of J n (z) and I n (z), ν for n an integer are non-single-valued : z = 0 is a branch point for these functions. The branches of these functions that lie on opposite sides of the cut (−∞, 0) are connected by the relations 8.476 WA 90(1) 1. J ν emπi z = emνπi J ν (z) mπi −mνπi 2. Yν e z =e Y ν (z) + 2i sin mνπ cot νπ J ν (z) WA 90(3) mπi 3. Y −ν e z = e−mνπi Y −ν (z) + 2i sin mνπ cosec νπ J ν (z) WA 90(4) mπi mνπi 4. Iν e z =e I ν (z) WA 95(17) sin mνπ I ν (z) 5. K ν emπi z = e−mνπi K ν (z) − iπ [ν not an integer] WA 95(18) sin νπ mπi −νπi sin mνπ J ν (z) e 6. H (1) z = e−mνπi H (1) ν ν (z) − 2e sin νπ sin mνπ (2) sin(1 − m)νπ (1) H ν (z) − e−νπi H ν (z) = sin νπ sin νπ −
WA 95(5)
9.
mπi νπi sin mνπ J ν (z) e H (2) z = e−mνπi H (2) ν ν (z) + 2e sin νπ sin mνπ (1) sin(1 + m)νπ (2) H ν (z) + eνπi H ν (z) = sin νπ sin νπ [m an integer] (2) eiπ z = − H −ν (z) = −e−iπν H (2) H (1) ν ν (z) (1) e−iπ z = − H −ν (z) = −eiπν H (1) H (2) ν ν (z)
10.8
H (2) ν (z) = H ν (z)
7.
8.
(1)
WA 90(6) MO 26 MO 26 MO 26
8.477 1.
J ν (z) Y ν+1 (z) − J ν+1 (z) Y ν (z) = −
2.
I ν (z) K ν+1 (z) + I ν+1 (z) K ν (z) =
2 πz
WA 91(12)
1 z
WA 95(20)
See also 3.863. • For a connection with Legendre functions, see 8.722. • For a connection with the polynomials C λn (t), see 8.936 4. • For a connection with a confluent hypergeometric function, see 9.235. 8.478
For ν > 0 and x > 0, the product x J 2ν (x) + Y 2ν (x) ,
considered as a function of x, decreases monotonically, if ν >
1 2
and increases monotonically if 0 < ν < 12 . MO 35
928
8.479 1.11 2.
Bessel Functions and Functions Associated with Them
1 1 π 2 J ν (x) + Y 2ν (x) ≥ > 2 2 x −ν |J n (nz)| ≤ 1 √
x2
x≥ν≥
1 2
8.479
MO 35
! & %! ! z exp √1 − z 2 ! ! ! √ ! < 1, n a natural number ! ! 1 + 1 − z2 !
MO 35
Relations between Bessel functions of the first, second, and third kinds Y −ν (z) − Y ν (z) cos νπ = H (1) ν (z) − i Y ν (z) sin νπ 1 (2) = H (2) H (1) ν (z) + i Y ν (z) = ν (z) + H ν (z) 2
8.481 J ν (z) =
(cf. 8.403 1, 8.405)
WA 89(1), JA
(cf. 8.403 1, 8.405)
WA 89(3), JA
J ν (z) cos νπ − J −ν (z) = i J ν (z) − i H (1) 8.482 Y ν (z) = ν (z) sin νπ i (1) H (2) = i H (2) ν (z) − i J ν (z) = ν (z) − H ν (z) 2 8.483 1. 2.
Y −ν (z) − e−νπi Y ν (z) J −ν (z) − e−νπi J ν (z) = = J ν (z) + i Y ν (z) i sin νπ sin νπ Y −ν (z) − eνπi Y ν (z) eνπi J ν (z) − J −ν (z) = = J ν (z) − i Y ν (z) H (2) (z) = ν i sin νπ sin νπ (cf. 8.405) H (1) ν (z) =
WA 89(5)
WA 89(6)
8.484 (1)
1.
H −ν (z) = eνπi H (1) ν (z)
2.
(2) H −ν (z)
8.4857
WA 89(7)
= e−νπi H (2) ν (z)
K ν (z) =
π I −ν (z) − I ν (z) 2 sin νπ
WA 89(7)
[ν not an integer]
(see also 8.407) WA 92(6)
8.486
Recursion formulas for the functions I ν (z) and K ν (z) and their consequences:
1.
z I ν−1 (z) − z I ν+1 (z) = 2ν I ν (z)
2.
I ν−1 (z) + I ν+1 (z) = 2
3.
z
4. 5.
d I ν (z) dz
d I ν (z) + ν I ν (z) = z I ν−1 (z) dz d z I ν (z) − ν I ν (z) = z I ν+1 (z) dz
m d {z ν I ν (z)} = z ν−m I ν−m (z) z dz
WA 93(1) WA 93(2) WA 93(3) WA 93(4) WA 93(5)
8.486(1)
Functional relations
6. 7.
d z dz
m
z −ν I ν (z) = z −ν−m I ν+m (z)
I −n (z) = ln (z)
10. 11.
K ν−1 (z) + K ν+1 (z) = −2
12.
z
9.
13. 14. 15.
K −ν (z) = K ν (z)
17.
K 2 (z) =
19.
WA 93(8)
WA 93(7) WA 93(1)
d K ν (z) dz
WA 93(2)
d K ν (z) + ν K ν (z) = −z K ν−1 (z) dz d z K ν (z) − ν K ν (z) = −z K ν+1 (z) dz
m d {z ν K ν (z)} = (−1)m z ν−m K ν−m (z) z dz
m −ν d z K ν (z) = (−1)m z −ν−m K ν+m (z) z dz
16.
18.
WA 93(6)
[n a natural number]
2 I 2 (z) = − l1 (z) + I 0 (z) z d I 0 (z) = I 1 (z) dz z K ν−1 (z) − z K ν+1 (z) = −2ν K ν (z)
8.
929
WA 93(3) WA 93(4) WA 93(5) WA 93(6) WA 93(8)
2 K 1 (z) + K 0 (z) z
d K 0 (z) = − K 1 (z) dz ∞ , ∂ J ν (z) + z (z/2)ν+1 (z/2)n J n+1 (z) = ln − ψ(ν + 1) J ν (z) + ∂ν 2 Γ(ν + 1) n=0 n!(ν + n + 1)2
WA 93(7) LUKE 360
8.486(1)7 1.
2.
3.
4.
Differentiation with respect to order ν+2k ∞ 1 1 ∂ J ν (z) ψ(ν + k + 1) k = J ν (z) ln z − z (−1) ∂ν 2 2 k! Γ(ν + k + 1) k=0 ν = n or n + 12 , −ν+2k ∞ 1 1 ψ(−ν + k + 1) ∂ J −ν (z) k = − J −ν (z) ln z + z (−1) ∂ν 2 2 k! Γ(−ν + k + 1) k=0 ν = n or n + 12 ,
n integer
MS 3.1.3
n integer
MS 3.1.3
∂ J ν (z) ∂ J −ν (z) ∂ Y ν (z) = cot πν − cosec πν − π cosec πν Y ν (z) ∂ν ∂ν ∂ν ν = n or n + 12 , n integer MS 3.1.3
ν+2k ∞ 1 1 ψ(ν + k + 1) ∂ I ν (z) = I ν (z) ln z − z ν = n or n + 12 , n integer ∂ν 2 2 k! Γ(ν + k + 1) k=0
MS 3.1.3
930
5.
6.
7.
Bessel Functions and Functions Associated with Them
8.486(1)
∂ I −ν (z) ∂ I ν (z) ∂ K ν (z) 1 = −π cot πν K ν (z) + π cosec πν − ∂ν 2 ∂ν ∂ν ν = n or n + 12 , n integer MS 3.1.3 n−1 1 z k−n J k (z) ∂ J ν (z) 1 n n 1 2 [n = 0, 1, . . .] = π (±1) Y n (z)±(±1) n! MS 3.2.3 ∂ν 2 2 k!(n − k) ν=±n k=0 n−1 1 z k−n Y k (z) ∂ Y ν (z) 1 n n 1 2 [n = 0, 1, . . .] = − π (±1) J n (z) ± (±1) n! ∂ν 2 2 k!(n − k) ν=±n k=0
MS 3.2.3
8.
∂ I ν (z) ∂ν
ν=±n
n−1 (−1)k 1 z k−n I k (z) n+1 n1 2 = (−1) K n (z) ± (−1) n! 2 k!(n − k)
[n = 0, 1, . . .]
k=0
MS 3.2.3
9.
∂ K ν (z) ∂ν
10.
(−1)n
11.11
12. 13. 14. 15. 16. 17. 18. 19.
ν=±n
1 = ± n! 2
∂ I ν (z) ∂ν ν=n
∂ K ν (z) ∂ν
= ν=n
1 n! 2
n−1 1 z k−n K k (z) 2
k=0
[n = 0, 1, . . .]
k!(n − k)
k−n 1 z I k (z) n−1 (−1) 1 2 = − K n (z) + n! 2 k!(n − k)
k
k=0
n−1 k=0
1 k−n K k (z) 2z k!(n − k)
[n = 0, 1, . . .]
AS 9.6.44
[n = 0, 1, . . .]
AS 9.6.45
Special cases ∂ J ν (z) = 12 π Y 0 (z) ∂ν ν=0 ∂ Y ν (z) = − 12 π J 0 (z) ∂ν ν=0 ∂ I ν (z) = − K 0 (z) ∂ν ν=0 ∂ K ν (z) =0 ∂ν ν=0 −1/2 ∂ J ν (x) = 12 πx [sin x Ci(3x) − cos x Si(2x)] ∂ν ν= 12 −1/2 ∂ J ν (x) = 12 πx [cos x Ci(2x) + sin x Si(2x)] ∂ν ν=− 12 −1/2 ∂ Y ν (x) = 12 πx {cos x Ci(2x) + sin x [Si(2x) − π]} ∂ν ν= 12 −1/2 ∂ Y ν (x) = − 12 πx {sin x Ci(2x) − cos x [Si(2x) − π]} ∂ν ν=− 1 2
MS 3.2.3
MS 3.2.3 MS 3.2.3 MS 3.2.3 MS 3.2.3 MS 3.3.3
MS 3.3.3
MS 3.3.3
MS 3.3.3
8.491
Differential equations leading to Bessel functions
20. 21. 8.487 1. 2. 3.
∂ I ν (x) ∂ν
∂ K ν (x) ∂ν
931
= (2πx)−1/2 ex Ei(−2x) ∓ e−x Ei(2x)
ν=± 12
=∓ ν=± 12
MS 3.3.3
π 12 ex Ei(−2x) 2x
MS 3.3.3
Continuity with respect to the order∗ : lim Y ν (z) = Y n (z)
[n an integer]
WA 76
lim H (1,2) (z) = H (1,2) (z) ν n
[n an integer]
WA 183
lim K ν (z) = K n (z)
[n an integer]
WA 92
ν→n ν→n ν→n
8.49 Differential equations leading to Bessel functions See also 8.401 8.491
1 d ν2 (zu ) + β 2 − 2 u = 0 1. z dz z νγ 2 1 d γ−1 2 (zu ) + βγz 2. − u=0 z dz z 2 α 2 − ν 2 γ 2 1 − 2α u + βγz γ−1 + 3. u + u=0 z z2 4ν 2 γ 2 − 1 γ−1 2 4. u + βγz − u=0 4z 2
4ν 2 − 1 5. u + β 2 − u=0 4z 2
1 − 2α α2 − ν 2 2 u + β + 6. u + u=0 z z2 7.
8. 9.
u + bz u = 0
1−ν 1u u + =0 z 4z
u +
11.
u + β 2 γ 2 z 2β−2 u = 0
∗ The
2
1 ν u + u + 4 z 2 − 2 u = 0 z z
1 1 ν2 u + u + 1− u=0 z 4z z
10.
JA
u = Z ν (βz γ )
JA
u = z α Z ν (βz γ )
JA
u=
√ z Z ν (βz γ )
JA
u=
√ z Z ν (βz)
JA
u = z α Z ν (βz)
JA
√ 1 u = z Z m+2
m
u = Z ν (βz)
√ 2 b m+2 z 2 m+2 JA 111(5)
u = Z ν z2 u = Zν
√ z
√ z β 1 γz u = z 1/2 Z 2β ν
u = z2 Zν
WA 111(6) WA 111(7) WA 111(9)a WA 110(3)
continuity of the functions J ν (z) and I ν (z) follows directly from the series representations of these functions.
932
12.
8.492 1. 2. 8.493 1. 2. 8.494 1. 2. 3. 4.10 5. 6.
Bessel Functions and Functions Associated with Them
z 2 u + (2α − 2βν + 1)zu + β 2 γ 2 z 2β + α(α − 2βν) u = 0
u = z βν−α Z ν γz β
u + e2z − ν 2 u = 0 u +
e
u = Z ν (ez ) u = z Z ν e1/z
−ν u=0 z4
2/z
2
2
1 ν tan z − 2 tan z u − + u=0 z z2 z
2
1 ν cot z + 2 cot z u − u + − u=0 z z2 z u +
1 ν2 u + u − 1 + 2 u = 0 z z ν 2 1 1 + u + u − u=0 z z 2z
1 1 − ν2 u = 0 u + u + 2 z 4
2ν + 1 2ν + 1 − k u − ku = 0 u + z 2z 1−ν 1u u − =0 z 4z u u ± √ = 0 z
u ± zu = 0
8.
ν(ν + 1) u − c + z2
9.
u −
10.
u − c2 z 2ν−2 u = 0
1.
2
u = cosec z Z ν (z)
JA
u = Z ν (iz) = C1 I ν (z) + C2 K ν (z)
JA
√ u = Z ν 2i z
JA
3 √ z Z 23 43 z 4 , √
z Z 13
2 32 3z
u=0
2ν u − c2 u = 0 z
√ −z ze 2 Z ν
,
u=
JA JA WA 111(8)
u=
√ 3 z Z 23 43 iz 4
WA 111(10)
u=
√ 3 z Z 13 23 iz 2
WA 111(10)
1 ν2 u + u + i− 2 u=0 z z
WA 112(22)
JA
iz 2
ikz −ν 12 kx u=z e Zν 2 √ ν u = z2 Zν i z
u=
8.495
WA 112(22)
u = sec z Z ν (z)
u=
u +
WA 112(21)
u= 7.
8.492
√ z Z ν+ 12 (icz) 1
u = z ν+ 2 Z ν+ 12 (icz) c √ u = z Z 2ν1 i z ν ν √ u = Zν z i
WA 108(1) WA 109(3, 4) WA 109(5, 6)
JA
8.511
2. 3. 4.
Series of Bessel functions
u +
d2 dz 2 2
2.
3.
4.
2
1 ν i ∓ 2i u − ± u=0 z z2 z
u = e±iz Z ν (z)
1 u + u + seiα u = 0 z
1 iα u + se + 2 u = 0 4z
8.496 1.
d dz 2 d2 dz 2
z4
d2 u dz 2
z
16 5
z 12
u=
− z2u = 0
d u dz 2 d2 u dz 2
u=
JA
√ i sze 2 α
JA
√ √ i z Z0 sze 2 α
JA
u = Z0
2
933
√ 1 √ Z 2 2 z + Z 2 2i z z WA 122(7)
3 3 −7/10 5 5 5 u=z Z 56 3 z + Z 56 3 iz 5
8
− z5u = 0
WA 122(8)
− z6u = 0
u = z −4 Z 10 2z −1/2 + Z 10 2iz −1/2
WA 122(9)
4
ν − 4ν 2 d4 u 2 d3 u 2ν 2 + 1 d2 u 2ν 2 + 1 du + + − + − 1 u = 0, dz 4 z dz 3 z 2 dz 2 z 3 dz z4 u = A1 J ν (z) + A2 Y ν (z) + A3 I ν (z) + A4 K ν (z), where A1 , A2 , A3 , A4 are constants
MO 29
8.51–8.52 Series of Bessel functions 8.511 1.
Generating functions for Bessel functions:
∞ ∞ k 1 1 t + (−t)−k J k (z) = exp J k (z)tk t− z = J 0 (z) + 2 t k=1
2.
k=−∞
∞ ∞
1 k m exp t − t J k (z) t J m (z) z= t m=−∞
[|z| < |t|]
KU 119(12) WA 40
k=−∞
3.
exp (±iz sin ϕ) = J 0 (z) + 2
4.
∞
J 2k (z) cos 2kϕ ± 2i
k=1
∞
∞ π exp (iz cos ϕ) = (2k + 1)ik J k+ 12 (z) P k (cos ϕ) 2z k=0 ∞ = ik J k (z)eikϕ k=−∞
= J 0 (z) + 2
∞ k=1
ik J k (z) cos kϕ
J 2k+1 (z) sin(2k + 1)ϕ
KU 120(13)
k=0
WA 401(1) MO 27
MO 27
934
Bessel Functions and Functions Associated with Them
5.
The series
i iz cos 2ϕ e π #
√
2z cos ϕ
e
8.512
∞
1 1 dt = J 0 (z) + e 4 kπi J k (z) cos kϕ 2 2
−it2
−∞
MO 28
k=1
J k (z)
8.512 1.
J 0 (z) + 2
∞
J 2k (z) = 1
WA 44
k=1
2.
∞ (n + 2k)(n + k − 1)!
k!
k=0
3.
∞ (4k + 1)(2k − 1)!!
J 2k+ 12 (z) =
2k k!
k=0
J n+2k (z) =
8.513 Notation: In formulas 8.513
(p) Qk
=
z n
[n = 1, 2, . . .]
2 2z π
k−1
2 (−1)m
m=0
1.
∞
(2k)2p J 2k (z) =
k=1
2.
∞
p
(2p)
Q2k z 2k
k m
(k − 2m)p
2k k! [p = 1, 2, 3, . . .]
WA 46(1)
[p = 0, 1, 2, 3, . . .]
WA 46(2)
k=0
(2k + 1)2p+1 J 2k+1 (z) =
k=0
3.
WA 45
p
(2p+1)
Q2k+1 z 2k+1
k=0
In particular: ∞ 1 z + z3 (2k + 1)3 J 2k+1 (z) = 2
WA 47(4)
k=0
4.
∞
(2k)2 J 2k (z) =
k=1
5.
∞
1 2 z 2
WA 47(4)
2k(2k + 1)(2k + 2) J 2k+1 (z) =
k=1
1 3 z 2
WA 47(4)
8.514 1.
∞ k=0
2.
sin z 2
WH
(−1)k J 2k (z) = cos z
WH
(−1)k J 2k+1 (z) =
J 0 (z) + 2
∞ k=1
3.
∞ k=1
(−1)k+1 (2k)2 J 2k (z) =
z sin z 2
WA 32(9)
8.518
4.
Series of Bessel functions ∞
(−1)k (2k + 1)2 J 2k+1 (z) =
k=0
5.
J 0 (z) + 2
∞
z cos z 2
935
WA 32(10)
J 2k (z) cos 2kθ = cos (z sin θ)
KU 120(14), WA 32
k=1
6.
∞
J 2k+1 (z) sin(2k + 1)θ =
k=0
7.
∞ k=0
8.515 1.
1 J 2k+1 (x) = 2
sin (z sin θ) 2
KU 120(15), WA 32
x
J 0 (t) dt
[x is real]
WA 638
0
k
ν ∞ z (−1)k tk 2z + t J ν+k (z) = J ν (z + t) k! 2z z+t
AD (9140)
k=0
2.
∞
J 2k− 12 x2 = S (x)
MO 127a
J 2k+ 12 x2 = C (x)
MO 127a
k=1
3.
∞ k=0
8.516
∞ (2n + 2k)(2n + k − 1)!
k!
k=0
The series
#
Ak J k (kx) and
#
1.
k=1
2.
∞
z J k (kz) = 2(1 − z) (−1)k J k (kz) = −
k=1
3.
∞
J 2k (2kz) =
k=1
WA 47
Ak J k (kx)
8.517 ∞
2n
J 2n+2k (2z sin θ) = (z sin θ)
z 2(1 + z)
z2 2 (1 − z 2 )
! & %! ! z exp √1 − z 2 ! ! ! √ ! 0, 0 ≤ t 0, t > 1, 2mπ < x(t − 1) < 2(m + 1)π, 2nπ < x(t + 1) < 2(n + 1)π, m + 1 and n + 1 are natural numbers. 8.525 ∞ n 1 1 < 1. (−1)k J 0 (kx) cos kxt = − + MO 61 2 2 2 k=1 l=m+1 x − [(2l − 1)π − tx]
8.526
2.
Series of Bessel functions ∞ k=1
(−1)k J 0 (kx) sin kxt =
939
m
n 1 1 1 < + 2π l 2 l=1 l=1 [(2l − 1)π − tx] − x2 ⎧ ⎫ ∞ ⎨ 1 1 ⎬ < + − ⎩ 2lπ ⎭ 2 l=1 [(2l − 1)π + tx] − x2 ⎧ ⎫ ∞ ⎨ 1 1 ⎬ < − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2
MO 61
3.
∞ k=1
n x 1 1 1 C + ln + π 4π 2π l l=1 m 1 < − 2 l=1 [(2l − 1)π − tx] − x2 ⎧ ⎫ ∞ ⎨ 1 1 ⎬ < − − ⎩ 2lπ ⎭ 2 l=1 [(2l − 1)π + tx] − x2 ⎧ ⎫ ∞ ⎨ 1 ⎬ 1 < − − ⎩ 2lπ ⎭ 2 l=n+1 [(2l − 1)π − tx] − x2
(−1)k Y 0 (kx) cos kxt = −
MO 61
In formulas 8.525, x > 0, t > 1, (2m − 1)π < x(t − 1) < (2m + 1)π, (2n − 1)π < x(t + 1) < (2n + 1)π, m and n are natural numbers. 8.526 ∞ ∞ x π 1 π 1 1 C + ln + √ 1. K 0 (kx) cos kxt = + − 2 4π 2 2lπ 2x 1 + t2 x2 + (2lπ − tx)2 k=1 l=1 ∞ π 1 1 + − 2 2 2 2lπ x + (2lπ + tx) l=1
2.
⎧ ⎫ ∞ ∞ ⎨ 1 x π 1 1 ⎬ < C + ln + (−1)k K 0 (kx) cos kxt = − ⎩ 2 4π 2 2lπ ⎭ 2 k=1 l=1 x2 + [(2l − 1)π − xt] ⎧ ⎫ ∞ 1 1 ⎬ π ⎨ < − + ⎩ 2 2lπ ⎭ 2 l=1 x2 + [(2l − 1)π + xt] [x > 0, t real] (see also 8.66)
MO 61
MO 62
940
Bessel Functions and Functions Associated with Them
8.530
8.53 Expansion in products of Bessel functions “Summation theorems” 8.530 Suppose that r > 0, > 0, ϕ > 0, and R = r2 + 2 − 2r cos ϕ; that is, suppose that r, , and R are the sides of a triangle such that the angle between the sides r and is equal to ϕ. Suppose also that 0, then xν > ν and xν > ν. Suppose also that yν is the smallest positive zero of the function Y ν (z). WA 534, 536 Then, xν < yν < xν . Suppose that zν,m (for m = 1, 2, 3, . . . ) are the zeros of the function z −ν J ν (z), numbered in order of the absolute value of their real parts. Here, we assume that ν = −1, −2, −3, . . . . Then, for arbitrary z z ν
∞ z2 2 WA 550 1− 2 . J ν (z) = Γ(ν + 1) m=1 zν,m 8.5458 The number of zeros of the function z −ν J ν (z) that occur between the imaginary axis and the line on which WA 497 Re z = m + 12 Re ν + 14 π, is exactly m. 8.546 For ν ≥0, the number of zeros of the function K ν (z) that occur in the region Re z − 12
8.554
Struve functions
ν π/2 2 z2 2ν Lν (z) = √ sinh (z cos ϕ) (sin ϕ) dϕ π Γ ν + 12 0
2.
8.552
1.6
2.6
943
Re ν > − 12
WA 360(11)
Special cases: z n−2m−1 n−1
2 Γ m + 1 2 1 2 Hn (z) = [n = 1, 2, . . .] − En (z) π m=0 Γ n + 12 − m z −n+2m+1 n−1
2 Γ n − m − 1 2 1 2 H−n (z) = (−1)n+1 − E−n (z) π m=0 Γ m + 32
EH II 40(66), WA 337(1)
[n = 1, 2, . . .]
EH II 40(67), WA 337(2)
[n = 0, 1, . . .]
EH II 39(64)
[n = 0, 1, . . .]
EH II 39(65)
L−(n+ 1 ) (z) = I n+ 12 (z) [n = 0, 1, . . .] 2 √ 2 H 12 (z) = √ (1 − cos z) πz
1/2 z 1/2 2 2 cos z H 32 (z) = 1+ 2 − sin z + 2π z πz z
EH II 39(65)
z 1 n 1 Γ m+ 2 2 Hn+ 12 (z) = Y n+ 12 (z) + π m=0 Γ(n + 1 − m)
−2m+n− 12
3.
n
4.
H−(n+ 1 ) (z) = (−1) J n+ 12 (z) 2
5. 6. 7. 8.553 1.
Functional relations: Hν zeimπ = eiπ(ν+1)m Hν (z)
[m = 1, 2, 3, . . .]
d ν [z Hν (z)] = z ν Hν−1 (z) dz −1 d −ν z Hν (z) = 2−ν π −1/2 Γ ν + 32 − z −ν Hν+1 (z) dz z ν −1 Γ ν + 32 Hν−1 (z) + Hν+1 (z) = 2νz −1 Hν (z) + π −1/2 2 ν −1 −1/2 z Γ ν + 32 Hν−1 (z) − Hν+1 (z) = 2 Hν (z) − π 2
2. 3. 4. 5.
Asymptotic representations:
ξ −2m+ν−1 1 p−1 Γ m + 2 1 2 ν−2p−1 Hν (ξ) = Y ν (ξ) + + O |ξ| π m=0 Γ ν + 12 − m [|arg ξ| < π] For the asymptotic representation of Y ν (ξ), see 8.451 2.
EH II 39, WA 364(3)
WA 364(3)
WA 362(5) WA 358 WA 359 WA 359(5) WA 359(6)
8.554
EH II 39(63), WA 363(2)
944
8.555
Bessel Functions and Functions Associated with Them
8.555
The differential equation for Struve functions: 2
z y + zy + z − ν 2
2
ν+1 1 4 z2 √ y= π Γ ν + 12
WA 359(10)
8.56 Thomson functions and their generalizations berν (z), beiν (z), herν (z), heiν (z), kerν (z), keiν (z) 8.561 1. 2. 8.562 1. 2.
3 berν (z) + i beiν (z) = J ν ze 4 πi 3 berν (z) − i beiν (z) = J ν ze− 4 πi .
WA 96(6) WA 96(6)
3 herν (z) + i heiν (z) = H ν(1) ze 4 πi 3 herν (z) − i heiν (z) = H ν(1) ze− 4 πi
(see also 8.567)
WA 96(7)
(see also 8.567)
WA 96(7)
8.563 1. 2.
ber0 (z) ≡ ber(z); bei0 (z) ≡ bei(z) π π ker(z) ≡ − hei0 (z); kei(z) ≡ hei0 (z) 2 2
WA 96(8) WA 96(8)
For integral representations, see 6.251, 6.536, 6.537, 6.772 4, 6.777. Series representation 8.564 1.
ber(z) =
∞ (−1)k z 4k
2.
bei(z) =
WA 96(3)
2
k=0 ∞
24k [(2k)!]
(−1)k z 4k+2 2
24k+2 [(2k + 1)!]
∞ 2k π z 4k 1 2 ker(z) = ln − C ber(z) + bei(z) + (−1)k 2 4k z 4 2 [(2k)!] m=1 m
WA 96(4)
k=0
3.
WA 96(9)a, DW
k=1
4.
kei(z) =
∞ 2k+1 1 π z 4k+2 2 ln − C bei(z) − ber(z) + (−1)k 2 z 4 24k+2 [(2k + 1)!] m=1 m
WA 96(10)a, DW
k=0
8.565 ber2ν (z) + bei2ν (z) =
∞ k=0
2ν+4k
(z/2) k! Γ(ν + k + 1) Γ(ν + 2k + 1)
WA 163(6)
8.570
Lommel functions
945
Asymptotic representation 8.566 1. 2. 3. 4.
eα(z) ber(z) = √ cos β(z) 2πz eα(z) sin β(z) bei(z) = √ 2πz π α(−z) e ker(z) = cos β(−z) 2z π α(−z) e kei(z) = sin β(−z) 2z
+ π, |arg z| < 4 + π, |arg z| < 4 5 |arg z| < π 4 5 |arg z| < π , 4
WA 227(1)
WA 227(1)
WA 227(2)
WA 227(2)
where z 1 25 13 √ − α(z) ∼ √ + √ − − ..., 4 3 128z 2 8z 2 384z 2 1 π 1 z 25 √ + ... β(z) ∼ √ − − √ − − 2 8 8z 2 16z 384z 3 2 2 8.567 1. 2.
Functional relations
√ ker(z) + i kei(z) = K 0 z i √ ker(z) − i kei(z) = K 0 z −i
(see 8.562)
WA 96(5), DW
(see 8.562)
WA 96(5), DW
For integrals of Thomson’s functions, see 6.87.
8.57 Lommel functions 8.570 1.
2.11
Definitions of the Lommel functions s μ,ν (z) and S μ,ν (z): (−1)m z μ+1+2m , + 2 [(μ + 1)2 − ν 2 ] [(μ + 3)2 − ν 2 ] . . . (μ + 2m + 1) − ν 2 2m+2 1 ∞ (−1)m z2 Γ 2 μ − 12 ν + 12 Γ 12 μ + 12 ν + 12 μ−1 =z Γ 12 μ − 12 ν + m + 32 Γ 12 μ + 12 ν + m + 32 m=0 [μ ± ν is not a negative odd integer]
s μ,ν (z) =
S μ,ν (z) = s μ,ν (z) + 2μ−1 Γ 12 μ − 12 ν + 12 Γ 12 μ + 12 ν + 12 cos 12 (μ − ν)π J −ν (z) − cos 12 (μ + ν)π J ν (z) × sin νπ = s μ,ν (z) + 2μ−1 Γ 12 μ − 12 ν + 12 Γ 12 μ + 12 ν + 12 × sin 12 (μ − ν)π J ν (z) − cos 12 (μ − ν)π Y ν (z)
EH II 40(69), WA 377(2)
EH II 40(71), WA 379(2)
EH II 41(71), WA 379(3)
946
Bessel Functions and Functions Associated with Them
8.571
Integral representations z z π μ μ 8.571 s μ,ν (z) = WA 378(9) Y ν (z) z J ν (z) dz − J ν (z) z Y ν (z) dz 2 0 0 8.572 s μ,ν (z)
π/2 z 12 (1+ν+μ) 1 1 1 1 (1+ν−μ) ν+μ + μ− ν Γ J 12 (1+μ−ν) (z sin θ) (sin θ) 2 (cos θ) dθ = 2μ 2 2 2 2 0 EH II 42(86) [Re(ν + μ + 1) > 0] 8.573 Special cases: 1. 2. 3. 4. 5. 6. 8.574 1. 2.
S 1,2n (z) = zO2n (z) z O 2n+1 (z) S 0,2n+1 (z) = 2n + 1 1 S 2n (z) S −1,2n (z) = 4n 1 S 0,2n+1 (z) = S 2n+1 (z) 2
1 √ ν−1 S ν,ν (z) = Γ ν + π2 Hν (z) 2 √ S ν,ν (z) = [Hν (z) − Y ν (z)] 2ν−1 π Γ ν + 12
WA 382(1) WA 382(1) WA 382(2) WA 382(2) EH II 42(84) EH II 42(84)
Connections with other special functions: 1 sin(νπ) [s 0,ν (z) − ν s −1,ν (z)] π 1 Eν (z) = − [(1 + cos νπ) s 0,ν (z) + ν (1 − cos νπ) s −1,ν (z)] π Jν (z) =
EH II 41(82) EH II 42(83)
A connection with a hypergeometric function 3.
s μ,ν (z) =
z μ+1 μ − ν + 3 μ + ν + 3 z2 , ; − 1; 1F 2 (μ − ν + 1)(μ + ν + 1) 2 2 4 EH II 40(69), WA 378(10)
8.575 1. 2.8 3. 4. 5.8
Functional relations: s μ+2,ν (z) = z μ+1 − (μ + 1)2 − ν 2 s μ,ν (z) ν s μ,ν (z) = (μ + ν − 1) s μ−1,ν−1 (z) s μ,ν (z) + z ν s μ,ν (z) = (μ − ν − 1) s μ−1,ν+1 (z) s μ,ν (z) − z ν s μ,ν (z) = (μ + ν − 1) s μ−1,ν−1 (z) − (μ − ν − 1) s μ−1,ν+1 (z) 2 z 2 s μ,ν (z) = (μ + ν − 1) s μ−1,ν−1 (z) + (μ − ν − 1) s μ−1,ν+1 (z)
In formulas 8.575 1–5, s μ,ν (z) can be replaced with S μ,ν (z).
EH II 41(73), WA 380(1) EH II 41(74), WA 380(2) EH II 41(75), WA 380(3) EH II 41(76), WA 380(4) EH II 41(77), WA 380(5)
8.579
Lommel functions
947
8.576 Asymptotic expansion of S μ,ν (z). In the case in which μ±ν is not a positive odd integer, S μ,ν (z) has the following asymptotic expansion:
∞ 1−μ+ν 1−μ−ν z −2m μ−1 m S μ,ν (z) ∼ z (−1) 2 2 m m 2 m=0
[|z| → ∞, |arg z| < π] WA 347, 352 The series terminates and is equal to S μ,ν (z) when μ ± ν is a positive odd integer. 8.577 Lommel functions satisfy the following differential equation: z 2 w + zw + z 2 − ν 2 w = z μ+1 WA 377(1), EH II 40(68) 8.578 Lommel functions of two variables U ν (w, z) and V ν (w, z): Definition 1.
U ν (w, z) =
∞
(−1)m
w ν+2m z
m=0
J ν+2m (z)
1 z2 + νπ + U −ν+2 (w, z) w+ 2 w Particular values:
2.
V ν (w, z) = cos
3.
U 0 (z, z) = V 0 (z, z) =
4.
U 1 (z, z) = − V 1 (z, z) = sin z n−1 (−1)n m U 2n (z, z) = (−1) ε2m J 2m (z) cos z − 2 m=0
5.
EH II 42(87), WA 591(5)
EH II 42(88), WA 591(6)
{J 0 (z) + cos z}
1 2
WA 591(9)
1 2
WA 591(10)
[n ≥ 1] ,
6.
(−1)n U 2n+1 (z, z) = 2
sin z −
n−1
(−1) ε2m+1 J 2m+1 (z)
m=0
V n (w, z) = (−1)n U n w 1/2 2
8.
U ν (w, 0) =
9.
V −ν+2 (w, 0) =
8.579 1. 2. 3.
z2 ,z w
S ν− 32 , 12
Γ(ν − 1) w 1/2 2
Γ(ν − 1)
WA 591(11)
m
[n ≥ 0] , 7.
εm
2, m > 0, = 1, m = 0
εm
2, m > 0, = 1, m = 0
WA 591(12)
w
WA 593(9)
2
S ν− 32 , 12
w 2
Functional relations: z 2 ∂ U ν (w, z) = U ν−1 (w, z) + U ν+1 (w, z) 2 ∂w w ∂ z 2 V ν (w, z) = V ν+1 (w, z) + 2 V ν−1 (w, z) ∂w w The function U ν (w, z) is a particular solution of the differential equation
WA 593(10)
WA 593(2) WA 593(4)
948
Bessel Functions and Functions Associated with Them
w ν−2 ∂2U z2U 1 ∂U + − = J ν (z) ∂z 2 z ∂z w2 z 4.
The function V ν (w, z) is a particular solution of the differential equation w −ν z2V ∂2V 1 ∂V + − = J −ν+2 (z) ∂z 2 z ∂z w2 z
8.580
WA 592(2)
WA 592(3)
8.58 Anger and Weber functions Jν (z) and Eν (z) 8.580 1.
Definitions: The Anger function Jν (z): Jν (z) =
2.
1 π
The Weber function Eν (z): Eν (z) =
8.581
1 π
π
cos (νθ − z sin θ) dθ
WA 336(1), EH II 35(32)
sin (νθ − z sin θ) dθ
WA 336(2), EH II 35(32)
0
π
0
Series representations: 2n ∞ (−1)n z2 νπ Jν (z) = cos 2 n=0 Γ n + 1 + 12 ν Γ n + 1 − 12 ν 2n+1 ∞ (−1)n z2 νπ + sin 2 n=0 Γ n + 32 + 12 ν Γ n + 32 − 12 ν
1.
EH II 36(36), WA 337(3)
2n ∞ (−1)n z2 νπ Eν (z) = sin 2 n=0 Γ n + 1 + 12 ν Γ n + 1 − 12 ν 2n+1 ∞ (−1)n z2 νπ − cos 2 n=0 Γ n + 32 + 12 ν Γ n + 32 − 12 ν
2.
EH II 36(37), WA 338(4)
8.582
Functional relations:
1.6
2 Jν (z) = Jν−1 (z) − Jν+1 (z)
2.6
2 Eν (z)
3.
6
4.6
EH II 36(40), WA 340(2)
= Eν−1 (z) − Eν+1 (z)
Jν−1 (z) + Jν+1 (z) = 2νz
−1
EH II 36(41), WA 340(6) −1
Jν (z) − 2(πz)
sin(νπ)
Eν−1 (z) + Eν+1 (z) = 2νz −1 Eν (z) − 2(πz)−1 (1 − cos νπ)
EH II 36(42), WA 340(1) EH II 36(43), WA 340(5)
8.591
8.583 1.6
Neumann’s and Schl¨ afli’s polynomials
Asymptotic expansions: ⎡ p−1 1+ν Γ n + 1−ν sin νπ ⎣ n 2n Γ n+ 2 2 1−ν z −2n Jν (z) = J ν (z) + (−1) 2 1+ν πz Γ Γ 2 2 n=0 ⎤ p−1 1 1 Γ n + 1 + ν Γ n + 1 − ν −2p 2 2 z −2n−1 + ν O |z|−2p−1 ⎦ −ν + O |z| (−1)n 22n 1 1 Γ 1 + ν Γ 1 − ν 2 2 n=0 [|arg z| < π]
2.
949
EH II 37(47), WA 344(1)
Eν (z) = − Y ν (z)
& % p−1 1+ν 1−ν 1 + cos(νπ) −2p n 2n Γ n + 2 Γ n + 2 −2n − (−1) 2 + O |z| z πz Γ 1−ν Γ 1+ν 2 2 n=0 % p−1 & 1 1 ν −2n−1 ν (1 − cos νπ) −2p−1 n 2n Γ n +1 + 2 ν Γ n + 1 − 2 (−1) 2 + O |z| − z zπ Γ 1 + 12 ν Γ 1 − 12 ν n=0 WA344(2), EH II 37(48)
For the asymptotic expansion of J ν (z) and Y ν (z), see 8.451. 8.584 The Anger and Weber functions satisfy the differential equation
ν2 −1 y + z y + 1 − 2 y = f (ν, z), z z−ν where f (ν, z) = sin νπ for Jν (z) πz 2 1 and f (ν, z) = − 2 [z + ν + (z − ν) cos νπ] for Eν (z) πz
WA 341(9), EH II 37(44) EH II 37(45), WA 341(10)
8.59 Neumann’s and Schl¨ afli’s polynomials: O n (z) and S n (z) 8.590
Definition of Neumann’s polynomials
1.
n2 1 n(n − m − 1)! z 2m−n−1 O n (z) = 4 m=0 m! 2
[n ≥ 1]
WA 299(2), EH II 33(6)
2.
O −n (z) = (−1)n O n (z)
[n ≥ 1]
WA 303(8)
3.
O 0 (z) =
4. 5.
1 z 1 O 1 (z) = 2 z 4 1 O 2 (z) = + 3 z z
WA 299(3), EH II 33(7) EH II 33(7) EH II 33(7)
In general, O n (z) is a polynomial in z −1 of degree n + 1. 8.591 Functional relations: 1.
O 0 (z) = − O 1 (z)
2.
2 O n (z)
= O n−1 (z) − O n+1 (z)
EH II 33(9), WA 301(3)
[n ≥ 1]
EH II 33(10), WA 301(2)
950
3.
4. 5. 8.592
Mathieu Functions
8.592
π 2 (n − 1) O n+1 (z) + (n + 1) O n−1 (z) − 2z −1 n2 − 1 O n (z) = 2nz −1 sin n 2 [n ≥ 1] EH II 33(11), WA 301(1) 2 2 π nz O n−2 (z) − n − 1 O n (z) = (n − 1)z O n (z) + n sin n EH II 33(12), WA 303(4) 2 π 2 nz O n+1 (z) − n2 − 1 O n (z) = −(n + 1)z O n (z) + n sin n EH II 33(13), WA 303(5)a 2 The generating function: ∞ 1 = J 0 (ξ)z −1 + 2 J n (ξ) O n (z) z−ξ n=1
[|ξ| < |z|]
The integral representation: √ √ n n ∞ u + u2 + z 2 + u − u2 + z 2 O n (z) = e−u du 2z n+1 0 See also 3.547 6, 8, 3.549 1, 2. 8.594 The inequality
EH II 32(1), WA 298(1)
8.593
−n−1
8.595
8.596 1. 2.
3. 8.597 1.
EH II 32(3), WA 305(1)
|O n (z)| ≤ 2n−1 n!|z| e 4 |z| [n > 1] EH II 33(8), WA 300(8) Neumann’s polynomial O n (z) satisfies the differential equation π 2 d2 y dy 2 π 2 + z + 1 − n2 y = z cos n z 2 2 + 3z + n sin n EH II 33(14), WA 303(1) dz dz 2 2 Schl¨ afli’s polynomials S n (z). These are the functions that satisfy the formulas 1
2
S 0 (z) = 0
1 π 2 S n (z) = 2zOn (z) − 2 cos n n 2 n
2 (n − m − 1)! z 2m−n = m! 2 m=0
EH II 34(18), WA 312(2)
[n ≥ 1]
EH II 34(19), WA 312(3)
[n ≥ 1]
EH II 34(18)
S −n (z) = (−1)n+1 S n (z)
WA 313(6)
Functional relations: S n−1 (z) + S n+1 (z) = 4 O n (z)
WA 313(7)
Other functional relations may be obtained from 8.591 by replacing O n (z) with the expression for S n (z) given by 8.596 2.
8.6 Mathieu Functions 8.60 Mathieu’s equation d2 y + a − 2k 2 cos 2z y = 0, 2 dz
k2 = q
MA
(2n)
8.621
(2n+1)
(2n+1)
(2n+2)
Recursion relations for the coefficients A2r , A2r+1 , B2r+1 , B2r+2
951
8.61 Periodic Mathieu functions 8.610 In general, Mathieu’s equation 8.60 does not have periodic solutions. If k is a real number, there exist infinitely many eigenvalues a, not identically equal to zero, corresponding to the periodic solutions y(z) = y(2π + z). If k is nonzero, there are no other linearly independent periodic solutions. Periodic solutions of Mathieu’s equations are called Mathieu’s periodic functions or Mathieu functions of the first kind, or, more simply, Mathieu functions. 8.611 Mathieu’s equation has four series of distinct periodic solutions: 1.
ce2n (z, q) =
∞
(2n)
A2r cos 2rz
MA
r=0
2.
ce2n+1 (z, q) =
∞
(2n+1)
A2r+1 cos(2r + 1)z
MA
r=0
3.
se2n+1 (z, q) =
∞
(2n+1)
B2r+1 sin(2r + 1)z
MA
r=0
4.
se2n+2 (z, q) =
∞
(2n+2)
B2r+2 sin(2r + 2)z
MA
r=0
5.
8.612
The coefficients A and B depend on q. The eigenvalues a of the functions ce2n , ce2n+1 , se2n , se2n+1 are denoted by a2n , a2n+1 , b2n , b2n+1 . The solutions of Mathieu’s equation are normalized so that 2π y 2 dx = π
MO 65
0
8.613 1. 2. 3.
1 lim ce0 (x) = √ 2 lim cen (x) = cos nx
q→0
[n = 0]
q→0
lim sen (x) = sin nx
MO 65
q→0
(2n)
(2n+1)
(2n+1)
(2n+2)
8.62 Recursion relations for the coefficients A2r , A2r+1 , B2r+1 , B2r+2 8.621 1. 2. 3.
(2n)
(2n)
− qA2
=0 (2n) (2n) (2n) =0 (a − 4)A2 − q A4 + 2A0 (2n) (2n) (2n) a − 4r2 A2r − q A2r+2 + A2r−2 = 0 aA0
MA MA
[r ≥ 2]
MA
952
Mathieu Functions
8.622
8.622 1. 2.
(2n+1)
(2n+1)
(a − 1 − q)A1 − qA3 =0 (2n+1) (2n+1) (2n+1) =0 a − (2r + 1)2 A2r+1 − q A2r+3 + A2r−1
MA
[r ≥ 1]
MA
8.623 1. 2.
(2n+1)
(2n+1)
− qB3 =0 (a − 1 + q)B1 (2n+1) (2n+1) (2n+1) =0 a − (2r + 1)2 B2r+1 − q B2r+3 + B2r−1
MA
[r ≥ 1]
MA
8.624 1. 2.11
(2n+2)
(2n+2)
− qB4 =0 (a − 4)B2 (2n+2) (2n+2) (2n+2) =0 a − 4r2 B2r − q B2r+2 + B2r−2
MA
[r ≥ 2]
MA
8.625 We can determine the coefficients A and B from equations 8.612, 8.613 and 8.621-8.624 pro(2n) vided a is known. Suppose, for example, that we need to determine the coefficients A2r for the function ce2n (z, q). From the recursion formulas, we have ! ! ! a −q 0 0 0 . . .!! ! !−2q a − 4 −q 0 0 . . .!! ! ! 0 −q a − 16 −q 0 . . .!! ! ST 1. ! =0 ! 0 0 −q a − 36 −q ! ! ! ! 0 0 0 −q a − 64 ! ! ! .. .. .. . . !! ! . . . . For given q in equation 8.625 1, we may determine the eigenvalues [|A0 | ≤ |A2 | ≤ |A4 | ≤ . . .]
2.
a = A0 , A2 , A4 , . . .
3.
If we now set a = A2n , we can determine the coefficients A2r from the recursion formulas 8.621 up to a proportionality coefficient. This coefficient is determined from the formula ∞ + + ,2 ,2 (2n) (2n) 2 A0 + = 1, MA A2r
(2n)
r=1
which follows from the conditions of normalization.
8.63 Mathieu functions with a purely imaginary argument 8.630 1.11
If, in equation 8.60, we replace z with iz, we arrive at the differential equation d2 y + (−a + 2q cosh 2z) y = 0 dz 2
We can find the solutions of this equation if we replace the argument z with iz in the functions cen (z, q) and sen (z, q). The functions obtained in this way are called associated Mathieu functions of the first kind and are denoted as follows:
8.652
1.
Mathieu functions for negative q
Ce2n (z, q),
Ce2n+1 (z, q),
Se2n+1 (z, q),
953
Se2n+2 (z, q)
8.631 1.
Ce2n (z, q) =
∞
(2n)
A2r cosh 2rz
MA
r=0
2.
Ce2n+1 (z, q) =
∞
(2n+1)
A2r+1 cosh(2r + 1)z
MA
r=0
3.
Se2n+1 (z, q) =
∞
(2n+1)
B2r+1 sinh(2r + 1)z
MA
r=0
4.
Se2n+2 (z, q) =
∞
(2n+2)
B2r+2 sinh(2r + 2)z
MA
r=0
8.64 Non-periodic solutions of Mathieu’s equation Along with each periodic solution of equation 8.60, there exists a second non-periodic solution that is linearly independent. The non-periodic solutions are denoted as follows: fe2n (z, q), fe2n+1 (z, q), ge2n+1 (z, q), ge2n+2 (z, q). Analogously, the second solutions of equation 8.630 1 are denoted by Fe2n (z, q), Fe2n+1 (z, q), Ge2n+1 (z, q), Ge2n+2 (z, q).
8.65 Mathieu functions for negative q 8.651
If we replace the argument z in equation 8.60 with±
π
2 d2 y + (a + 2q cos 2z) y = 0. dz 2 This equation has the following solutions: 8.652 1. ce2n (z, −q) = (−1)n ce2n 12 π − z, q 2. ce2n+1 (z, −q) = (−1)n se2n+1 12 π − z, q 3. se2n+1 (z, −q) = (−1)n ce2n+1 12 π − z, q 4. se2n+2 (z, −q) = (−1)n se2n+2 12 π − z, q 5. fe2n (z, −q) = (−1)n+1 fe2n 12 π − z, q 6. fe2n+1 (z, −q) = (−1)n ge2n+1 12 π − z, q 7. ge2n+1 (z, −q) = (−1)n fe2n+1 12 π − z, q 8. ge2n+2 (z, −q) = (−1)n ge2n+2 12 π − z, q
± z , we get the equation MA
MA MA MA MA MA MA MA MA
954
8.653
Mathieu Functions
Analogously, if we replace z with
3. 4. 5. 6.11 7.
11
8.11
π
i + z, q 2 n+1 Ce2n+1 (z, −q) = (−1) i Se2n+1 12 πi + z, q Se2n+1 (z, −q) = (−1)n+1 i Ce2n+1 12 πi + z, q Se2n+2 (z, −q) = (−1)n+1 Se2n+2 12 πi + z, q Fe2n (z, −q) = (−1)n Fe2n 12 πi + z, q Fe2n+1 (z, −q) = (−1)n+1 i Ge2n+1 12 πi + z, q Ge2n+1 (z, −q) = (−1)n+1 i Fe2n+1 12 πi + z, q Ge2n+2 (z, −q) = (−1)n+1 Ge2n+2 12 πi + z, q Ce2n (z, −q) = (−1)n Ce2n
2.
π i + z in equation 8.630 1, we get the equation 2
d2 y − (a + 2q cosh z) y = 0. dz 2
It has the following solutions: 8.654 1.
8.653
MA MA MA MA MA MA MA MA
8.66 Representation of Mathieu functions as series of Bessel functions 8.661 1.
ce2n (z, q) = =
2.
ce2n
π
2,q (2n) A0
ce2n (0, q) (2n) A0
ce2n+1 (z, q) = −
∞ (2n) (−1)r A2r J 2r (2k cos z)
r=0 ∞
(2n)
(−1)r A2r I 2r (2k sin z)
r=0
ce2n+1
π
2,q (2n+1) kA1
∞
(2n+1)
(−1)r A2r+1 J 2r+1 (2k cos z)
4.
∞ ce2n+1 (0, q) (2n+1) cot z (−1)r (2r + 1)A2r+1 I 2r+1 (2k sin z) kA1 (2n + 1) r=0 π ∞ se2n+1 2 , q (2n+1) se2n+1 (z, q) = tan z (−1)r (2r + 1)B2r+1 J 2r+1 (2k cos z) (2n+1) kB1 r=0 ∞ se2n+1 (0, q) (2n+1) = (−1)r B2r+1 I 2r+1 (2k sin z) (2n+1) kB1 r=0 π ∞ − se2n+2 2 , q (2n+2) se2n+2 (z, q) = tan z (−1)r (2r + 2)B2r+2 J 2r+2 (2k cosz ) (2n+2) k 2 B2 r=0 ∞ se2n+2 (0, q) (2n+2) = cot z (−1)r (2r + 2)B2r+2 I 2r+2 (2k sin z) (2n+2) 2 k B2 r=0
8.662 1.
MA
MA
r=0
=
3.
MA
MA
MA MA
MA MA
∞
fe2n (z, q) = −
π fe2n (0, q) (2n) π (−1)r A2r Im J r keiz Y r ke−iz 2 ce2n 2 , q r=0
MA
8.663
2.
Representation of Mathieu functions
fe2n+1 (z, q) =
955
πk fe2n+1 (0, q) 2 ce2n+1 π2 , q ∞ (2n+1) (−1)r A2r+1 Im J r keiz Y r+1 ke−iz + J r+1 keiz Y r ke−iz × r=0
MA
3.
πk ge2n+1 (0, q) 2 se2n+1 π2 , q ∞ (2n+1) (−1)r B2r+1 Re J r keiz Y r+1 ke−iz − J r+1 keiz Y r ke−iz ×
ge2n+1 (z, q) = −
r=0
MA
4.
πk 2 ge2n+2 (0, q) 2 se2n+2 12 π, q ∞ (−1)r Re J k keiz Y r+2 ke−iz − J r+2 keiz Y r ke−iz ×
ge2n+2 (z, q) = −
r=0
MA
The expansions of the functions Fen and Gen as series of the functions Y ν are denoted, respectively, by Feyn and Geyn , and the expansions of these functions as series of the functions K ν are denoted, respectively, by Fekn and Gekn . 8.663 ∞ ce2n (0, q) (2n) 1. Fey2n (z, q) = A2r Y 2r (2k sinh z) (2n) A0 r=0 k 2 = q [|sinh z| > 1, Re z > 0] MA ∞ ce2n π2 , q r (2n) = (−1) A2r Y 2r (2k cosh z) (2n) A0 r=0 [|cosh z| > 1] MA ∞ ce2n (0, q) ce2n π2 , q (2n) = (−1)r A2r J r ke−z Y r (kez ) + ,2 (2n) r=0 A0 MA
956
2.
Mathieu Functions
Fey2n+1 (z, q) =
8.663
∞ ce2n+1 (0, q) coth z (2n+1) (2r + 1)A2r+1 Y 2r+1 (2k sinh z) , kA1 (2n + 1) r=0
k 2 = q, =−
ce2n+1
π
2,q (2n+1) kA1
∞
[|sinh z| > 1,
Re z > 0] MA
r
(−1)
(2n+1) A2r+1
Y 2r+1 (2k cosh z)
r=0
[|cosh z| > 1] ce2n+1 (0, q) ce2n+1 =− ,2 + (2n+1) k A1 ×
∞
(2n+1)
(−1)r A2r+1
π
2,q
MA
J r ke−z Y r+1 (kez ) + J r+1 ke−z Y r (kez )
r=0
MA ∞
3.
Gey2n+1 (z, q) =
se2n+1 (0, q) (2n+1)
kB1
(2n+1)
B2r+1 Y 2r+1 (2k sinh z)
r=0
[|sinh z| > 1, =
se2n+1
π
2,q
(2n+1)
tanh z
kB1
∞
Re z > 0] MA
(−1)r (2r +
(2n+1) 1)B2r+1
Y 2r+1 (2k cosh z)
r=0
[|cosh z| > 1] se2n+1 (0, q) se2n+1 = ,2 + (2n+1) k B1
π
2,q
∞
MA r
(−1)
(2n+1) B2r+1
r=0
× J r ke−z Y r+1 (kez ) J r+1 ke−z Y r (kez ) MA
8.671
4.
The general theory
Gey2n+2 (z, q) =
se2n+2 (0, q)
coth z
(2n+2)
k 2 B2
957
∞ (2n+2) (2r + 2)B2r+2 Y 2r+2 (2k sinh z) r=0
[|sinh z| > 1, =−
se2n+2
π
2,q (2n+2) 2 k B2
Re z > 0] MA
∞ (2n+2) tanh z (−1)r (2r + 2)B2r+2 Y 2r+2 (2k cosh z) r=0
[|cosh z| > 1] MA
∞ se2n+2 (0, q) se2n+2 π ,q (2 ) (2n+2) = (−1)r B2r+2 + ,2 (2n+2) r=0 k 2 B2
× J r ke−z Y r+2 (kez ) − J r+2 ke−z Y r (kez ) MA
8.664 ∞
1.
Fek2n (z, q) =
ce2n (0, q) (2n) πA0
(2n)
(−1)r A2r K 2r (−2ik sinh z)
r=0
k 2 = q,
[|sinh z| > 1,
Re z > 0] MA
2.
3.
4.
Fek2n+1 (z, q) =
Gek2n+1 (z, q) =
Gek2n+2 (z, q) =
ce2n+1 (0, q) (2n+1)
coth z
πkA1
se2n+1
r=0
k2 = q π
2,q (2n+1) πkB1
se2n+2
∞ (2n+1) (−1)r (2r + 1)A2r+1 K 2r+1 (−2ik sinh z)
π
Re z > 0] MA
tanh z
∞ (2n+1) (2r + 1)B2r+1 K 2r+1 (−2ik cosh z)
MA
r=0
2,q (2n+2) 2 πk B2
[|sinh z| > 1,
tanh z
∞ (2n+2) (2r + 2)B2r+2 K 2r+2 (−2ik cosh z)
MA
r=0
8.67 The general theory If iμ is not an integer, the general solution of equation 8.60 can be found in the form 8.671 ∞ ∞ 1. y = Aeμz c2r e2rzi + Be−μz c2r e−2rzi r=−∞
MA
r=−∞
The coefficients c2r can be determined from the homogeneous system of linear algebraic equations 2.11
c2r + ξ2r (c2r+2 + c2r−2 ) = 0, where
r = . . . , −2, −1, 0, 1, 2, . . . ,
MA
958
Associated Legendre Functions
ξ2r =
3.7
The condition that this ! !· · · ! !· ξ−4 1 ! !· 0 ξ−2 Δ (iμ) = !! 0 0 !· !· 0 0 ! !· · ·
8.700
q 2
(2r − iμ) − a
system be compatible yields an equation that μ must satisfy: ! · · · · · ·!! ξ−4 0 0 0 0 ·!! 1 ξ−2 0 0 0 ·!! =0 1 ξ0 0 0 ·!! ξ0 1 ξ2 0 ·!! 0 ξ2 · · · · · ·!
MA
4.
This equation can also be written in the form √
π a 2 cosh μπ = 1 − 2Δ(0) sin , where Δ(0) is the value that is assumed by the determinant 2 of the preceding article if we set μ = 0 in the expressions for ξ2r .
5.
If the pair (a, q) is such that |cosh μπ| < 1, then μ = iβ, Im β = 0, and the solution 8.671 1 is bounded on the real axis.
6.
If |cosh μπ| > 1, μ may be real or complex, and the solution 8.671 1 will not be bounded on the real axis.
7.
If cosh μπ = ±1, then iμ will be an integer. In this case, one of the solutions will be of period π or 2π (depending on whether n is even or odd). The second solution is non-periodic (see 8.61 and 8.64).
8.7–8.8 Associated Legendre Functions 8.70 Introduction 8.700 1.
An associated Legendre function is a solution of the differential equation d2 u μ2 du 1 − z2 + ν(ν + 1) − − 2z u = 0, dz 2 dz 1 − z2
in which ν and μ are arbitrary complex constants. This equation is a special case of (Riemann’s) hypergeometric equation (see 9.151). The points +1, −1, ∞ are, in general, its singular points, specifically, its ordinary branch points. We are interested, on the one hand, in solutions of the equation that correspond to real values of the independent variable z that lie in the interval [−1, 1] and, on the other hand, in solutions corresponding to an arbitrary complex number z such that Re z > 1. These are multiple-valued in the z-plane. To separate these functions into single-valued branches, we make a cut along the real axis from −∞ to +1. We are also interested in those solutions of equation 8.700 1 for which ν or μ or both are integers. Of special significance is the case in which μ = 0. 8.701 In connection with this, we shall use the following notations: The letter z will denote an arbitrary complex variable; the letter x will denote a real variable that varies over the interval [−1, +1]. We shall sometimes set x = cos ϕ, where ϕ is a real number. We shall use the symbols P μν (z), Q μν (z) to denote those solutions of equation 8.700 1 that are singlevalued and regular for |z| 1. When these functions cannot be unrestrictedly extended without violating their single-valuedness, we make a cut along the real axis to the left of the point z = 1. The values of the functions P μν (z) and Q μν (z) on the upper and lower boundaries of that portion of the cuts lying between the points −1 and +1 are denoted, respectively, by P μν (x ± i0) , Q μν (x ± i0) . The letters n and m denote natural numbers or zero. The letters ν and μ denote arbitrary complex numbers unless the contrary is stated. The upper index will be omitted when it is equal to zero. That is, we set P 0ν (z) = P ν (z), Q 0ν (z) = Q ν (z) The linearly independent functions
μ
z+1 2 1 1−z μ 8.702 P ν (z) = F −ν, ν + 1; 1 − μ; Γ(1 − μ) z − 1 2 z+1 = 0, if z is real and greater than 1 and MO 80, WH arg z−1
μ2 −ν−μ−1 eμπi Γ(ν + μ + 1) Γ 12 2 ν+μ+2 ν+μ+1 3 1 z , ; ν + ; 8.703 Q μν (z) = − 1 z F 2 2 2 z2 2ν+1 Γ ν + 32 2 [arg z − 1 = 0 when z is real and greater than 1; arg z = 0 when z is real and greater than zero] which are solutions of the differential equation 8.700 1, are called associated Legendre functions (or spherical functions) of the first and second kinds, respectively. They are uniquely defined, respectively, in the intervals |1 − z| < 2 and |z| > 1, with the portion of the real axis that lies between −∞ and +1 excluded. They can be extended by means of hypergeometric series to the entire z-plane where the above-mentioned cut was made. These expressions for P μν (z) and Q μν (z) lose their meaning when 1 − μ and ν + 32 are nonMO 80 positive integers, respectively. When z is a real number lying on the interval [−1, +1], so that (z = x = cos ϕ), we take the following functions as linearly+ independent solutions of the equation: , e 2 μπi P μν (cos ϕ + i0) + e− 2 μπi P μν (cos ϕ − i0) EH I 143(1)
μ2
1+x 1 1−x = F −ν, ν + 1; 1 − μ; EH I 143(6) Γ(1 − μ) 1 − x 2 , + 1 1 8.705 Q μν (x) = 12 e−μπi e− 2 μπi Q μν (x + i0) + e 2 μπi Q μν (x − i0) EH I 143(2) π Γ (ν + μ + 1) −μ P (x) = (cf. 8.732 5) P μν (x) cos μπ − 2 sin μπ Γ(ν − μ + 1) ν If μ = ±m is an integer, the last equation loses its meaning. In this case, we get the following formulas by passing to the limit: 8.706 m dm m 1 − x2 2 Q (x) (cf. 8.752 1) EH I 149(7) 1. Qm ν (x) = (−1) dxm ν Γ(ν − m + 1) m Q (x) 2.11 Q −m EH I 144(18) ν (x) = Γ(ν + m + 1) ν 8.704 P μν (x) =
1 2
1
1
The functions Q μν (z) are not defined when ν + μ is equal to a negative integer. Therefore, we must exclude the cases when ν + μ = −1, −2, −3, . . . for these formulas. The functions
960
Associated Legendre Functions
8.707
P ±μ Q ±μ P ±μ Q ±μ ν (±z) , ν (±z) , −ν−1 (±z) , −ν−1 (±z) are linearly independent solutions of the differential equation for ν + μ = 0, ±1, ±2, . . . . 8.707 Nonetheless, two linearly independent solutions can always be found. Specifically, for ν ± μ not an integer, the differential equation 8.700 1 has the following solutions: 1.
P ±μ ν (±z) ,
P ±μ −ν−1 (±z) ,
Q ±μ ν (±z) ,
Q ±μ −ν−1 (±z)
respectively, for z = x = cos ϕ, 2.
P ±μ ν (±x) ,
P ±μ −ν−1 (±x) ,
Q ±μ ν (±x) ,
Q ±μ −ν−1 (±x) .
If ν ± μ is not an integer, the solutions 3.
P μν (z),
Q μν (z), respectively, and P μν (x),
Q μν (x)
are linearly independent. If ν ±μ is an integer but μ itself is not an integer, the following functions are linearly independent solutions of equation 8.700 1: 4.
μ P −μ ν (z), respectively, and P ν (x),
P μν (z),
P −μ ν (x).
If μ = ±m, ν = n, or ν = −n − 1, the following functions are linearly independent solutions of equation 8.700 1 for n ≥m: 5.
Pm n (z),
m Qm n (z), respectively, and P n (x),
Qm n (x),
and for n <m, the following functions will be linearly independent solutions 6.
P −m n (z),
−m Qm n (z), respectively, and P n (x),
Qm n (x).
8.71 Integral representations 8.711 1.
P −μ ν (z)
μ− 12 μ 1 z2 − 1 2 1 − t2 = μ√ dt √ 2 π Γ μ + 12 −1 z + t z 2 − 1 μ−ν
Re μ > − 12 ,
|arg (z ± 1)| < π
MO 88
2.
3.
4.
,ν (ν + 1)(ν + 2) . . . (ν + m) π + z + z 2 − 1 cos ϕ cos mϕ dϕ π 0 π cos mϕ dϕ m ν(ν − 1) . . . (ν − m + 1) = (−1) √ ν+1 π 0 z + z 2 − 1 cos ϕ, + π π |arg z| < , arg z + z 2 − 1 cos ϕ = arg z for ϕ = (cf. 8.822 1) SM 483(15), WH 2 2 2 μ2 ∞ √ sinh2μ t dt eμπi Γ(ν + μ + 1) z Q μν (z) = π μ − 1 √ ν+μ+1 1 2 Γ μ + 2 Γ(ν − μ + 1) 0 z + z 2 − 1 cosh t [Re (ν ± μ) > −1, |arg (z ± 1)| < π] (cf. 8.822 2) MO 88 Pm ν (z) =
Q μν (z) =
eμπi Γ(ν + 1) Γ(ν − μ + 1)
0
∞
√
cosh μt dt
ν+1 z + z 2 − 1 cosh t [Re(ν + μ) > −1, ν = −1, −2, −3, . . . ,
|arg (z ± 1)| < π]
WH, MO 88
8.714
Integral representations
5. 8.712
8.713 1.
961
1 l(l − 1) l! =− (l − 2)! 2l + 1 2l + 1 −1 1 ν − μ eμπi Γ(ν + μ + 1) 2 z −1 2 1 − t2 (z − t)−ν−μ−1 dt Q μν (z) = ν+1 2 Γ(ν + 1) −1 [Re(ν + μ) > −1, Re μ > −1, |arg (z ± 1)| < π] (cf. 8.821 2) 1
P 2l (x) P 0l (x) dx = −
MO 88a, EH I 155(5)a
1 ∞ eμπi Γ μ + 1 π −(ν+ 12 )t μ t dt cos ν + e dt 2 2 √ z2 − 1 2 Q μν (z) = − cos νπ μ+ 12 μ+ 12 2π 0 (z − cos t) 0 (z + cosh t) Re μ > − 12 , Re(ν + μ) > −1, |arg (z ± 1)| < π
MO 89
2.
P −μ ν (z)
μ ∞ z2 − 1 2 sinh2ν+1 t = ν dt 2 Γ(μ − ν) Γ(ν + 1) 0 (z + cosh t)ν+μ+1 [Re z > −1, |arg (z ± 1)| < π, Re(ν + 1) > 0,
3.
P −μ ν (z)
=
8.714 1.
2.
μ ∞ cosh ν + 12 t dt 2 Γ μ + 12 z 2 − 1 2 π Γ(ν + μ + 1) Γ(μ − ν) 0 (z + cosh t)μ+ 12 [Re z > −1, |arg (z ± 1)| < π, Re(ν + μ) > −1,
P μν
(cos ϕ) =
P −μ ν (cos ϕ) =
2 sinμ ϕ π Γ 12 − μ
0
ϕ
cos ν + 12 t dt (cos t − cos ϕ) μ
μ+ 12
Γ(2μ + 1) sin ϕ 2μ Γ(μ + 1) Γ(ν + μ + 1) Γ(μ − ν)
0 < ϕ < π,
Re(μ − ν) > 0]
MO 89
Re(μ − ν) > 0]
MO 89
Re μ
−1, Re(μ − ν) > 0] MO 89
μ
3.
4.
Q μν (cos ϕ) =
P μν (cos ϕ) =
1 Γ(ν + μ + 1) sin ϕ 2μ+1 Γ(ν − μ + 1) Γ μ + 12 & ∞% sinh2μ t sinh2μ t × dt ν+μ+1 + ν+μ+1 (cos ϕ + i sin ϕ cosh t) (cos ϕ − i sin ϕ cosh t) 0 Re(ν + μ + 1) > 0, Re(ν − μ + 1) > 0, Re μ > − 12
MO 89
i Γ(ν + μ + 1) sinμ ϕ 2μ Γ (ν − μ + 1) Γ μ + 12 % & ∞ sinh2μ t sinh2μ t × dt ν+μ+1 − ν+μ+1 (cos ϕ + i sin ϕ cosh t) (cos ϕ − i sin ϕ cosh t) 0 Re (ν ± μ + 1) > 0, Re μ > − 12
MO 89
962
Associated Legendre Functions
8.715 1.
P μν
√ α cosh ν + 12 t dt 2 sinhμ α (cosh α) = √ 1 π Γ 12 − μ 0 (cosh α − cosh t)μ+ 2
2.
Q μν (cosh α) =
π eμπi sinhμ α 2 Γ 12 − μ
∞
α
8.715
1 2
α > 0,
Re μ
0,
Re μ < 12 ,
MO 87
e−(ν+ 2 )t dt 1
(cosh t − cosh α)
μ+ 12
Re(ν + μ) > −1
MO 87
See also 3.277 1, 4, 5, 7, 3.318, 3.516 3, 3.518 1, 2, 3.542 2, 3.663 1, 3.894, 3.988 3, 6.622 3, 6.628 1, 4–7, and also 8.742.
8.72 Asymptotic series for large values of |ν| 8.7216 1.
For real values of μ, |ν| 1, |ν| |μ|, |arg ν| < π, we have: + , cos ν + k + 1 ϕ + π (2k − 1) + μπ ∞ 1 Γ μ+k+ 2 2 2 4 2 P μν (cos ϕ) = √ Γ(ν + μ + 1) 1 k+ 12 3 π Γ μ − k + (2 sin ϕ) k! Γ ν + k + 2 k=0 2 5π π 3 ν + μ = −1, −2, −3, . . . ; ν = − 2 , − 52 , 72 . . . ; for < ϕ < 6 6 This series also converges for complex values of ν and μ. In the remaining cases, it is an asymptotic expansion for |ν| |μ|, |ν| 1, if ν > 0, μ > 0 and 0 < ε ≤ ϕ ≤ π − ε MO 92
2.6
√ Q μν (cos ϕ) = π Γ(ν + μ + 1) + μπ , π 1 ∞ 1 cos ν + k + (2k − 1) + ϕ − Γ μ+k+ 2 2 4 2 × (−1)k 1 k+ 12 3 Γ μ − k + (2 sin ϕ) k! Γ ν + k + 2 k=0 2 5 π ν + μ = −1, −2, −3, . . . ; ν = − 32 , − 52 , − 72 , . . . ; for < ϕ < π 6 6 This series also converges for complex values of ν and μ. In the remaining cases, it is an asymptotic expansion for |ν| |μ|, |ν| 1, if ν > 0, μ > 0, 0 < ε ≤ ϕ ≤ π − ϕ
3.
+ π cos ν + 12 ϕ − + 2 Γ(ν + μ + 1) μ 4 √ P ν (cos ϕ) = √ π Γ ν + 32 2 sin ϕ
μπ 2
,
1 1+O ν 0 < ε ≤ ϕ ≤ π − ε,
EH I 147(6), MO 92
1 |ν| ε For ν > 0, μ > 0 and ν > μ, it follows from formulas 8.721 1 and 8.721 2 that
MO 92
8.724
Asymptotic series
4. 5.
1 ν+ ν (cos ϕ) = ϕ− 2
1 π cos ν + ν −μ Q μν (cos ϕ) = ϕ+ 2ν sin ϕ 2 −μ
P μν
2 cos νπ sin ϕ
963
1 π μπ + +O √ 4 2 ν3
1 π μπ + O √ 4 2 ν3 0 < ε ≤ ϕ ≤ π − ε;
1 ν ε
MO 92
8.722 If ϕ is sufficiently close to 0 or π that νϕ or ν(π −ϕ) is small in comparison with 1, the asymptotic formulas 8.721 become unsuitable. In this case, the following asymptotic representation is applicable for μ ≤ 0, ν 1, and small values of ϕ:
μ ϕ η 1 ϕ 2 ϕ J μ+1 (η) − J J P −μ (cos ϕ) = J (η)+sin (η) + (η) +O sin4 1. ν+ cos μ μ+2 μ+3 ν 2 2 2 2η 6 2 where η = (2ν + 1) sin ϕ2 . In particular, it follows that x 1. lim ν μ P −μ cos = J μ (x) ν ν→∞ ν 8.723 1.
2.
[x ≥ 0, μ ≥ 0]
MO 93
We can see how the functions P μν (z) and Q μν (z) behave for large |ν| and real values of z >
Γ −ν − 12 e(μ−ν)α sinhμ α 1 3 1 1 F μ + , −μ + ; ν + ; Γ(−ν − μ) (e2α − 1)μ+ 12 2 2 2 1 − e2α
Γ ν + 12 1 1 1 e(ν+μ+1)α sinhμ α 1 F μ + , −μ + ; −ν + ; + Γ(ν − μ + 1) (e2α − 1)μ+ 12 2 2 2 1 − e2α ν = ± 12 , ± 32 , ± 52 , . . . ; a > 12 ln 2
3 √ : 2 2
2μ P μν (cosh α) = √ π
√ Γ(ν + μ + 1) e−(ν+μ+1)α Q μν (cosh α) = eμπi 2μ π sinhμ α μ+ 12 −2α Γ ν + 32 (1 − e )
1 3 1 1 × F μ + , −μ + ; ν + ; 2 2 2 1 − e2α μ + ν + 1 = 0, −1, −2, . . . ;
α>
1 2
ln 2
MO 94
MO 94
See also 8.776. 8.724 For the inequalities in 8.776 1–4, ν and μ are arbitrary real numbers satisfying the inequalities ν ≥ 1, ν − μ + 1 > 0, and μ ≥ 0: ! ±μ ! 1 8 Γ (ν ± μ + 1) !P ν (cos ϕ)! < 1. MO 91-92 νπ Γ(ν + 1) sinμ+ 12 ϕ ! ±μ ! 1 2π Γ (ν ± μ + 1) ! ! Q ν (cos ϕ) < 2. MO 91-92 μ+ ν Γ(ν + 1) sin 12 ϕ ! ±μ ! 1 !P ν (cos ϕ)! < √2 Γ (ν ± μ + 1) 3. MO 91-92 νπ Γ(ν + 1) sinμ+ 12 ϕ
964
4. 5.8
Associated Legendre Functions
! ±μ ! !Q ν (cos ϕ)!
− 2 MO 88 π cos ν + 12 (t − π) dt 2 sinμ ϕ 2 μ μ P ν (cos ϕ) cos (ν + μ) π − Q ν (cos ϕ) sin(ν + μ)π = μ+ 1 π π Γ 12 − μ ϕ (cos ϕ − cos t) 2 Re μ < 12 MO 88 Γ(ν − μ − 1) Γ(ν + μ + 1)
cos μπ P μν (cos ϕ) −
2 sin μπ Q μν (cos ϕ) π
π sin2μ t dt 1 Γ(ν + μ + 1) sinμ ϕ = μ√ 1 2 π Γ(ν − μ + 1) Γ μ + 2 0 (cos ϕ ± i sin ϕ cos t)ν−μ Re μ > − 12 , 0 < ϕ < π MO 38
For integrals of Legendre functions, see 7.11–7.21.
968
Associated Legendre Functions
8.751
8.75 Special cases and particular values 8.751 1. 2.
3.8
m
+ m + 1) 1 − x2 2 1−x F m − ν, m + ν + 1; m + 1; = (−1) MO 84 2m Γ(ν − m + 1)m! 2 m
Γ(ν + m + 1) z 2 − 1 2 1−z m F m − ν, m + ν + 1; m + 1; P ν (z) = MO 84 2m m! Γ(ν − m + 1) 2
3
eμπi Γ μ + n + μ2 1/2 −n−μ−3/2 μ + n + 52 μ + n + 32 1 2 2 μ , ; n + 2; 2 Q n+ 1 (z) = F z −1 π z 3 2 2 2 z 2n+ 2 (n + 1)! Pm ν (x)
m Γ(ν
MO 84
8.752 1. 2.
3.
m dm m 1 − x2 2 Pm P ν (x) WH, MO 84, EH I 148(6) ν (x) = (−1) dxm 1 m 1 m Γ(ν − m + 1) 2 −2 Pm P −m ... P ν (x)(dx)m ν (x) = (−1) ν (x) = 1 − x Γ(ν + m + 1) x x 2 − m2 P −m ν (z) = z − 1
z
4. 5.
P ν (z)(dz)m
... 1
[m ≥ 1]
HO 99a, MO 85, EH I 149(10)a
[m ≥ 1]
MO 85, EH I 149(8)
z
1
2 m2 dm Qm Q (z) ν (z) = z − 1 dz m ν 2 − m2 m z Q −m (z) = (−1) − 1 ν
WH, MO 85, EH I 148(5)
∞
∞
...
z
Q ν (z)(dz)m
z
[m ≥ 1]
MO 85, EH I 149(9)
Special values of the indices 8.753 ϕ 1 cotμ Γ(1 − μ) 2
1.
P μ0 (cos ϕ) =
2.
P −1 ν (cos ϕ) = −
3.
Pm n (z) ≡ 0,
8.754 1. 2. 3.
MO 84
d P ν (cos ϕ) 1 ν(ν + 1) dϕ
Pm n (x) ≡ 0
MO 84
for m > n
MO 85
2 cosh να (cosh α) = π sinh α 2 1/2 cos νϕ P ν− 1 (cos ϕ) = 2 π sin ϕ 2 sin νϕ −1/2 P ν− 1 (cos ϕ) = 2 π sin ϕ ν 1/2 P ν− 1 2
MO 85 MO 85 MO 85
8.762
Derivatives with respect to the order
1/2 Q ν− 1 2
4. 8.755
(cosh α) = i
ν sin ϕ 2
ν sinh α 1 −ν P ν (cosh α) = Γ(1 + ν) 2 1 Γ(1 + ν)
P −ν ν (cos ϕ) =
1. 2.
π e−να 2 sinh α
969
MO 85
MO 85 MO 85
Special values of Legendre functions 8.756
√ 2μ π −ν−μ+1 Γ 2 +1 Γ 2 1 μ μ+1 2 sin 2 (ν + μ)π Γ ν+μ d P ν (0) 2 +1 = √ dx π Γ ν−μ+1 2 √ Γ ν+μ+1 1 μ μ−1 2 Q ν (0) = −2 π sin (ν + μ)π ν−μ 2 Γ 2 +1 √ Γ ν+μ d Q μν (0) 1 μ 2 +1 = 2 π cos (ν + μ)π ν−μ+1 dx 2 Γ 2 P μν (0) =
1. 2. 3. 4.
ν−μ
MO 84
MO 84
MO84
MO 84
8.76 Derivatives with respect to the order
μ2 ∞ (−ν)(1 − ν) . . . (n − 1 − ν)(ν + 1)(ν + 2) . . . (ν + n) (μ + 1)(μ + 2) . . . (μ + n)1 · 2 . . . n n=1
n 1−x × [ψ(ν + n + 1) − ψ(ν − n + 1)] 2 [ν = 0, ±1, ±2, . . . ; Re μ > −1] MO 94
8.761
1 ∂P −μ ν (x) = ∂ν Γ(μ + 1)
8.762
1. 2. 3.
∂ P ν (cos ϕ) ∂ν
1−x 1+x
∂P −1 ν (cos ϕ) ∂ν ∂P −1 ν (cos ϕ) ∂ν
= 2 ln cos ν=0
= − tan ν=0
ν=1
ϕ 2
MO 94
ϕ ϕ ϕ − 2 cot ln cos 2 2 2
MO 94
ϕ ϕ ϕ 1 = − tan sin2 + sin ϕ ln cos 2 2 2 2
• For a connection with the polynomials C λn (x), see 8.936. • For a connection with a hypergeometric function, see 8.77.
MO 94
970
Associated Legendre Functions
8.771
8.77 Series representation For a representation in the form of a series, see 8.721. It is also possible to represent associated Legendre functions in the form of a series by expressing them in terms of a hypergeometric function. 8.771
μ
z+1 2 1−z 1 F −ν, ν + 1; 1 − μ; MO 15 1. P μν (z) = z−1 Γ(1 − μ) 2 μ
ν+μ ν+μ+1 3 1 eμπi Γ(ν + μ + 1) Γ 12 z 2 − 1 2 μ 8 + 1, ;ν + ; 2 2. Q ν (z) = ν+1 F MO 15 2 z ν+μ+1 2 2 2 z Γ ν + 32 See also 8.702, 8.703, 8.704, 8.723, 8.751, 8.772. The analytic continuation for |z| > 1 The formulas are consequences of theorems on the analytic continuation of hypergeometric series (see 9.154 and 9.155): 8.772
μ2 −ν−μ−1 ν+μ ν+μ+1 3 1 sin(ν + μ)π Γ(ν + μ + 1) 2 μ z −1 z + 1, ;ν + ; 2 F 1. P ν (z) = ν+1 √ 2 2 2 z 2 π cos νπ Γ ν + 32
1 ν μ 2 Γ ν+2 μ−ν+1 μ−ν 1 1 , ; − ν; 2 z 2 − 1 2 z ν−μ F +√ 2 2 2 z π Γ(ν − μ + 1) [2ν = ±1, ±3, ±5, . . . ; |z| > 1; |arg (z ± 1)| < π] MO 85
2.
3.
P μν (z) =
P μν (z) =
− ν+1
2 Γ −ν − 12 z 2 − 1 ν−μ+1 ν+μ+1 3 1 √ , ; ν + ; F 2 2 2 1 − z2 2ν+1 π Γ(−ν − μ)
1 ν 2 ν 2 Γ ν+2 μ−ν μ+ν 1 1 +√ ,− ; − ν; z −1 2 F 2 2 ! 2 ! 1 − z2 π Γ(ν − μ + 1) 2ν = ±1, ±3, ±5; . . . ; !1 − z 2 ! > 1; |arg (z ± 1)| < π 1 Γ(1 − μ)
8.773 1.
2.
Q μν (z)
=e
Q μν (z) =
μπi
z−1 z+1
− μ2
z+1 2
ν F
z−1 −ν, −ν − μ; 1 − μ; z+1 ! ! !z − 1! ! 1
MO 86
1−z F −ν, ν + 1; 1 − μ; Γ(μ) 2
μ2 Γ(−μ) Γ(ν + μ + 1) z − 1 + F −ν, ν + 1; 1 + μ; Γ (ν − μ + 1) z+1 1 μπi e 2
MO 85
1−z 2
[|arg (z ± 1)| < π,
|1 − z| < 2]
MO 86
8.777
Series representation
971
8.774
P μν
1 ϕ ν+ 12 1 3 sin ϕ Γ −ν − 12 −i(ν+1) π ϕ 2 tan e + μ, − μ; ν + ; sin2 (i cot ϕ) = F 2 2 2 2 2 2π Γ(−ν − μ)1
ν+ 12 1 π ϕ 1 1 sin ϕ Γ ν + 2 ϕ eiν 2 cot + μ, − μ; − ν; sin2 F + 2π Γ(ν − μ + 1) 2 + 2 2 2 2 , π MO 86 2ν = ±1, ±3, ±5, . . . , 0 < ϕ < 2
8.775 1.
6
2.6
P μν (x) =
μ (ν + μ) π Γ ν+μ+1 ν+μ+1 μ−ν 1 2 2 ν−μ , ; ;x 1 − x2 2 F √ 2 2 2 πΓ 2 +1 ν+μ
μ ν + μ −ν +μ+1 3 2 2μ+1 sin 12 (ν + μ)π Γ 2 + 1 ν−μ+1 + 1, ; ;x + √ x 1 − x2 2 F 2 2 2 π Γ 2 2μ cos
1 2
ν+μ+1
√ μ ν+μ+1 μ−ν 1 2 π sin 12 (ν + μ)π Γ μ 2 2 2 , ; ;x 1−x Q ν (x) = − 1−μ F 2 2 2 2 Γ ν−μ 1 2 + 1 ν+μ
μ √ cos 2 (ν + μ)π Γ 2 + 1 ν+μ μ−ν+1 3 2 μ 2 2 + 1, ; ;x F +2 π x 1−x 2 2 2 Γ ν−μ+1 2
MO 87
MO 87
8.776 1.
2.
For |z| 1 P μν (z)
=
Q μν (z) =
2ν Γ ν + 12 Γ −ν − 12 1 ν −ν−1 √ z + ν+1 √ z 1+O z2 π Γ(ν − μ + 1) 2 π Γ(−ν − μ) [2ν = ±1, ±3, ±5, . . . ,
√ e Γ(μ + ν + 1) −ν−1 z π ν+1 1+O 2 Γ ν + 32 μπi
1 z2
|arg z| < π] MO 87
[2ν = −3, −5, −7, . . . ;
|arg z| < π] MO 87
√ 8.777 Set ζ = z + z 2 − 1. The variable ζ is uniquely defined by this equation on the entire z-plane in which a cut is made from −∞ to +1. Here, we are considering that branch of the variable ζ for which values of ζ exceeding 1 correspond to real values of z exceeding 1. In this case, μ
2μ Γ −ν − 12 z 2 − 1 2 1 3 1 μ + μ, ν + μ + 1; ν + ; 1. P ν (z) = √ F 2 2 ζ2 π Γ(−ν − μ) ζ ν+μ+1 μ
z2 − 1 2 1 1 1 2μ Γ ν + 12 + μ, μ − ν; − ν; +√ F 2 2 ζ2 π Γ(ν − μ + 1) ζ μ−ν [2ν = ±1, ±3, ±5, . . . ; |arg(z − 1)| < π] MO 86
2.
Q μν (z)
μ μπi
=2 e
μ
√ Γ(ν + μ + 1) z 2 − 1 2 1 3 1 + μ, ν + μ + 1; ν + ; π F ζ ν+μ+1 2 2 ζ2 Γ ν + 32 [|arg(z − 1)| < π]
MO 86
972
Associated Legendre Functions
8.781
8.78 The zeros of associated Legendre functions 8.781 The function P −μ ν (cos ϕ), considered as a function of ν, has infinitely many zeros for μ ≥0. These are all simple and real. If a number ν0 is a zero of the function P −μ ν (cos ϕ), the number −ν0 − 1 is also MO 91 a zero of this function. μ 8.782 If ν and μ are both real and μ ≤0, or if ν and μ are integers, the function P ν (t) has no real zeros exceeding 1. If ν and μ are both real with ν < μ 0, but does have one such zero when sin μπ sin(μ − ν)π − 32 and ν + μ + 1 > 0, the function Q μν (t) has no real zeros exceeding 1. 8.784 The function P − 12 +iλ (z) has infinitely many zeros for real λ. All these zeros are real and greater than unity. 8.785 For n a natural number, the function P n (x) has exactly n real zeros which lie in the closed interval −1, +1. 8.786 The function Q n (z) has no zeros for which |arg(z − 1)| < π if n is a natural number. The function MO 91 Q n (cos ϕ) has exactly n + 1 zeros in the interval 0 ≤ ϕ ≤ π. 8.787 The following approximate formula can be used to calculate the values of ν for which the equation P −μ ν (cos ϕ) = 0 holds for given small values of ϕ:
sin2 ϕ2 1 jμ 4μ2 − 1 4 ϕ ν+ =− MO 93 1− + O sin 1− . 2 2 sin ϕ2 6 jμ2 2 Here, jμ denotes an arbitrary nonzero root of the equation J μ (z) = 0 (for μ ≥ 0). If ϕ is close to π then, instead of this formula, we can use the following formulas:
2μ π−ϕ Γ(2μ + k + 1) [μ > 0, k = 0, 1, 2, . . .] MO 93 1. ν ≈μ+k+ Γ(μ) Γ(μ + 1) Γ(k + 1) 3 1 2. ν ≈k+ MO 93 [μ = 0, k = 0, 1, 2, . . .] 2 2 ln π−ϕ
8.79 Series of associated Legendre functions 8.791 1.
! ! !, +! ! ! ! ! !t + t2 − 1! < !z + z 2 − 1!
∞
1 = (2k + 1) P k (t) Q k (z) z−t k=0
2.
8.792
Here, t must lie inside an ellipse passing through the point z with foci at the points ±1. √ ∞ 1 z − t + 1 − 2tz + t2 √ √ ln tk Q k (z) = z2 − 1 1 − 2tz + t2 k=0 [Re z > 1, |t| < 1] MO 78 ∞ 1 1 sin νπ −β −β k − P −α (cos ϕ) P (cos ψ) = (−1) P −α ν ν k (cos ϕ) P k (cos ψ) π ν−k ν+k+1 k=0
[a ≥ 0,
β ≥ 0,
ν real,
−π < ϕ ± ψ < π]
MO 94
8.796
Series of associated Legendre functions
8.793 P −μ ν (cos ϕ) =
973
∞ 1 1 sin νπ − (−1)k P −μ k (cos ϕ) π ν−k ν+k+1
[μ ≥ 0,
0 < ϕ < π]
k=0
MO 94
Addition theorems 8.794 1.11
P ν (cos ψ1 cos ψ2 + sin ψ1 sin ψ2 cos ϕ) = P ν (cos ψ1 ) P ν (cos ψ2 ) + 2 = P ν (cos ψ1 ) P ν (cos ψ2 ) + 2 [0 ≤ ψ1 < π,
2.
0 ≤ ψ2 < π,
∞ k=1 ∞ k=1
k (−1)k P −k ν (cos ψ1 ) P ν (cos ψ2 ) cos kϕ
Γ(ν − k + 1) k P (cos ψ1 ) P kν (cos ψ2 ) cos kϕ Γ(ν + k + 1) ν
ψ1 + ψ2 < π,
Q ν (cos ψ1 ) cos ψ2 + sin ψ1 sin ψ2 cos ϕ
8.795 1.
0 < ψ2 < π,
(cf. 8.814, 8.844 1)
∞
∞ < < 2 2 (−1)k P kν (z1 ) P −k P ν z1 z2 − z1 − 1 z2 − 1 cos ϕ = P ν (z1 ) P ν (z2 ) + 2 ν (z2 ) cos kϕ k=1
[Re z1 > 0, 2.
MO 90
(−1)k P −k ν (cos ψ1 ) Q ν k(cos ψ2 ) cos kϕ k=1 , 0 < ψ1 + ψ2 < π; ϕ real (cf. 8.844 3) MO 90
= P ν (cos ψ1 ) Q ν (cos ψ2 ) + 2 + π 0 < ψ1 < , 2
ϕ real]
|arg (z1 − 1)| < π,
Re z2 > 0,
|arg (z2 − 1)| < π]
MO 91
∞ < < k Q ν x1 x2 − x21 − 1 x22 − 1 cos ϕ = P ν (x1 ) Q ν (x2 ) + 2 (−1)k P −k ν (x1 ) Q ν (x2 ) cos kϕ k=1
[1 < x1 < x2 , 3.
∞ < < Q n x1 x2 + x21 + 1 x22 + 1 cosh α = k=n+1
ν = −1, −2, −3, . . . ,
MO 91
1 Q k (ix1 ) Q kn (ix2 ) e−kα (k − n − 1)!(k + n)! n [x1 > 0,
x2 > 0,
8.796 P ν (− cos ψ1 cos ψ2 − sin ψ1 sin ψ2 cos ϕ) = P ν (− cos ψ1 ) P ν (cos ψ2 ) + 2
∞
α > 0] (−1)k
k=1
× P −k ν
(− cos ψ1 ) P −k ν
[0 < ψ2 < ψ1 < π, See also 8.934 3.
ϕ real]
MO 91
Γ(ν + k + 1) Γ(ν − k + 1)
(cos ψ2 ) cos kϕ
ϕ real]
(cf. 8.844 2)
MO 91
974
Associated Legendre Functions
8.810
8.81 Associated Legendre functions with integer indices 8.810 For integer values of ν and μ, the differential equation 8.700 1. (with |ν| > |μ|) has a simple solution in the real domain, namely: m dm m 1 − x2 2 P n (x). u = Pm n (x) = (−1) dxm m The functions P n (x) are called associated Legendre functions (or spherical functions) of the first kind. The number n is called the degree, and the number m is called the order of the function P m n (x). The sin mϑ P m functions {cos mϑ P m n (cos ϕ) , n (cos ϕ)}, which depend on the angles ϕ and ϑ, are also called Legendre functions of the first kind, or, more specifically, tesseral harmonics for m < n and sectoral harmonics for m = n. These last functions are periodic with respect to the angles ϕ and ϑ. Their periods are, respectively, π and 2π. They are single-valued and continuous everywhere on the surface of the unit sphere x21 + x22 + x23 = 1 (where x1 = sin ϕ cos ϑ, x2 = sin ϕ sin ϑ, x3 = cos ϕ), and they are solutions of the differential equation
1 ∂ ∂Y 1 ∂2Y + n(n + 1)Y = 0. sin ϕ + sin ϕ ∂ϕ ∂ϕ sin2 ϕ ∂ϑ2 8.811 The integral representation ϕ 2 (−1)m (n + m)! m− 1 −m sin (cos ϕ) = ϕ (cos t − cos ϕ) 2 cos n + 12 t dt MO 75 Pm n 1 Γ m + 2 (n − m)! π 0 8.812 The series representation:
m (−1)m (n + m)! (n − m)(m + n + 1) 1 − x 2 2 1 − x Pm (x) = 1− n 2m m!(n − m)! 1!(m + 1) 2
2 (n − m)(n − m + 1)(m + n + 1)(m + n + 2) 1 − x + − ... MO 73 2!(m + 1)(m + 2) 2
m (−1)m (2n − 1)!! (n − m)(n − m − 1) n−m−2 x 1 − x2 2 xn−m − = (n − m)! 2(2n − 1) (n − m)(n − m − 1)(n − m − 2)(n − m − 3) n−m−4 x + − ... MO 73 2 · 4(2n − 1)(2n − 3)
m m−n m−n+1 1 1 (−1)m (2n − 1)!! , ; − n; 2 1 − x2 2 xn−m F = MO 73 (n − m)! 2 2 2 x 8.813 Special cases: 1. 2. 3. 4. 5. 6.
1/2 P 11 (x) = − 1 − x2 = − sin ϕ 1/2 P 12 (x) = −3 1 − x2 x = − 32 sin 2ϕ P 22 (x) = 3 1 − x2 = 32 (1 − cos 2ϕ) 1/2 2 5x − 1 = − 38 (sin ϕ + 5 sin 3ϕ) P 13 (x) = − 32 1 − x2 P 23 (x) = 15 1 − x2 x = 15 4 (cos ϕ − cos 3ϕ) 3/2 P 33 (x) = −15 1 − x2 = − 15 4 (3 sin ϕ − sin 3ϕ)
MO 73 MO 73 MO 73 MO 73 MO 73 MO 73
8.820
Legendre functions
975
Functional relations For recursion formulas, see 8.731. 8.814 P n (cos ϕ1 cos ϕ2 + sin ϕ1 sin ϕ2 cos Θ) = P n (cos ϕ1 ) P n (cos ϕ2 ) + 2 [0 ≤ ϕ1 ≤ π, 8.815
n (n − m)! m P n (cos ϕ1 ) P m n (cos ϕ2 ) cos mΘ (n + m)! m=1
0 ≤ ϕ2 ≤ π]
(“addition theorem”)
MO 74
If Y n1 (ϕ, ϑ) = A0 P n1 (cos ϕ) + Z n2 (ϕ, ϑ) = α0 P n2 (cos ϕ) +
n1 m=1 n2
(am cos mϑ + bm sin mϑ) P m n1 (cos ϕ) , (αm cos mϑ + βm sin mϑ) P m n2 (cos ϕ) ,
m=1
then
2π
π
dϑ
2π
sin ϕ dϕ Y n1 (ϕ, ϑ) Y n2 (ϕ, ϑ) = 0, 0
π
sin ϕ dϕ Y n (ϕ, ϑ) P n [cos ϕ cos ψ + sin ϕ sin ψ cos(ϑ − θ)] =
dϑ 0
0
0
n
4π Y n (ψ, θ) 2n + 1
n! cos mϑ P m n (cos ϕ) (n + m)! m=1 For integrals of the functions, P m n (x), see 7.112 1, 7.122 1. n
8.816 (cos ϕ + i sin ϕ cos ϑ) = P n (cos ϕ) + 2
(−1)m
MO 75 MO 75
8.82–8.83 Legendre functions 8.820
The differential equation d 2 du 1−z + ν(ν + 1)u = 0 (cf. 8.700 1), dz dz where the parameter ν can be an arbitrary number, has the following two linearly independent solutions:
1−z 1. P ν (z) = F −ν, ν + 1; 1; 2 1
Γ(ν + 1) Γ 2 −ν−1 ν + 2 ν + 1 2ν + 3 1 , ; ; 2 F SM 518(137) 2. Q ν (z) = ν+1 z 2 2 2 z 2 Γ ν + 32 The functions P ν (z) and Q ν (z) are called Legendre functions of the first and second kind respectively. If ν is not an integer, the function P ν (z) has singularities at z = −1 and z = ∞. However, if ν = n = 0, 1, 2, . . . , the function P ν (z) becomes the Legendre polynomial P n (z) (see 8.91) For ν = −n = −1, −2, . . . , we have P −n−1 (z) = P n (z). 3.
If ν = 0, 1, 2, . . . , the function Q ν (z) has singularities at the points z = ±1 and z = ∞. These points are branch points of the function. On the other hand, if ν = n = 0, 1, 2, . . . , the function Q n (z) is single-valued for |z| > 1 and regular for z = ∞.
976
4.
Associated Legendre Functions
In the right half-plane,
P ν (z) = 5.
1+z 2
ν
z−1 F −ν, −ν; 1; z+1
8.821
[Re z > 0]
The function P ν (z) is uniquely determined by equations 8.820 1 and 8.820 4 within a circle of radius 2 with its center at the point z = 1 in the right half-plane.
6.
For z = x = cos ϕ, a solution of equation 8.820 is the function ϕ P ν (x) = P ν (cos ϕ) = F −ν, ν + 1; 1; sin2 ; 2 In general,
7.
P ν (z) = P −ν−1 (z) = P ν (x) = P −ν−1 (x), for z = x
8.
The function Q ν (z) for |z| > 1 is uniquely determined by equation 8.820 2 everywhere in the z-plane in which a cut is made from the point z = −∞ to the point z = 1. By means of a hypergeometric series, the function can be continued analytically inside the unit circle. On the cut (−1 ≤ x ≤ +1) of the real axis, the function Q ν (x) is determined by the equation
9.
Q ν (x) =
1 2
[Q ν (x + i0) + Q ν (x − i0)]
HO 52(53), WH
Integral representations 8.821 1.
2.
3.
ν (1+,z+) 2 t −1 1 dt 2πi A 2ν (t − z)ν+1 Here, A is a point on the real axis to the right of the point t = 1 and to the right of z if z is real. At the point A, we set
P ν (z) =
arg(t − 1) = arg(t + 1) = 0 and [|arg(t − z)| < π] WH ν (1−,1+) 2 t −1 1 Q ν (z) = dt 4i sin νπ A 2ν (z − t)ν+1 [ν is not an integer; the point A is at the end of the major axis of an ellipse to the right of t = 1 drawn in the t-plane with foci at the points ±1 and with a minor axis sufficiently small that the point z lies outside it. The contour begins at the point A, follows the path (1−, −1+), and returns to A; |arg z| ≤ π and |arg(z − t)| → arg z as t → 0 on the contour; arg(t + 1) = arg(t − 1) = 0 at the point A; z does not lie on the real axis between −1 and 1.] For ν = n an integer, 1 n 1 1 − t2 (z − t)−n−1 dt Q n (z) = n+1 2 −1
8.822 1.
1 P ν (z) = π
0
π
z+
√
SM 517(134), WH
ν z + z 2 − 1 cos ϕ dϕ 0 z 2 − 1 cos+ϕ π, Re z > 0 and arg z + z 2 − 1 cos ϕ = arg z for ϕ = 2 dϕ
ν+1
1 = π
π
WH
8.827
Legendre functions
977
2.
∞ dϕ Q ν (z) = √ ν+1 , 2 0 z + z − 1 cosh ϕ + , Re ν > −1; if ν is not an integer, z + z 2 − 1 cosh ϕ for ϕ = 0 has its principal value
WH
2 8.823 P ν (cos θ) = π n 8.824 Q n (z) = 2 n!
θ
cos ν + 12 ϕ
dϕ 2 (cos ϕ − cos θ) ∞ ∞ (dz)n+1 (t − z)n n ... =2 n+1 n+1 dt 2 2 z z % (z − 1) z (t − & 1) n ∞ dt (−1)n dn 2 = z − 1 n+1 n 2 (2n − 1)!! dz z (t − 1)
WH, MO 78
P n (t) 1 dt 2 −1 z − t See also 6.622 3, 8.842. 8.826 Fourier series:
2.
[Re z > 1]
1
8.825 Q n (z) =
1.
WH
0 ∞
[|arg(z − 1)| < π]
WH, MO 78
n! 2n+2 1 n+1 sin(n + 3)ϕ P n (cos ϕ) = sin(n + 1)ϕ + π (2n + 1)!! 1 2n + 3 1 · 3(n + 1)(n + 2) + sin(n + 5)ϕ + . . . 1 · 2(2n + 3)(2n + 5) [0 < ϕ < π] ⎡ n! ⎣ cos(n + 1)ϕ + 1 n + 1 cos(n + 3)ϕ Q n (cos ϕ) = 2n+1 (2n + 1)!! 1 2n + 3 ⎤ 1 · 3 (n + 1)(n + 2) + cos(n + 5)ϕ + . . .⎦ 1 · 2 (2n + 3)(2n + 5) [0 < ϕ < π]
MO 79
MO 79
The expressions for Legendre functions in terms of a hypergeometric function (see 8.820) provide other series representations of these functions. Special cases and particular values 8.827 1. 2. 3. 4.
1 1+x ln = arctanh x 2 1−x x 1+x −1 Q 1 (x) = ln 2 1−x 1+x 3 1 2 3x − 1 ln − x Q 2 (x) = 4 1−x 2 1+x 5 2 2 1 3 5x − 3x ln − x + Q 3 (x) = 4 1−x 2 3 Q 0 (x) =
JA JA JA JA
978
5. 6.
Associated Legendre Functions
8.828
1 + x 35 3 55 1 35x4 − 30x2 + 3 ln − x + x 16 1−x 8 24 63 1 1 + x 49 8 63x5 − 70x3 + 15x ln − x4 + x2 − Q 5 (x) = 16 1−x 8 8 15 Q 4 (x) =
JA JA
8.828 1.
P ν (1) = 1
MO 79
ν+1 1 sin νπ ν 2. P ν (0) = − √ Γ Γ − 2 π3 2 2
ν+1 1 ν 8.829 Q ν (0) = √ (1 − cos νπ) Γ Γ − 2 2 4 π
MO 79
MO 79
Functional relationships 8.831 1. 2.
3.
π [cos νπ P ν (x) − P ν (−x)] [ν = 0, ±1, ±2, . . .] 2 sin νπ 1 1+x − Wn−1 (x) Q n (x) = P n (x) ln [n = 0, 1, 2, . . .] , 2 1−x where n−1
n 2 2(n − 2k) − 1 1 P n−2k−1 (x) = P k−1 (x) P n−k (x) Wn−1 (x) = (2k + 1)(n − k) k Q ν (x) =
k=0
MO 76
k=1
and 4. 5.
W−1 (x) ≡ 0 (see also 8.839)
∞ 1 1 π − P ν (cos ϕ) (−1)k P k (cos ϕ) = ν−k ν+k+1 sin νπ
SM 516(131), MO 76
k=0
[ν not an integer; 6.
∞
(−1)k
k=0
1 1 − ν−k ν+k+1
P k (cos ϕ) P k (cos ψ) = [ν not an integer,
See also 8.521 4. 8.832 d 2 P ν (z) = (ν + 1) [P ν+1 (z) − z P ν (z)] 1. z −1 dz 2. (2ν + 1)z P ν (z) = (ν + 1) P ν+1 (z) + ν P ν−1 (z) 3. 4.
d Q ν (z) = (ν + 1) Q ν+1 (z) − z Q ν (z) dz (2ν + 1)z Q ν (z) Q ν+1 (z) + ν Q ν−1 (z) z2 − 1
0 ≤ ϕ < π]
MO 77
π P ν (cos ϕ) P ν (cos ψ) sin νπ
−π < ϕ + ψ < π,
−π < ϕ − ψ < π]
MO 77
WH WH WH WH
8.838
Legendre functions
979
8.833 [Im z < 0]
MO 77
[Im z > 0]
MO 77
3.
2 sin νπ Q ν (z) π 2 P ν (−z) = e−νπi P ν (z) − sin νπ Q ν (z) π Q ν (−z) = −e−νπi Q ν (z)
[Im z < 0]
MO 77
4.
Q ν (−z) = −eνπi Q ν (z)
[Im z > 0]
MO 77
1. 2.
P ν (−z) = eνπi P ν (z) −
8.834 1. 2.
πi P ν (x) 2 1 z+1 − W n−1 (z) Q n (z) = P n (z) ln 2 z−1 Q ν (x ± i0) = Q ν (x) ∓
(see 8.831 3)
MO 77
[sin νπ = 0]
MO 77
Q −ν−1 (cos ϕ) = Q ν (cos ϕ) − π cot νπ P ν (cos ϕ) [sin νπ = 0] π Q ν (− cos ϕ) = − cos νπ Q ν (cos ϕ) − sin νπ P ν (cos ϕ) 2
MO 77
8.835 1. Q ν (z) − Q −ν−1 (z) = π cot νπ P ν (z) 2. 3. 8.836 1. 2.
MO 77
n z + 1 1 dn 2 z+1 1 z − 1 ln Q n (z) = n − P n (z) ln 2 n! dz n z−1 2 z−1 n 1 + x 1 dn 2 1+x 1 x − 1 ln Q n (x) = n − P n (x) ln n 2 n! dx 1−x 2 1−x
8.837 ϕ 1. P ν (x) = P ν (cos ϕ) = F −ν, ν + 1; 1; sin2 2 ν ν+1 tan νπ Γ(ν + 1) −ν−1 z + 1, ;ν + 2. P ν (z) = ν+1 √ F 2 2 2 π Γ ν + 32
1−ν ν 1 1 2ν Γ ν + 12 ν z F , − ; − ν; 2 +√ 2 2 2 z π Γ(ν + 1)
(cf. 8.820 6)
3 1 ; 2 z2
MO 77
MO 79 MO 79
MO 76
MO 78
See also 8.820. For integrals of Legendre functions, see 7.1–7.2. 8.838 Inequalities (0 ≤ ϕ ≤ π, ν > 1, and C0 is a number that does not depend on the values of ν or ϕ): 1 MO 78 1. |P ν (cos ϕ) − P ν+2 (cos ϕ)| ≤ 2C0 νπ ! ! !Q ν (cos ϕ) − Q ν+2 (cos ϕ)! < C0 π 2. MO 78 ν With regard to the zeros of Legendre functions of the second kind, see 8.784, 8.785, and 8.786. For the expansion of Legendre functions in series of associated Legendre functions, see 8.794, 8.795, and 8.796.
980
8.839
Associated Legendre Functions
A differential equation leading to the functions Wn−1 (see 8.831 3): d2 Wn−1 dWn−1 d Pν + (n + 1)nWn−1 = 2 1 − x2 − 2x 2 dx dx dx
8.839
MO 76
8.84 Conical functions 8.840
Let us set
ν = − 12 + iλ, where λ is a real parameter, in the defining differential equation 8.700 1 for associated Legendre functions. We then obtain the differential equation of the so-called conical functions. A conical function is a special case of the associated Legendre function. However, the Legendre functions P − 12 +iλ (x), Q − 12 +iλ (x) have certain peculiarities that make us distinguish them as a special class—the class of conical functions. The most important of these peculiarities is the following 8.841 The functions 2 4λ + 12 4λ2 + 32 ϕ 4λ2 + 12 2 ϕ P − 12 +iλ (cos ϕ) = 1 + + sin sin4 + . . . 2 2 2 2 2 2 4 2 are real for real values of ϕ. Also, P − 12 +iλ (x) ≡ P − 12 −iλ (x) MO 95 8.842 1. 2.6
Integral representations: ∞ cosh λu du cos λu du 2 2 ϕ P − 12 +iλ (cos ϕ) = = cosh λπ π 0 π 2 (cos u − cos ϕ) 2 (cos ϕ + cosh u) 0 ∞ ∞ cos λu du cos λu du + Q − 12 ∓λi (cos ϕ) = ±i sinh λπ 2 (cosh u + cos ϕ) 2 (cosh u − cos ϕ) 0 0
MO 95
MO 95
Functional relations (See also 8.73) 8.843 P − 12 +iλ (− cos ϕ) =
, cosh λπ + Q − 12 +iλ (cos ϕ) + Q − 12 −iλ (cos ϕ) π
MO 95
8.844 1.
P − 12 +iλ (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ)
k ∞ (−1)k 22k P k − 1 +iλ (cos ψ) P − 1 +iλ (cos ϑ) cos kϕ
2 2 = P − 12 +iλ (cos ψ) P − 12 +iλ (cos ϑ) + 2 (4λ2 + 12 ) (4λ2 + 32 ) · · · [4λ2 + (2k − 1)2 ] k=1 , + π (cf. 8.794 1) MO 95 0 < ϑ < , 0 < ψ < π, 0 < ψ + ϑ < π 2
2.
P − 12 +iλ (− cos ψ cos ϑ − sin ψ sin ϑ cos ϕ)
k ∞ (−1)k 22k P k − 12 +iλ (cos ψ) P − 12 +iλ (− cos ϑ) cos kϕ = P − 12 +iλ (cos ψ) P − 12 +iλ (− cos ϑ) + 2 (4λ2 + 1) (4λ2 + 32 ) · · · [4λ2 + (2k − 1)2 ] , + k=1 π (cf. 8.796) MO 95 0 < ψ < < ϑ, ψ + ϑ < π 2
8.852
3.
Toroidal functions
981
Q − 12 +iλ (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ)
k ∞ (−1)k 22k P k − 12 +iλ (cos ψ) Q − 12 +iλ (cos ϑ) cos kϕ , + = P − 12 +iλ (cos ψ) Q − 12 +iλ (cos ϑ) + 2 2 (4λ2 + 1) (4λ2 + 32 ) · · · 4λ2 + (2k − 1) k=1 + , π 0 < ψ < < ϑ, ψ + ϑ < π (cf. 8.794 2) MO 96 2
Regarding the zeros of conical functions, see 8.784.
8.85 Toroidal functions 8.850 1.
Solutions of the differential equation
m2 d2 u cosh η du 1 2 − + n + − u = 0, dη 2 sinh η dη 4 sinh2 η
are called toroidal functions. They are equivalent (under a coordinate transformation) to associated Legendre functions. In particular, the functions Qm MO 96 Pm n− 12 (cosh η) , n− 12 (sinh η) are solutions of equation 8.850 1. The following formulas, obtained from the formulas obtained earlier for associated Legendre functions, are valid for toroidal functions: 8.851 Integral representations: π m Γ n + m + 12 (sinh η) sin2m ϕ dϕ m 1. P n− 1 (cosh η) = √ 1 2 Γ n − m + 12 2m π Γ m + 12 0 (cosh η + sinh η cos ϕ)n+m+ 2 2π cos mϕ dϕ (−1)m Γ n + 12 = 2π Γ n − m + 12 0 (cosh η + sinh η cos ϕ)n+ 12
2.
1 2
MO 96
∞
cosh mt dt m Γ n + Qm 1 n− 12 (cosh η) = (−1) 1 Γ n − m + 2 0 (cosh η + sinh η cosh t)n+ 2 η ln coth 2 Γ n + m + 12 n− 1 = (−1)m (cosh η − sinh η cosh t) 2 cosh mt dt 1 Γ n+ 2 0
[n ≥ m]
MO 96
8.852 1.
Functional relations:
√ π 2m Γ n + m + 12 1 sinhm ηe−(n+m+ 2 )η Γ(n + 1) 1 × F m + 2 , n + m + 12 ; n + 1; e−2η
m Qm n− 1 (cosh η) = (−1) 2
MO 96 ∗ Sometimes
called torus functions
982
2.
Orthogonal Polynomials
P −m (cosh η) = n− 1 2
8.853
m 1 2−2m 1 − e−2η e−(n+ 2 )η F m + 12 , n + m + 12 ; 2m + 1; 1 − e−2η Γ(m + 1) MO 96
8.853
An asymptotic representation P n− 12 (cosh η) for large values of n: 1 Γ(n)e(n− 2 )η P n− 12 (cosh η) = √ 1 π Γ n + 2 % &
2 Γ2 n + 12 1 1 η −2nη −2η × ln (4e ) e , n + ; n + 1; e +A+B , F πn! Γ(n) 2 2 where
2 (2n − 1)!! 1 1 · (2n − 1) −2η 1 1 · 3 · (2n − 1)(2n − 3) −4η 1 e e A=1+ 2 + 4 + · · · + 2n−2 e−2(n−1)η 2 1 · (n − 1) 2 1 · 2 · (n − 1)(n − 2) 2 (n − 1)!
1 1 ∞ Γ k+ 2 Γ n+k+ Γ n + 12 2 un+k + uk − vn+k− 12 − vk− 12 e−2(n+k)η B= √ Γ(n + k + 1) Γ(k + 1) π 3 Γ(n) k=1
Here, ur =
r 1 s=1
s
,
vr− 12 =
r s=1
2 2s − 1
[r is a natural number]
MO 97
8.9 Orthogonal Polynomials 8.90 Introduction 8.901 Suppose that w(x) is a nonnegative real function of a real variable x. Let (a, b) be a fixed interval on the x-axis. Let us suppose further that, for n = 0, 1, 2, . . . , the integral b xn w(x) dx exists and that the integral
a
b
w(x) dx a
is positive. In this case, there exists a sequence of polynomials p0 (x), p1 (x), . . . , pn (x), . . . , that is uniquely determined by the following conditions: 1.
pn (x) is a polynomial of degree n and the coefficient of xn in this polynomial is positive.
2.
The polynomials p0 (x), p1 (x), . . . are orthonormal; that is, b 0 for n = m, pn (x)pm (x)w(x) dx = 1 for n = m. a We say that the polynomials pn (x) constitute a system of orthogonal polynomials on the interval (a, b) with the weight function w(x).
8.910
8.902 1.
Legendre polynomials
983
If qn is the coefficient of xn in the polynomial pn (x), then n
pk (x)pk (y) =
k=0
qn pn+1 (x)pn (y) − pn (x)pn+1 (y) qn+1 x−y
(Darboux–Christoffel formula) EH II 159(10)
2.11
n
2
[pk (x)] =
k=0
qn qn+1
pn (x)pn+1 (x) − pn (x)pn+1 (x)
EH II 159(11)
8.903
Between any three consecutive orthogonal polynomials, there is a dependence pn (x) = (An x + Bn ) pn−1 (x) − Cn pn−2 (x) [n = 2, 3, 4, . . .] In this formula, An , Bn , and Cn are constants and qn qn qn−2 , Cn = 2 An = qn−1 qn−1 8.904 Examples of normalized systems of orthogonal polynomials:
n+
1 1/2 2
Notation and name
P n (x) 1/2 (n + λ) n! 2λ Γ(λ) C λn (x) 2π Γ (2λ + n) εn T n (x), ε0 = 1, εn = 2 for n = 1, 2, 3, . . . π −1/2 −n 2 2 π −1/4 (n!) H n (x) 1/2 Γ(n + 1) Γ(α + β + 1 + n)(α + β + 1 + 2n) P (α,β) (x) n Γ(α + 1 + n) Γ(β + 1 + n)2α+β+1 1/2 Γ(n + 1) (−1)n Lα n (x) Γ (α + n + 1)
MO 102
Interval
Weight
see 8.91
(−1, +1)
see 8.93
(−1, +1)
1 λ− 12 1 − x2
see 8.94
(−1, +1)
see 8.95
(−∞, ∞)
e−x
see 8.96
(−1, +1)
(1 − x)α (1 + x)β
see 8.97
(0, ∞)
xα e−x
1 − x2
−1/2 2
Cf. 7.221 1, 7.313, 7.343, 7.374 1, 7.391 1, 7.414 3.
8.91 Legendre polynomials 8.910 Definition. The Legendre polynomials P n (z) are polynomials satisfying equation 8.700 1 with μ = 0 and ν = n: that is, they satisfy the differential equation 1.
2.
d2 u du + n(n + 1)u = 0 − 2z dz 2 dz This equation has a polynomial solution if, and only if, n is an integer. Thus, Legendre polynomials constitute a special type of associated Legendre function.
1 − z2
Legendre polynomials of degree n are of the form n 1 dn 2 P n (z) = n z −1 n 2 n! dz
984
8.911
1.
Orthogonal Polynomials
8.911
Legendre polynomials written in expanded form:
n2 1 (−1)k (2n − 2k)! n−2k z P n (z) = n 2 k!(n − k)!(n − 2k)! k=0
(2n)! n(n − 1) n−2 n(n − 1)(n − 2)(n − 3) n−4 n z z = + − ... z − 2 2(2n − 1) 2 · 4(2n − 1)(2n − 3) 2n (n!)
(2n − 1)!! n 1 n 1−n 1 = z F − , ; − n; 2 n! 2 2 2 z HO 13, AD (9001), MO 69
2.
3.
2n(2n + 1) 2 2n(2n − 2)(2n + 1)(2n + 3) 4 n (2n − 1)!! z + z − ... P 2n (z) = (−1) 1− 2n n! 2! 4!
1 1 (2n − 1)!! = (−1)n F −n, n + ; ; z 2 n 2 n! 2 2
AD (9002), MO 69
+ 1)!! 2n(2n + 3) 3 2n(2n − 2)(2n + 3)(2n + 5) 5 z + z − ... P 2n+1 (z) = (−1) z− 2n n! 3! 5!
3 3 (2n + 1)!! = (−1)n z F −n, n + ; ; z 2 n 2 n! 2 2
n (2n
AD (9002), MO 69
⎛ 4.
P n (cos ϕ) =
1 n (2n − 1)!! ⎝ cos(n − 2)ϕ cos nϕ + 2n n! 1 2n − 1 +
1·3 n(n − 1) cos(n − 4)ϕ 1 · 2 (2n − 1)(2n − 3)
⎞ 1·3·5 n(n − 1)(n − 2) + cos(n − 6)ϕ − . . .⎠ 1 · 2 · 3 (2n − 1)(2n − 3)(2n − 5) WH
5.
(2n − 1)!! P 2n (cos ϕ) = (−1)n 2n n!
2n n! (2n)2 2n sin2n−2 ϕ cos2 ϕ + · · · + (−1)n cos2n ϕ × sin ϕ − 2! (2n − 1)!! AD (9011)
6.
(2n + 1)!! cos ϕ P 2n+1 (cos ϕ) = (−1)n 2n n!
2n n! (2n)2 2n 2n−2 2 n 2n sin cos ϕ × sin ϕ − ϕ cos ϕ + · · · + (−1) 3! (2n + 1)!! AD (9012)
7.
P n (z) =
n k=0
k
(−1) (n + k)! (n − k)! (k!)
2
2k+1
(1 − z)k + (−1)n (1 + z)k
WH
8.914
8.912
Legendre polynomials
985
Special cases:
1.
P 0 (x) = 1
JA
2.
P 1 (x) = x = cos ϕ 1 1 2 3x − 1 = (3 cos 2ϕ + 1) P 2 (x) = 2 4 1 1 3 5x − 3x = (5 cos 3ϕ + 3 cos ϕ) P 3 (x) = 2 8 1 1 35x4 − 30x2 + 3 = (35 cos 4ϕ + 20 cos 2ϕ + 9) P 4 (x) = 8 64 1 1 (63 cos 5ϕ + 35 cos 3ϕ + 30 cos ϕ) 63x5 − 70x3 + 15x = P 5 (x) = 8 128 1 1 (231 cos 6ϕ + 126 cos 4ϕ + 105 cos 2ϕ + 50) 231x6 − 315x4 + 105x2 − 5 = P 6 (x) = 16 512 1 P 7 (x) = 429x7 − 693x5 + 315x3 − 35x 16 1 (429 cos 7ϕ + 231 cos 5ϕ + 189 cos 3ϕ + 175 cos ϕ) = 1024 1 6435x8 − 12012x6 + 6930x4 − 1260x2 + 35 P 8 (x) = 128 1 (6435 cos 8ϕ − 3432 cos 6ϕ + 2772 cos 4ϕ − 2520 cos 2ϕ + 1225) = 16384
JA
3. 4. 5. 6. 7.10 8.
9.
8.913
π
sin n + 12 t
2 (cos ϕ − cos t) See also 3.611 3, 3.661 3, 4.
2.7
dt
JA
WH
Schl¨ afli’s integral formula: C
2 n t −1 dt, 2n (t − z)n+1
with C a simple contour containing z. 3.
JA
ϕ
1 P n (z) = 2πi 10
JA
Integral representations: 2 P n (cos ϕ) = π
1.
JA
Laplace integral formula: P n (z) =
1 π
π
,n + 1/2 cos ϕ dϕ x + x2 − 1
SA 175(9)
[|x| ≤ 1]
SA 180(19)
0
Functional relations 8.914 1.
Recurrence formulas: (n + 1) P n+1 (z) − (2n + 1)z P n (z) + n P n−1 (z) = 0
WH
986
Orthogonal Polynomials
2.
z2 − 1
8.915
d Pn n(n + 1) = n [z P n (z) − P n−1 (z)] = [P n+1 (z) − P n−1 (z)] dz 2n + 1
WH
8.915 n
1.10
(2k + 1) P k (x) P k (y) = (n + 1)
k=0
P n (x) P n+1 (y) − P n (y) P n+1 (x) y−x (Christoffel summation formula) MO 70
1(1)10 .
(y − x)
n
(2k + 1) P k (x) Q k (y) = 1 − (n + 1) P n+1 (x) Q n (y) − P n (x) Q n+1 (y)
k=0
AS 335(8.9.2) n−1
2
2.7
(2n − 4k − 1) P n−2k−1 (z) = P n (z)
(summation theorem)
MO 70
k=0 n−2
2
3.7
(2n − 4k − 3) P n−2k−2 (z) = z P n (z) − n P n (z)
SM 491(42), WH
k=0
4.
n
2
10
(2n − 4k + 1)[k(2n − 2k + 1) − 2] P n−2k (z) = z 2 P n (z) − n(n − 1) P n (z)
WH
k=1
m am−k ak an−k 2n + 2m − 4k + 1 P n+m−2k (z) = P n (z) P m (z) an+m−k 2n + 2m − 2k + 1 k=0 (2k − 1)!! , ak = k!
5.11
8.916 1. 2. 3. 4. 5.
1 1 (2n − 1)!! ∓inϕ ±2iϕ e , −n; − n; e F 2n n! 2 2 ϕ P n (cos ϕ) = F n + 1, −n; 1; sin2 2 ϕ P n (cos ϕ) = (−1)n F n + 1, −n; 1; cos2 2
1 1 1 n 2 P n (cos ϕ) = cos ϕ F − n, − n; 1; − tan ϕ 2 2 2 ϕ ϕ P n (cos ϕ) = cos2n F −n, −n; 1; − tan2 2 2 P n (cos ϕ) =
m≤n
AD (9036)
MO 69 MO 69 WH HO 23 HO 23, 29, WH
See also 8.911 1, 8.911 2, 8.911 3. For a connection with other functions, see 8.936 3, 8.836, 8.962 2. • For integrals of Legendre polynomials, see 7.22–7.25. • For the zeros of Legendre polynomials, see 8.785.
8.918
8.917
Legendre polynomials
Inequalities:
1.
P 0 (x) < P 1 (x) < P 2 (x) < · · · < P n (x) < . . .
2.
For x > −1, P 0 (x) + P 1 (x) + · · · + P n (x) > 0.
3.
[P n (cos ϕ)] >
4.
√ n sin ϕ|P n (cos ϕ)| ≤ 1.
2
[x > 1]
MO 71 MO 71
sin(2n + 1)ϕ (2n + 1) sin ϕ
MO 71 MO 71
|P n (cos ϕ)| ≤ 1.
5. 6.
987
10
WH
Let n ≥ 2. The successive relative maxima of |P n (x)|, when x decreases from 1 to 0, form a decreasing sequence. More precisely, if μ1 , μ2 , . . . , μ n/2 denote these maxima corresponding to decreasing values of x, we have 1 > μ1 > μ2 > · · · > μ n/2
7.10 8.
10
SZ 162(7.3.1)
1/2
Let n ≥ 2. The successive relative maxima of (sin θ) π/2, form an increasing sequence.
|P n (cos θ)| when θ increases from 0 to SZ 163(7.3.2)
We have 1/2
(sin θ)
|P n (cos θ)| < (2/π)1/2 n−1/2
[0 ≤ qθ ≤ qπ]
SZ 163(7.3.8)
Here the constant (2/π)1/2 cannot be replaced by a smaller one. 9.10
1/2
max (sin θ)
0≤qθ≤qπ
1 |P n (cos θ)| ∼ = (2/π)1/2 n− 2
[n → ∞]
SZ 164(7.3.12)
10.10 Stieltjes’ first theorem: |P n (cos θ)| ≤
1/2 2 4 √ π n sin θ
[n = 1, 2, . . . , 0 < θ < π]
SA 197(8)
11.10 Stieltjes’ second theorem:
12.10 13.10 8.91810
4 |P n (x) − P n+2 (x)| < √ √ π n+2 ! ! √ ! d P n (x) ! 2 n ! ! ! dx ! < √π 1 − x2 12 2 |P n+1 (x) + P n (x)| < 6 (1 − x)−1/2 πn
SA 199(15)
[|x| < 1,
n = 1, 2, . . .]
SA 201(18)
[|x| < 1,
n = 0, 1, . . .]
SA 201(19)
Asymptotic approximations:
1.
[|x| ≤ 1]
P n (cos θ) =
2 πn sin ϕ
1/2
1 π θ− + O n−3/2 2 4 [ε ≤ θ ≤ π − ε, 0 < ε < π/2m]
cos
n+
(Laplace’s formula)
SA 208(1)
988
Orthogonal Polynomials
8.921
1/2
2 1 1 1 π 1 π cos θ sin n + 1− cos n + θ− + θ− πn sin θ 4n 2 4 8n 2 4 −5/2 +O n
2.
P n (cos θ) =
[ε ≤ θ ≤ π − ε,
0 < ε < π/2]
(Bonnet–Heine formula)
SA 208(2)
8.91910 Series of products of Legendre and Chebyshev polynomials 1.
i+j=n 1
1
2 −1
T n (x) P n (x) dx =
i,j=0
−1
P i (x) P j (x) P n (x) dx
8.92 Series of Legendre polynomials 8.921
The generating function: ∞ 1 = tk P k (z) √ 1 − 2tz + t2 k=0 ∞ 1 = P k (z) tk+1
! !, + ! ! |t| < min !z ± z 2 − 1!
SM 489(31), WH
! !, + ! ! |t| > max !z ± z 2 − 1!
MO 70
k=0
8.922 ∞
1.
z 2n =
1 2n(2n − 2) . . . (2n − 2k + 2) P 0 (z) + P 2k (z) (4k + 1) 2n + 1 (2n + 1)(2n + 3) . . . (2n + 2k + 1)
MO 72
k=1
∞
2.
3.
3 2n(2n − 2) . . . (2n − 2k + 2) P 1 (z) + P 2k+1 (z) (4k + 3) 2n + 3 (2n + 3)(2n + 5) . . . (2n + 2k + 3) k=1
2 ∞ (2k − 1)!! 1 π √ = (4k + 1) P 2k (x) [|x| < 1, (−1)!! ≡ 1] 2 2k k! 1 − x2
z 2n+1 =
MO 72
k=0
MO 72, LA 385(15)
4.
5.
√
∞ x π (2k − 1)!!(2k + 1)!! P 2k+1 (x) = (4k + 3) 2k+1 2 2 2 k!(k + 1)! 1−x k=0
π 1 − x2 = 2
6.10
∞
[|x| < 1,
(−1)!! ≡ 1]
LA 385(17)
[|x| < 1,
(−1)!! ≡ 1]
LA 385(18)
1 (2k − 3)!!(2k − 1)!! − P 2k (x) (4k + 1) 2k+1 2 2 k!(k + 1)! k=1
∞
2 1−x 1 = P 0 (x) − 2 P n (x) 2 3 (2n − 1)(2n + 3) n=1
[−1 ≤ x ≤ 1]
8.925
7.10
8.923
Series of Legendre polynomials
1 − ρ2 (1 − 2ρx +
1/2 ρ2 )
=1+
∞
(2n + 1)ρn P n (x),
[|ρ| < 1,
|x| ≤ 1]
SA 170(4)
n=0
2 ∞ π (2k − 1)!! arcsin x = [P 2k+1 (x) − P 2k−1 (x)] + πx/2 2 2k k! k=1
[|x| < 1, 8.924
989
(−1)!! ≡ 1]
WH
∞ 1 + cos nπ 1 + cos nπ (4k + 5)n2 n2 − 22 . . . n2 − (2k)2 P P 2k+2 (cos θ) (cos θ) − 0 2 (n2 − 1) 2 (n2 − 12 ) (n2 − 32 ) . . . [n2 − (2k + 3)2 ] k=0 3 (1 − cos nπ) P 1 (cos θ) − 2 (n2 − 22 ) ∞ 1 − cos nπ (4k + 3) n2 − 12 . . . n2 − (2k − 1)2 − P 2k+1 (cos θ) = cos nθ 2 (n2 − 22 ) (n2 − 42 ) . . . [n2 − (2k + 2)2 ]
−
1.
k=1
AD (9060.1)
∞
(4k + 5)n2 n2 − 22 . . . n2 − (2k)2 P 2k+2 (cos θ) (n2 − 12 ) (n2 − 32 ) . . . [n2 − (2k + 3)2 ]
− sin nπ sin nπ P 0 (cos θ) − 2 (n2 − 1) 2 k=0 3 sin nπ P 1 (cos θ) + 2 (n2 − 22 ) ∞ sin nπ (4k + 3) n2 − 12 n2 − 32 . . . n2 − (2k − 1)2 + P 2k+1 (cos θ) = sin nθ 2 (n2 − 22 ) (n2 − 42 ) . . . [n2 − (2k + 2)2 ]
2.
k=1
AD (9060.2)
3.
3
n/2
2n−1 n! 2n−2k−1 (n − k − 1)!(2k − 3)!! P n (cos θ) − n P n−2k (cos θ) (2n − 4k + 1) (2n − 1)!! (2n − 2k + 1)!!k! k=1
= cos nθ AD (9061.1)
4.
∞ n (2n + 2k − 1)!!(2k − 1)!! (2n + 4k + 3) (2n − 1)!!Pn−1 (cos θ) − P n+2k+1 (cos θ) 2n−1 (n − 1)! 2n+1 22k (n + k + 1)!(k + 1)! k=0 4 sin nθ = π
AD (9061.2)
8.925 1.
2.
3.
2 (2k − 1)!! 4k − 1 2θ P 2k−1 (cos θ) = 1 − 22k (2k − 1)2 k! π k=1 2 ∞ 4k + 1 (2k − 1)!! 1 2 sin θ P 2k (cos θ) = − 22k+1 (2k − 1)(k + 1) k! 2 π k=1 2 ∞ (2k − 1)!! k(4k − 1) 2 cot θ P 2k−1 (cos θ) = 22k−1 (2k − 1) k! π ∞
k=1
AD (9062.2)
AD (9062.3)
990
Orthogonal Polynomials
2 ∞ 4k + 1 (2k − 1)!!
4.
22k
k=1
8.926 1.
k!
P 2k (cos θ) =
8.926
2 −1 π sin θ
AD (9062.4)
∞ 2 tan π−θ θ θ 1 4 P n (cos θ) = ln = − ln sin − ln 1 + sin n sin θ 2 2 n=1 ∞
1 + sin θ2 1 P n (cos θ) = ln −1 n+1 sin θ2 n=1
2.
8.927
AD (9063.2)
AD (9063.1)
∞
1 cos k + 12 β P k (cos ϕ) = 2 (cos β − cos ϕ) k=0 =0
[0 ≤ β < ϕ < π] [0 < ϕ < β < π] MO 72
8.928 1.
∞ 3 (−1)n (4k + 1) [(2n − 1)!!]
23n
n=1
2.
∞
3
(n!)
P 2n (cos θ) =
4 K (sin θ) −1 π2
3
(−1)n+1
n=1
(4n + 1) [(2n − 1)!!]
3
(2n − 1)(2n + 2)23n (n!)
P 2n (cos θ) =
4 E (sin θ) 1 − π2 2
AD (9064.1)
AD (9064.2)
• For series of products of Bessel functions and Legendre polynomials, see 8.511 4, 8.531 3, 8.533 1, 8.533 2, and 8.534. • For series of products of Legendre and Chebyshev polynomials, see 8.919.
8.93 Gegenbauer polynomials Cnλ (t) 8.930 Definition. The polynomials C λn (t) of degree n are the coefficients of αn in the power-series expansion of the function ∞ −λ 1 − 2tα + α2 = C λn (t)αn WH n=0
Thus, the polynomials C λn (t) are a generalization of the Legendre polynomials. 1.10
C λ0 (t) = 1
2.10
C λ1 (t) = 2λt
3.10
C λ2 (t) = 2λ(λ + 1)t2 − λ C λ3 (t) = 13 λ 4λ2 + 12λ + 8 t3 − 2λ(λ + 1)t C λ4 (t) = 23 λ λ3 + 6λ2 + 11λ + 6 t4 − 2λ λ2 + 3λ + 2 t2 + 12 λ(λ + 1) 1 C λ5 (t) = 15 λ 4λ4 + 40λ3 + 140λ2 + 200λ + 96 t5 − 13 λ 4λ3 + 24λ2 + 44λ + 24 t3 + λ λ2 + 3λ + 2 t
4.10 5.11 6.10
Gegenbauer polynomials Cnλ (t)
8.934
7.10
8.931
C λ6 (t) =
991
5 λ + 60λ4 + 340λ3 + 900λ2 + 1096λ + 480 t6 − 13 λ 2λ4 + 20λ3 + 70λ2 + 100λ + 48 t4 +λ λ3 + 6λ2 + 11λ + 6 t2 + 16 λ λ2 + 3λ + 2 1 45 λ
Integral representation:
π n 1 Γ(2λ + n) Γ 2λ+1 2 √ t + t2 − 1 cos ϕ sin2λ−1 ϕ dϕ = Γ(λ) π n! Γ(2λ) 0 See also 3.252 11, 3.663 2, 3.664 4. C λn (t)
MO 99
Functional relations 8.932 1.
2. 3. 8.933 1. 2. 3. 4.
Expressions in terms of hypergeometric functions:
∗ 1 1−t Γ(2λ + n) λ F 2λ + n, −n; λ + ; C n (t) = Γ(n + 1) Γ(2λ) 2 2
n 1 2 Γ(λ + n) n n 1−n t F − , ; 1 − λ − n; 2 = n! Γ(λ) 2 2 t
n 1 (−1) F −n, n + λ; ; t2 C λ2n (t) = (λ + n) B(λ, n + 1) 2
n 3 2 (−1) 2t λ F −n, n + λ + 1; ; t C 2n+1 (t) = B(λ, n + 1) 2
MO 97 MO 99 MO 99 MO 99
Recursion formulas: (n + 2) C λn+2 (t) = 2(λ + n + 1)t C λn+1 (t) − (2λ + n) C λn (t) + , λ+1 n C λn (t) = 2λ t C λ+1 (t) − C (t) n−1 n−2 + , λ+1 (2λ + n) C λn (t) = 2λ C λ+1 (t) − t C (t) n n−1 n C λn (t) = (2λ + n − 1)t C λn−1 (t) − 2λ 1 − t2 C λ+1 n−2 (t)
8.934
1 −λ λ+n− 12 , 1 − t2 2 dn + (−1)n Γ(2λ + n) Γ 2λ+1 2 2λ+1 1 − t2 = n n 2 n! dt Γ(2λ) Γ 2 + n
1.
C λn (t)
2.
C λn (cos ϕ) =
n Γ(λ + k) Γ(λ + l) k,l=0 k+l=n
∗ Equation
k!l! [Γ(λ)]
2
cos(k − l)ϕ
8.932.1 defines the generalized functions C λ n (t), where the subscript n can be an arbitrary number.
Mo 98 WH WH WH
WH
MO 99
992
3.
Orthogonal Polynomials
8.935
C λn (cos ψ cos ϑ + sin ψ sin ϑ cos ϕ) n 2 Γ(2λ − 1) 22k (n − k)! [Γ(λ + k)] (2λ + 2k − 1) sink ψ si nk ϑ = 2 Γ(2λ + n + k) [Γ(λ)] k=0 λ− 1
4.
ψ, ϑ, ϕ real;
lim Γ(λ) C λn (cos ϕ) =
λ→0
λ+k 2 × C λ+k (cos ϕ) n−k (cos ψ) C n−k (cos ϑ) C k 1 λ = 2 [“summation theorem”] (see also 8.794–8.796)
2 cos nϕ n
WH
MO 98
For orthogonality, see 8.904, 7.313. 8.935 Derivatives: 1.
2.11
Γ(λ + k) λ+k dk λ C n−k (t) C n (t) = 2k k dt Γ(λ) In particular,
MO 99
d C λn (t) = 2λ C λ+1 n−1 (t) dt
WH
For integrals of the polynomials C λn (x) see 7.31–7.33. 8.936 Connections with other functions: 1 λ 4 − 2 12 −λ Γ(2λ + n) Γ λ + 12 12 λ t −1 1. C n (t) = P λ+n− 1 (t) 2 Γ(2λ) Γ(n + 1) 4 m 2 −2 m!2m m dm P n (t) 1 m+ 1 m 1−t P n (t) 2. C n−m2 (t) = = (−1) m (2m − 1)!! dt (2m)! [m + 1 a natural number] 3.
C 1/2 n (t) = P n (t)
4.
J λ− 12 (r sin ϑ sin α) (r sin ϑ sin α)
5.
MO 98
MO 98, WH
−λ+ 12
λ
lim λ− 2 C n2 n
λ→∞
e−ir cos ϑ cos α ∞ λ λ √ Γ(λ) −k Jλ+k (r) C k (cos ϑ) C k (cos α) = 2 (λ + k)i Γ λ + 12 k=0 rλ C λk (1)
n 2− 2 2 = H n (t) t λ n!
MO 99 MO 99a
See also 8.932. 8.937 Special cases and particular values: sin(n + 1)ϕ sin ϕ
1.
C 1n (cos ϕ) =
2.
C 00 (cos ϕ) = 1
MO 98
3.
C λ0 (t) ≡ 1
MO 98
MO 99
8.940
The Chebyshev polynomials
4.
C λn (1)
≡
2λ + n − 1 n
993
MO 98
A differential equation leading to the polynomials C λn (t): (2λ + 1)t n(2λ + n) y − y=0 (cf. 9.174) y + 2 t −1 t2 − 1 For series of products of Bessel functions and the polynomials C λn (x), see 8.532, 8.534. 8.93910 Differentiation and Rodrigues’ formulas and orthogonality relation
8.938
1. 2. 3. 4. 5.
WH
d λ C (t) = 2λ C λ+1 MS n−1 (t) dt n dm λ C (t) = 2m λ(λ + 1)(λ + 2) . . . (λ + m − 1) C λ+m MS n−m (t) dtm n d λ d C (t) = t C λn (t) − n C λn (t) MS dt n−1 dt d λ d C (t) = t C λn (t) + (2λ + n) C λn (t) MS dt n+1 dt d λ C (t) = (n + 2λ − 1) C λn−1 (t) − nt C λn (t) = (n + 2λ)t C λn (t) − (n + 1) C λn+1 (t) 1 − t2 dt n = 2λ 1 − t2 C λ+1 n−1 (t)
5.3.2 5.3.2 5.3.2 5.3.2
MS 5.3.2
6.
, d + λ C n+1 (t) − C λn−1 (t) = 2(n + λ) C λn (t) dt
7.
C λn (t) =
MS 5.3.2
1 −λ 1, (−1)n 2λ(2λ + 1)(2λ + 2) . . . (2λ + n − 1) 1 − t2 2 dn + 2 n+λ− 2 1 − t dtn 2n n! λ + 12 λ + 32 . . . λ + n − 12 1 −λ 1, (−1)n Γ λ + 12 Γ(n + 2λ) 1 − t2 2 dn + 2 n+λ− 2 1 − t = dtn 2n n! Γ(2λ) Γ n + λ + 12 [Rodrigues’ formula]
1
8. −1
λ− 12 C λn (t) C λm (t) 1 − t2 dt = 0 =
π2
MS 5.3.2
n = m 1−2λ
Γ(n + 2λ)
n!(λ + n) [Γ(λ)]
2
n=m [λ = 0]
[Orthogonality relation]
MS 5.3.2
8.94 The Chebyshev polynomials Tn (x) and Un (x) 8.940 1.
Definition Chebyshev’s polynomials of the first kind n n , 1 + T n (x) = cos (n arccos x) = x + i 1 − x2 + x − i 1 − x2 2 n n n−4 2 n n−6 3 n n−2 1 − x2 + x 1 − x2 − 1 − x2 + . . . =x − x x 4 2 6 NA 66, 71
994
2.
Orthogonal Polynomials
8.941
Chebyshev’s polynomials of the second kind: n+1 n+1 1 sin [(n + 1) arccos x] 2 2 x+i 1−x = √ U n (x) = − x−i 1−x 2 x] sin [arccos
2i 1 − x 2 n+1 n n + 1 n−4 n + 1 n−2 = x − 1 − x2 + x 1 − x2 − . . . x 1 5 3
Functional relations 8.941
Recursion formulas:
1.
T n+1 (x) − 2x T n (x) + T n−1 (x) = 0
2.
U n+1 (x) − 2x U n (x) + U n−1 (x) = 0
3.
T n (x) = U n (x) − x U n−1 (x) 1 − x2 U n−1 (x) = x T n (x) − T n+1 (x)
4.
NA 358
EH II 184(3) EH II 184(4)
For the orthogonality, see 7.343 and 8.904. 8.942 Relations with other functions:
1 1−x 1. T n (x) = F n, −n; ; 2 2 √ n− 12 1 − x2 dn 1 − x2 2. T n (x) = (−1)n n (2n − 1)!! dx n+ 12 (−1)n (n + 1) dn 1 − x2 3. U n (x) = √ n 1 − x2 (2n + 1)!! dx
MO 104
MO 104 EH II 185(15)
See also 8.962 3. 8.94310 Special cases 1.
T 0 (x) = 1
10.
U 0 (x) = 1
2.
T 1 (x) = x
11.
U 1 (x) = 2x
3.
T 2 (x) = 2x2 − 1
12.
U 2 (x) = 4x2 − 1
4.
T 3 (x) = 4x3 − 3x
13.
U 3 (x) = 8x3 − 4x
5.
T 4 (x) = 8x4 − 8x2 + 1
14.
U 4 (x) = 16x4 − 12x2 + 1
6.
T 5 (x) = 16x5 − 20x3 + 5x
15.
U 5 (x) = 32x5 − 32x3 + 6x
7.
T 6 (x) = 32x − 48x + 18x − 1
16.
U 6 (x) = 64x6 − 80x4 + 24x2 − 1
8.
T 7 (x) = 64x7 − 112x5 + 56x3 − 7x
17.
U 7 (x) = 128x7 − 192x5 + 80x3 − 8x
9.
T 8 (x) = 128x8 − 256x6 + 160x4 − 32x2 + 1
18.
U 8 (x) = 256x8 − 448x6 + 240x4 − 40x2 + 1
6
4
2
8.949
8.944 1.
The Chebyshev polynomials
Particular values: T n (1) = 1 n
2.
T n (−1) = (−1)
3.
n
T 2n (0) = (−1)
4.
T 2n+1 (0) = 0
8.945
995
5.
U 2n+1 (0) = 0
6.
U 2n (0) = (−1)n
The generating function: ∞
1.11
1 − t2 = T 0 (x) + 2 T k (x)tk 2 1 − 2tx + t
[|t| < 1]
MO 104
[|t| < 1]
MO 104a, EH II 186(31)
k=1
∞
2.11
1 = U k (x)tk 2 1 − 2tx + t k=0
8.946 Zeros. The polynomials T n (x) and U n (x) only have real simple zeros. All these zeros lie in the interval (−1, +1). √ 8.947 The functions T n (x) and 1 − x2 U n−1 (x) are two linearly independent solutions of the differential equation d2 y dy + n2 y = 0. 1 − x2 −x NA 69(58) dx2 dx 8.948 Of all polynomials of degree n with leading coefficient equal to 1, the one that deviates the least from zero on the interval [−1, +1] is the polynomial 2−n+1 T n (x). 8.94910 Differentiation and Rodrigues’ formulas and orthogonality relations 1. 2. 3. 4. 5. 6.
d T n (x) = n U n−1 (x) MS dx dm T n (x) = 2m−1 Γ(m)n C m MS n−m (x) dxm d T n (x) = n [T n−1 (x) − x T n (x)] = n [x T n (x) − T n+1 (x)] 1 − x2 MS dx d U n (x) = 2 C 2n−1 (x) MS dx dm m+1 U n (x) = 2m m!Cn−m (x) MS dxm d U n (x) = (n + 1) U n−1 (x) − nx U n (x) = (n + 2)x U n (x) − (n + 1) U n+1 (x) 1 − x2 dx
5.7.2 5.7.2 5.7.2 5.7.2 5.7.2
MS 5.7.2
7.
8.
c 1 1, (−1)n π 1/2 1 − x2 2 dn + 2 n− 2 1 − x T n (x) = [Rodrigues’ formula] dxn 2n+1 Γ n + 12 −1/2 1, (−1)n π 1/2 (n + 1) 1 − x2 dn + 2 n+ 2 1 − x U n (x) = dxn 2n+1 Γ n + 32 [Rodrigues’ formula]
MS 5.7.2
MS 5.7.2
996
Orthogonal Polynomials
1
9. −1
1
10. −1
8.950
−1/2 T m (x) T n (x) 1 − x2
⎧ ⎪ m = n ⎨0, dx = π/2, m = n = 0 ⎪ ⎩ π, m=n=0
−1/2 U m (x) U n (x) 1 − x2
0, m = n dx = π/8, m = n
[Orthogonality relation]
MS 5.7.2
[Orthogonality relation]
MS 5.7.2
8.95 The Hermite polynomials H n (x) 8.950
Definition 2
H n (x) = (−1)n ex
1.
dn −x2 e dxn
or H n (x) = 2n xn − 2n−1
2. 3.10
SM 567(14)
n n n xn−2 + 2n−2 · 1 · 3 · xn−4 − 2n−3 · 1 · 3 · 5 · xn−6 + . . . 2 4 6
MO 105a
H 0 (x) = 1
4.
10
H 1 (x) = 2x
5.
10
H 2 (x) = 4x2 − 2
6.10
H 3 (x) = 8x3 − 12x
7.10
H 4 (x) = 16x4 − 48x2 + 12
8.10
H 5 (x) = 32x5 − 160x3 + 120x
9.10
H 6 (x) = 64x6 − 480x4 + 720x2 − 120
10.10 H 7 (x) = 128x7 − 1344x5 + 3360x3 − 1680x 11.10 H 8 (x) = 256x8 − 3584x6 + 13440x4 − 13440x2 + 1680 8.951
The integral representation: 2n H n (x) = √ π
∞
(x + it)n e−t dt 2
MO 106a
−∞
Functional relations 8.952
Recursion formulas: d H n (x) = 2n H n−1 (x) dx H n+1 (x) = 2x H n (x) − 2n H n−1 (x)
1. 2.
SM 569(22) SM 570(23)
For the orthogonality, see 7.374 1 and 8.904. 3.
10
n H n (x) = −n H n−1 (x) + x H n (x)
MS 5.6.2
8.957
Hermite polynomials
H n (x) = 2x H n−1 (x) − H n−1 (x)
4.10 8.953
(2n)! Φ −n, 12 ; x2 n! (2n + 1)! x Φ −n, 32 ; x2 H 2n+1 (x) = (−1)n 2 n!
2.
MS 5.6.2
The connection with other functions: H 2n (x) = (−1)n
1.
997
MO 106a MO 106a
• For a connection with the polynomials C λn (x), see 8.936 5. • For a connection with the Laguerre polynomials, see 8.972 2 and 8.972 3. • For a connection with functions of a parabolic cylinder, see 9.253. 8.954
Inequalities: √ n! e2x n/2
n/2 !
1.10
|H n (x)| ≤ 2 2 − 2
2.10
√ 2 |H n (x)| < k n!2n/2 ex /2 ,
8.955 1. 2. 8.956
n
n
k ≈ 1.086435
MO 106a SA 324
Asymptotic representation:
√ 1 cos 4n + 1x + O √ H 2n (x) = (−1) 2 (2n − 1)!!e 4 n
√ √ 2 1 1 H 2n+1 (x) = (−1)n 2n+ 2 (2n − 1)!! 2n + 1ex /2 sin 4n + 3x + O √ 4 n x2 /2
n n
SM 579 SM 579
Special cases and particular values:
1.
H 0 (x) = 1
2.
H 1 (x) = 2x
3.
H 2 (x) = 4x2 − 2
4.
H 3 (x) = 8x3 − 12x
5.
H 4 (x) = 16x4 − 48x2 + 12
6.
H 2n (0) = (−1)n 2n (2n − 1)!!
7.
H 2n+1 (0) = 0
SM 570(24)
Series of Hermite polynomials 8.957 1.
The generating function: ∞ k t H k (x) exp −t2 + 2tx = k!
SM 569(21)
k=0
∞
2.
1 1 sinh 2x = H 2k+1 (x) e (2k + 1)! k=0
MO 106a
998
Orthogonal Polynomials
8.958
∞
3.
1 1 cosh 2x = H 2k (x) e (2k)!
MO 106a
k=0
4.
∞
e sin 2x =
(−1)k
k=0
5.
e cos 2x =
∞
(−1)k
k=0
8.958
1.11
2.
1 H 2k+1 (x) (2k + 1)!
MO 106a
1 H 2k (x) (2k)!
MO 106a
“The summation theorem”: r n2 ⎛ r ⎞ 2 ak ak xk ⎟ ⎜ r mk ⎜ k=1 ⎟ ak k=1 ⎟= Hn ⎜ H (x ) # mk k ⎜ n! a2k ⎟ ⎝ ⎠ m1 +m2 +···+mr =n k=1 mk ! A special case: n
2 2 H n (x + y) =
n n k=0
8.959 1.
2. 3.
2.
k
√ √ H n−k x 2 H k y 2
MO 107a
Hermite polynomials satisfy the differential equation d2 u n dun + 2nun = 0; − 2x SM 566(9) 2 dx dx A second solution of this differential equation is provided by the functions (A and B are arbitrary constants): u2n = Ax Φ 12 − n; 32 ; x2 , u2n+1 = B Φ − 12 − n; 12 ; x2 MO 107
8.959(1)10 1.
MO 106a
Rodrigues’ formula and orthogonality relation
dn + −x2 , e dxn ∞ 0 −x2 e H m (x) H n (x) dx = π 1/2 2n n! −∞ 2
H n (x) = (−1)n ex
[Rodrigues’ formula] for m = n for m = n
MS 5.6.2 MS 5.6.2
8.96 Jacobi’s polynomials 8.960 1.
Definition n (−1)n −α −β d (1 − x) (1 − x)α+n (1 + x)β+n (1 + x) n n 2 n!
dx n n+β 1 n+α = n (x − 1)n−m (x + 1)m m n−m 2 m=0
P (α,β) (x) = n
EH II 169(10), CO EH II 169(2)
8.962
8.961
Jacobi’s polynomials
999
Functional relations:
1.11
P (α,α) (−x) = (−1)n P (α,α) (x) n n
2.
2(n + 1)(n + α + β + 1)(2n + α + β) P n+1 (x) = (2n + α + β + 1) (2n + α + β)(2n + α + β + 2)x + α2 − β 2 P (α,β) (x) n
EH II 169(13) (α,β)
(α,β)
−2(n + α)(n + β)(2n + α + β + 2) P n−1 (x) EH II 169(11)
3.
d (α,β) P (2n + α + β) 1 − x2 (x) = n[(α − β) − (2n + α + β)x] P (α,β) (x) n dx n (α,β) +2(n + α)(n + β) P n−1 (x) EH II 170(15) m
4.11
d dxm
+ , 1 Γ(n + m + α + β + 1) (α+m,β+m) P (α,β) P n−m (x) = m (x) n 2 Γ(n + α + β + 1) [m = 1, 2, . . . , n]
EH II 170(17) (α,β)
EH II 173(32)
6.
n + 12 α + 12 β + 1 (1 − x) P (α+1,β) (x) = (n + α + 1) P (α,β) (x) − (n + 1) P n+1 (x) n n (α,β) n + 12 α + 12 β + 1 (1 + x) P (α,β+1) (x) = (n + β + 1) P (α,β) (x) + (n + 1) P n+1 (x) n n
7.
(1 − x) P (α+1,β) (x) + (1 + x) P (α,β+1) (x) = 2 P (α,β) (x) n n n
EH II 173(34)
8.
(2n + α + β) P (α−1,β) (x) = (n + α + β) P (α,β) (x) − (n + β) P n−1 (x) n n
9.
(2n + α + β) P (α,β−1) (x) = (n + α + β) P (α,β) (x) + (n + α) P n−1 (x) n n
10.
P (α,β−1) (x) − P (α−1,β) (x) = P n−1 (x) n n
5.
8.962 1.
(α,β)
(α,β)
(α,β)
EH II 173(33)
EH II 173(35) EH II 173(36) EH II 173(37)
Connections with other functions:
(−1)n Γ(n + 1 + β) 1+x F n + α + β + 1, −n; 1 + β; n! Γ(1 + β)
2 Γ(n + 1 + α) 1−x = F n + α + β + 1, −n; 1 + α; n! Γ(1 + α) 2
n
Γ(n + 1 + α) 1 + x x−1 = −n, −n − β; α + 1; F n! Γ(1 + α) 2
x + 1
n Γ(n + 1 + β) x − 1 x+1 = F −n, −n − α; β + 1; n! Γ(1 + β) 2 x−1
P (α,β) (x) = n
2.
P n (x) = P (0,0) (x) n
3.
T n (x) =
CO, EH II 170(16) EH II 170(16) EH II 170(16) EH II 170(16) CO, EH II 179(3)
2
4.
22n (n!) (− 1 ,− 1 ) P n 2 2 (x) (2n)! Γ(n + 2ν) Γ ν + 12 ν P (ν−1/2,ν−1/2) C n (x) = (x) n Γ(2ν) Γ n + ν + 12
CO, EH II 184(5)a
MO 108a, EH II 174(4)
1000
Orthogonal Polynomials
8.963
8.963 The generating function: ∞ P (α,β) (x)z n = 2α+β R−1 (1 − z + R)−α (1 + z + R)−β , n n=0
R=
1 − 2xz + z 2
[|z| < 1] EH II 172(29)
8.964 The Jacobi polynomials constitute the unique rational solution of the differential (hypergeometric) equation EH II 169(14) 1 − x2 y + [β − α − (α + β + 2)x]y + n(n + α + β + 1)y = 0. 8.965 Asymptotic representation (cos θ) = P (α,β) n cos n + 12 (α + β + 1) θ − 12 α + 14 π [Im α = Im β = 0, 0 < θ < π] EH II 198(10) + O n−3/2 α+ 12 β+ 12 √ 1 1 cos 2 θ πn sin 2 θ 8.966 A limit relationship: + z , z −α J α (z) EH II 173(41) lim n−α P (α,β) cos = n n→∞ n 2 8.967 If α > −1 and β > −1, all the zeros of the polynomial P (α,β) (x) are simple, and they lie in the n interval (−1, 1).
8.97 The Laguerre polynomials 8.970
1 x −α dn −x n+α e x e x n! dx n
n n + α xm m = (−1) n − m m! m=0
Lα n (x) =
1.
[Rodrigues’ formula]
EH II 188(5), MO 108 MO 109, EH II 188(7)
L0n (x) = Ln (x)
2. 3.
Definition.
10
4.10 5.10 6.10 7.10
Lα 0 (x) Lα 1 (x)
=1
= −x + α + 1 1 2 x − 2(α + 2)x + (α + 1)(α + 2) Lα 2 (x) = 2 1 α L3 (x) = − x3 − 3(α + 3)x2 + 3(α + 2)(α + 3)x − (α + 1)(α + 2)(α + 3) 6 1 Lα (x) = x4 − 4(α + 4)x3 + 6(α + 3) (α + 4) x2 − 4(α + 2)(α + 3)(α + 4)x 4 24 + (α + 1)(α + 2)(α + 3)(α + 4)
8.10
Lα 5 (x) = −
1 x5 − 5(α + 5)x4 + 10(α + 4)(α + 5)x3 − 10(α + 3)(α + 4)(α + 5)x2 120
+ 5(α + 2)(α + 3)(α + 4)(α + 5)x − (α + 1)(α + 2)(α + 3)(α + 4)(α + 5)
ET I 369
8.973
8.971 1. 2.11 3.
The Laguerre polynomials
1001
Functional relations: d α α L (x) − Lα n+1 (x) = Ln (x) dx n α n Lα d α n (x) − (n + α) Ln−1 (x) Ln (x) = − Lα+1 n−1 (x) = dx x d α x Lα (x) = n Lα n (x) − (n + α) Ln−1 (x) dx n α = (n + 1) Lα n+1 (x) − (n + α + 1 − x) Ln (x)
EH II 189(16) EH II 189(15), SM 575(42)a
EH II 189(12), MO 109 α x Lα+1 (x) = (n + α + 1) Lα n n (x) − (n + 1) Ln+1 (x)
4.
α = (n + α) Lα n−1 (x) − (n − x) Ln (x)
SM 575(43)a, EH II 190(23)
Lnα−1 (x)
5. 6.
(n +
=
Lα n (x)
1) Lα n+1 (x)
−
Lα n−1 (x)
− (2n + α + 1 −
SM 575(44)a, EH II 190(24)
x) Lα n (x)
+ (n +
α) Lα n−1 (x)
=0
[n = 1, 2, . . .] 7.10 8.
10
α (n + α) Lnα−1 (x) = (n + 1) Lα n+1 (x) − (n + 1 − x) Ln (x)
n Lα n (x)
= (2n + α − 1 −
x) Lα n−1 (x)
− (n + α −
MS 5.5.2
1) Lα n−2 (x) [n = 2, 3, . . .]
8.972 1. 2. 3. 8.973
Connections with other functions:
n+α α Ln (x) = Φ(−n, α + 1; x) n 2 x H 2n (x) = (−1)n 22n n! L−1/2 n 2 x H 2n+1 (x) = (−1)n 22n+1 n!x L1/2 n
MO 109, EH II 190(25, 24)
MS 5.5.2
MO 109, FI II 189(14) EH II 193(2), SM 576(47) EH II 193(3), SM 577(48)
Special cases:
1.
Lα 0 (x) = 1
EH II 188(6)
2.
Lα 1 (x) = α + 1 − x
n+α α Ln (0) = n
EH II 188(6)
3.
xn n!
4.
n L−n n (x) = (−1)
5.
L1 (x) = 1 − x
6.
L2 (x) = 1 − 2x +
x2 2
EH II 189(13) MO 109
MO 109
1002
8.974 1.
Orthogonal Polynomials
8.974
Finite sums: n
α m! (n + 1)! α α α Lα Ln (x) Lα m (x) Lm (y) = n+1 (y) − Ln+1 (x) Ln (y) Γ(m + α + 1) Γ(n + α + 1)(x − y) m=0 EH II 188(9)
2.11
3.
n Γ(α − β + m) β Ln−m (x) = Lβn (x) Γ(α − β)m! m=0 n
MO 110, EH II 192(39)
α+1 Lα (x) m (x) = Ln
EH II 192(38)
β α+β+1 Lα (x + y) m (x) Ln−m (y) = Ln
EH II 192(41)
m=0
4.11
n m=0
8.975 1.
2.
Arbitrary functions: (1 − z)−α−1 exp e
−xz
α
(1 + z) =
∞ xz = Lα (x)z n z − 1 n=0 n ∞
Lnα−n (x)z n
[|z| < 1]
EH II 189(17), MO 109
[|z| < 1]
MO 110, EH II 189(19)
[α > −1]
EH II 189(18), MO 109
n=0
3. 8.976 1.
∞ √ 1 J α 2 xz ez (xz)− 2 α =
zn Lα n (x) Γ(n + α + 1) n=0
Other series of Laguerre polynomials:
√
1 α n xyz (xyz)− 2 α x+y Lα n (x) Ln (y)z = exp −z n! Iα 2 Γ(n + α + 1) 1−z 1−z 1−z n=0 ∞
[|z| < 1] 2.
∞ Lα n (x) = ex x−α Γ(α, x) n + 1 n=0
3.6
2 Lα n (x) =
Γ(n + α + 1) 22n n! n
k=0
4.6
α Lα n (x) Ln (y) =
[α > −1,
2n − 2k n−k
EH II 189(20)
x > 0]
EH II 215(19)
(2k)! 1 L2α (2x) k! Γ(α + k + 1) 2k
n α+2k Γ(1 + α + n) Ln−k (x + y) (xy)k n! Γ(1 + α + k) k!
MO 110
MO 110, EH II 192(42)
k=0
8.977 1.
Summation theorems: Lnα1 +α2 +···+αk +k−1 (x1 + x2 + · · · + xk ) =
αk α2 1 Lα i1 (x1 ) Li2 (x2 ) · · · Lik (xk )
MO 110
i1 +i2 +···+i2 =n
2.
y Lα n (x + y) = e
∞ (−1)k k=0
k!
y k Lnα+k (x)
MO 110
8.982
8.978 1. 2. 3.
The Laguerre polynomials
1003
Limit relations and asymptotic behavior:
2x (α,β) Lα (x) = lim P 1 − n n β→∞ β + x , √ 1 = x− 2 α J α 2 x lim n−α Lα n n→∞ n + √ 1 3 1 1 1 1 1 1 απ π , 2 x x− 2 α− 4 n 2 α− 4 cos 2 − Lα nx − + O n 2 α− 4 n (x) = √ e 2 4 π [Im α = 0,
EH II 191(35) EH II 191(36)
x > 0]
Laguerre polynomials satisfy the following differential equation: du d2 u + nu = 0 x 2 + (α − x + 1) dx dx 8.98011 Orthogonality relation ∞ 0, m = n α n+α e−x xα Lα n (x) Lm (x) dx = Γ(1 + α) , m=n 0 n
EH II 199(1)
8.979
8.98110
EH II 188(10), SM 574(34)
MS 5.5.2
Behavior of relative maxima of |Lα n (x)|
1.
Let α be arbitrary and real. The sequence formed by the relative maxima of |Lα n (x)| and by the value of this function at x = 0, is decreasing for x < α + 12 , and increasing for x > α + 12 . The successive relative maxima of |Lα n (x)| form a decreasing sequence for x ≤ 0, and an increasing SZ 174(7.6.1) sequence for x ≥ 0.
2.
Let α be an arbitrary real number. The successive relative maxima of −x/2 α/2+ 4 x |Lα e−x/2 x(α+1)/2 |Lα n (x)| and e n (x)| 1
form an increasing sequence, provided x > x0 . In the first case ⎧ ⎨0 if α2 ≤ 1, 2 x0 = ⎩ α −1 if α2 > 1 2n + α + 1 In the second case,
0 x0 = 2 1 12 α −4
if α2 ≤ q 14 , if α2 >
1 4
SZ 174(7.6.2)
In the first case, we take n so large that 2n + α + 1 > 0. 8.98210 1.
Asymptotic and limiting behavior of Lα n (x) Let α be arbitrary and real, c and w fixed positive constants, and let n → ∞. Then −α/2− 14 α/2− 14 if cn−1 ≤ qx ≤ qω O n x Lα n (x) = O (nα ) if 0 ≤ qx ≤ qcn−1 These bounds are precise as regards their orders in n. For α ≥ q − 12 , both bounds hold in both intervals, that is,
1004
Orthogonal Polynomials
Lα n (x) = 2.
1 1 x−α/2− 4 O nα/2− 4 , O (nα ) ,
0 < x ≤ qω,
Let α be arbitrary and real. Then for an arbitrary complex z −α/2 lim n−α Lα J α 2z 1/2 , n (x) = z n→∞
uniformly if z is bounded.
8.982
α≥q−
1 2
SZ 175(7.6.4)
SZ 191(8.1.3)
9.114
Integral representations
1005
9.1 Hypergeometric Functions 9.10 Definition 9.100
A hypergeometric series is a series of the form α(α + 1)β(β + 1) 2 α(α + 1)(α + 2)β(β + 1)(β + 2) 3 α·β z+ z + z + ... F (α, β; γ; z) = 1 + γ·1 γ(γ + 1) · 1 · 2 γ(γ + 1)(γ + 2) · 1 · 2 · 3 9.101 A hypergeometric series terminates if α or β is equal to a negative integer or to zero. For γ = −n (n = 0, 1, 2, . . .), the hypergeometric series is indeterminate if neither α nor β is equal to −m (where m < n and m is a natural number). However, 1.
lim
γ→−n
F (α, β; γ; z) α(α + 1) . . . (α + n)β(β + 1) . . . (β + n) = Γ(γ) (n + 1)! ×z n+1 F (α + n + 1, β + n + 1; n + 2; z) EH I 62(16)
9.102 If we exclude these values of the parameters α, β, γ, a hypergeometric series converges in the unit circle |z| Re(α + β − γ) ≥0. point z = 1.
2.
Re(α + β − γ) Re β > 0] B(β, γ − β) 0 2π z −n Γ(p)n! cos nt dt 9.1128 F p, n + p; n + 1; z 2 = 2π Γ(p + n) 0 (1 − 2z cos t + z 2 )p [n = 0, 1, 2, . . . ; p = 0, −1, −2, . . . ; |z| < 1]
WH
WH, MO 16
∞i 1 Γ(α + t) Γ(β + t) Γ(−t) Γ(γ) (−z)t dt Γ(α) Γ(β) 2πi −∞i Γ(γ + t) Here, |arg(−z)| < π and the path of integration are chosen in such a way that the poles of the functions Γ(α + t) and Γ(β + t) lie to the left of the path of integration and the poles of the function Γ(−t) lie to the right of it.
p+m (−2)m (p + m) π p+m ;1 − ; −1 = cosm ϕ cos pϕ dϕ 9.114 F −m, − 2 2 sin pπ 0 [m + 1 is a natural number; p = 0, ±1, . . . ] EH I 80(8), MO 16
9.113 F (α, β; γ; z) =
See also 3.194 1, 2, 5, 3.196 1, 3.197 6, 9, 3.259 3, 3.312 3, 3.518 4–6, 3.665 2, 3.671 1, 2, 3.681 1, 3.984 7.
1006
Hypergeometric Functions
9.121
9.12 Representation of elementary functions in terms of a hypergeometric functions 9.121 1.8 2. 3. 4. 5. 6. 7. 8.
F (−n, β; β; −z) = (1 + z)
n n − 1 1 z2 (t + z)n + (t − z)n ; ; 2 = F − ,− 2 2 2 t 2tn z z n = 1+ lim F −n, ω; 2ω; − ω→∞ t 2t
n − 1 n − 2 3 z2 (t + z)n − (t − z)n ,− ; ; 2 = F − 2 2 2 t 2nztn−1 z (t + z)n − tn = F 1 − n, 1; 2; − t nztn−1 ln(1 + z) F (1, 1; 2; −z) = z
1+z ln 1 3 2 1−z , 1; ; z = F 2 2 2z z z = 1 + z lim F 1, k; 2; lim F 1, k; 1; k→∞ k→∞ k k z z2 lim F 1, k; 3; = · · · = ez =1+z+ 2 k→∞ k n
9. 10.
lim F
k→∞ k →∞
lim F k→∞
k →∞
11. 12. 13. 14. 15. 16. 17.
lim F
k→∞ k →∞
2
1 z k, k ; ; 2 4kk 3 z2 k, k ; ; 2 4kk
z
GA 127 IIIa GA 127 IV GA 127 V GA 127 VI GA 127 VII
−z
=
e +e 2
= cosh z
GA 127 IX
=
sinh z ez − e−z = 2z z
GA 127 X
=
sin z z
z2 1 = cos z k, k ; ; − 2 4kk k →∞
1 1 3 z , ; ; sin2 z = F 2 2 2 sin z
3 z 2 F 1, 1; ; sin z = 2 sin z cos z
3 1 z , 1; ; − tan2 z = F 2 2 tan z
n+1 n−1 3 sin nz 2 ,− ; ; sin z = F 2 2 2 n sin z
n+2 n−2 3 sin nz ,− ; ; sin2 z = F 2 2 2 n sin z cos z lim F k→∞
GA 127 II
GA 127 VIII
z2 3 k, k ; ; − 2 4kk
EH I 101(4), GA 127 Ia
GA 127 XI
GA 127 XII
GA 127 XIII GA 127 XIV GA 127 XV GA 127 XVI GA 127 XVII
9.121
Elementary functions as hypergeometric function
18.
F
19.
F
20.
F
21.
F
22.
F
23.
F
24.
F
25.
F
26.
F
27.
F
28.
F
29.
F
30.
F
31.
F
32.
F
n−2 n−1 3 sin nz 2 ,− ; ; − tan z = − 2 2 2 n sin z cosn−1 z
n+2 n+1 3 sin nz cosn+1 z , ; ; − tan2 z = 2 2 2 n sin z
n n 1 , − ; ; sin2 z = cos nz 2 2 2
n+1 n−1 1 cos nz ,− ; ; sin2 z = 2 2 2 cos z
n n−1 1 cos nz ; ; − tan2 z = − ,− 2 2 2 cosn z
n+1 n 1 , ; ; − tan2 z = cos nz cosn z 2 2 2
1 1 , 1; 2; 4z(1 − z) = 2 1−z
1 , 1; 1; sin2 z = sec z 2
1 1 3 2 arcsin z , ; ;z = 2 2 2 z
1 3 arctan z , 1; ; −z 2 = 2 2 z
1 1 3 arcsinh z , ; ; −z 2 = 2 2 2 z
1+n 1−n 3 2 sin (n arcsin z) , ; ;z = 2 2 2 nz
n n 3 sin (n arcsin z) √ 1 + , 1 − ; ; z2 = 2 2 2 nz 1 − z 2
n n 1 2 ,− ; ;z = cos (n arcsin z) 2 2 2
1+n 1−n 1 2 cos (n arcsin z) √ , ; ;z = 2 2 2 1 − z2
1007
GA 127 XVIII GA 127 XIX EH I 101(11), GA 127 XX EH I 101(11), GA 127 XXI EH I 101(11), GA 127 XXII GA 127 XXIII
|z| ≤ 12 ;
|z(1 − z)| ≤
1 4
(cf. 9.121 13) (cf. 9.121 15) (cf. 9.121 26) (cf. 9.121 16) (cf. 9.121 17) (cf. 9.121 20) (cf. 9.121 21)
The representation of special functions in terms of a hypergeometric function: • for complete elliptic integrals, see 8.113 1 and 8.114 1; • for integrals of Bessel functions, see 6.574 1, 3, 6.576 2–5, 6.621 1–3; • for Legendre polynomials, see 8.911 and 8.916. (All these hypergeometric series terminate; that is, these series are finite sums); • for Legendre functions, see 8.820 and 8.837; • for associated Legendre functions, see 8.702, 8.703, 8.751, 8.77, 8.852, and 8.853; • for Chebyshev polynomials, see 8.942 1; • for Jacobi’s polynomials, see 8.962;
1008
Hypergeometric Functions
9.122
• for Gegenbauer polynomials, see 8.932; • for integrals of parabolic cylinder functions, see 7.725 6. 9.122 1.
Particular values: F (α, β; γ; 1) =
Γ(γ) Γ(γ − α − β) Γ(γ − α) Γ(γ − β)
[Re γ > Re(α + β)] GA 147(48), FI II 793
2.
3.
F (α, β; γ; 1) = F (−α, −β; γ − α − β; 1) 1 = F (−α, β; γ − α; 1) 1 = F (α, −β; γ − β; 1)
3 1 π = F 1, 1; ; 2 2 2
[Re γ > Re(α + β)]
GA 148(49)
[Re γ > Re(α + β)]
GA 148(50)
[Re γ > Re(α + β)]
GA 148(51)
(cf. 9.121 14)
9.13 Transformation formulas and the analytic continuation of functions defined by hypergeometric series 9.130 The series F (α, β; γ; z) defines an analytic function that, speaking generally, has singularities at the points z = 0, 1, and ∞. (In the general case, there are branch points.) We make a cut in the z-plane along the real axis from z = 1 to z = ∞; that is, we require that |arg(−z)| < π for |z| ≥1. Then, the series f (α, β; γ; z) will, in the cut plane, yield a single-valued analytic continuation, which we can obtain by means of the formulas below (provided γ + 1 is not a natural number and α − β and γ − α − β are not integers). These formulas make it possible to calculate the values of F in the given region, even in the case in which |z| > 1. There are other closely related transformation formulas that can also be used to get the analytic continuation when the corresponding relationships hold between α, β, γ. Transformation formulas 9.131 1.11
z z − 1
z −β = (1 − z) F β, γ − α; γ; z−1 −α
F (α, β; γ; z) = (1 − z)
F
α, γ − β; γ;
GA 218(91) GA 218(92)
= (1 − z)γ−α−β F (γ − α, γ − β; γ; z) 2.
F (α, β; γ; z) =
Γ(γ) Γ(γ − α − β) F (α, β; α + β − γ + 1; 1 − z) Γ(γ − α) Γ(γ − β) Γ(γ) Γ(α + β − γ) +(1 − z)γ−α−β F (γ − α, γ − β; γ − α − β + 1; 1 − z) Γ(α) Γ(β) EH I 94, MO 13
9.136
Transformation formulas for hypergeometric series
9.132 1.
1009
(1 − z)−α Γ(γ) Γ(β − α) 1 F (α, β; γ; z) = F α, γ − β; α − β + 1; Γ(β) Γ(γ − α) 1−z
1 −β Γ(γ) Γ(α − β) +(1 − z) F β, γ − α; β − α + 1; Γ(α) Γ(γ − β) 1−z
Γ(γ) Γ(β − α) 1 −α (−z) F α, α + 1 − γ; α + 1 − β; F (α, β; γ; z) = Γ(β) Γ(γ − α) z
Γ(γ) Γ(α − β) 1 (−z)−β F β, β + 1 − γ; β + 1 − α; + Γ(α) Γ(γ − β) z [|arg z| < π, α − β = ±m, m = 0, 1, 2, . . .]
2.11
F 2α, 2β; α + β + 12 ; z = F α, β; α + β + 12 ; 4z(1 − z) |z| ≤ 12 ,
9.133 9.134
|z(1 − z)| ≤
2 z 1 α α+1 1. , ;β + ; 2 2 2 2−z
4z 1 −2α 2. F (2α, 2α + 1 − γ; γ; z) = (1 + z) F α, α + ; γ; 2 (1 + z)2
1 2 1 4z 3. = (1 + z)−2α F α, β; 2β; F α, α + − β; β + ; z 2 2 (1 + z)2
1 1 2 2 ϕ 9.135 F α, β; α + β + ; sin ϕ = F 2α, 2β; α + β + ; sin 2 2 % 2
z −α F (α, β; 2β; z) = 1 − F 2
1 4
MO 13
GA 220(93)
WH
ϕ real; x = sin 2 2
MO 13, EH I 111(4)
GA 225(100) GA 225(101)
& √ 1− 2 1 <x< 2 2 MO 13
8
9.136
then
We set
√ Γ α + β + 12 π , A= 1 Γ α + 2 Γ β + 12
√ − Γ α + β + 12 2 π ; B= Γ(α) Γ(β)
1.
√
√ 1 1− z 1 3 1 1 = AF α, β; ; z + B z F α + , β + ; ; z F 2α, 2β; α + β + ; 2 2 2 2 2 2
2.
√
√ 1 3 1 1+ z 1 1 = AF α, β; ; z − B z F α + , β + ; ; z F 2α, 2β; α + β + ; 2 2 2 2 2 2
3.
1
α− 2 β− α + β − 12
1 2
√ A zF
√
1 1+ z 3 α, β; ; z = F 2α − 1, 2β − 1; α + β − ; 2 2 2 √
1 1− z − F 2α − 1, 2β − 1; α + β − ; 2 2
GA 227(106)
GA 227(107)
GA 229(110)
1010
9.1377
Hypergeometric Functions
9.137
Gauss’ recursion functions:
1.
γ[γ − 1 − (2γ − α − β − 1)z] F (α, β; γ; z) + (γ − α)(γ − β)z F (α, β; γ + 1; z) + γ(γ − 1)(z − 1) F (α, β; γ − 1; z) = 0
2.
(2α − γ − αz + βz) F (α, β; γ; z) + (γ − α) F (α − 1, β; γ; z) + α(z − 1) F (α + 1, β; γ; z) = 0
3.
(2β − γ − βz + αz) F (α, β; γ; z) + (γ − β) F (α, β − 1; γ; z) + β(z − 1) F (α, β + 1; γ; z) = 0 γ F (α, β − 1; γ; z) − γ F (α − 1, β; γ; z) + (α − β)z F (α, β; γ + 1; z) = 0
4. 5.
γ(α − β) F (α, β; γ; z) − α(γ − β) F (α + 1, β; γ + 1; z) + β(γ − α) F (α, β + 1; γ + 1; z) = 0
6.
γ(γ + 1) F (α, β; γ; z) − γ(γ + 1) F (α, β; γ + 1; z) − αβz F (α + 1, β + 1; γ + 2; z) = 0
7.
γ F (α, β; γ; z) − (γ − α) F (α, β + 1; γ + 1; z) − α(1 − z) F (α + 1, β + 1; γ + 1; z) = 0
8.
γ F (α, β; γ; z) + (β − γ) F (α + 1, β; γ + 1; z) − β(1 − z) F (α + 1, β + 1; γ + 1; z) = 0
9.
γ(γ − βz − α) F (α, β; γ; z) − γ(γ − α) F (α − 1, β; γ; z) + αβz(1 − z) F (α + 1, β + 1; γ + 1; z) = 0
10.
γ(γ − αz − β) F (α, β; γ; z) − γ(γ − β) F (α, β − 1; γ; z) + αβz(1 − z) F (α + 1, β + 1; γ + 1; z) = 0
11.
γ F (α, β; γ; z) − γ F (α, β + 1; γ; z) + αz F (α + 1, β + 1; γ + 1; z) = 0
12.8
γ F (α, β; γ; z) − γ F (α + 1, β; γ; z) + βz F (α + 1, β + 1; γ + 1; z) = 0
13.
γ[α − (γ − β)z] F (α, β; γ; z) − αγ(1 − z) F (α + 1, β; γ; z) + (γ − α)(γ − β)z F (α, β; γ + 1; z) = 0
14.
γ[β − (γ − α)z] F (α, β; γ; z) − βγ(1 − z) F (α, β + 1; γ; z) + (γ − α)(γ − β)z F (α, β; γ + 1; z) = 0
8
15.
γ(γ + 1) F (α, β; γ; z) − γ(γ + 1) F (α, β + 1; γ + 1; z) + α(γ − β)z F (α + 1, β + 1; γ + 2; z) = 0
16.
γ(γ + 1) F (α, β; γ; z) − γ(γ + 1) F (α + 1, β; γ + 1; z) + β(γ − α)z F (α + 1, β + 1; γ + 2; z) = 0
17.
γ F (α, β; γ; z) − (γ − β) F (α, β; γ + 1; z) − β F (α, β + 1; γ + 1; z) = 0
8
8
18.
γ F (α, β; γ; z) − (γ − α) F (α, β; γ + 1; z) − α F (α + 1, β; γ + 1; z) = 0
MO 13–14
9.14 A generalized hypergeometric series The series 1.
pF q
(α1 , α2 , . . . , αp ; β1 , β2 , . . . , βq ; z) =
∞ (α1 )k (α2 )k . . . (αp )k z k (β1 )k (β2 )k . . . (βq )k k!
MO 14
k=0
is called a generalized hypergeometric series (see also 9.210). 2.
2 F 1 (α, β; γ; z)
≡ F (α, β; γ; z)
MO 15
For integral representations, see 3.254 2, 3.259 2, and 3.478 3.
9.15 The hypergeometric differential equation 9.151
A hypergeometric series is one of the solutions of the differential equation du d2 u − αβu = 0, z(1 − z) 2 + [γ − (α + β + 1)z] dz dz which is called the hypergeometric equation.
WH
9.153
The hypergeometric differential equation
1011
The solution of the hypergeometric differential equation 9.152 The hypergeometric differential equation 9.151 possesses two linearly independent solutions. These solutions have analytic continuations to the entire z-plane, except possibly for the three points 0, 1, and ∞. Generally speaking, the points z = 0, 1, ∞ are branch points of at least one of the branches of each solution of the hypergeometric differential equation. The ratio w(z) of two linearly independent solutions satisfies the differential equation 2 w 1 − a21 1 − a22 a2 + a22 − a23 − 1 w , = + + 1 2 −3 2 2 w w z (z − 1) z(z − 1) where a21 = (1 − γ)2 , a22 = (γ − α − β)2 , a23 = (α − β)2 . If α, β, γ are real, the function w(z) maps the upper (Im z > 0) or the lower (Im z 0. However, if α − γ − m ≤0, the right member of this expression is a polynomial taken to the (1 − z)th power. 4.
If α, β, and γ are integers, the hypergeometric differential equation always has a solution that is regular for z = 0 and that is of the form R1 (z) + ln(1 − z)R2 (z), where R1 (z) and R2 (z) are rational functions of z. To get a solution of this form, we need to apply formulas 9.137 1–9.137 3 to the function F (α, β; γ; z). However, if γ = −λ, where λ + 1 is a natural number, formulas 9.137 1 and 9.137 2 should be applied not to F (α, β; γ; z) but to the function z λ+1 F (α + λ + 1, β + λ + 1; λ + 2, z). By successive applications of these formulas, we can reduce the positive values of the parameters to the pair, unity and zero. Furthermore, we can obtain the desired form of the solution from the formulas −1 F (1, 1; 2; z) = −z ln(1 − z), F (0, β; γ; z) = F (α, 0; γ; z) = 1
MO 19–20
1014
Hypergeometric Functions
9.160
9.16 Riemann’s differential equation 9.160 1.11
The hypergeometric differential equation is a particular case of Riemann’s differential equation 1 − α − α 1 − β − β 1 − γ − γ du d2 u + + + dz 2 z−b z−c dz ⎤ ⎡z − a ββ γγ u αα (a − b)(a − c) (b − c)(b − a) (c − a)(c − b) ⎦ + =0 +⎣ z−a z−b z−c (z − a)(z − b)(z − c) WH
The coefficients of this equation have poles at the points a, b, and c, and the numbers α, α ; β, β ; γ, γ are called the indices corresponding to these poles. The indices α, α ; β, β ; γ, γ are related by the following equation: α + α + β + β + γ + γ − 1 = 0 2. 3.
WH
The differential equations 9.160 1 are written diagramatically as follows: ⎧ ⎫ ⎨a b c ⎬ u=P α β γ z ⎩ ⎭ α β γ
The singular points of the equation appear in the first row in this scheme, the indices corresponding to WH them appear beneath them, and the independent variable appears in the fourth column. 9.161 The two following transformation formulas are valid for Riemann’s P -equation: ⎧ ⎫ ⎧ ⎫
k
l ⎨ a b c a b c ⎬ ⎨ ⎬ z−a z−c P α β γ z =P α+k β−k−1 γ+l z WH 1. ⎩ ⎭ ⎩ ⎭ z−b z−b α β γ α + k β − k − l γ + l ⎧ ⎫ ⎧ ⎫ ⎨a b c ⎬ ⎨a1 b1 c1 ⎬ 2. P α β γ z = P α β γ z1 WH ⎩ ⎭ ⎩ ⎭ α β γ α β γ The first of these formulas means that if
⎧ ⎨a u=P α ⎩ α
then the function
b β β
c γ γ
⎫ ⎬ z
⎭
,
k
l z−a z−c u z−b z−b satisfies a second-order differential equation having the same singular points as equation 9.161 2 and indices equal to α + k, α + k; β − k − l, β − k − l; γ + l, γ + l. The second transformation formula converts a differential equation with singularities at the points a,b, and c, indices α, α ; β, β ; γ, γ , and an independent variable z into a differential equation with the same indices, singular points a1 , b1 , and c1 , and independent variable z1 . The variable z1 is connected with the variable z by the fractional transformation u1 =
9.164
Riemann’s differential equation
1015
Az1 + B [AD − BC = 0] Cz1 + D The same transformation connects the points a1 , b1 , and c1 with the points a, b, and c. z=
WH, MO 20
9.162 By the successive application of the two transformation formulas 9.161 1 and 9.161 2, we can convert Riemann’s differential equation into the hypergeometric differential equation. Thus, the solution of Riemann’s differential equation can be expressed in terms of a hypergeometric function. For k = −α, l = −γ, and z1 = (z−a)(c−b) (z−b)(c−a) , we have 1.
⎧ ⎧ ⎫ b ⎨a b c ⎬ z − a α z − c γ ⎨ a 0 β+α+γ u= P α β γ z = P ⎩ ⎩ ⎭ z−b z−b α β γ α − α β + α + γ ⎧ ⎫
α
γ ⎨ 0 ∞ 1 ⎬ z−a z−c (z−a)(c−b) 0 β+α+γ 0 = P (z−b)(c−a) ⎭ ⎩ z−b z−b α − α β + α + γ γ − γ
c 0 γ − γ
⎫ ⎬ z
⎭
MO 23
2.
Thus, this solution can be expressed as a hypergeometric series as follows:
α
γ
z−a z−c (z − a)(c − b) u= F α + β + γ, α + β + γ; 1 + α − α ; z−b z−b (z − b)(c − a)
If the constants a, b, c; α, α ; β, β ; γ, γ are permuted in a suitable manner, Riemann’s equation remains unchanged. Thus, we obtain a set of 24 solutions of differential equations having the following form WH, MO 23 (provided none of the differences α − α , β − β , γ − γ is an integer): 9.163
1.
u1
2.
u2
3.
u3
4.
u4
9.164 1.10 2. 3.
α
γ z−c (c − b)(z − a) = F α + β + γ, α + β + γ; 1 + α − α ; z−b (c − a)(z − b)
α
γ z−a z−c (c − b)(z − a) = F α + β + γ, α + β + γ; 1 + α − α; z−b z−b (c − a)(z − b)
α
γ z−a z−c (c − b)(z − a) = , α + β + γ ; 1 + α − α ; α + β + γ F z−b z−b (c − a)(z − b)
α
γ z−a z−c (c − b)(z − a) = F α + β + γ , α + β + γ; 1 + α − α; z−b z−b (c − a)(z − b)
z−a z−b
β
α z−a (a − c)(z − b) F β + γ + α, β + γ + α; 1 + β − β ; z−c (a − b)(z − c)
β
α z−b z−a (a − c)(z − b) u6 = + γ + α, β + γ + α; 1 + β − β; β F z−c z−c (a − b)(z − c)
β
α z−b z−a (a − c)(z − b) u7 = F β + γ + α ,β + γ + α ;1 + β − β ; z−c z−c (a − b)(z − c) u5 =
z−b z−c
1016
Hypergeometric Functions
4. 9.165 1. 2. 3. 4.
u8 =
z−b z−c
β
z−a z−c
α
(a − c)(z − b) F β + γ + α , β + α + γ ; 1 + β − β; (a − b)(z − c)
γ
β z−c z−b (b − a)(z − c) u9 = F γ + α + β, γ + α + β; 1 + γ − γ ; z−a z−a (b − c)(z − a)
γ
β z−c z−b (b − a)(z − c) u10 = + α + β, γ + α + β; 1 + γ − γ; γ F z−a z−a (b − c)(z − a)
γ
β z−c z−b (b − a)(z − c) u11 = F γ + α + β ,γ + α + β ;1 + γ − γ ; z−a z−a (b − c)(z − a)
γ
β z−c z−b (b − a)(z − c) u12 = F γ + α + β , γ + α + β ; 1 + γ − γ; z−a z−a (b − c)(z − a)
9.166
u13 =
2.
u14
3.
u15
4.
u16
9.167
z−a z−c
γ
α z−a (a − b)(z − c) F γ + β + α, γ + β + α; 1 + γ − γ ; z−b (a − c)(z − b)
γ
α z−c z−a (a − b)(z − c) = + β + α, γ + β + α; 1 + γ − γ; γ F z−b z−b (a − c)(z − b)
γ
α z−c z−a (a − b)(z − c) = F γ + β + α ,γ + β + α ;1 + γ − γ ; z−b z−b (a − c)(z − b)
γ
α z−c z−a (a − b)(z − c) = F γ + β + α , γ + β + α ; 1 + γ − γ; z−b z−b (a − c)(z − b)
1.
u17 =
2.
u18
3.
u19
4.
u20
9.168
α
β z−b (b − c)(z − a) + β; 1 + α − α ; α + γ + β, α + γ F z−c (b − a)(z − c)
α
β z−a z−b (b − c)(z − a) = F α + γ + β, α + γ + β; 1 + α − α; z−c z−c (b − a)(z − c)
α
β z−a z−b (b − c)(z − a) = F α + γ + β ,α + γ + β ;1 + α − α ; z−c z−c (b − a)(z − c)
α
β z−a z−b (b − c)(z − a) = F α + γ + β , α + γ + β ; 1 + α − α; z−c z−c (b − a)(z − c)
1.
z−c z−b
β
γ z−c (c − a)(z − b) + γ; 1 + β − β ; β + α + γ, β + α F z−a (c − b)(z − a)
β
γ z−b z−c (c − a)(z − b) = F β + α + γ, β + α + γ; 1 + β − β; z−a z−a (c − b)(z − a)
1.
u21 =
2.
u22
z−b z−a
9.165
9.175
Second-order differential equations
3.
u23
4.
u24
β
γ z−c (c − a)(z − b) = F β + α + γ ,β + α + γ ;1 + β − β ; z−a (c − b)(z − a)
β
γ z−b z−c (c − a)(z − b) = F β + α + γ , β + α + γ ; 1 + β − β; z−a z−a (c − b)(z − a) z−b z−a
1017
WH
9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme 9.171
The hypergeometric equation (see 9.151): ⎧ ⎫ ∞ 1 ⎨ 0 ⎬ 0 α 0 z WH u=P ⎩ ⎭ 1−γ β γ−α−β 9.172 The associated Legendre’s equation defining the functions P m n (z) for n and m integers (see 8.700 1): ⎧ ⎫ ∞ 1 ⎪ ⎪ ⎨ 0 ⎬ 1−z 1 1 WH 1. u=P m n + 1 m 2 2 ⎪ 2 ⎪ ⎩ 1 ⎭ 1 − 2 m −n − 2 m ⎫ ⎧ 0 ∞ 1 ⎪ ⎪ ⎪ ⎬ ⎨ 1 1 ⎪ 1 0 2m 2. u = P −2n WH 2 1 − z ⎪ ⎪ ⎪ ⎪ ⎭ ⎩n + 1 −1m 1 2 2 2
z2 m 9.173 The function P n 1 − 2 satisfies the equation 2n ⎧ 2 ⎫ ∞ 0 ⎨ 4n ⎬ 1 1 2 m n + 1 m z WH u=P 2 ⎩ 21 ⎭ − 2 m −n − 12 m The function J m (z) satisfies the limiting form of this equation obtained as n → ∞. 9.174 The equation defining the Gegenbauer polynomials C λn (z) (see 8.938): ⎧ ⎫ ∞ 1 ⎨ −1 ⎬ WH u = P 12 − λ n + 2λ 12 − λ z ⎩ ⎭ 0 −n 0 9.175 Bessel’s equation (see 8.401) is the limiting form of the equations: ⎧ ⎫ ∞ c ⎨ 0 ⎬ 1 n ic + ic z WH 1. u=P 2 ⎩ ⎭ −n −ic 12 − ic ⎧ ⎫ ∞ c ⎨ 0 ⎬ 1 n 0 z 2. u = eiz P WH 2 ⎩ ⎭ −n 32 − 2ic 2ic − 1 ⎧ ⎫ ∞ c2 ⎨ 0 ⎬ 1 1 2 n (c − n) 0 z 3. u=P WH 2 ⎩ 21 ⎭ − 2 n − 12 (c + n) n + 1 as c → ∞.
1018
Hypergeometric Functions
9.180
9.18 Hypergeometric functions of two variables 9.180 1.
F 1 (α, β, β , γ; x, y) =
∞ ∞ (α)m+n (β)m (β )n m n x y (γ)m+n m!n! m=0 n=0
[|x| < 1,
|y| < 1] EH I 224(6), AK 14(11)
2.
F 2 (α, β, β , γ, γ ; x, y) =
∞ ∞
(α)m+n (β)m (β )n m n x y (γ)m (γ )n m!n! m=0 n=0 [|x| + |y| < 1]
3.
F 3 (α, α , β, β , γ; x, y) =
∞ ∞
EH I 224(7), AK 14(12)
(α)m (α )n (β)m (β )n m n x y (γ)m+n m!n! m=0 n=0 [|x| < 1,
|y| < 1] EH I 224(8), AK 14(13)
4.
F 4 (α, β, γ, γ ; x, y) =
∞ ∞ (α)m+n (β)m+n m n x y (γ)m (γ )n m!n! m=0 n=0
!√ ! √ ! x! + | y| < 1 EH I 224(9), AK 14(14)
9.181 for z:
The functions F 1 , F 2 , F 3 , and F 4 satisfy the following systems of partial differential equations
1.
System of equations for z = F 1 :
2.
∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, + y(1 − x) 2 ∂x ∂x ∂y ∂x ∂y ∂2z ∂2z ∂z ∂z + [γ − (α + β + 1) y] − βx − αβ z = 0 y(1 − y) 2 + x(1 − y) ∂y ∂x ∂y ∂x ∂x System of equations for z = F 2 :
3.
∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, − xy 2 ∂x ∂x ∂y ∂x ∂y ∂2z ∂2z ∂z ∂z + [γ − (α + β + 1) y] − βx − αβ z = 0 y(1 − y) 2 − xy ∂y ∂x ∂y ∂y ∂x System of equations for z = F 3 :
x(1 − x)
x(1 − x)
EH I 233(9)
EH I 234(10)
∂z ∂2z ∂2z + [γ − (α + β + 1)x] − αβz = 0, + y 2 ∂x ∂x ∂y ∂x ∂2z ∂2z ∂z y(1 − y) 2 + x + [γ − (α + β + 1) y] − α β z = 0 ∂y ∂x ∂y ∂y x(1 − x)
EH I 234(11)
9.182
4.
Hypergeometric functions of two variables
1019
System of equations for z = F 4 : x(1 − x)
2 ∂z ∂z ∂2z ∂2z 2∂ z + [γ − (α + β + 1)x] − (α + β + 1)y − αβz = 0, − y − 2xy 2 2 ∂x ∂y ∂x ∂y ∂x ∂y
EH I 234(12)
y(1 − y)
2
2
2
∂z ∂ z ∂ z ∂ z ∂z + [γ − (α + β + 1)y] − (α + β + 1)x − αβz = 0 − x2 2 − 2xy ∂y 2 ∂x ∂x ∂y ∂y ∂x AK 44
9.182 For certain relationships between the parameters and the argument, hypergeometric functions of two variables can be expressed in terms of hypergeometric functions of a single variable or in terms of elementary functions:
−α x−y EH I 238(1), AK 24(28) 1. F 1 (α, β, β , β + β ; x, y) = (1 − y) F α, β; β + β ; 1−y
y −α 2. (α, β, β , β, γ ; x, y) = (1 − x) ; γ ; EH I 238(2), AK 23 α, β F2 F 1−x
xy −β −β 3. EH I 238(3) F 2 (α, β, β , α, α; x, y) = (1 − x) (1 − y) F β, β ; α; (1 − x)(1 − y) 4. 5.
F 3 (α, γ − α, β, γ − β, γ; x, y) = (1 − y)
α+β−γ
F (α, β; γ; x + y − xy)
EH I 238(4), AK 25(35)
F 4 (α, γ + γ − α − 1, γ, γ ; x(1 − y), y(1 − x)) = F (α, γ + γ − α − 1; γ; x) F (α, γ + γ − α − 1; γ ; y)
6. 7.
−y x (1 − x)β (1 − y)α , EH I 238(6) = F 4 α, β, α, β; − (1 − x)(1 − y) (1 − x)(1 − y) (1 − xy)
y x ,− = (1 − x)α (1 − y)α F (α, 1 + α − β; β; xy) F 4 α, β, β, β; − (1 − x)(1 − y) (1 − x)(1 − y)
8.
EH I 238(5)
F 4 α, β, 1 + α − β, β; −
y x ,− (1 − x)(1 − y) (1 − x)(1 − y)
EH I 238(7)
x(1 − y) = (1 − y) F α, β; 1 + α − β; − 1−x α
9.
1 1 1 √ −2α F 4 α, α + , γ, ; x, y = (1 + y) F 2 2 2
EH I 238(8)
x 1 α, α + ; γ; √ 2 2 1+ y 1 x 1 √ −2α + (1 − y) F α, α + ; γ; √ 2 2 2 1− y AK 23
10.
F 1 (α, β, β , γ; x, 1) =
Γ(γ) Γ (γ − α − β ) F (α, β : γ − β ; x) Γ(γ − α) Γ (γ − β )
EH I 239(10), AK 22(23)
1020
11. 9.183 1.
Hypergeometric Functions
9.183
F 1 (α, β, β , γ; x, x) = F (α, β + β ; γ; x)
EH I 239(11), AK 23(25)
Functional relations between hypergeometric functions of two variables:
y x −β −β , F 1 (α, β, β , γ; x, y) = (1 − x) (1 − y) F 1 γ − α, β, β , γ; x−1 y−1 = (1 − x)−α F 1 α, γ − β − β , β , γ;
y−x x , x−1 1−x
EH I 239(1)
EH I 239(2)
y y−x −α , = (1 − y) F 1 α, β, γ − β − β , γ; y−1 y−1
= (1 − x)γ−α−β (1 − y)−β F 1 γ − α, γ − β − β , β , γ; x,
= (1 − x)−β (1 − y)γ−α−β F 1 γ − α, β, γ − β − β , γ;
x−y 1−y
x−y ,y x−1
EH I 239(3)
EH I 240(4)
EH I 240(5), AK 30(5)
2.8
−α F 2 (α, β, β , γ, γ ; x, y) = (1 − x) F 2 α, γ − β, β , γ, γ ;
= (1 − y)−α F 2 α, β, γ − β , γ, γ ;
= (1 − x − y)−α F 2
y x , x−1 1−x
y x , 1−y y−1
EH I 240(6)
EH I 240(7)
y x , α, γ − β, γ − β , γ, γ ; x+y−1 x+y−1
EH I 240(8), AK 32(6)
3.7
Γ(γ ) Γ(β − α) x 1 −α (−y) F 4 α, α + 1 − γ , γ, α + 1 − β; , F 4 (α, β, γ, γ ; x, y) = Γ (γ − α) Γ(β) y y
x 1 Γ(γ (Γ(α − β) β (−y) F 4 β + 1 − γ , β, γ, β + 1 − α; , + Γ (γ − β) Γ(α) y y EH I 240(9), AK 26(37)
9.185
9.184 1.
Hypergeometric functions of two variables
Integral representations: Double integrals of the Euler type F 1 (α, β, β , γ; x, y) =
Γ(γ) ) Γ (γ − β − β ) Γ(β) Γ (β −α uβ−1 v β −1 (1 − u − v)γ−β−β −1 (1 − ux − vy) du dv × u≥0,v≥0 u+v≤1
Re β > 0,
[Re β > 0, 2.
F 2 (α, β, β , γ, γ ; x, y) =
EH I 230(1), AK 28(1)
Γ(γ) Γ (γ ) − β) Γ (γ − β )
1 1 uβ−1 v β −1 (1 − u)γ−β−1 (1 − v)γ −β −1 (1 − ux − vy)−α du dv × 0
0
Re β > 0,
Re (γ − β ) > 0]
Re (γ − β) > 0,
EH I 230(2), AK 28(2)
F 3 (α, α , β, β , γ; x, y)
=
Γ(γ) Γ(β) Γ (β ) Γ (γ
×
u≥0,v≥0 u+v≤1
− β − β)
uβ−1 v β
−1
(1 − u − v)−γ−β−β
Re β > 0,
[Re β > 0, 4.
Re (γ − β − β ) > 0]
Γ(β) Γ (β ) Γ(γ
[Re β > 0, 3.
1021
−1
(1 − ux)−α (1 − vy)−α du dv
Re (γ − β − β ) > 0]
EH I 230(3), AK 28(3)
F 4 (α, β, γ, γ ; x(1 − y), y(1 − x)) 1 1 Γ(γ) Γ (γ ) uα−1 v β−1 (1 − u)γ−α−1 (1 − v)γ −β−1 = Γ(α) Γ(β) Γ(γ − α) Γ (γ − β) 0 0
×(1 − ux)α−γ−γ [Re α > 0,
+1
(1 − vy)β−γ−γ
Re β > 0,
+1
(1 − ux − vy)γ+γ
Re (γ − α) > 0,
−α−β−1
Re (γ − β) > 0]
du dv EH I 230(4)
9.185 Integral representations: Integrals of the Mellin–Barnes type The functions F 1 , F 2 , F 3 , and F 4 can be represented by means of double integrals of the following form: i∞ i∞ Γ(γ) Ψ(s, t) Γ(−s) Γ(−t)(−x)s (−y)t ds dt F (x, y) = Γ(α) Γ(β)(2πi)2 −i∞ −i∞
1022
Confluent Hypergeometric Functions
9.201
Ψ(s, t)
F (x, y)
Γ(α + s + t) Γ(β + s) Γ (β + t) Γ (β ) Γ(γ + s + t)
F1 (α, β, β , γ; x, y)
Γ(α + s + t) Γ(β + s) Γ (β + t) Γ (γ ) Γ (β ) Γ(γ + s) Γ (γ + t)
F2 (α, β, β , γ, γ ; x, y)
Γ(α + s) Γ (α + t) Γ(β + s) Γ (β + t) Γ (α ) Γ (β ) Γ(γ + s + t)
F 3 (α, α , β, β , γ; x, y)
Γ(α + s + t) Γ(β + s + t) Γ (γ ) Γ(γ + s) Γ (γ + t) [α, α , β, β may not be negative integers]
F4 (α, β, γ, γ ; x, y) EH I 232(9–13), AK 41(33)
9.19 A hypergeometric function of several variables F A (α; β1 , . . . , βn ; γ1 , . . . , γn ; z1 , . . . , zn ) ∞ ∞ ∞ (α)m1 +···+mn (β1 )m1 · · · (βn )mn m1 m2 z1 z2 · · · znmn ... = (γ ) · · · (γ ) m ! · · · m ! 1 n 1 n m1 mn m =0m =0 m =0 1
2
n
ET I 385
9.2 Confluent Hypergeometric Functions 9.20 Introduction 9.20110 A confluent hypergeometric function is obtained by taking the limit as c → ∞ in the solution of Riemann’s differential equation ⎧ ⎫ ∞ c ⎨ 0 ⎬ WH u = P 12 + μ −c c − λ z ⎩1 ⎭ − μ 0 λ 2 9.202 The equation obtained by means of this limiting process is of the form 1 2
λ d2 u du 4 −μ 1. + + + WH u=0 dz 2 dz z z2 Equation 9.202 1 has the following two linearly independent solutions: 1 2. z 2 +μ e−z Φ 12 + μ − λ, 2μ + 1; z 1 3. z 2 −μ e−z Φ 12 − μ − λ, −2μ + 1; z which are defined for all values of μ = ± 12 , ± 22 , ± 32 , . . .
MO 111
9.214
The functions Φ(α, γ; z) and Ψ(α, γ; z)
1023
9.21 The functions Φ(α, γ; z) and Ψ(α, γ; z) 9.21010 1.
The series α(α + 1) z 2 α(α + 1)(α + 2) z 3 αz + + + ... γ 1! γ(γ + 1) 2! γ(γ + 1)(γ + 2) 3! is also called a confluent hypergeometric function. Φ(α, γ; z) = 1 +
A second notation: Φ(α, γ; z) =
1F 1
(α; γ; z).
Γ(1 − γ) Γ(γ − 1) 1−γ Φ(α, γ; z) + z Φ(α − γ + 1, 2 − γ; z) Γ(α − γ + 1) Γ(α)
2.
Ψ(α, γ; z) =
3.
Bateman’s function kν (x) is defined by 2 π/2 kν (x) = cos (x tan θ − νθ) dθ π 0
9.211
2.
3.
4.8
[x, ν real]
EH I 267
[0 < Re α < Re γ]
MO 114
Integral representation: 1
1.
EH I 257(7)
21−γ e 2 z Φ(α, γ; z) = B(α, γ − α)
Φ(α, γ; z) =
1
−1
1 Γ(α)
1
(1 − t)γ−α−1 (1 + t)α−1 e 2 zt dt
1 z 1−γ B(α, γ − α)
Φ(−ν, α + 1; z) =
Ψ(α, γ; z) =
z
et tα−1 (z − t)γ−α−1 dt 0
Γ(α + 1) z − α e z 2 Γ(α + ν + 1)
∞
e−t tν+ 2
0
α
[0 < Re α < Re γ] √ J α 2 zt dt + Re(α + ν + 1) > 0,
MO 114
|arg z|
0,
Re z > 0]
EH I 255(2)
0
Functional relations 9.212 Φ(α, γ; z) = ez Φ(γ − α, γ; −z) z Φ(α + 1, γ + 1; z) = Φ(α + 1, γ; z) − Φ(α, γ; z) γ
MO 112
3.
α Φ(α + 1, γ + 1; z) = (α − γ) Φ(α, γ + 1; z) + γ Φ(α, γ; z)
MO 112
4.
α Φ(α + 1, γ; z) = (z + 2a − γ) Φ(α, γ; z) + (γ − α) Φ(α − 1, γ; z)
MO 112
1. 2.
α dΦ = Φ(α + 1, γ + 1; z) dz γ
1 n+1 α + n Φ(α, γ; z) = z 9.214 lim Φ(α + n + 1, n + 2; z) γ→−n Γ(γ) n+1
MO 112
9.213
MO 112
[n = 0, 1, 2, . . .]
MO 112
1024
Confluent Hypergeometric Functions
9.215
9.21510 Φ(α, α; z) = ez
1.
MO 15
π 1 Φ(α, 2α; 2z) = 2 exp 4 (1 − 2α)πi Γ α + 12 ez z 2 −α J α− 12 ze 2 i z −p Φ p + 12 , 2p + 1; 2iz = Γ(p + 1) eiz J p (z) 2 α− 12
2. 3.
1
MO 112 MO 15
For a representation of special functions in terms of a confluent hypergeometric function Φ(α, γ; z), see: • • • • • • 9.216 1.
for for for for for for
the probability integral, 9.236; integrals of Bessel functions, 6.631 1; Hermite polynomials, 8.953 and 8.959; Laguerre polynomials, 8.972 1; parabolic cylinder functions, 9.240; the Whittaker functions M λ,μ (z), 9.220 2 and 9.220 3.
The function Φ(α, γ; z) is a solution of the differential equation d2 F dF − αF = 0 + (γ − z) dz 2 dz This equation has two linearly independent solutions: z
2.
Φ(α, γ; z)
3.
z 1−γ Φ(α − γ + 1, 2 − γ; z)
MO 111
MO 112
9.22–9.23 The Whittaker functions Mλ,μ (z) and Wλ,μ (z) 9.220 1.
2. 3.11
If we make the change of variable u = e− 2 W in equation 9.202 1, we obtain the equation 1 2
d2 W 1 λ 4 −μ + + + − MO 115 W =0 dz 2 4 z z2 Equation 9.220 1 has the following two linearly independent solutions: 1 M λ,μ (z) = z μ+ 2 e−z/2 Φ μ − λ + 12 , 2μ + 1; z 1 M λ,−μ (z) = z −μ+ 2 e−z/2 Φ −μ − λ + 12 , −2μ + 1; z MO 115 z
To obtain solutions that are also suitable for 2μ = ±1, ±2, . . . , we introduce Whittaker’s function 4.
W λ,μ (z) =
Γ
Γ(−2μ) Γ(2μ) 1 M λ,μ (z) + 1 M λ,−μ (z) − μ − λ Γ 2 2 +μ−λ
WH
which, for 2μ approaching an integer, is also a solution of equation 9.220 1. For the functions M λ,μ (z) and W λ,μ (z), z = 0 is a branch point and z = ∞ is an essential singular point. Therefore, we shall examine these functions only for |arg z| < π. These functions W λ,μ (z) and W −λ,μ (−z) are linearly independent solutions of equation 9.220 1.
9.226
The Whittaker functions Mλ,μ (z) and Wλ,μ (z)
1025
Integral representations 1 1 1 1 1 z μ+ 2 (1 + t)μ−λ− 2 (1 − t)μ+λ− 2 e 2 zt dt, 1 1 2μ 2 B μ + λ + 2 , μ − λ + 2 −1 if the integral converges. See also 6.631 1 and 7.623 3. 9.222 ∞ 1 1 1 z μ+ 2 e−z/2 11 W λ,μ (z) = e−zt tμ−λ− 2 (1 + t)μ+λ− 2 dt 1. 1 Γ μ−λ+ 2 0 +
9.221 M λ,μ (z) =
Re(μ − λ) > − 12 ,
WH
|arg z|
− 12 , |arg z| < π WH z i∞ Γ(u − λ) Γ −u − μ + 12 Γ −u + μ + 12 u e− 2 9.223 W λ,μ (z) = z du 2πi −i∞ Γ −λ + μ + 12 Γ −λ − μ + 12 [the path of integration is chosen in such a way that the poles of the function Γ(u − λ) are separated from the poles of the functions Γ −u − μ + 12 and Γ −u + μ + 12 .] See also 7.142. MO 118 ∞ ∞ 1 1 9.224 W μ, 12 +μ (z) = z μ+1 e− 2 z (1 + t)2μ e−zt dt = z −μ e 2 z t2μ e−t dt [Re z > 0] WH
0
9.225 1.
z
W λ,μ (x) W −λ,μ (x) = −x
∞
tanh2λ
0
t {J 2μ (x sinh t) sin(μ − λ)π 2
+ Y 2μ (x sinh t) cos(μ − λ)π} dt
(z1 z2 ) exp − 12 (z1 + z2 ) W κ,μ (z1 ) W λ,μ (z2 ) = Γ(1 − κ − λ)
|Re μ| − Re λ < 12 ;
x>0
MO 119
μ+ 12
2.
∞
× 0
×F Θ=
t (z1 + z2 + t) , (z1 + t) (z2 + t)
− 12 +κ−μ
e−t t−κ−λ (z1 + t)
1 2
− 12 +λ−μ
(z2 + t)
− κ + μ, 12 − λ + μ; 1 − κ − λ; Θ dt [z1 = 0,
z2 = 0,
|arg z1 | < π,
|arg z2 | < π,
Re(κ + λ) < 1] MO 119
See also 3.334, 3.381 6, 3.382 3, 3.383 4, 8, 3.384 3, 3.471 2. 9.226 Series representations ∞ 1 z 2k +μ M 0,μ (z) = z 2 1+ 24k k!(μ + 1)(μ + 2) . . . (μ + k) k=1
WH
1026
Confluent Hypergeometric Functions
9.227
Asymptotic representations For large values of |z| + ⎛ 2 , + 2 2 , + 2 2 , ⎞ ∞ μ2 − λ − 12 μ − λ − 32 . . . μ − λ − k + 12 ⎠ W λ,μ (z) ∼ e−z/2 z λ ⎝1 + k!z k
9.2277
k=1
[|arg z| ≤ π − α < π] 9.228
WH
For large values of |λ|
√ 1 1 1 M λ,μ (z) ∼ √ Γ(2μ + 1)λ−μ− 4 z 1/4 cos 2 λz − μπ − π 4 π
9.229
MO 118
1 √ 4z 4 −λ+λ ln λ π 1. W λ,μ ∼ − e sin 2 λz − λπ − λ 4 z 14 √ 2. W −λ,μ ∼ eλ−λ ln λ−2 λz 4λ Formulas 9.228 and 9.229 are applicable for √ |λ| 1, |λ| |z|, |λ| |μ|, z = 0, |arg z| < 3π 4 and |arg λ|
−1;
for x < 0,
arg xp = pπi] MO 122
2. 9.242 1.10 2.
3.
D p (z) =
2 − z4
e Γ(−p)
∞
e−xz−
x2 2
x−p−1 dx
[Re p < 0]
(cf. 3.462 1)
MO 122
0
Γ(p + 1) − 1 z2 (0+) −zt− 1 t2 2 e 4 e (−t)−p−1 dt [|arg(−t)| ≤ π] 2πi ∞ p (−1+) 1 1 2 1 1 (p−1) Γ 2 + 1 2 D p (z) = 2 e 4 z t (1 + t)− 2 p−1 (1 − t) 2 (p−1) dt iπ −∞ , + π |arg z| < ; |arg(1 + t)| ≤ π 4 1 − 1 z2 ∞i Γ 12 t − 12 p Γ(−t) √ t−p−2 t e 4 D p (z) = 2 z dt 2πi Γ(−p) −∞i |arg z| < 34 π; p is not a positive integer D p (z) = −
WH
WH
WH
9.246
4.
Parabolic cylinder functions Dp (z)
1 − 1 z2 e 4 D p (z) = 2πi
(0−)
Γ
1
2t
∞
1029
− 12 p Γ(−t) √ t−p−2 t 2 z dt Γ(−p)
[for all values of arg z; also, the contours encircle the poles of the function Γ(−t), but they do not encircle the poles of the function Γ 12 t − 12 p ]. WH 9.243 ⎧ ⎨ ∞ π −1/2 √ √ 2 2 cos n+1 1 z − 1 n 2 zt n dt n e4 e−n(t−1) 1. D n (z) = (−1)μ ⎩ −∞ 2 sin ⎫ ∞+ 0 ⎬ , √ √ 2 2 2 cos 1 cos zt n dt − + zt n dt e−n(t−1) e 2 n(1−t ) tn − e−n(t−1) ⎭ sin sin 0 −∞
2.
[n is a natural number]
WH
∞
2 cos 1 2 (2zt) dt D n (z) = (−1)μ 2n+2 (2π)−1/2 e 4 z tn e−2t sin :n; 0 , and the cosine or sine is chosen accordingly as n is even or odd] [n is a natural number, μ = 2
WH
9.244 1.
D −p−1 [(1 + i)z] =
e− 2
p−1 2
p+1
2.
D p [(1 + i)z] =
2 2 Γ − p2
Γ
iz 2 2
p+1
∞
∞
0
2
e−ix
2 2
(1 +
z
xp
1+ p x2 ) 2
dx
p−1
e− 2 z i
2
x
1
(x + 1) 2 p dx (x − 1)1+ 2
Re p > −1, Re p < 0;
Re iz 2 ≥ 0 Re iz 2 ≥ 0
MO 122
MO 122
See also 3.383 6, 7, 3.384 2, 6, 3.966 5, 6. 9.245 2
∞ x sinh t + pπ t 1 1 p+ 12 10 √ sin D p (x) D −p−1 (x) = − √ coth 1. dt 2 2 π 0 sinh t
2.
π π D p ze 4 i D p ze− 4 i =
1 Γ(−p)
∞ 0
[x is real, Re p < 0]
dt z p coth t exp − sinh 2t 2 sinh t , + π |arg z| < ; Re p < 0 4
MO 122
2
MO 122
See also 6.613. 9.246 Asymptotic expansions. If |z| 1 and |z| |p|, then
2 p(p − 1) p(p − 1)(p − 2)(p − 3) − z4 p 1. D p (z) ∼ e z 1− + − ... 2z 2 2 · 4z 4 |arg z| < 34 π MO 121
2 p(p − 1) p(p − 1)(p − 2)(p − 3) 2.11 D p (z) ∼ e−z /4 z p 1 − + − ... 2z 2 ⎛ 2 · 4z 4 ⎞ √ 2π pπi z2 /4 −p−1 ⎝ (p + 1)(p + 2) (p + 1)(p + 2)(p + 3)(p + 4) e e 1+ − z + + . . .⎠ Γ(−p) 2z 2 2 · 4z 4 1 5 MO 121 4 π < arg z < 4 π
1030
3.
11
Confluent Hypergeometric Functions
9.247
p(p − 1) p(p − 1)(p − 2)(p − 3) D p (z) ∼ e z 1− + − ... 2z 2 2 · 4z 4 ⎛ ⎞ √ 2π −pπi z2 /4 −p−1 ⎝ (p + 1)(p + 2) (p + 1)(p + 2)(p + 3)(p + 4) − e 1+ e z + + . . .⎠ Γ(−p) 2z 2 2 · 4z 4 1 − 4 π > arg z > − 54 π MO 121 −z 2 /4 p
Functional relations 9.247 1. 2. 3. 9.248 1.
Recursion formulas: D p+1 (z) − z D p (z) + p D p−1 (z) = 0 d D p (z) + dz d D p (z) − dz
WH
1 z D p (z) − p D p−1 (z) = 0 2 1 z D p (z) + D p+1 (z) = 0 2
WH MO 121
Linear relations:
, Γ(p + 1) + π/2 √ D −p−1 (iz) + e−πpi/2 D −p−1 (−iz) e 2π √ 2π −π(p+1)i/2 −pπi =e e D p (−z) + D −p−1 (iz) Γ(−p) √ 2π π(p+1)i/2 e D −p−1 (−iz) = epπi D p (−z) + Γ(−p)
D p (z) =
MO 121
∞ i 2 cos xt −it2 /4 21+p/2 π exp − x +p 9.24910 D p [(1 + i)x] + D p [−(1 + i)x] = e dt Γ(−p) 2 2 tp+1 0 MO 122 [x real; −1 < Re p < 0] n 2 2 d [n = 0, 1, 2, . . .] 9.25110 D n (z) = (−1)n ez /4 n e−z /2 WH dz
p k ∞ 2 p a (bx − ay) b √ D p−k 9.252 D p (ax+by) = exp a2 + b 2 x D k a2 + b 2 y 2 2 k 4 a a +b k=0
[a > b > 0,
x > 0,
y > 0,
Re p ≥ 0]
Connections with other functions
n z2 z 9.25311 D n (z) = 2− 2 e− 4 H n √ 2 9.254
z2 z π 4 1. D −1 (z) = e 1−Φ √ 2 2
z2 z π 2 − z2 11 4 2 e 2. D −2 (z) = e −z 1−Φ √ 2 π 2
“summation theorem”
MO 124
MO 123a
MO 123
MO 123
9.262
9.255 1.
Confluent hypergeometric series of two variables
1031
Differential equations leading to parabolic cylinder functions:
d2 u 1 z2 + p+ − u=0 dz 2 2 4 The solutions are u = D p (z), D p (−z), D −p−1 (iz), and D −p−1 (−iz). (These four solutions are linearly dependent. See 9.248.)
2.
d2 u 2 + z + λ u = 0, 2 dz
u = D − 1+iλ [±(1 + i)z] 2
EH II 118(12,13)a, MO 123 2
3.7
d u du + (p + 1)u = 0, +z 2 dz dz
u = e−
z2 4
D p (z)
MO 123
9.26 Confluent hypergeometric series of two variables 9.261 ∞ (α)m+n (β)m m n x y (γ)m+n m!n! m,n=0
1.6
Φ1 (α, β, γ, x, y) =
2.
Φ2 (β, β , γ, x, y) =
3.
Φ3 (β, γ, x, y) =
[|x| < 1]
∞
(β)m (β )m m n x y (γ)m+n m!n! m,n=0
EH I 225(20)
EH I 225(21)a, ET I 385
∞
(β)m xm y n (γ) m!n! m+n m,n=0
EH I 225(22)
The functions Φ1 , Φ2 , Φ3 satisfy the following systems of partial differential equations: 9.262 1.
z = Φ1 (α, β, γ, x, y)
EH I 235(23)
∂z ∂z ∂2z ∂2z + [γ − (α + β + 1)x] − βy − αβz = 0, + y(1 − x) ∂x2 ∂x ∂y ∂x ∂y ∂2z ∂z ∂z ∂2z y 2 +x + (γ − y) −x − αz = 0 ∂y ∂x ∂y ∂y ∂x
x(1 − x)
2.
z = Φ2 (β, β , γ, x, y)
EH I 235(24)
∂z ∂2z ∂2z + (γ − x) − βz = 0, + y 2 ∂x ∂x ∂y ∂x ∂2z ∂z ∂2z y 2 +x + (γ − y) − βz = 0 ∂y ∂x ∂y ∂y x
1032
3.
Meijer’s G-Function
9.301
z = Φ3 (β, γ, x, y)
EH I 235(25)
∂z ∂2z ∂2z + (γ − x) − βz = 0, + y 2 ∂x ∂x ∂y ∂x ∂2z ∂z ∂2z y 2 +x +γ −z =0 ∂y ∂x ∂y ∂y x
9.3 Meijer’s G-Function 9.30 Definition m -
9.301
m,n G p,q
!
! a1 , . . . , ap 1 x !! = b1 , . . . , bq 2πi
j=1 q j=m+1
Γ (bj − s)
n -
Γ (1 − aj + s)
j=1
Γ (1 − bj + s)
p -
xs ds Γ (aj − s)
j=n+1
[0 ≤ m ≤ q, 0 ≤ n ≤ p, and the poles of Γ (bj − s) must not coincide with the poles of Γ (1 − ak + s) for any j and k (where j = 1, . . . , m; k = 1, . . . , n]). Besides 9.301, the following notations are also used: !
! ar mn G(x) EH I 207(1) , Gmn G pq x !! pq (x), bs 9.302 Three types of integration paths L in the right member of 9.301 can be exhibited: 1.
The path L runs from −∞ to +∞ in such a way that the poles of the functions Γ (1 − ak + s) lie to the left, and the poles of the functions Γ (bj − s) lie to the right of L (for j = 1, 2, . . . , m and k = 1, 2, . . . , n). In this case, the conditions under which the integral 9.301 converges are of the form EH I 207(2) p + q < 2(m + n), |arg x| < m + n − 12 p − 12 q π.
2.
L is a loop, beginning and ending at +∞, that encircles the poles of the functions Γ (bj − s) (for j = 1, 2, . . . , m) once in the negative direction. All the poles of the functions Γ (1 − ak + s) must remain outside this loop. Then, the conditions under which the integral 9.301 converges are: q ≥ 1 and either p < q or p = q and |x| < 1.
3.
EH I 207(3)
L is a loop, beginning and ending at −∞, that encircles the poles of the functions Γ (1 − ak + s) (for k = 1, 2, . . . , n) once in the positive direction. All the poles of the functions Γ (bj − s) (for j = 1, 2, . . . , m) must remain outside this loop. The conditions under which the integral in 9.301 converges are EH I 207(4) p ≥ 1 and either p > q or p = q and |x| > 1. !
! ar The function G mn x !! is analytic with respect to x; it is symmetric with respect to the pq bs parameters a1 , . . . , an and also with respect to an+1 , . . . , ap ; b1 , . . . , bm ; bm+1 , . . . , bq . EH I 208
9.304
Functional relations
1033
9.30311 If no two bj (for j = 1, 2, . . . , n) differ by an integer, then, under the conditions that either p < q or p = q and |x| < 1, n m Γ (bj − bh ) Γ (1 + bh − aj ) ! m ! ar j=1 j=1 mn ! = xbh G pq x ! q p bs h=1 Γ (1 + bh − bj ) Γ (aj − bh ) j=n+1 × p F q−1 1 + bh − a1 , . . . , 1 + bh − ap ; j=m+1
. . . , ∗, . . . , 1 + bh − bq ;
1 + bh − b1 , . . . p−m−n (−1) x EH I 208(5)
The prime by the product symbol denotes the omission of the product when j = h. The asterisk in the function p F q−1 denotes the omission of the hth parameter. 9.3047 If no two ak (for k = 1, 2, . . . , n) differ by an integer then, under the conditions that q < p or q = p and |x| > 1, n m Γ (a − a ) Γ (bj − ah + 1) h j ! n ! a j=1 j=1 r mn = xah −1 G pq x !! p q bs h=1 Γ (aj − ah + 1) Γ (ah − bj ) j=m+1 × q F p−1 1 + b1 − ah , . . . , 1 + bq − ah ; j=n+1
. . . , ∗, . . . , 1 + ap − ah ;
1 + a1 − ah , . . . (−1)q−m−n x−1 EH I 208(6)
9.31 Functional relations If one of the parameters aj (for j = 1, 2, . . . , n) coincides with one of the parameters bj (for j = m + 1, m + 2, . . . , q), the order of the G-function decreases. For example, !
!
! a1 , . . . , ap ! a2 , . . . , ap m,n−1 mn ! ! = G p−1,q−1 x ! 1. G pq x ! b1 , . . . , bq−1 , a1 b1 , . . . , bq−1 [n, p, q ≥ 1] An analogous relationship occurs when one of the parameters bj (for j = 1, 2, . . . , m) coincides with one of the aj (for j = n + 1, . . . , p). In this case, it is m and not n that decreases by one unit.
2.
The G-function with p > q can be transformed into the G-function with p < q by means of the relationships: !
!
! ! 1 − bs mn −1 ar ! EH I 209(9) = G nm x G pq x !! qp ! 1 − ar bs
1034
3.
4.
5.
Meijer’s G-Function
9.304
!
!
!
! ar ! a1 − 1, a2 , . . . , ap ! ar d mn mn mn ! ! x G pq x ! = G pq x ! + (a1 − 1) G pq x !! dx bs b1 , . . . , bq bs [n ≥ 1] !
!
! ap , 1 − r ! 1 − r, ap m+1,n ! = (−1)r G m,n+1 z G p+1,q+1 z !! p+1,q+1 ! bq , 1 0, bq !
!
! ap ! ap + k k mn mn ! ! z G pq z ! = G pq z ! bq + k bq
EH I 210(13)
[r = 0, 1, 2, . . .]
MS2 6 (1.2.2) MS2 7 (1.2.7)
9.32 A differential equation for the G-function mn G pq
!
! ar x !! satisfies the following linear q th -order differential equation: bs ⎡ ⎤
p q d d ⎣(−1)p−m−n x − aj + 1 − − bj ⎦ y = 0 x x dx dx j=1 j=1
[p ≤ q]
EH I 210(1)
9.33 Series of G-functions mn G pq
!
!
∞ ! a1 , . . . , ap ! a1 , . . . , ap 1 r b1 mn ! ! (1 − λ) G pq x ! λx ! =λ b1 , . . . , bq r! b1 + r, b2 , . . . , bq r=0 [|λ − 1| < 1, m ≥ 1, if m = 1 and p < q, λ may be arbitrary] !
∞ ! a1 , . . . , ap 1 bq r mn ! (λ − 1) G pq x ! =λ r! b1 , . . . , bq−1 , bq + r r=0 [m < q,
EH I 213(1)
|λ − 1| < 1] EH I 213(2)
!
r ∞ ! a1 − r, a2 , . . . , ap 1 1 a1 −1 mn =λ λ− G pq x !! r! λ b1 , . . . , bq r=0 1 n ≥ 1, Re λ > 2 , (if n = 1 and p > q, then λ may be arbitrary) =λ
ap −1
r !
∞ ! a1 , . . . , ap−1 , ap − r 1 1 ! − 1 G mn x pq ! b1 , . . . , bq r! λ r=0 n < p,
EH I 213(3)
Re γ >
1 2
EH I 213(4)
For integrals of the G-function, see 7.8.
9.34 Connections with other special functions
!
1 2 !! x J ν (x)x = 2 4 ! 12 ν + 12 μ, 12 μ − 12 ν ! 1 2 !! 12 μ − 12 ν − 12 μ μ 20 x ! Y ν (x)x = 2 G 13 4 ! 21 μ − 12 ν, 12 μ + 12 ν, 12 μ − 12 ν −
1. 2.
μ
μ
10 G 02
EH I 219(44)
1 2
EH I 219(46)
9.304
Functional relations
1035
!
1 2 !! x K ν (x)x = 2 4 ! 12 μ + 12 ν, 12 μ − 12 ν !1
! √ x 20 2 ! K ν (x) = e π G 12 2x ! ν, −ν ! !1 1 1 1 μ μ 11 2 ! 2 + 2ν + 2μ x ! Hν (x)x = 2 G 13 4 ! 12 + 12 ν + 12 μ, 12 μ − 12 ν, 12 μ + 12 ν ! !1 1 1 1 μ−1 31 2 ! 2 + 2μ 1−μ+ν G 13 x ! S μ,ν (x) = 2 4 ! 12 + 12 μ, 12 ν, − 12 ν Γ Γ 1−μ−ν 2 2 !
! −a, −b Γ(c)x 12 ! x! G 2 F 1 (a, b; c; −x) = −1, −c Γ(a) Γ(b) 22 $q !
! 1 − a1 , . . . , 1 − ap j=1 Γ (bj ) 1,p ! −x ! G p F q (a1 , . . . , ap ; b1 , . . . , bq ; x) = $p 0, 1 − b1 , . . . , 1 − bq Γ (aj ) p,q+1 $j=1 !
q ! Γ (b ) 1, b1 , . . . , bq j 1 j=1 p,1 = $p G q+1,p − !! x a1 , . . . , ap j=1 Γ (aj )
3. 4. 5.
6. 7.7 8.
9.
μ
μ−1
20 G 02
√ 1 2k xe 2 x 40 x2 W k,m (x) = √ G 24 4 2π
EH I 219(47)
EH I 219(49)
EH I 220(51)
EH I 220(55)
EH I 222(74)a
EH I 215(1)
! ! 1 − 1 k, 3 − 1 k !4 2 4 2 !1 1 ! 2 + 2 m, 12 − 12 m, 12 m, − 12 m
EH I 221(70)
9.4 MacRobert’s E-Function 9.41 Representation by means of multiple integrals E (p; αr : q; s : x) =
Γ (αq+1 ) Γ (1 − α1 ) Γ (2 − α2 ) · · · Γ (q − αq ) q ∞ p−q−1 - −μ μ −αμ −1 × λμ (1 − λμ ) dλμ μ=1 0 ∞ × e−λp λpαp −1 0
1+
ν=2
λq+2 λq+3 · · · λp (1 + λ1 ) · · · (1 + λq ) x
∞
q+ν e−λq+ν λq+ν
α
−1
dλq+ν
0
−αq+1 dλp
[|arg x| < π, p ≥ q + 1, αr and s are bounded by the condition that the integrals on the right be EH I 204(3) convergent.]
9.42 Functional relations 1.
α1 x E (α1 , . . . , αp : 1 , . . . , q : x) = x E (α1 + 1, α2 , . . . , αp : 1 , . . . , q : x) + E (α1 + 1, α2 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) EH I 205(7)
1036
2.
Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s)
9.511
(1 − 1) x E (α1 , . . . , αp : 1 , . . . , q : x) = x E (α1 , . . . , αp : 1 − 1, 2 , . . . , q : x) + E (α1 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) EH I 205(9)
3.
d E (α1 , . . . , αp : 1 , . . . , q : x) = x−2 E (α1 + 1, . . . , αp + 1 : 1 + 1, . . . , q + 1 : x) dx EH I 205(8)
9.5 Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s) 9.51 Definition and integral representations 9.511
ζ(z, q) =
1 Γ(z)
0
∞ z−1 −qt
t e dt; 1 − e−t
1 q 1−z +2 = q −z + 2 z−1
∞
2
q +t
z 2 −2
0
dt t sin z arctan q e2πt − 1
WH
[0 < q < 1,
Re z > 1] WH
Γ(1 − z) (0+) (−θ)z−1 e−qθ dθ 2πi 1 − e−θ ∞ This equation is valid for all values of z, except for z = 1, 2, 3, . . . . It is assumed that the path of integration (see drawing below) does not pass through the points 2nπi (where n is a natural number).
9.512 ζ(z, q) = −
See also 4.251 4, 4.271 1, 4, 8, 4.272 9, 12, 4.294 11. 9.513 ∞ z−1 1 t 1. ζ(z) = dt [Re z > 0] (1 − 21−z ) Γ(z) 0 et + 1 ∞ z−1 t t e 2z dt [Re z > 1] 2. ζ(z) = z (2 − 1) Γ(z) 0 e2t − 1 & % ∞ z ∞ 1−z 2 z π2 1 −1 −k πt 11 t 2 + t2 t + 3. ζ(z) = z e dt z(z − 1) Γ 2 1 k=1 ∞ − z 2z−1 dt − 2z 1 + t2 2 sin (z arctan t) πt 4. ζ(z) = z−1 e +1 0
−z/2 ∞ 1 z 2 2z−1 dt + z + t2 5. ζ(z) = z sin (z arctan 2t) 2πt 2 −1z−1 2 −1 0 4 e −1 See also 3.411 1, 3.523 1, 3.527 1, 3, 4.271 8.
WH WH
WH
WH
WH
9.532
Functional relations
1037
9.52 Representation as a series or as an infinite product 9.521 1.
2.
∞
1 [Re z > 1, (q + n)z n=0 % & ∞ ∞ 2 Γ(1 − z) zπ sin 2πqn zπ cos 2πqn ζ(z, q) = + cos sin (2π)1−z 2 n=1 n1−z 2 n=1 n1−z ζ(z, q) =
[Re z < 0, 3.8
ζ(z, q) =
q = 0, −1, −2, . . .]
WH
0 < q ≤ 1]
WH
∞ 1 1 − − F n (z) , (q + n)z (1 − z)(N + q)z−1 n=0 N
n=N
where
1 1 1 1 − − F n (z) = 1 − z (n + 1 + q)z−1 (n + q)z−1 (n + 1 + q)z n+1 (t − n) dt =z (t + q)z+1 n
WH
9.522 1.
ζ(z) =
∞ 1 z n n=1
[Re z > 1]
WH
2.
ζ(z) =
∞ 1 1 (−1)n+1 z 1 − 21−z n=1 n
[Re z > 0]
WH
9.523 1.7
The following product and summation are taken over all primes p: ζ(z) =
p
2.
ln ζ(z) =
1 1 − p−z
∞ p k=1
1 kpkz
[Re z > 1]
WH
[Re z > 1]
WH
∞ ζ (z) Λ(k) =− , [Re z > 1] ζ(z) kz k=1 where Λ(k) = 0 when k is not a power of a prime and Λ(k) = ln p when k is a power of a prime p.
9.52411
WH
9.53 Functional relations 9.531
9.532
ζ(−n, q) = −
∞ (−1)k−1 k=2
k
B n+2 (q) − B n+1 (q) = (n + 1)(n + 2) n+1 [n is a nonnegative integer]
see EH I 27 (11)
WH
∞
z k ζ(k, q) = ln
qz e−Cz Γ(q) z − + Γ(z + q) q k(q + k) k=1
[|z| < q]
WH
1038
Riemann’s Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s)
9.533
9.533 1. 2. 3.
ζ(z, q) = −1 − z)
1 lim ζ(z, q) − = − Ψ(q) z→1 z−1
d 1 ζ(z, q) = ln Γ(q) − ln 2π dz 2 z=0 lim
9.534 ζ(z, 1) = ζ(z) 9.535 1 ζ z, 12 1. ζ(z) = z 2 −1 2.11
WH
z→1 Γ(1
2z Γ(1 − z) ζ(1 − z) sin
WH WH
[Re z > 1] zπ 2
= π 1−z ζ(z)
WH WH
zπ = π z ζ(1 − z) WH 2
z z−1 z 1−z π − 2 ζ(z) = Γ 4. Γ WH π 2 ζ(1 − z) 2 2
1 9.536 lim ζ(z) − =C z→1 z−1 (z − 1) Γ z2 + 1 1 √ ζ(z) = Ξ(−t) is an even function of t with real 9.537 Set z = 2 + it. Then, Ξ(t) = πz 2 JA coefficients in its expansion in powers of t . 3.
21−z Γ(z) ζ(z) cos
9.54 Singular points and zeros 9.5417 1.
z = 1 is the only singular point of the function ζ(z)
2.
The function ζ(z) has simple zeros at the points −2n, where n is a natural number. All other zeros of the function ζ(z) lie in the strip 0 ≤ Re z 0, or |z| ≤ 1, z = 1, Re s > 0, or z = 1,
EH I 30(9) EH I 30(11)
∞ s−1 −vt 0
Re s > 1]
EH I 27(3)
Limit relationships 9.557 9.558
lim (1 − z)1−s Φ(z, s, v) = Γ(1 − s)
z→1
lim
[Re s < 1]
Φ(z, 1, v) =1 − z)
EH I 30(12) EH I 30(13)
z→1 − ln(1
A connection with a hypergeometric function 9.559 ∗ In
Φ(z, 1, v) = v −1 2 F 1 (1, v; 1 + v; z)
9.554 the prime on the symbol
#
[|z| < 1]
means that the term corresponding to n = m − 1 is omitted.
EH I 30(10)
1040
Bernoulli Numbers and Polynomials, Euler Numbers
9.561
9.56 The function ξ (s)
Γ 12 s 1 9.561 ξ(s) = s(s − 1) 1 s ζ(s) 2 π2 9.562 ξ(1 − s) = ξ (s)
EH III 190(10) EH III 190(11)
9.6 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), λ(x, y) and Euler Polynomials 9.61 Bernoulli numbers 9.610
The numbers Bn , representing the coefficients of ∞ t tn = B n et − 1 n=0 n!
are called Bernoulli numbers. Thus, the function bers. 9.611 1.
[0 < |t| < 2π] ,
t is a generating function for the Bernoulli numet − 1 GE 48(57), FI II 520
Integral representations ∞ 2n−1 x dx B2n = (−1)n−1 4n 2πx − 1 0 e
[n = 1, 2, . . .]
B2n = (−1)n−1 π −2n
3.
B2n
Bn = lim
d
n
x→0 dxn
ex
2n
x −1
[n = 1, 2, . . .]
See also 3.523 2, 4.271 3. Properties and functional relations 9.6128
A symbolic notation: (B + α)[n] =
n n k=0
in particular Bn = (B + 1)[n] =
k
Bk αn−k
n n k=0
hence by recursion
(cf. 3.411 2, 4) FI II 721a
∞
x dx [n = 1, 2, . . .] 2 0 sinh x 2n(1 − 2n) ∞ 2n−2 = (−1)n−1 x ln 1 − e−2πx dx π 0
2.
4.∗
tn in the expansion of the function n!
k
Bk
[n ≥ 2]
[n ≥ 2]
9.622
Bernoulli polynomials
Bn = −n!
n−1 k=0
Bk k!(n + 1 − k)!
1041
[n ≥ 2]
9.613 9.614
All the Bernoulli numbers are rational numbers. Every number Bn can be represented in the form 1 Bn = Cn − , k+1 where Cn is an integer and the sum is taken over all k > 0 such that k + 1 is a prime and k is a divisor GE 64 of n. 1 11 9.615 All the Bernoulli numbers with odd index are equal to zero, except that B1 = − 2 ; that is, GE 52, FI II 521 B2n+1 = 0 for n a natural number. n−1 1 2n(2n − 1) . . . (2n − 2k + 2) 1 + − Bk/2 B2n = − [n ≥ 1] 2n + 1 2 (2k)! k=1; k even
(−1)n−1 (2n)! 9.616 B2n = ζ(2n) [n ≥ 0] 22n−1 π 2n 2(2n)! 1 9.6177 B2n = (−1)n−1
∞ 2n (2π) 1 1 − 2n p p=2
(cf. 9.542)
[n ≥ 1]
GE 56(79), FI II 721a
(cf. 9.523)
(where the product is taken over all primes p). • For a connection with Riemann’s zeta function, see 9.542. • For a connection with the Euler numbers, see 9.635. • For a table of values of the Bernoulli numbers, see 9.71 9.619
An inequality
! ! ! ! !(B − θ)[n] ! ≤ |Bn |
[0 < θ < 1]
9.62 Bernoulli polynomials 9.620
The Bernoulli polynomials B n (x) are defined by n n Bk xn−k B n (x) = k
GE 51(62)
k=0
or symbolically, B n (x) = (B + x)[n] . 9.621 The generating function ∞ tn−1 ext = B (x) n et − 1 n=0 n! 9.622 1.
7
GE 52(68)
[0 < |t| < 2π]
(cf. 1.213)
GE 65(89)a
Series representation ∞ n! cos 2πkx − 12 πn B n (x) = −2 (2π)n kn k=1
[n > 1,
1 ≥ x ≥ 0;
n = 1,
1 > x > 0]
AS 805(23.1.16)
1042
Bernoulli Numbers and Polynomials, Euler Numbers
9.623
∞
2.7
B 2n−1 (x) = 2
(−1)n 2(2n − 1)! sin 2kπx (2π)2n−1 k 2n−1 k=1
[n > 1,
1 ≥ x ≥ 0;
n = 1,
1 > x > 0]
AS 805(23.1.17)
∞
3.10
B 2n (x) =
(−1)n−1 2(2n)! cos 2kπx (2π)2n k 2n
[0 ≤ x ≤ 1,
n = 1, 2, . . .]
GE 71
k=1
9.623
Functional relations and properties: B m+1 (n) = Bm+1 + (m + 1)
1.
n−1
km
k=1
[n and m are natural numbers] 2.
B n (x + 1) − B n (x) = nxn−1
3.
B n (x)
4.
B n (1 − x) = (−1) B n (x)
5.
(see also 0.121)
GE 51(65) GE 65(90)
= n B n−1 (x)
[n = 1, 2, . . .]
GE 66
n
10
9.6247
n
GE 66 n−1
(−1) B n (−x) = B n (x) + nx m−1
B n (mx) = mn−1
k=0
[n = 0, 1, . . .]
k Bn x + m [m = 1, 2, . . . n = 0, 1, . . .] ;
9.625
AS 804(23.1.9)
“summation theorem”
GE 67
For n odd, the differences
B n (x) − Bn vanish on the interval [0, 1] only at the points 0, 12 , and 1. They change sign at the point x = 12 . For n even, these differences vanish at the end points of the interval [0, 1]. Within this interval, they do not change sign, and their greatest absolute value occurs at the point x = 12 . 9.626 The polynomials B 2n (x) − B2n and B 2n+2 (x) − B2n+2 have opposite signs in the interval (0, 1). GE 87 9.627 Special cases: 1.
B 1 (x) = x −
2.
B 2 (x) = x − x +
3.
B 3 (x) = x −
4.
B 4 (x) = x4 − 2x + x −
GE 70
5.
B 5 (x) = x −
GE 70
9.628
1 2
GE 70
2 3
5
1 6
3 2 2x 3
+
5 4 2x
+
GE 70 1 2x 2
GE 70
1 30 5 3 1 3x − 6x
Particular values:
1.
B n (0) = Bn
2.
B 1 (1) = −B1 = 12 ,
B n (1) = Bn
[n = 1]
GE 76
9.640
The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y)
1043
9.63 Euler numbers 9.630
n
The numbers En , representing the coefficients of tn! in the expansion of the function ∞ + 1 tn π, = En |t| < , cosh t n=0 n! 2
are known as the Euler numbers. Thus, the function
1 cosh t
is a generating function for the Euler numbers. CE 330
9.631
A recursion formula (E + 1)[n] + (E − 1)[n] = 0
[n ≥ 1] ,
E0 = 1
CE 329
Properties of the Euler numbers 9.632 The Euler numbers are integers. 9.633 The Euler numbers of odd index are equal to zero; the signs of two adjacent numbers of even indices are opposite; that is, E2n+1 = 0, E4n > 0, E4n+2 < 0. CE 329 9.634 If α, βγ, . . . are the divisors of the number n − m, the difference E2n − E2m is divisible by those of the numbers 2α + 1, 2β + 1, 2γ + 1, . . . that are primes. 9.635 A connection with the Bernoulli numbers (symbolic notation):
3.6
(4B − 1)[n] − (4B − 3)[n] + 4(−1)n+1 3n−1 − 1 B1 En−1 + 4(−1)n 3n−1 − 1 B1 = 2n n(E + 1)[n−1] [n ≥ 2] Bn = n n 2 (2 − 1) [2n+1] B + 14 = −4−2n−1 (2n + 1)E2n [n ≥ 0]
4.
En−1 =
1.11 2.
(4B + 3)[n] − (4B + 1)[n] 2n
CE 330 CE 330 CE 341
[n ≥ 1]
For a table of values of the Euler numbers, see 9.72.
9.64 The functions ν(x), ν(x, α), μ(x, β), μ(x, β, α), and λ(x, y) 9.640 1.
∞
ν(x) = 0
2.
xt dt Γ(t + 1) ∞
ν(x, α) = 0
3.
∞
μ(x, β) = 0
4.
xα+t dt Γ(α + t + 1)
EH III 217(1)
xt tβ dt Γ(β + 1) Γ(t + 1)
EH III 217(2)
μ(x, β, α) =
5.
λ(x, y) = 0
0 y
EH III 217(1)
∞
xα+t tβ dt Γ(β + 1) Γ(α + t + 1)
Γ(u + 1) du xu
EH III 217(2) MI 9
1044
Bernoulli Numbers and Polynomials, Euler Numbers
9.650
9.6510 Euler polynomials 9.650
The Euler polynomials are defined by
n−k n n Ek 1 E n (x) = x− k 2k 2
AS 804 (23.1.7)
k=0
9.651
The generating function: ∞ tn 2ext = E n (x) t e + 1 n=0 n!
9.652 1.
AS 804 (23.1.1)
Series representation: ∞ n! sin (2k + 1)πx − 12 πn π n+1 (2k + 1)n+1
E n (x) = 4
k=0
[n > 0,
1 ≥ x ≥ 0,
n = 1,
1 > x > 0]
AS 804 (23.1.16)
∞
2.
10
(−1)n 4(2n − 1)! cos(2k + 1)πx E 2n−1 (x) = π 2n (2k + 1)2n
[n = 1, 2, . . . ,
1 ≥ x ≥ 0]
k=0
AS 804 (23.1.17) ∞
3.
E 2n (x) =
(−1)n 4(2n)! sin(2k + 1)πx π 2n+1 (2k + 1)2n+1 k=0
[n > 0,
9.653 1.
1 ≥ x ≥ 0,
n = 0,
1 > x > 0]
AS 804 (23.1.18)
Functional relations and properties: E m (n + 1) = 2
n
(−1)n−k k m + (−1)n+1 E m (0),
[m and n are natural numbers]
k=1
AS 804 (23.1.4)
2.
En (x) = nEn−1 (x). n
[n = 1, 2, . . .]
AS 804 (23.1.5) AS 804 (23.1.6)
3.
E n (x + 1) + E n (x) = 2x
[n = 0, 1, . . .]
4.8
m−1 k n k E n (mx) = m (−1) En x − m
[n = 0, 1, . . . , m = 1, 3, . . .]
k=0
AS 804 (23.1.10)
5.
E n (mx) =
m−1 −2 k mn (−1)k Bn+1 x + n+1 m
[n = 0, 1, . . . , m = 2, 4, . . .]
k=0
AS 804 (23.1.10)
9.654
Special cases:
1.
E 1 (x) = x −
2.
E 2 (x) = x2 − x
1 2
9.655
Euler numbers
3.
E 3 (x) = x3 − 32 x2 +
4.
E 4 (x) = x4 − 2x3 + x
5.
E 5 (x) = x5 − 52 x4 + 52 x2 −
9.655 1.
1045
1 4
1 2
Particular values: E2n+1 = 0.
[n = 0, 1, . . .] E n (0) = − E n (1) = −2(n + 1)−1 2n+1 − 1 Bn+1 [n = 1, 2, . . .] 1 −n E n 2 = 2 En [n = 0, 1, . . .] 1 2 E 2n−1 3 = − E 2n−1 3 = −(2n)−1 1 − 31−2n 22n − 1 B2n
2. 3. 4.
[n = 1, 2, . . .]
AS 805 (23.1.19) AS 805 (23.1.20) AS 805 (23.1.21)
AS 806 (23.1.22)
9.7 Constants 9.71 Bernoulli numbers • • • • • • • • • •
B0 = 1 B1 = − 1/2 B2 = 1/6 B4 = − 1/30 B6 = 1/42 B8 = − 1/30 B10 = 5/66 B12 = − 691/2730 B14 = 7/6 B16 = − 3617/510
• • • • • • • • •
B18 B20 B22 B24 B26 B28 B30 B32 B34
= 43 867/798 = − 174 611/330 = 854 513/138 = − 236 364 091/2730 = 8 553 103/6 = − 23 749 461 029/870 = 8 615 841 276 005/14 322 = − 7 709 321 041 217/510 = 2 577 687 858 367/6
• • • • •
E12 E14 E16 E18 E20
= 2 702 765 = −199 360 981 = 19 391 512 145 = −2 404 879 675 441 = 370 371 188 237 525
9.72 Euler numbers • • • • • •
E0 = 1 E2 = −1 E4 = 5 E6 = −61 E8 = 1385 E10 = −50 521
The Bernoulli and Euler numbers of odd index (with the exception of B 1 ) are equal to zero.
1046
Constants
9.740
9.73 Euler’s and Catalan’s constants Euler’s constant C = 0.577 215 664 901 532 860 606 512 . . .
(cf. 8.367)
Catalan’s constant ∞ (−1)k = 0.915 965 594 . . . G= (2k + 1)2 k=0
9.7410 Stirling numbers (m)
(m)
9.740 The Stirling number of the first kind Sn is defined by the requirement that (−1)n−m Sn is the number of permutations of n symbols which have exactly m cycles. AS 824 (23.1.3) 9.741 Generating functions: 1.
x(x − 1) · · · (x − n + 1) =
n
Sn(m) xm
AS 824 (24.1.3)
m=0
2.
{ln(1 + x)}
m
∞
= m!
Sn(m)
n=m
9.742 1.8
2.
9.743 1.
2.
3.
xn n!
[|x| < 1]
AS 824 (24.1.3)
Recurrence relations: (m)
Sn+1 = Sn(m−1) − nSn(m) ; m r
Sn(m) =
n−r k=m−r
Sn(0) = δ0n ;
Sn(1) = (−1)n−1 (n − 1)!;
n (r) (m+r) S S k n−k k
Sn(n) = 1
[n ≥ m ≥ 1]
AS 824 (24.1.3)
[n ≥ m ≥ r]
AS 824 (24.1.3)
Functional relations and properties m k hm Γ hx + 1 x (m) = hm x(x − h)(x − 2h) · · · (x − mh + h) = x Sk h Γ h −m+1 k=1 &−1
−1 % p x+m −1 k (m) [(x + 1)(x + 2) · · · (x + m)] = m! = (x + m) Sk m k=1 &−1 % m k Γ hx + 1 x (m) −1 m = h + m Sk [(x + h)(x + 2h) · · · (x + mh)] = m x h h Γ h +m+1 k=1
(m)
9.744 The Stirling number of the second kind Sn elements into m non-empty subsets.
is the number of ways of partitioning a set of n
9.748
9.745 1.
Stirling numbers
1047
Generating functions: xn =
n
S(m) n x(x − 1) · · · (x − m + 1)
AS 824 (24.1.4)
m=0
2.
m
(ex − 1)
∞
= m!
S(m) n
n=m
3.
xn n!
[(1 − x)(1 − 2x) · · · (1 − mx)]
AS 824 (24.1.4) −1
∞
=
n−m S(m) n x
|x| < m−1
AS 824 (24.1.4)
n=m
9.746
Closed form expression: m 1 kn (−1)m−k k m! m
1.
S(m) = n
AS 824 (24.1.4)
k=0
9.747 1.8
2.
3.
Recurrence relations: (m)
Sn+1 = mS(m) + S(m−1) , n n m r
k
k=m−r
Sn(m) =
n−m
(−1)k
k=0
9.7487
n
n−r
S(m) = n
S(0) n = δ0n ,
(r)
(n) S(1) n = Sn = 1
(m−r)
Sn−k Sk
n−1+k n−m+k
[n ≥ m ≥ 1]
AS 825(24.1.4)
[n ≥ m ≥ r]
AS 825 (24.1.4)
2n − m (k) Sn−m+k n−m−k
Particular values:
AS 824 (24.1.3)
(m)
Stirling numbers of the first kind Sn m 1 2 3 4 5 6 7 8 9
(m)
S1
1
(m)
S2
-1 1
(m)
S3
2 -3 1
(m)
S4
-6 11 -6 1
(m)
S5
24 -50 35 -10 1
(m)
S6 -120 274 -225 85 -15 1
(m)
S7 720 -1764 1624 -735 175 -21 1
(m)
S8 -5040 13068 -13132 6769 -1960 332 -28 1
(m)
S9 40320 -109584 118121 -67284 22449 -4536 546 -36 1
1048
Constants
9.749
(m)
Stirling numbers of the second kind Sn m 1 2 3 4 5 6 7 8 9 9.7498 1.
(m)
S1
(m)
S2
1
1 1
(m)
S3
(m)
S4
1 3 1
1 7 6 1
(m)
S5
1 15 25 10 1
(m)
S6
1 31 90 65 15 1
(m)
S7
1 63 301 350 140 21 1
(m)
(m)
S8
S9
1 127 966 1701 1050 266 28 1
1 255 3025 7770 6951 2646 462 36 1 −m
Relationship between Stirling numbers of the first kind and derivatives of (ln x) dn dxn
1 lnm x
=
1 lnm x
:
n (k) (−1)k (m)k Sn k=1
xn lnk x where (m)k = Γ(m + k)/ Γ(m),
[m, n are positive integers]
10 Vector Field Theory 10.1–10.8 Vectors, Vector Operators, and Integral Theorems 10.11 Products of vectors Let a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ), and c = (c1 , c2 , c2 ) be arbitrary vectors, and i, j, k be the set of orthogonal unit vectors in terms of which the components of a, b, and c are expressed. Two different products involving pairs of vectors are defined, namely, the scalar product, written a · b, and the vector product, written either a × b or a ∧ b. Their properties are as follows: 1. 2.
3. 4.
a · b = a1 b 1 + a2 b 2 + a3 b 3 ! ! !i j k !! ! a × b = !!a1 a2 a3 !! ! b1 b2 b3 ! ! ! ! a1 a2 a3 ! ! ! a × b · c = !! b1 b2 b3 !! ! c1 c2 c3 !
(scalar product) (vector product)
(triple scalar product)
a × (b × c) = (a · c) b − (a · b) c
(triple vector product)
10.12 Properties of scalar product 1.
a·b=b·a
2.
a × b · c = b × c · a = c × a · b = −a × c · b = −b × a · c = −c × b · a.
(commutative)
Note: a × b · c is also written [a, b, c]; thus (2) may also be written 3.
[a, b, c] = [b, c, a] = [c, a, b] = − [a, c, b] = − [b, a, c] = − [c, b, a]
10.13 Properties of vector product 1.
a × b = −b × a
2.
a × (b × c) = −a × (c × b) = − (b × c) × a
3.
a × (b × c) + b × (c × a) + c × (a × b) = 0
(anticommutative)
1049
1050
Vectors, Vector Operators, and Integral Theorems
10.14 Differentiation of vectors If a(t) = (a1 (t), a2 (t), a3 (t)), b(t) = (b1 (t), b2 (t), b3 (t)), c(t) = (c1 (t), c2 (t), c3 (t)), φ(t) is a scalar and all functions of t are differentiable, then 1. 2. 3. 4. 5. 6. 7.
da1 da2 da3 da = i+ j+ k dt dt dt dt d da db (a + b) = + dt dt dt dφ da d (φa) = a+φ dt dt dt da db d (a · b) = ·b+a· dt dt dt da db d (a × b) = ×b+a× dt dt dt da db dc d (a × b · c) = ×b·c+a× ·c+a×b· dt dt dt dt
db da dc d {a × (b × c)} = × (b × c) + a × ×c +a× b× dt dt dt dt
10.21 Operators grad, div, and curl In cartesian coordinates O {x1 , x2 , x3 }, in which system it is convenient to denote the triad of unit vectors by e1 , e2 , e3 , the vector operator ∇, called either “del” or “nabla,” has the form 1.
2.
3.
4.
∂ ∂ ∂ + e2 + e3 ∂x1 ∂x2 ∂x3 If Φ(x, y, z) is any differentiable scalar function, the gradient of Φ, written grad Φ, is
∇ ≡ e1
∂Φ ∂Φ ∂Φ e1 + e2 + e3 ∂x1 ∂x2 ∂x3 The divergence of the differentiable vector function f = (f1 , f2 , f3 ), written div f , is grad Φ ≡ ∇ Φ =
∂f1 ∂f2 ∂f3 + + ∂x1 ∂x2 ∂x3 The curl, or rotation, of the differentiable vector function f = (f1 , f2 , f3 ), written either curl f or rot f , is
∂f1 ∂f2 ∂f3 ∂f2 ∂f3 ∂f1 − − − e1 + e2 + e3 , curl f ≡ rot f ≡ ∇ ×f = ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 or equivalently, ! ! ! e1 e2 e2 !! ! ∂ ∂ ∂ ! curl f = !! ∂x1 ∂x ∂x3 ! 2 ! f1 f2 f3 ! div f ≡ ∇ ·f =
Properties of the operator ∇
1051
10.31 Properties of the operator ∇ Let Φ (x1 , x2 , x3 ), Ψ (x1 , x2 , x3 ) be any two differentiable scalar functions, f (x1 , x2 , x3 ), g (x1 , x2 , x3 ) any two differentiable vector functions, and a an arbitrary vector. Define the scalar operator ∇2 , called the Laplacian, by ∂2 ∂2 ∂2 + + ∇2 ≡ 2 2 ∂x1 ∂x2 ∂x23 Then, in terms of the operator ∇, we have the following: MF I 114 1.
∇(Φ + Ψ) = ∇ Φ + ∇ Ψ
2.
∇(ΦΨ) = Φ ∇ Ψ + Ψ ∇ Φ
3.
∇ (f · g) = (f · ∇) g + (g · ∇) f + f × (∇ ×g) + g × (∇ ×f )
4.
∇ · (Φf ) = Φ (∇ ·f ) + f · ∇ Φ
5.
∇ · (f × g) = g · (∇ ×f ) − f · (∇ ×g)
6.
∇ × (Φf ) = Φ (∇ ×f ) + (∇ Φ) × f
7.
∇ × (f × g) = f (∇ ·g) − g (∇ ·f ) + (g · ∇) f − (f · ∇) g
8.
∇ × (∇ ×f ) = ∇ (∇ ·f ) − ∇2 f
9.
∇ × (∇ Φ) ≡ 0
10.
∇ · (∇ ×f ) ≡ 0
11.10 ∇2 (ΦΨ) = Φ ∇2 Ψ + 2 (∇ Φ) · (∇ Ψ) + Ψ ∇2 Φ The equivalent results in terms of grad, div, and curl are as follows: 1.
grad(Φ + Ψ) = grad Φ + grad Ψ
2.
grad(ΦΨ) = Φ grad Ψ + Ψ grad Φ
3.
grad (f · g) = (f · grad) g + (g · grad) f + f × curl g + g × curl f
4.
div (Φf ) = Φ div f + f · grad Φ
5.
div (f × g) = g · curl f − f · curl g
6.
curl (Φf ) = Φ curl f + grad Φ × f
7.
curl (f × g) = f div g − g div f + (g · grad) f − (f · grad) g
8.
curl (curl f ) = grad (div f ) − ∇2 f
9.
curl (grad Φ) ≡ 0
10.
div (curl f ) ≡ 0
11.
∇2 (ΦΨ) = Φ ∇2 Ψ + 2 grad Φ · grad Ψ + Ψ ∇2 Φ
The expression (a · ∇) or, equivalently (a · grad), defined by ∂ ∂ ∂ + a2 + a3 , (a · ∇) ≡ a1 ∂x1 ∂x2 ∂x3 is the directional derivative operator in the direction of vector a.
1052
Vectors, Vector Operators, and Integral Theorems
10.411
10.41 Solenoidal fields A vector field f is said to be solenoidal if div f ≡ 0. We have the following representation: 10.411 Representation theorem for vector Helmholtz equation. If u is a solution of the scalar Helmholtz equation ∇2 u + λ2 u = 0, and m is a constant unit vector, then the vectors 1 X = curl (mu) , Y = curl X λ are independent solutions of the vector Helmholtz equation ∇2 H + λ2 H = 0 involving a solenoidal vector H. The general solution of the equation is 1 H = curl (mu) + curl curl (mu) . λ
10.51–10.61 Orthogonal curvilinear coordinates Consider a transformation from the cartesian coordinates O {x1 , x2 , x3 } to the general orthogonal curvilinear coordinates O {u1 , u2 , u3 }: x1 = x1 (u1 , u2 , u3 ) , x2 = x2 (u1 , u2 , u3 ) , x3 = x3 (u1 , u2 , u3 ) Then, 1.
2.
∂xi ∂xi ∂xi du1 + du2 + du3 ∂u1 ∂u2 ∂u3 and the length element dl may be determined from dxi =
(i = 1, 2, 3) ,
dl2 = g11 du21 + g22 du22 + g33 du23 + 2g23 du2 du3 + 2g31 du3 du1 + 2g12 du1 du2 , where
3.3
4.
∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 + + = gji , ∂ui ∂uj ∂ui ∂uj ∂ui ∂uj provided the Jacobian of the transformation ! ! ! ∂x1 ∂x2 ∂x3 ! ! ∂u1 ∂u1 ∂u1 ! ! ∂x1 ∂x2 ∂x3 ! J = ! ∂u ∂u2 !! 2 ! ∂x12 ∂u ∂x2 ∂x3 ! ! ∂u ∂u ∂u
gij =
3
3
gij = 0,
i = j,
3
does not vanish (see 14.313). 5.
Define the metrical coefficients √ √ √ h1 = g11 , h2 = g22 , h3 = g33 ; then the volume element dV in orthogonal curvilinear coordinates is
6.
dV = h1 h2 h3 du1 du2 du3 , and the surface elements of area dsi on the surfaces ui = constant, for i = 1, 2, 3, are
7.
ds1 = h2 h3 du2 du3 ,
ds2 = h1 h3 du1 du3 ,
ds3 = h1 h2 du1 du2
Denote by e1 , e2 , and e3 the triad of orthogonal unit vectors that are tangent to the u1 , u2 , and u3 coordinate lines through any given point P , and choose their sense so that they form a right-handed set in this order. Then in terms of this triad of vectors and the components fu1 , fu2 , and fu3 of f along the coordinate line,
10.613
8. 10.611 1. 2.3
3.
4.
Orthogonal curvilinear coordinates
f = fu1 e1 + fu2 e2 + fu3 e3
1053
MF I 115
∇ Φ, div f , curl f , and ∇2 in general orthogonal curvilinear coordinates. e1 ∂Φ e2 ∂Φ e3 ∂Φ + + h1 ∂u1 h2 ∂u2 h3 ∂u3
∂ 1 ∂ ∂ div f = (h2 h3 fu1 ) + (h3 h1 fu2 ) + (h1 h2 fu3 ) h1 h2 h3 ∂u1 ∂u2 ∂u3 ! ! ! h1 e 1 h2 e 2 h3 e 3 ! ! ∂ ! 1 ∂ ∂ ! ! curl f = ∂u1 ∂u2 ∂u3 ! ! h1 h2 h3 ! h1 fu1 h2 fu2 h3 fu3 !
∂ h2 h3 ∂ h3 h1 ∂ h1 h2 ∂ 1 ∂ ∂ 2 ∇ ≡ + + h1 h2 h3 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 ∂u3 h3 ∂u3 grad Φ =
MF I 21-31
10.612 Cylindrical polar coordinates. In terms of the coordinates O{r, φ, z}, that is, u1 = r, u2 = φ, u3 = z, where x1 = r cos φ, x2 = r sin φ, x3 = z for −π < φ ≤ π, it follows that 1.
h1 = 1,
h2 = r,
h3 = 1,
and 2. 3.
4.
5.
1 ∂Φ ∂Φ ∂Φ er + eφ + ez , ∂r r ∂φ ∂z ∂fz 1 ∂fφ 1 ∂ (rfr ) + + , div f = r ∂r r ∂φ ∂z ! ! !e reφ ez !! 1 !! ∂r ∂ ∂ ! curl f = ! ∂r ∂φ ∂z ! , r! fr rfφ fz !
1 ∂ ∂2 ∂ 1 ∂2 2 ∇ ≡ r + 2 2+ 2 r ∂r ∂r r ∂φ ∂z grad Φ =
MF I 116
10.613 Spherical polar coordinates. In terms of the coordinates O{r, θ, φ}, that is, u1 = r, u2 = θ, u3 = φ, where x1 = r sin θ cos φ, x2 = r sin θ sin φ, x3 = r cos θ, for 0 ≤ θ ≤ π, −π < φ ≤ π, we have 1.
h1 = 1,
h2 = r,
h3 = r sin θ,
and 2.10 3.
4.
5.
∂Φ 1 ∂Φ 1 ∂Φ er + eθ + eφ , ∂r r ∂θ r sin θ ∂φ ∂ 1 1 ∂fφ 1 ∂ 2 r fr + (fθ sin θ) + , div f = 2 r ∂r r sin θ ∂θ r sin θ ∂φ ! ! ! er reθ r sin θeφ ! ! ! 1 ∂ ∂ !, !∂ curl f = 2 ∂r ∂θ ∂φ ! ! r sin θ ! fr rfθ r sin θfφ !
∂2 ∂ ∂ 1 ∂ 1 ∂ 1 ∇2 ≡ 2 r2 + 2 sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2
grad Φ =
MF I 116
1054
Vectors, Vector Operators, and Integral Theorems
10.614
Special Orthogonal Curvilinear Coordinates and their Metrical Coefficients h1 , h2 , h3 10.614 1. 2. 10.615 1. 2. 10.616 1.
Elliptic cylinder coordinates O{u1 , u2 , u3 }. < x1 = u1 u2 , x2 = (u21 − c2 ) (1 − u22 ), . . u21 − c2 u22 u21 − c2 u22 h1 = , h2 = , 2 2 u1 − c 1 − u22
10.617 1. 2. 10.618 1. 2. 10.619 1. 2.
h3 = 1
MF I 657
Parabolic cylinder coordinates O{u1 , u2 , u3 }. 1 2 u1 − u22 , 2 < h1 = u21 + u22 , x1 =
x2 = u1 u2 , x3 = u3 < h2 = u21 + u22 , h3 = 1
Conical coordinates O{u1 , u2 , u3 }. < < u1 u1 x1 = (a2 − u22 ) (a2 + u23 ), x2 = (b2 + u22 ) (b2 − u23 ), a b .
2.
x3 = u3
h1 = 1,
h2 = u 1
(a2
u22 + u23 , − u22 ) (b2 + u22 )
MF I 658
x3 =
u1 u2 u3 ab
with a2 + b2 = 1 . u22 + u23 h3 = u 1 2 (a + u23 ) (b2 − u23 )
Rotational parabolic coordinates O{u1 , u2 , u3 }. < 1 2 u1 − u22 x2 = u1 u2 1 − u23 , x3 = x1 = u1 u2 u3 , 2 < < u1 u2 h2 = u21 + u22 , h3 = h1 = u21 + u22 , 1 − u23 Rotational prolate spheroidal coordinates O{u1 , u2 , u3 }. < < x1 = (u21 − a2 ) (1 − u22 ), x2 = (u21 − a2 ) (1 − u22 ) (1 − u23 ), x3 = u1 u2 . . . u21 − a2 u22 u21 − a2 u22 (u21 − a2 ) (1 − u22 ) h1 = , h = , h = 2 3 u21 − a2 1 − u22 1 − u23 Rotational oblate spheroidal coordinates O{u1 , u2 , u3 }. < < x2 = (u21 + a2 ) (1 − u22 ) (1 − u23 ), x3 = u1 u2 x1 = u3 (u21 + a2 ) (1 − u22 ), . . . u21 + a2 u22 u21 + a2 u22 (u21 + a2 ) (1 − u22 ) h1 = , h2 = , h3 = 2 2 2 u1 + a 1 − u2 1 − u23
MF I 659
MF I 660
MF I 661
MF I 662
10.713
10.620 1.
2.
Vector integral theorems
1055
Ellipsoidal coordinates O{u1 , u2 , u3 }. . . (u21 − a2 ) (u22 − a2 ) (u23 − a2 ) (u21 − b2 ) (u22 − b2 ) (u23 − b2 ) u1 u2 u3 x1 = , x , x3 = = 2 a2 (a2 − b2 ) b2 (b2 − a2 ) ab . . . (u21 − u22 ) (u21 − u23 ) (u22 − u21 ) (u22 − u23 ) (u23 − u21 ) (u23 − u22 ) , h , h h1 = = = 2 3 (u21 − a2 ) (u21 − b2 ) (u22 − a2 ) (u22 − b2 ) (u23 − a2 ) (u23 − b2 ) MF I 663
10.621 1.
2.
Paraboloidal coordinates O{u1 , u2 , u3 }. . . (u21 − a2 ) (u22 − a2 ) (u23 − a2 ) (u21 − b2 ) (u22 − b2 ) (u23 − b2 ) , x2 = , x1 = 2 2 a −b b 2 − a2 x3 = 12 u21 + u22 + u23 − a2 − b2 . . . (u21 − u22 ) (u21 − u23 ) (u23 − u21 ) (u23 − u22 ) (u23 − u21 ) (u23 − u22 ) h1 = , h , h = u = u 2 2 3 3 2 2 2 2 (u1 − a2 ) (u1 − b2 ) (u2 − a2 ) (u2 − b2 ) (u23 − a2 ) (u23 − b2 ) MF I 664
10.622 1. 2.
Bispherical coordinates O{u1 , u2 , u3 }. 1 − u22 (1 − u22 ) (1 − u23 ) x1 = au3 , x2 = a , u1 − u2 u1 − u2 a h1 = , (u1 − u2 ) u21 − 1
x3 =
a , h2 = (u1 − u2 ) 1 − u22
u21 − 1 u1 − u2
h3 =
a u1 − u2
.
1 − u22 1 − u23
MF I 665
10.71–10.72 Vector integral theorems 10.711 Gauss’s divergence theorem. Let V be a volume bounded by a simple closed surface S and let f be a continuously differentiable vector field defined in V and on S. Then, if dS is the outward drawn vector element of area, f · dS = div f dV KE 39 S
V
10.712 Green’s theorems. Let Φ and Ψ be scalar fields which, together with ∇2 Φ and ∇2 Ψ, are defined both in a volume V and on its surface S, which we assume to be simple and closed. Then, if ∂/∂n denotes differentiation along the outward drawn normal to S, we have 10.713 Green’s first theorem ∂Ψ Φ ∇2 Ψ + grad Φ · grad Ψ dV dS = Φ KE 212 S ∂n V
1056
Vectors, Vector Operators, and Integral Theorems
Green’s second
theorem ∂Φ ∂Ψ −Ψ Φ ∇2 Ψ − Ψ ∇2 Φ dV Φ dS = ∂n ∂n S V 10.715 Special cases 2 (Φ grad Φ) · dS = 1. Φ ∇2 Φ + (grad Φ) dV S V ∂Φ dS = 2. ∇2 Φ dV S ∂n V
10.714
10.714
10.716 3.
Green’s reciprocal theorem. If Φ and Ψ are harmonic, so that ∇2 Φ = ∇2 Ψ = 0, then ∂Ψ ∂Φ dS = Ψ dS Φ S ∂n S ∂n
KE 215
MV 81
MM 105
10.717 Green’s representation theorem. If Φ and ∇2 Φ are defined within a volume V bounded by a simple closed surface S, and P is an interior point of V , then in three dimensions
1 1 2 1 ∂Φ 1 ∂ 1 1 ∇ Φ dV + dS − Φ KE 219 4. Φ(P ) = − dS 4π V r 4π S r ∂n 4π S ∂n r
5.
6.
If Φ is harmonic within V , so that ∇2 Φ = 0, then the previous result becomes
1 ∂Φ 1 ∂ 1 1 dS − Φ Φ(P ) = dS 4π S r ∂n 4π S ∂n r In the case of two dimensions, result (4) takes the form 1 Φ(p) = ∇2 Φ(q) ln |p − q| dS 2π S 1 ∂ 1 ∂ + Φ(q) ln |p − q| dq − Φ(q) dq ln |p − q| 2π C ∂nq 2π ∂nq MM 116
7.
where C is the boundary of the planar region S, and result (5) takes the form ∂ 1 ∂ 1 Φ(q) ln |p − q| dq − ln |p − q| Φ(q) dq Φ(p) = 2π C ∂nq 2π C ∂nq
VL 280
10.718 Green’s representation theorem in Rn . If Φ is twice differentiable within a region Ω in Rn bounded by the surface Σ with outward drawn unit normal n, then for p ∈ Σ and n > 3 −1 ∇2 Φ(q) ∂ Φ(q) 1 1 ∂ 1 Φ(p) = dΩq + − Φ(q) dΣq , (n − 2)σn Ω |p − q|n−2 (n − 2)σn Σ |p − q|n−2 ∂nq ∂nq |p − q|n−2 where 2π n/2 σn = VL 279 Γ(n/2) is the area of the unit sphere in Rn . 10.719 Green’s theorem of the arithmetic mean. If Φ is harmonic in a sphere, then the value of Φ at KE 223 the center of the sphere is the arithmetic mean of its value on the surface. 10.720 Poisson’s integral in three dimensions. If Φ is harmonic in the interior of a spherical volume V of radius R and is continuous on the surface of the sphere on which, in terms of the spherical polar coordinates (r, θ, φ), it satisfies the boundary condition Φ (R, θ, φ) = f (θ, φ), then
10.811
where
Integral rate of change theorems
1057
π π R R2 − r 2 f (θ , φ ) sin θ dθ dφ Φ(r, θ, φ) = , 3/2 4π 0 −π (r 2 + R2 − 2rR cos γ)
cos γ = cos θ cos θ + sin θ sin θ cos (φ − φ ) . KE 241 10.721 Poisson’s integral in two dimensions. If Φ is harmonic in the interior of a circular disk S of radius R and is continuous on the boundary of the disk on which, in terms of the polar coordinates (r, θ), it satisfies the boundary condition Φ(R, θ) = f (θ), then π 2 R − r2 f (φ) dφ . Φ(r, θ) = 2 2 2π −π r + R − 2rR cos(θ − φ) 10.722 Stokes’ theorem. Let a simple closed curve C be spanned by a surface S. Define the positive normal n to S, and the positive sense of description of the curve C with line element dr, such that the positive sense of the contour C is clockwise when we look through the surface S in the direction of the normal. Then, if f is continuously differentiable vector field defined on S and C with vector element S = n dS, C f · dr = curl f · dS, MM 143 C
S
where the line integral around C is taken in the positive sense. 10.723 Planar case of Stokes’ theorem. If a region R in the (x, y)-plane is bounded by a simple closed curve C, and f1 (x, y), f2 (x, y) are any two functions having continuous first derivatives in R and on C, then
C ∂f2 ∂f1 − (f1 dx + f2 dy) = MM 143 dx dy, ∂x ∂y C R where the line integral is taken in the counterclockwise sense.
10.81 Integral rate of change theorems 10.811 Rate of change of volume integral bounded by a moving closed surface. Let f be a continuous scalar function of position and time t defined throughout the volume V (t), which is itself bounded by a simple closed surface S (t) moving with velocity v. Then the rate of change of the volume integral of f is given by ∂f D dV + f dV = f v · dS, Dt V (t) V (t) ∂t S(t) where dS is the outward drawn vector element of area, and D ∂ ≡ + v · ∇. Dt ∂t By virtue of Gauss’s theorem, this also takes the form
D Df + f div v dV. f dV = MV 88 Dt V (t) Dt V (t)
1058
Vectors, Vector Operators, and Integral Theorems
10.812
10.812 Rate of change of flux through a surface. Let q be a vector function that may also depend on the time t, and n be the unit outward drawn normal to the surface S that moves with velocity v. Defining the flux of q through S as m = q · n dS, then
S
∂q Dm = + v div q + curl (q × v) · n dS. MV 90 Dt ∂t S 10.813 Rate of change of the circulation around a given moving curve. Let C be a closed curve, moving with velocity v, on which is defined a vector field q. Defining the circulation ζ of q around C by q · dr, ζ=
C
then Dζ = Dt
C
∂q + (curl q) × v · dr. ∂t
MV 94
11 Algebraic Inequalities 11.1–11.3 General Algebraic Inequalities 11.11 Algebraic inequalities involving real numbers 11.111
Lagrange’s identity. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two sets of real numbers; then n 2 n n 2 ak b k = a2k b2k − BB 3 (ak bj − aj bk ) k=1
k=1
k=1
11.112 Cauchy–Schwarz–Buniakowsky inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of real numbers; then n 2 n n 2 2 ak b k ≤ ak bk . k=1
k=1
k=1
The equality holds if, and only if, the sequences a1 , a2 , . . . , an and b1 , b2 , . . . , bn are proportional. MT 30
11.113 Minkowski’s inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two sets of nonnegative real numbers, and let p > 1; then 1/p n 1/p n 1/p n p p p (ak + bk ) ≤ ak + bk . k=1
k=1
k=1
The equality holds if, and only if, the sequences a1 , a2 , . . . , an and b1 , b2 , . . . , bn are proportional. MT 55
11.114 H¨ older’s inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two sets of nonnegative real numbers, and let p1 + 1q = 1, with p > 1; then 1/p n 1/q n n p q ak bk ≥ ak b k . k=1
k=1
k=1
The equality holds if, and only if, the sequences ap1 , ap2 , . . . , apn and bq1 , bq2 , . . . , bqn are proportional. MT 50
11.115 Chebyshev’s inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be two arbitrary sets of real numbers such that either a1 ≥ a2 ≥ · · · ≥ an and b1 ≥ b2 ≥ · · · ≥ bn , or a1 ≤ a2 ≤ · · · ≤ an and b1 ≤ b2 ≤ · · · ≤ bn ; then
n a1 + a 2 + · · · + a n b1 + b2 + · · · + bn 1 ak b k . ≤ n n n k=1 The equality holds if, and only if, either a1 = a2 = · · · = an or b1 = b2 = · · · = bn . 1059
1060
General Algebraic Inequalities
11.116
11.116 Arithmetic-geometric inequality. Let a1 , a2 , . . . , an be any set of positive numbers, with arithmetic mean
a1 + a2 + · · · + an An = n and geometric mean 1/n
then An ≥ Gn or, equivalently,
Gn = (a1 a2 . . . an )
;
a1 + a2 + · · · + an 1/n . ≥ (a1 a2 . . . an ) n The equality holds only in the event that all of the numbers ai are equal. 11.117 Carleman’s inequality. If a1 , a2 , . . . , an is any finite set of non-negative numbers, then n 1/r (a1 a2 . . . ar ) ≤ e (a1 + a2 + · · · + an ) ,
BB 4
r=1
where e is the best possible constant in this inequality. The inequality is strict except for the trivial case MT 131 when ar = 0 for r = 1, 2, . . . , n. 11.118 An inequality involving absolute values. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be two arbitrary sets of real numbers; then n p p p p {|ai − bj | + |bi − aj | − |ai − aj | − |bi − bj | } ≥ 0, 0 < p ≤ 2. i,j=1
11.21 Algebraic inequalities involving complex numbers If α, β are any two real numbers, the complex number z = α + iβ with real part α and imaginary part β has for its modulus |z| the nonnegative number |z| = α2 + β 2 , and for its argument (amplitude)arg z the angle arg z = θ such that α β cos θ = and sin θ = , |z| |z| where −π < θ ≤ π. The complex number z = α−iβ is said to be the complex conjugate of z = α+iβ. If z = reiθ = r (cos θ + i sin θ) , then z n = rn einθ = rn (cos nθ + i sin nθ) , and, setting r = 1, we have de Moivre’s theorem n
(cos θ + i sin θ) = cos nθ + i sin nθ. It follows directly that, if z = eiθ , then
1 1 i 1 cos θ = z+ , sin α = − z− , 2 z 2 z and
1 1 i 1 r r sin rθ = − cos rθ = z + r , z − r . 2 z 2 z p/q iθ If w = z with p, q integral, and z = re , then the q roots of w0 , w1 , . . . , wq−1 of z are
11.313
Inequalities for sets of complex numbers
wk = r
p/q
1061
pθ + 2kπ pθ + 2kπ cos + i sin , q q
with k = 0, 1, 2, . . . , q − 1. 11.2117 Simple properties and inequalities involving the modulus and the complex conjugate. If the real part of z is denoted by Re z and the imaginary part by Im z, then z + z = 2 Re z = 2α, z − z = 2 Im z = 2iβ, z = (z),
1 1 = , z z n
(z n ) = (z) , ! ! ! z1 ! |z1 | ! != ! z2 ! |z2 | , (z1 + z2 + · · · + zn ) = z1 + z2 + · · · + zn , z1 z2 · · · zn = z1 z2 · · · zn . 11.212
Inequalities for pairs of complex numbers.
(i)
|a + b| ≤ |a| + |b|
(ii)
|a − b| ≥ ||a| − |b||.
If a,b are any two complex numbers, then
(triangle inequality),
11.31 Inequalities for sets of complex numbers 11.311 Complex Cauchy–Schwarz–Buniakowsky inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of complex numbers; then !2 ! n n n ! ! ! ! 2 2 ak b k ! ≤ |ak | |bk | . ! ! ! k=1 k=1 k=1 The equality holds if, and only if, the sequences a1 , a2 , . . . , an and b1 , b2 , . . . , bn are proportional. MT 42
11.312 Complex Minkowski inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of complex numbers, and let the real number p be such that p > 1; then n 1/p n 1/p n 1/p p p p |ak + bk | ≤ |ak | + |bk | . MT 56 k=1
11.313
k=1
k=1
Complex H¨ older inequality. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be any two arbitrary sets of 1 complex numbers, and let the real numbers p,q be such that p > 1 and + 1q = 1; then p ! 1/p n 1/p ! n n ! ! ! ! p q |ak | |bk | ≥! ak b k ! . ! ! k=1 k=1 k=1 The equality holds if, and only if, the sequences p p p p p p |a1 | , |a2 | , . . . , |an | and |b1 | , |b2 | , . . . |bn | , MT 53 are proportional and arg ak bk is independent of k for k = 1, 2, . . . , n.
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12 Integral Inequalities 12.11 Mean Value Theorems 12.111 First mean value theorem Let f (x) and g(x) be two bounded functions integrable in [a, b], and let g(x) be of one sign in this interval. Then b b f (x)g(x) dx = f (ξ) g(x) dx, CA 105 a
with a ≤ ξ ≤ b.
a
12.112 Second mean value theorem (i)
Let f (x) be a bounded, monotonic decreasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then, ξ b f (x)g(x) dx = f (a) g(x) dx, a
a
with a ≤ ξ ≤ b. (ii)
Let f (x) be a bounded, monotonic increasing, and nonnegative function in [a, b], and let g(x) be a bounded integrable function. Then, b b f (x)g(x) dx = f (b) g(x) dx, a
η
with a ≤ η ≤ b. (iii)
Let f (x) be bounded and monotonic in [a, b], and let g(x) be a bounded integrable function which experiences only a finite number of sign changes in [a, b]. Then, b ξ b f (x)g(x) dx = f (a + 0) g(x) dx + f (b − 0) g(x) dx, CA 107 a
a
ξ
with a ≤ ξ ≤ b.
12.113 First mean value theorem for infinite integrals Let f (x) be bounded for" x ≥ a, and integrable in the arbitrary interval [a, b], and let g(x) be of one sign ∞ in x ≥ a and such that a g(x) dx is finite. Then, 1063
1064
Differentiation of Definite Integral Containing a Parameter
∞
f (x)g(x) dx = μ
a
∞
g(x) dx,
CA 123
a
where m ≤ μ ≤ M and m, M are, respectively, the lower and upper bounds of f (x) for x ≥ a.
12.114 Second mean value theorem for infinite integrals Let f (x) be bounded and monotonic when x ≥ a, and g(x) be bounded and integrable in the" arbitrary ∞ interval [a, b] in which it experiences only a finite number of changes of sign. Then, provided a g(x) dx is finite, ξ ∞ ∞ f (x)g(x) dx = f (a + 0) g(x) dx + f (∞) g(x) dx, CA 123 with a ≤ ξ ≤ ∞.
a
a
ξ
12.21 Differentiation of Definite Integral Containing a Parameter 12.211 Differentiation when limits are finite Let φ(α) and ψ(α) be twice differentiable functions in some interval c ≤ α ≤ d, and let f (x, α) be both integrable with respect to x over the interval φ(α) ≤ x ≤ ψ(α) and differentiable with respect to α. Then,
ψ(α) dψ d ψ(α) dφ ∂f dx. f (x, α) dx = FI II 680 f (ψ(α), α) − f (φ(α), α) + dα φ(α) dα dα φ(α) ∂α
12.212 Differentiation when a limit is infinite Let f (x, α) and ∂f /∂α both be integrable with respect to x over the semi-infinite region x ≥ a, b ≤ α < c. Then, if the integral ∞ f (x, α) dx f (α) = a " ∞ ∂f exists for all b ≤ α ≤ c, and if a ∂α dx is uniformly convergent for α in [b, c], it follows that ∞ ∂f d ∞ dx f (x, α) dx = dα a ∂α a
12.31 Integral Inequalities 12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals Let f (x) and g(x) be any two real integrable functions on [a, b]. Then, 2 b b b 2 2 f (x)g(x) dx ≤ f (x) dx g (x) dx , a
a
a
and the equality will hold if, and only if, f (x) = kg(x), with k real.
BB 21
12.312 H¨ older’s inequality for integrals p
q
Let f (x) andg(x) be any two real functions for which |f (x)| and |g(x)| are integrable on [a, b] with p > 1 and p1 + 1q = 1; then
Gram’s inequality for integrals
b
1/p
b
f (x)g(x) dx ≤
1065
|f (x)| dx
a
1/q
b
p
q
|g(x)| dx
a
.
a
p
q
The equality holds if, and only if, α|f (x)| = β|g(x)| , where α and β are positive constants.
BB 21
12.313 Minkowski’s inequality for integrals p
p
Let f (x) and g(x)be any two real functions for which |f (x)| and |g(x)| are integrable on [a, b] for p > 0; then 1/p 1/p 1/p b b b p p p |f (x) + g(x)| dx ≤ |f (x)| dx + |g(x)| dx . a
a
a
The equality holds if, and only if, f (x) = kg(x) for some real k ≥ 0.
BB 21
12.314 Chebyshev’s inequality for integrals Let f1 , f2 , . . . , fn be nonnegative integrable functions on [a, b] which are all either monotonic increasing or monotonic decreasing; then b b b b n−1 f1 (x) dx f2 (x) dx . . . fn (x) dx ≤ (b − a) f1 (x)f2 (x) . . . fn (x) dx MT 39 a
a
a
a
12.315 Young’s inequality for integrals Let f (x) be a real-valued continuous strictly monotonic increasing function on the interval [0, a], with f (0) = 0 and b ≤ f (a). Then b a f (x) dx + f −1 (y) dy, ab ≤ 0
0
where f −1 (y) denotes the function inverse to f (x). The equality holds if, and only if, b = f (a).
BB 15
12.316 Steffensen’s inequality for integrals Let f (x) be nonnegative and monotonic decreasing in [a, b], and g(x) be such that 0 ≤ g(x) ≤ 1 in [a, b]. Then b a+k b f (x) dx ≤ f (x)g(x) dx ≤ f (x) dx, b−k a a "b where k = a g(x) dx. MT 107
12.317 Gram’s inequality for integrals Let f1 (x), f2 (x), . . . , fn (x) be real square integrable functions on [a, b]; then ! "b ! "b "b ! ! 2 f (x)f (x) dx · · · f (x)f (x) dx 1 2 1 n ! " a f1 (x) dx ! a" "ab ! b ! b 2 ! a f2 (x)f1 (x) dx ! f (x) dx · · · f (x)f (x) dx n a 2 a 2 ! ! ≥ 0. .. .. .. .. ! ! . ! ! . . . !" b ! "b "b 2 ! fn (x)f1 (x) dx f (x)f2 (x) dx · · · f (x) dx ! a a n a n
MT 47
1066
Convexity and Jensen’s Inequality
12.318 Ostrowski’s inequality for integrals Let f (x) be a monotonic function integrable on [a, b], and let f (a)f (b) ≥ 0, |f (a)| ≥ |f (b)|. Then, if g is a real function integrable on [a, b], ! ! ! ! ! b ! ξ ! ! ! ! ! ! ! f (x)g(x) dx! ≤ |f (a)| max ! g(x) dx!. ! a ! ! a≤ξ≤b! a
12.41 Convexity and Jensen’s Inequality A function f (x) is said to be convex on an interval [a, b] if for any two points x1 , x2 in [a, b]
x1 + x2 f (x1 ) + f (x2 ) f . ≤ 2 2 A function f (x) is said to be concave on an interval [a, b] if for any two points x1 , x2 in [a, b] the function −f (x) is convex in that interval. If the function f (x) possesses a second derivative in the interval [a, b], then a necessary and sufficient condition for it to be convex on that interval is that f (x) ≥ 0 for all x in [a, b]. A function f (x) is said to be logarithmically convex on the interval [a, b] if f > 0 and log f (x) is concave on [a, b]. If f (x) and g(x) are logarithmically convex on the interval [a, b], then the functions f (x) + g(x) and MT 17 f (x)g(x) are also logarithmically convex on [a, b].
12.411 Jensen’s inequality Let f (x), p(x) be two functions defined for a ≤ x ≤ b such that α ≤ f (x) ≤ β and p(x) ≥ 0, with p(x) ≡ 0. Let φ(u) be a convex function defined on the interval α ≤ u ≤ β; then "b " b f (x)p(x) dx φ (f ) p(x) dx a ≤ a"b . φ HL 151 "b p(x) dx p(x) dx a a
12.412 Carleman’s inequality for integrals If f (x) ≥ 0 and the integrals exist, then x
∞ ∞ 1 exp f (t) dt dx ≤ e f (x) dx. x 0 0 0
12.51 Fourier Series and Related Inequalities The trigonometric Fourier series representation of the function f (x) integrable on [−π, π] is ∞ a0 + f (x) ∼ (an cos nx + bn sin nx) , 2 n=1 where the Fourier coefficients an and bn of f (x) are given by 1 π 1 π an = f (x) cos nx dx, bn = f (x) sin nx dx. 2π −π 2π −π (See 0.320–0.328 for convergence of Fourier series on (−l, l).)
TF 1
Generalized Fourier series
1067
12.511 Riemann-Lebesgue lemma If f (x) is integrable on [−π, π], then
lim
π
t→∞ −π
and
π
lim
t→∞ −π
12.512 Dirichlet lemma
in which sin n +
1 2
x
D
π 0
2 sin
1 2x
f (x) sin tx dx → 0
f (x) cos tx dx → 0.
TF 11
sin n + 12 x π dx = , 2 2 sin 12 x
is called the Dirichlet kernel.
ZY 21
12.513 Parseval’s theorem for trigonometric Fourier series If f (x) is square integrable on [−π, π], then ∞ 1 π 2 a20 2 + ar + b2r = f (x) dx. 2 π −π r=1
Y 10
12.514 Integral representation of the nth partial sum If f (x) is integrable on [−π, π], then the nth partial sum n a0 + sn (x) = (ar cos rx + br sin rx) 2 r=1 has the following integral representation in terms of the Dirichlet kernel: sin n + 12 t 1 π f (x − t) sn (x) = dt. π −π 2 sin 12 t
12.515 Generalized Fourier series ∞
Let the set of functions {φn }n=0 form an orthonormal set over [a, b], so that b 1 for m = n, φm (x)φn (x) dx = 0 for m = n. a Then the generalized Fourier series representation of an integrable function f (x) on [a, b] is ∞ f (x) ∼ cn φn (x), n=0
where the generalized Fourier coefficients of f (x) are given by b cn = f (x)φn (x) dx. a
Y 20
1068
Fourier Series and Related Inequalities
12.516 Bessel’s inequality for generalized Fourier series For any square integrable function defined on [a, b], b ∞ c2n ≤ f 2 (x) dx, n=0
a
where the cn are the generalized Fourier coefficients of f (x).
12.517 Parseval’s theorem for generalized Fourier series ∞
If f (x) is a square integrable function defined on [a, b] and {φn (x)}n=0 is a complete orthonormal set of continuous functions defined on [a, b], then b ∞ 2 cn = f 2 (x) dx, n=0
a
where the cn are generalized Fourier coefficients of f (x).
13 Matrices and Related Results 13.11–13.12 Special Matrices 13.111 Diagonal matrix A square matrix A of the form
⎡ λ1 ⎢0 ⎢ ⎢ A=⎢0 ⎢ .. ⎣.
0 λ2 0 .. .
0 0 λ3
... ... ..
0 0 0
.
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
0 0 0 λn in which all entries away from the leading diagonal are zero.
13.112 Identity matrix and null matrix The identity matrix is a diagonal matrix I in which all entries in the leading diagonal are unity. The null matrix is all zeros.
13.113 Reducible and irreducible matrices The n × n matrix A = [aij ] is said to be reducible, if the indices 1, 2, . . . , n can be divided into two disjoint non-empty sets i1 , i2 , . . . , iμ ; j1 , j2 , . . . , jν with (μ + ν = n), such that aiα jβ = 0 (α = 1, 2, . . . , μ; β = 1, 2, . . . , ν) . GA 61 Otherwise, A will be said to be irreducible.
13.114 Equivalent matrices An m×n matrix A is equivalent to an m×n matrix B if, and only if, B = PAQ for suitable non-singular m × m and n × n matrices P and Q, respectively.
13.115 Transpose of a matrix If A = [aij ] is an m × n matrix with element aij in the ith row and the j th column, then the transpose AT of A is the n × m matrix AT = [bij ] with bij = aji , that is, the matrix derived from A by interchanging rows and columns.
1069
1070
Special Matrices
13.116 Adjoint matrix If A is an n × n matrix, then its adjoint, denoted by adj A, is the transpose of the matrix of cofactors Aij of A, so that adj A = [Aij ]
T
(see 14.13).
13.117 Inverse matrix If A = [aij ] is an n × n matrix with a nonsingular determinant |A|, then its inverse A−1 is given by adj A . A−1 = |A|
13.118 Trace of a matrix The trace of an n × n matrix A = [aij ], written tr A, is defined to be the sum of the terms on the leading diagonal, so that tr A = a11 + a22 + . . . + ann .
13.119 Symmetric matrix The n × n matrix A = [aij ] is symmetric if aij = aji for i, j = 1, 2, . . . , n.
13.120 Skew-symmetric matrix The n × n matrix A = [aij ] is skew-symmetric if aij = −aji for i, j = 1, 2, . . . , n.
13.121 Triangular matrices An n × n matrix A = [aij ] is of upper triangular type if aij = 0 for i > j and of lower triangular type if aij = 0 for j > i.
13.122 Orthogonal matrices A real n × n matrix A is orthogonal if, and only if, AAT = I.
13.123 Hermitian transpose of a matrix If A = [aij ] is an n × n matrix with complex elements, then its hermitian transpose AH is defined to be AH = [aji ] , with the bar denoting the complex conjugate operation.
13.124 Hermitian matrix An n × n matrix A is hermitian if A = AH , or equivalently, if A = AT , with the bar denoting the complex conjugate operation.
Diagonally dominant
1071
13.125 Unitary matrix An n × n matrix A is unitary if AAH = AH A = I.
13.126 Eigenvalues and eigenvectors If A is an n × n matrix, each eigenvector x corresponding to λ satisfies the equation AX = λx, while the eigenvalues λ satisfy the characteristic equation |A − λI| = 0
(see 15.61).
13.127 Nilpotent matrix An n × n matrix A is nilpotent if Ak = 0 for some k.
13.128 Idempotent matrix An n × n matrix A is idempotent if A2 = A.
13.129 Positive definite An n × n matrix A is positive definite if xT Ax > 0, for x = 0 an n element column vector.
13.130 Non-negative definite An n × n matrix A is non-negative definite if xT Ax ≥ 0, for x = 0 an n element column vector.
13.131 Diagonally dominant An n × n matrix A is diagonally dominant if |aii | >
# j=i
|aij | for all i.
13.21 Quadratic Forms A quadratic form involving the n real variables x1 , x2 , . . . , xn that are associated with the real n × n matrix A = [aij ] is the scalar expression n n Q(x1 , x2 , . . . , xn ) = aij xi xj . i=1 j=1
In terms of matrix notation, if x is the n × 1 column vector with real elements x1 , x2 , . . . , xn , and xT is the transpose of x, then Q(x) = xT Ax. Employing the inner product notation, this same quadratic form may also be written Q(x) ≡ (x, Ax) . If the n × n matrix A is hermitian, so that AT = A, where the bar denotes the complex conjugate operation, then the quadratic form associated with the hermitian matrix A and the vector x, which may have complex elements, is the real quadratic form
1072
Quadratic Forms
Q(x) = (x, Ax) . It is always possible to express an arbitrary quadratic form n n αij xi xj Q(x) = i=1 j=1
in the form Q(x) = (x, Ax) , where A = [aij ] is a symmetric matrix, by defining aii = αii for i = 1, 2, . . . , n and for i, j = 1, 2, . . . , n and i = j. aij = 12 (αij + αji )
13.211 Sylvester’s law of inertia When a quadratic form Q in n variables is reduced by a nonsingular linear transformation to the form 2 2 − yp+2 − . . . − yr2 , Q = y12 + y22 + . . . + yp2 − yp+1 the number p of positive squares appearing in the reduction is an invariant of the quadratic form Q, and it does not depend on the method of reduction itself. ML 377
13.212 Rank The rank of the quadratic form Q in the above canonical form is the total number r of squared terms ML 360 (both positive and negative) appearing in its reduced form.
13.213 Signature The signature of the quadratic form Q above is the number s of positive squared terms appearing in its ML 378 reduced form. It is sometimes also defined to be 2s − r.
13.214 Positive definite and semidefinite quadratic form The quadratic form Q(x) = (x, Ax) is said to be positive definite when Q(x) > 0 for x = 0. It is said ML 394 to be positive semidefinite if Q(x) ≥ 0 for x = 0.
13.215 Basic theorems on quadratic forms 1.
Two real quadratic forms are equivalent under the group of linear transformations if, and only if, they have the same rank and the same signature.
2.
A real quadratic form in n variables is positive definite if, and only if, its canonical form is Q = z12 + z22 . + . . . + zn2 .
3.
A real symmetric matrix A is positive definite if, and only if, there exists a real nonsingular matrix M such that A = MMT .
4.
Any real quadratic form in n variables may be reduced to the diagonal form
Basic theorems on quadratic forms
1073
Q = λ1 z12 + λ2 z22 + . . . + λn zn2 , λ1 ≥ λ2 ≥ . . . ≥ λn by a suitable orthogonal point-transformation. 5.
The quadratic form Q = (x, Ax) is positive definite if, and only if, every eigenvalue of A is positive; it is positive semidefinite if, and only if, all the eigenvalues of A are nonnegative, and it is indefinite if the eigenvalues of A are of both signs.
6.
The necessary conditions for an hermitian matrix A to be positive definite are aii > 0 for all i, aii aij > |aij |2 for i = j, the element of largest modulus must lie on the leading diagonal, |A| > 0.
(i) (ii) (iii) (iv) 7.
The quadratic form Q = (x, Ax) with A hermitian will be positive definite if all the principal minors in the top left-hand corner of A are positive, so that ! ! !a11 a12 a13 ! ! ! ! ! !a11 a12 ! ! > 0, !a21 a22 a23 ! > 0, . . . . a11 > 0, !! ML 353-379 ! ! ! a21 a22 !a31 a32 a33 !
13.31 Differentiation of Matrices If the n × m matrices A(t) and B(t) have elements that are differentiable functions of t, so that B(t) = [bij (t)] A(t) = [aij (t)] , then d d A(t) = aij (t) 1. dt dt d d d [A(t) ± B(t)] = aij (t) ± bij (t) 2. dt dt dt d d = A(t) ± B(t). dt dt 3. If the matrix product A(t)B(t) is defined, then
d d d [A(t)B(t)] = A(t) B(t) + A(t) B(t) . dt dt dt 4.
If the matrix product A(t)B(t) is defined, then
T
T d d d T [A(t)B(t)] = B(t) A(t) AT (t) + BT (t) . dt dt dt
5.
If the square matrix A is nonsingular, so that |A| = 0, then
d −1 d −1 A A(t) A−1 (t) = −A (t) dt dt % & T
T
A(τ ) dτ =
6. t0
aij (τ ) dτ t0
1074
The Matrix Exponential
13.41 The Matrix Exponential If A is a square matrix, and z is any complex number, then the matrix exponential eAz is defined to be ∞ 1 r r An z n + ... = A z . eAz = I + Az + . . . + n! r! r=0
3.411 Basic properties 1.
e−Az 2.
eIz = Iez , eA(z1 +z2 ) = eAz1 · eAz2 , −1 = eAz , eAz · eBz = e(A+B)z
e0 = I,
[when A + B is defined and AB = BA]
dr Az e = Ar eAz = eAz Ar . dz r ML 340
3.
B 0 If the square matrix A can be expressed in the form A = , with B and C square matrices, 0 C then Bz e 0 Az e = . 0 eCz
14 Determinants 14.11 Expansion of Second- and Third-Order Determinants 1.
2.
! !a11 ! !a21 ! !a11 ! !a21 ! !a31
! a12 !! = a11 a22 − a12 a21 . a22 ! ! a12 a13 !! a22 a23 !! = a11 a22 a33 − a11 a23 a32 + a12 a23 a31 − a12 a21 a33 + a13 a21 a32 − a13 a22 a31 . a32 a33 !
14.12 Basic Properties Let A = [aij ] and B = [bij ] be n × n matrices. Then the following results are true: 1.
If any two adjacent rows (or columns) of a square matrix are interchanged, then the sign of the associated determinant is changed.
2.
If any two rows (or columns) of a determinant are identical, the determinant is zero.
3.
A determinant is not changed in value if any multiple of a row (or column) is added to any other row (or column).
4.
|kA| = kn |A| ! T! !A ! = |A|
5.
for any scalar k. where AT is the transpose of A.
7.
|AB| = |A||B|. ! −1 ! !A ! = 1
8.
If the elements aij of A are functions of x, then
6.
|A|
when the inverse exists. n d|A| daij = Aij dx dx i,j=1
(see 14.13).
14.13 Minors and Cofactors of a Determinant The minor Mij of the element aij in the nth -order determinant |A| associated with the square n × n matrix A is the (n − 1)th -order determinant derived from A by deletion of the ith row and j th column. The cofactor Aij of the element aij is defined to be Aij = (−1)i+j Mij . 1075
ML 20
1076
Principal Minors
14.14 Principal Minors A principal minor is one whose elements are situated symmetrically with respect to the leading diagonal ML 197 of A.
14.15* Laplace Expansion of a Determinant The nth -order determinant denoted by |A|, or det A, associated with the n × n matrix A = [aij ] may be expanded either by elements of the ith row as n aij Aij , |A| = j=1
or by elements of the j th column as |A| =
n
aij Aij ,
i=1
where Aij is the cofactor of element aij . The cofactors Aij satisfy the following n linear equations: n n aij Akj = δik |A| , aij Aik = δjk |A| , j=1
i=1
1 for i = j for i, j, k = 1, 2, . . . , n and δij = 0 for i = j.
ML 21
14.16 Jacobi’s Theorem Let Mr be an r-rowed minor of the nth -order determinant |A|, associated with the n × n matrix A = [aij ], in which the rows i1 , i2 , . . . , ir are represented together with the columns k1 , k2 , . . . , kr . Define the complementary minor to Mr to be the (n−k)-rowed minor obtained from |A| by deleting all the rows and columns associated with Mr , and the signed complementary minor M (r) to Mr to be M (r) = (−1)i1 +i2 +···+ir +k1 +k2 +···+kr × (complementary minor to Mr ). Then, if Δ is the matrix of cofactors given by ! ! ! A11 A12 · · · A1n ! ! ! ! A21 A22 · · · A2n ! ! ! Δ=! . .. .. ! , .. ! .. . . . !! ! !An1 An2 · · · Ann ! and Mr and Mr are corresponding r-rowed minors of |A| and Δ, it follows that Corollary. If |A| = 0, then
Mr = |A|r−1 M (r) . Apk Anq = Ank Apq .
ML 25
Cramer’s Rule
1077
14.17 Hadamard’s Theorem If |A| is an n × n determinant with elements aij that may be complex, then |A| = 0 if n |aii | > |aij | . j=1,j=i
14.18 Hadamard’s Inequality Let A = [aij ] be an arbitrary n × n nonsingular matrix with real elements and determinant |A|. Then n n 2 2 |A| ≤ aik . i=1
k=1
This result is also true when A is hermitian. Deductions. 1.
ML 418
If M = max |aij |, then |A| ≤ M n nn/2 .
2.
ML 419
If the n × n matrix A = [aij ] is positive definite, then |A| ≤ a11 a22 . . . ann .
3.
If the real n × n matrix A is diagonally dominant, so that then |A| = 0.
BL 126
#n j=1
|aij | < |aii | for i = 1, 2, . . . , n,
14.21 Cramer’s Rule If the n linear equations a11 x1 a21 x1 .. .
+ +
a12 x2 a22 x2 .. .
+ ··· + ··· .. .
+ +
a1n xn a2n xn .. .
= =
b1 , b2 , .. .
an1 x1 + an2 x2 + · · · + ann xn = bn , have a nonsingular coefficient matrix A = [aij ], so that |A| = 0, then there is a unique solution A1j b1 + A2j b2 + · · · + Anj bj xj = |A| for j = 1, 2, . . . , n, where Aij is the cofactor of element aij in the coefficient matrix A. ML 134
1078
Some Special Determinants
14.31 Some Special Determinants 14.311 Vandermonde’s determinant (alternant) Third order.
! ! !1 1 1 !! ! !x1 x2 x3 ! = (x3 − x2 ) (x3 − x1 ) (x2 − x1 ) , ! ! 2 !x1 x22 x23 ! and, in general, the nth -order Vandermonde’s determinant is ! ! ! 1 1 ··· 1 !! ! ! x1 x2 ··· xn !! ! 2 2 ! x1 x · · · x2n !! = 2 (xj − xi ) , ! ! .. .. .. ! 1≤i<j≤n .. ! . ! . . . ! ! n−1 !x ! xn−1 · · · xn−1 n 1 2 where the right-hand side is the continued product of all the differences that can be formed from the 1 2 n(n − 1) pairs of numbers taken from x1 , x2 , . . . , xn , with the order of the differences taken in the ML 17 reverse order of the suffixes that are involved.
14.312 Circulants Second order.
! !x1 ! !x2
! x2 !! = (x1 + x2 ) (x1 − x2 ) . x1 !
Third order.
! ! !x1 x2 x3 ! ! ! !x3 x1 x2 ! = (x1 + x2 + x3 ) x1 + ωx2 + ω 2 x3 x1 + ω 2 x2 + ωx3 , ! ! !x2 x3 x1 ! where ω and ω 2 are the complex cube roots of 1. In general, the nth -order circulant determinant is ! ! ! x1 x2 x3 · · · xn !! ! ! xn x1 x2 · · · xn−1 !! n ! !xn−1 xn x1 · · · xn−2 ! - x1 + x2 ωj + x3 ωj2 + · · · + xn ωjn−1 , ! != ! .. .. .. .. ! j=1 .. ! . . . . . !! ! ! x2 x3 x4 · · · x1 ! where ωj is an nth root of 1. The eigenvalues λ (see 15.61) of an n × n circulant matrix are λj = x1 + x2 ωj + x3 ωj2 + · · · + xn ωjn−1 , th where ωj is again an n root of 1.
ML 36
14.313 Jacobian determinant If f1 , f2 , . . . , fn are n real-valued functions which are differentiable with respect to x1 , x2 , . . . , xn , then the Jacobian Jf (x) of the fi with respect to the xj is the determinant ! ! ∂f ! 1 ∂f1 · · · ∂f1 ! ! ∂x1 ∂x2 ∂xn ! ! ∂f2 ∂f2 ∂f2 ! · · · ∂x ! ! ∂x ∂x2 n! . Jf (x) = !! . 1 . . . .. .. .. !! ! .. ! ∂fn ∂fn ∂fn !! ! ∂x · · · ∂x ∂x 1
2
n
Properties
1079
The notation ∂(f1 , f2 , . . . , fn ) ∂(x1 , x2 , . . . , xn ) is also used to denote the Jacobian Jf (x).
14.314 Hessian determinants ∂φ ∂φ ∂φ , ,..., of a function φ(x1 , x2 , . . . , xn ) with respect to x1 , x2 , ∂x1 ∂x2 ∂xn . . . , xn is called the Hessian H of φ, so that ! 2 ! 2 ∂2φ ∂2φ φ ! ! ∂ φ · · · ∂x∂1 ∂x ! ∂x21 ∂x1 ∂x2 ∂x1 ∂x3 n! ! ∂2φ ! ∂2φ ∂2φ ∂ 2 d2φ ! ! · · · ∂x ∂x2 ∂x3 ! ∂x2 ∂x1 ∂x22 2 ∂xn ! H=! . .. .. .. !! . .. ! . . . . . . ! ! 2 2 2 ! ∂2φ ∂ φ ∂ φ ∂ φ !! ! ∂x ∂x ··· ∂x ∂x ∂x ∂x ∂x2
The Jacobian of the derivatives
n
1
n
2
n
3
n
14.315 Wronskian determinants Let f1 , f2 , . . . , fn be n functions each n times differentiable with respect to x in some open interval (a, b). Then the Wronskian W (x) of f1 , f2 , . . . , fn is defined by ! ! ! f1 f2 ··· fn !! ! (1) (1) (1) ! f f2 ··· fn !! ! 1 ! (2) (2) (2) ! f2 ··· fn ! , W (x) = !! f1 .. .. !! .. ! .. . ! . . . ! ! ! (n−1) (n−1) (n−1) ! !f f · · · f n 1 2 dr fi (r) where fi = . dxr
14.316 Properties 1.
2.
dW follows from W (x) by replacing the last row of the determinant defining W (x) by the nth dx (n) (n) (n) derivatives f1 , f2 , . . . , fn . If constants k1 , k2 , . . . , kn exist, not all zero, such that k1 f1 + k2 f2 + · · · + kn fn = 0 for all x in (a, b), then W (x) = 0 for all x in (a, b).
3.
The vanishing of the Wronskian throughout (a, b) is necessary, but not sufficient, for the linear dependence of f1 , f2 , . . . , fn .
1080
Some Special Determinants
14.318
14.317 Gram-Kowalewski theorem on linear dependence A necessary and sufficient condition for n functions f1 , f2 , . . . , fn square integrable over a ≤ n ≤ b to be linearly dependent in this interval is the vanishing of the Gram determinant ! ! "b "b "b ! 2 f (x)f2 (x) dx · · · f1 (x)fn (x) dx!! ! " a f1 (x) dx a "1 a "b ! ! b b 2 ! a f2 (x)f1 (x) dx f (x) dx ··· f (x)fn (x) dx! a 2 a 2 !. ! G(f1 , f2 , . . . , fn ) = ! SA 2 (Theorem 3) .. .. .. .. ! . ! ! . . . ! !" b "b "b 2 ! fn (x)f1 (x) dx f (x)f2 (x) dx · · · f (x) dx ! a a n a n 14.318 If the n functions f1 , f2 , . . . , fn are square integrable over a ≤ n ≤ b, then the Gram determinant G (f1 , f2 , . . . , fn ) ≥ 0, and the equality sign holds only when the functions are linearly dependent in a ≤ n ≤ b. SA 4 (Corollary 1)
14.319 The rank of the matrix corresponding to the Gram determinant G(f1 , f2 , . . . , fn ) gives the maximum number of linearly independent functions f1 , f2 , . . . , fn in a ≤ x ≤ b. If the rank is r, then r of the functions are linearly independent, and the other n − r functions are linearly dependent on these. SA 3 (Theorem 4)
15 Norms 15.1–15.9 Vector Norms 15.11 General Properties The vector norm ||x|| of an n × 1 column vector x is a nonnegative number having the property that 1.
||x|| > 0 when x = 0 and ||x|| = 0 if, and only if, x = 0;
2.
||kx|| = |k|||x|| for any scalar k;
3.
||x + y|| ≤ ||x|| + ||y||.
15.21 Principal Vector Norms 15.211 The norm ||x||1 If x is a vector with complex components x1 , x2 , . . . , xn , then n ||x||1 = |xr |.
VA 15
r=1
15.212 The norm ||x||2 (Euclidean or L2 norm) If x is a vector with complex components x1 , x2 , . . . , xn , then n 1/2 2 ||x||2 = |xr | .
VA 8
r=1
15.213 The norm ||x||∞ If x is a vector with complex components x1 , x2 , . . . , xn , then ||x||∞ = max|xi |. i
1081
VA 15
1082
Matrix Norms
15.31 Matrix Norms 15.311 General properties The matrix norm ||A|| of a square matrix A is a nonnegative number associated with A having the properties that 1.
||A|| > 0 when A = 0 and ||A|| = 0 if, and only if, A = 0;
2.
||kA|| = |k| ||A|| for any scalar k;
3.
||A + B|| ≤ ||A|| + ||B||;
4.
||AB|| ≤ ||A|| ||B||.
VA 9
The matrix norm ||A|| associated with A = [aij ], and the vector norm ||x|| associated with the column vector x for which the matrix product Ax is defined, are said to be compatible if ||Ax|| ≤ ||A|| ||x||.
15.312 Induced norms When a vector z with norm ||z|| exists such that the maximum is attained in the expression ||A|| = max ||Az||, ||z||=1
then ||A|| is a matrix norm and is said to be the natural norm induced by, or subordinate to, the NO 428 vector norm ||z||.
15.313 Natural norm of unit matrix If I is the unit matrix, then for any natural norm ||I|| = 1.
NO 429
15.41 Principal Natural Norms The natural matrix norms induced on matrix A = [aij ] by the 1, 2, and ∞ vector norms are as follows:
15.411 Maximum absolute column sum norm ||A||1 = max j
n
|aij |
NO 429
i=1
15.412 Spectral norm If AH denotes the Hermitian transpose of the square matrix A = [aij ], so that AH = [aji ] with a bar denoting the complex conjugate operation, then ||A||2 = maximum eigenvalue of AH A, or, equivalently, ||Ax||2 . NO 429 ||A||2 = max ||x||2 =0 ||x||2
Deductions from Gerschgorin’s theorem (see 15.814)
1083
15.413 Maximum absolute row sum norm ||A||∞ = max i
n
|aij |
NO 429
j=1
15.51 Spectral Radius of a Square Matrix Let A = [aij ] be an n×n matrix with elements that may be complex, and with eigenvalues λ1 , λ2 , . . . , λn . Then the spectral radius ρ(A) of A is the number ρ(A) = max |λi |. VA 9 1≤i≤n
15.511 Inequalities concerning matrix norms and the spectral radius 1.
||A||2 2 ≤ ||A||1 ||A||∞ .
2.
If A is any arbitrary n × n matrix with elements that may be complex, and the n × n matrix U is unitary, so that UH = U−1 , with H denoting the Hermitian transpose of A (see 13.123), then
NO 431
||AU|| = ||UA|| = ||A||. 3.
If A is any nonsingular n × n matrix with elements that may be complex with eigenvalues λ1 , λ2 , λn , then 1 ≤ |λ| ≤ ||A||. ||A−1 ||
4.
VA 16
For any square matrix A with spectral radius ρ(A) and any natural norm ||A||, ρ (A) ≤ ||A||.
5.
VA 15
NO 430
If the square matrix A is Hermitian, then ρ (A) = ||A||.
6.
If the square matrix A is Hermitian and Pm (x) is any polynomial of degree m with real coefficients, then ||Pm (A)|| = ρ (Pm (A)) .
7.
If A is any arbitrary n × n matrix with elements that may be complex, then the sequence of matrices A, A2 , A3 , . . . converges to the null matrix as n → ∞ if, and only if, ρ(A) < 1. NO 303
15.512 Deductions from Gerschgorin’s theorem (see 15.814) 1.
Let ⎛ A be any arbitrary n × n matrix with elements that may be complex; then ⎞ n n |aij |, max |aij |⎠ . min ⎝ max 1≤i≤n
j=1
1≤j≤n
i=1
ρ(A) ≤ VA 17
1084
2.
Inequalities Involving Eigenvalues of Matrices
Let A be any arbitrary n×n matrix with elements , x2 , . . . , xn be any thatmay and x1 #nbe complex, n |aij | j=1 |aij |xj set of n positive numbers; then ρ(A) ≤ min max , max xj . 1≤i≤n 1≤j≤n xi xi i=1 VA 18
15.61 Inequalities Involving Eigenvalues of Matrices The eigenvalues (characteristic values or latent roots) λ of an n × n matrix A = [aij ] are the solutions to the characteristic equation |A − λI| = 0. When expanded, the determinant |A − λI| is called the characteristic polynomial, and it has the form |A − λI| = (−1)n λn + cn−1 λn−1 + cn−2 λn−2 + · · · + c1 λ + c0 . The zeros of this polynomial satisfy the characteristic equation and so are the eigenvalues of A. In the characteristic polynomial the coefficients have the form cn−r = (−1)n−r It then follows that
(sum of all principal minors of |A| of order r) .
bn−1 = (−1)n (a11 + a22 + · · · + ann ) , bn−2 = (−1)n (aii ajj − aij aji ) , i<j
b0 = |A|. Since the sum of the elements of the leading diagonal of A is called the trace of A, written tr A, it follows that bn−1 = (−1)n tr A. ML 198
15.611 Cayley-Hamilton theorem Every square matrix A satisfies its characteristic equation, so that (−1)n An + cn−1 An−1 + cn−2 An−2 + · · · + c1 A + c0 I = 0.
ML 206
15.612 Corollaries 1.
If A is nonsingular, then its adjoint, denoted by adj A, is adj A = − (−1)n An−1 + cn−1 An−2 + cn−2 An−3 + · · · + c2 A + c1 I .
2.
If A is nonsingular, then the characteristic polynomial of A−1 is
c1 n−1 c2 n−2 (−1)n λ λ + + ··· + (−1)n λn + . |A| |A| |A|
15.71 Inequalities for the Characteristic Polynomial The first group of inequalities that follow, which relate to the characteristic polynomial of an n × n matrix A whose elements may be complex, refer directly to the coefficients of the polynomial when written in the form
Named and unnamed inequalities
1085
P (λ) ≡ |λI − A| = λn + b1 λn−1 + b2 λn−2 + · · · + bn−1 λ + bn , and only implicitly to the coefficients aij of A that give rise to the bi .
15.711 Named and unnamed inequalities The first group of inequalities relating to the eigenvalues λ satisfying P (λ) = 0 are unnamed and are as follows: 1.
All the eigenvalues λ lie within or on the circle ||z|| ≤ r, where r is the positive root of . |bn | + |bn−1 |z + |bn−2 |z 2 + · · · + |b1 |z n−1 − z n = 0
2.
All the eigenvalues λ lie within the circle |z| < 1 + max|bi |.
MG 123
i
3.
When bn = 0 the eigenvalue λ of smallest modulus lies in the annulus R ≤ |z| ≤ R is the positive root of |bn | − |bn−1 |z − |bn−2 |z 2 − · · · − z n = 0.
4.
5.
MG 122
All the eigenvalues λ lie on or outside the circle |bn | |z| = min . k (|bn | + |bk |)
R , where 21/n − 1 MG 126
MG 126
If the eigenvalues λ are ordered so that |λ1 | ≥ |λ2 | ≥ · · · ≥ |λp | > 1 ≥ |λp+1 | ≥ · · · ≥ |λn |, then 1
|z1 z2 . . . zp | ≤ N,
|zp | ≤ N p ,
where 2
2
2
N 2 = 1 + |b1 | + |b2 | + · · · + |bn | . 6.
MG 129
All the eigenvalues λ lie in or on the circle |z| ≤
n
|bj |
1/j
.
MG 126
j=1
7.
8.
All the eigenvalues λ lie on the disk ! ! ! ! ! ! ! ! !z + b1 ! ≤ ! b1 ! + |b2 |1/2 + |b3 |1/3 + · · · + |bn |1/n . ! 2! !2! All the eigenvalues λ lie in the annulus m ≤ ||z|| ≤ M , where
+ , 2 2 m = max 0, min 1 − |bj |, |bn | 1≤j≤n−1
and
MG 145
1086
Inequalities for the Characteristic Polynomial
⎧ ⎫ n−1 ⎨ ⎬ 2 2 M 2 = max 1 + |bj |, |bn | + 2 |bj | . ⎩ ⎭ j=1
The next group of inequalities are named theorems that apply to the explicit form of the characteristic MG 145 polynomial P (λ).
15.712 Parodi’s theorem The eigenvalues λ satisfying P (λ) = 0 lie in the union of the disks n |z| ≤ 1, |z + b1 | ≤ |bj |.
MG 143
j=1
15.713 Corollary of Brauer’s theorem If |b1 | > 1 +
n
|bj |,
j=2
then one and only one eigenvalue satisfying P (λ) = 0 lies on the disk n |bj |. |z + b1 | ≤
MG 141
j=2
15.714 Ballieu’s theorem For any set μ = (μ1 , μ2 , . . . , μn ) of positive numbers, let μ0 = 0 and μk + μn |bn−k | Mμ = max . 0≤k≤n−1 μk+1 Then all the eigenvalues satisfying P (λ) = 0 lie on the disk ||z|| ≤ Mμ .
MG 144
15.715 Routh-Hurwitz theorem Consider the characteristic equation |λI − A| = λn + b1 λn−1 + · · · + bn−1 λ + bn = 0 determining the n eigenvalues λ of the real n × n matrix A. Then the eigenvalues λ all have negative real parts if Δ2 > 0, ... , Δn > 0, Δ1 > 0, where ! ! ! b1 1 0 0 0 0 . . . 0 !! ! ! b3 b2 b1 1 0 0 . . . 0 !! ! ! b5 b4 b3 b2 b1 0 . . . 0 !!. Δk = ! GM 230 ! .. ! .. .. .. .. ! . ! . . . . ! ! !b2k−1 b2k−2 b2k−3 b2k−4 b2k−5 b2k−6 . . . bk !
Poincare’s separation theorem
1087
15.81–15.82 Named Theorems on Eigenvalues In the following theorems involving eigenvalue inequalities the elements aij of matrix A enter directly, and not in the form of the coefficients of the characteristic polynomial.
15.811 Schur’s inequalities If A = [aij ] is an n × n matrix with elements that may be complex, and eigenvalues λ1 , λ2 , . . . , λn , then 1.
n
2
|λi | ≤
i=1
2.
n
n
n
2
i,j=1 2
|Re λi | ≤
i=1
3.
|aij |
2
|Im λi | ≤
i=1
! n ! ! aij + aji !2 ! ! ! ! 2 i,j=1
! n ! ! aij − aji !2 ! ! ! ! 2 i,j=1
ML 309
15.812 Sturmian separation theorem Let Ar = [aij ] with i, j = 1, 2, . . . , r and r = 1, 2, . . . , N be a sequence of N symmetric matrices of increasing order. Then if λk (Ar ) for k = 1, 2, . . . , r denotes the k th eigenvalue of Ar , where the ordering is such that λ1 (Ar ) ≥ λ2 (Ar ) ≥ · · · ≥ λr (Ar ) , it follows that BL 115 λk+1 (Ai+1 ) ≤ λk (Ai ) ≤ λk (Ai+1 ) .
15.813 Poincare’s separation theorem
Let yk , with k = 1, 2, . . . , K, be a set of orthonormal vectors so that the inner product (yk , yk ) = 1. Set K x= uk y k , k=1
so that for any square matrix A for which the product Ax is defined, the quadratic form K (x, Ax) = uk ul yk , Ayl . k,l=1
Then if
bK = yk , Ayl for k, l = 1, 2, . . . , K,
it follows that λi (bK ) ≤ λi (A) λK−j (bK ) ≥ λN −j (A)
for i = 1, 2, . . . , K, for j = 0, 1, 2, . . . , K − 1.
BL 117
1088
Named Theorems on Eigenvalues
15.814 Gerschgorin’s theorem Let A = [aij ] be any arbitrary n × n matrix with elements that may be complex, and let n |aij | for i = 1, 2, . . . , n. Λi ≡ j=1,i=j
Then all of the eigenvalues λi of A lie in the union of the n disks Γi , where Γi : |z − aii | ≤ Λi for i = 1, 2, . . . , n.
VA 16
15.815 Brauer’s theorem If in Gerschgorin’s theorem for a given m |ajj − amm | ≥ Λj + Λm for all j = m, then one and only one eigenvalue of A lies in the disk Γm .
MG 141
15.816 Perron’s theorem If μ = (μ1 , μ2 , . . . , μn ) is an arbitrary set of positive numbers, then all the eigenvalues λ of the n × n matrix A = [aij ] lie on the disk |z| ≤ Mμ , where n μj Mμ = max |aij |. MG 141 1≤i≤n μ j=1 i
15.817 Frobenius theorem If A = [aij ] is a matrix with positive coefficients, so that aij > 0 for all i, j = 1, 2, . . . , n, then A has a positive eigenvalue λ0 , and all its eigenvalues lie on the disk MG 142 |z| ≤ λ0 .
15.818 Perron–Frobenius theorem If all elements aij of an irreducible matrix A are nonnegative, then R = min Mλ is a simple eigenvalue of A, and all the eigenvalues of A lie on the disk |z| ≤ R, where, if λ = (λ1 , λ2 , . . . , λn ) is a set of nonnegative numbers, not all zero, ⎧ ⎫ n ⎨ ⎬ Mλ = inf μ : μλi > |aij |λj , 1 ≤ i ≤ n ⎩ ⎭ j=1
and R = min Mλ . Furthermore, if A has exactly p eigenvalues (p ≤ n) on the circle |z| = R, then the set of all its GM 69 eigenvalues is invariant under rotations 2π/p about the origin.
15.819 Wielandt’s theorem If the n × n matrix A satisfies the conditions of the Perron–Frobenius theorem and if in the n × n matrix C = [cij ] |cij | ≤ aij , i, j = 1, 2, . . . , n, then any eigenvalue λ0 of C satisfies the inequality |λ0 | ≤ R. The equality sign holds only when there exists an n × n matrix D = [±δij ] such that δii = 1 for all i, δij = 0 for all i = j, and
Hermitian matrices and diophantine relations
C = (λ0 /R) DAD−1 .
1089
GM 69
15.820 Ostrowski’s theorem If A = [aij ] is a matrix with positive coefficients and λ0 is the positive eigenvalue in Frobenius’ theorem, then the n − 1 eigenvalues λj = λ0 satisfy the inequality M 2 − m2 |λj | ≤ λ0 2 , M + m2 where M = max aij , m = min aij for i, j = 1, 2, . . . , n. MG 145
15.821 First theorem due to Lyapunov In order that all the eigenvalues of the real n × n matrix A have negative real parts, it is necessary and sufficient that if V is an n × n matrix, the equation AT V + VA = −I has as a solution the matrix of coefficients V of some positive-definite quadratic form (x, Vx) (see 13.21). GM 224
15.822 Second theorem due to Lyapunov If all the eigenvalues of the real matrix A have negative real parts, then to an arbitrary negative-definite quadratic form (x, Wx) with x = x(t) there corresponds a positive-definite quadratic form (x, Vx) such that if one takes dx = Ax dt then (x, Vx) and (x, Wx) satisfy d (x, Vx) = (x, Wx) . dt Conversely, if for some negative-definite form (x, Wx) there exists a positive-definite form (x, Vx) connected to (x, Wx) by the preceding two equations, then all the eigenvalues of A have negative real parts (see 13.21, 13.31). GM 222
15.823 Hermitian matrices and diophantine relations involving circular functions of rational angles due to Calogero and Perelomov 1.
The off-diagonal Hermitian matrix A of rank n whose elements are given by
(j − k) π ajk = (1 − δjk ) 1 + i cot , n has the integer eigenvalues λ(a) s = 2s − n − 1 for s = 1, 2, . . . , n, and the corresponding eigenvectors v (s) have the components
1090
Named Theorems on Eigenvalues
(s) vj
2.
2πisj = exp − n
for j = 1, 2, . . . , n.
The two off-diagonal Hermitian matrices B and C whose elements are defined by the formulas (j − k) π , bjk = (1 − δjk ) sin−2 n (j − k) π , cjk = (1 − δjk ) sin−4 n are related to the matrix A in (1) by the equations 1 2 A + 2A − σn(1) I , 2 1 2 B − 2 2 + σn(1) B − σn(2) I , C=− 6 B=
where I is the unit matrix and 1 2 n −1 , σn(1) = 3
σn(2) =
1 2 n − 1 n2 + 11 . 45 (s)
The eigenvalues of B and C corresponding to the eigenvector vj (1) λ(b) s = σn − 2s(n − s) (2) λ(c) s = σn − 2s(n − s)
3.
in (1) have the form
for s = 1, 2, . . . , n, s(n − s) + 2 3
for s = 1, 2, . . . , n.
Together, the above two results imply the following diophantine summation rules: (a)
(b)
(c)
(d)
kπ 2skπ sin = n − 2s n n k=1
n−1 kπ 2skπ sin−2 cos = bs n n k=1
n−1 kπ 2skπ −4 sin cos = cs n n k=1
n−1 kπ −2p sin = σn(p) , n n−1
for s = 1, 2, . . . , n − 1
cot
for s = 1, 2, . . . , n − 1, for s = 1, 2, . . . , n − 1,
k=1
(1)
(2)
with σn and σn as defined in (2), and σn(3) = σn(1)
2n4 + 23n2 + 191 , 315
bs = σn(1) − 2s(n − s),
σn(4) = σn(2)
3n4 + 10n2 + 227 315
2 cs = σn(2) − s(n − s)[s(n − s) + 2]. 3
Basic theorems
1091
15.91 Variational Principles 15.911 Rayleigh quotient If A is an Hermitian matrix, the Rayleigh quotient ρ(x) is the expression (x, Ax) . ρ (x) = (x, x)
NO 407
15.912 Basic theorems 1.
If the n × n matrix A is Hermitian and has eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn , then λ1 ≤ ρ ≤ λn , where ρ is the Rayleigh quotient for any x = 0, and λ1 = min x=0
2.
(x, Ax) (x, x)
and
λn = max x=0
(x, Ax) . (x, x)
NO 407
If the n × n matrix A is Hermitian and has eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn corresponding to the eigenvectors x1 , x2 , . . . , xn , respectively, and x = 0 is such that (x, x1 ) = (x, x2 ) = · · · = (x, xn ) = 0, then λj = min x
(x, Ax) , (x, x)
and λj ≤ 3.
(x, Ax) ≤ λn . (x, x)
NO 410
If the n × n matrix A is Hermitian, then the eigenvalue
(x, Ax) λr = max min , (x, x) where first the minimum over x is taken subject to (bi , x) = 0, i = 1, 2, . . . , r − 1, with the bi regarded as fixed vectors, and then the maximum over all possible bi . Also, the eigenvalue
(x, Ax) λr = min max , (x, x) where now the maximum over x is taken first subject to (bi , x) = 0, i = r + 1, r + 2, . . . , n for NO 414 fixed bi , and then the minimum over all possible bi .
4.
The (n − 1) eigenvalues λ1 , λ2 , . . . , λn−1 obtained from the (n − 1) × (n − 1) matrix derived from an Hermitian matrix A from which the last row and column have been omitted separate the n eigenvalues of A, so that λ1 < λ1 < λ2 < λ2 < · · · < λn−1 < λn
(see 15.812).
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16 Ordinary Differential Equations 16.1–16.9 Results Relating to the Solution of Ordinary Differential Equations 16.11 First-Order Equations 16.111 Solution of a first-order equation Consider the real function f (t, x) that is defined and continuous in an open set D ⊂ R2 . Then a solution to the first-order differential equation dx = f (t, x) dt in the open interval I ⊂ R is a real function u(t) that is defined and is both continuous and differentiable in I, with the property that (i)
(t, u(t)) ∈ D for t ∈ I,
(ii)
du = f (t, u(t)) for t ∈ I. dt
16.112 Cauchy problem The Cauchy problem for the differential equation dx = f (t, x) dt is the problem of existence and uniqueness of the solution to this equation satisfying the initial condition u(t0 ) = x0 , where (t0 , u(t0 )) ∈ D, the open set defined above. The solution to the initial value problem may be expressed in the form of the integral equation t f (τ, u(τ )) dτ (see 16.316). u(t) = x0 + t0
16.113 Approximate solution to an equation The real function φ(t) is said to be an approximate solution, to within the error , of the differential equation
1093
1094
Fundamental Inequalities and Related Results
dx = f (t, x) dt if φ is piecewise continuous, and for a given > 0 and an open interval I ⊂ R, |φ (t) − f (t, φ(t))| ≤ , except at points of discontinuity of the derivative.
HU 3
16.114 Lipschitz continuity of a function The real function f (t, x) defined and continuous in some open set D ⊂ R2 is said to be Lipschitz continuous with respect to x for some constant k > 0 if, for all points (t, x1 ) and (t, x2 ) belonging to D |f (t, x1 ) − f (t, x2 )| ≤ k|x1 − x2 |. HU 5
16.21 Fundamental Inequalities and Related Results 16.211 Gronwall’s lemma Let the three piecewise continuous, non-negative functions u, v, and w be defined in the interval [0, a] and satisfy the inequality t v(τ )w(τ ) dτ, w(t) ≤ u(t) + 0
except at points of discontinuity of the functions. Then, except at these same points, t
t u(τ )v(τ ) exp v (σ) dσ dτ. w(t) ≤ u(t) + 0
BB 135
τ
16.212 Comparison of approximate solutions of a differential equation Let f be a real function that is defined in an open set D ⊂ R2 , in which it is both continuous and Lipschitz continuous. In addition, let u1 and u2 be two approximate solutions of dx = f (t, x) dt in an open set I ⊂ R in the sense already defined, with |u1 (t) − f (t, u1 (t))| ≤ 1 , |u2 (t) − f (t, u2 (t))| ≤ 2 , except where the derivatives are discontinuous. Then, if for all t0 ∈ I |u1 (t0 ) − u2 (t0 )| ≤ δ, it follows that
1 + 2 |u1 (t) − u2 (t)| ≤ δ exp {|t − t0 |} + [exp {k|t − t0 |} − 1] . k
16.31 First-Order Systems 16.311 Solution of a system of equations The system of n first-order differential equations
HU 6
Lipschitz continuity of a vector
1095
dx1 = f1 (t, x1 , x2 , . . . , xn ) , dt dx2 = f2 (t, x1 , x2 , . . . , xn ) , dt .. . dxn = fn (t, x1 , x2 , . . . , xn ) , dt in which the functions f1 , f2 , . . . , fn are real and continuous in an open set D ⊂ Rn+1 , may be written in the concise matrix form dx = f (t, x) , dt where x and f are n × 1 column vectors. Its solution in the open interval I ⊂ R is the vector u(t) with elements u1 (t), u2 (t), . . . , un (t) with the property that (i)
(t, u(t)) ∈ D for t ∈ I,
(ii)
du = f (t, u(t)) for t ∈ I. dt
HU 24
16.312 Cauchy problem for a system The Cauchy problem for the system dx = f (t, x) dt is the problem of existence and uniqueness of the solution to this system satisfying the initial vector condition u (t0 ) = x0 , where (t0 , u(t0 ) ∈ D, the open set defined above in connection with the system. The solution to the initial value problem may be expressed in the form of the vector integral equation t f (τ, u(τ )) dτ. u(t) = x0 + t0
16.313 Approximate solution to a system The real vector φ(t) is said to be an approximate vector solution, to within the order , of the system dx = f (t, x) , dt if the elements of φ are piecewise continuous, and for a given > 0 and open interval I ⊂ R, ||φ (t) − f (t, φ(t))|| ≤ , except at points of discontinuity of the derivative, where ||w|| denotes the supremum norm ||w|| = sup (|w1 |, |w2 |, . . . , |wn |) .
HU 25
16.314 Lipschitz continuity of a vector The real vector f (t, x) defined and continuous in some open set D ⊂ Rn is said to be Lipschitz continuous with respect to x for some constant k > 0 if, for all points (t, x1 ), (t, x2 ) belonging to D, ||f (t, x1 ) − f (t, x2 )|| ≤ k||x1 − x2 ||. HU 26
1096
First-Order Systems
16.315 Comparison of approximate solutions of a system Let f be a real vector defined in an open set D ⊂ R × Rn in which it is both continuous and Lipschitz continuous. In addition, let u1 and u2 be two approximate solutions of the system dx = f (t, x) dt in an open set I ⊂ R in the sense already defined, with |u1 (t) − f (t, u1 (t))| ≤ 1 , |u2 (t) − f (t, u2 (t))| ≤ 2 , except where the derivatives are discontinuous. Then, if for all t0 ∈ I ||u1 (t0 ) − u2 (t0 )|| ≤ δ, it follows that
1 + 2 ||u1 (t) − u2 (t)|| ≤ δ exp {k|t − t0 |} + [exp {k|t − t0 |} − 1] . k
HU 27
16.316 First-order linear differential equation The first-order linear differential equation when expressed in the canonical form dy + P (t)y = Q(t) dt has an integrating factor
μ(t) = exp P (t) dt , and a general solution 1 y(t) = μ(t)
t μ(ξ)Q(ξ) dξ , μ (t0 ) y0 + t0
where y0 = y(t0 ).
16.317 Linear systems of differential equations Consider the homogeneous system of linear differential equations dx = A(t)x, dt where x is an n×1 column vector and A(t) an n×n matrix. Then a fundamental system of solutions of this system is a set of n linearly independent solution vectors φ1 (t), φ2 (t), . . . , φn (t), The square matrix K(t) whose columns comprise the vectors φ1 (t), φ2 (t), . . . , φn (t) is called the fundamental matrix of the differential equation, and we have the representation t
tr A(τ ) dτ . |K(t)| = |K (t0 )| exp t0
Using the fundamental matrix K(t) defined in terms of the homogeneous system, the unique solution to the inhomogeneous system dx = A(t)x + b(t), dt assuming the initial value x(t0 ) = x0 , is t φ(t) = K(t)[K (t0 )]−1 x0 + K(t) [K(τ )]−1 b(τ ) dτ, HU 43 where b(t) is an n × 1 column vector.
t0
CL 69
Homogeneous differential equations
1097
16.41 Some Special Types of Elementary Differential Equations 16.411 Variables separable A first-order differential equation is said to be variables separable if it is of the form dy = M (x)N (y), dx or P (x)Q(y) dx + R(x)S(y) dy = 0. It may then be written in the form 1 M (x) dx − dy = 0, N (y) or P (x) S(y) dx + dy = 0, R(x) Q(y) provided R(x)Q(y) = 0.
16.412 Exact differential equations A differential equation M (x, y) dx + N (x, y) dy = 0 is said to be exact if there exists a function h(x, y) such that d [h(x, y)] = M (x, y) dx + N (x, y) dy.
IN 16
16.413 Conditions for an exact equation A necessary and sufficient condition that an equation of this form is exact is that the functions M (x, y) and N (x, y) together with their partial derivatives ∂M/∂y and ∂N/∂x exist and are continuous in a region in which ∂N ∂M = . IN 16 ∂y ∂x
16.414 Homogeneous differential equations A differential equation M (x, y) dx + N (x, y) dy = 0 is said to be algebraically homogeneous if, for arbitrary k, M (kx, ky) M (x, y) = . N (kx, ky) N (x, y) Setting y = sx, it may then be expressed in the form [M (1, s) + sN (1, s)] dx + xN (1) dx = 0, in which the variables s and x are separable.
IN 18
1098
Second-Order Equations
16.51 Second-Order Equations 16.511 Adjoint and self-adjoint equations The linear second-order differential equation du d2 u + c(x)u = 0 L(u) ≡ a(x) 2 + b(x) dx dx has associated with it the adjoint equation d2 d M (v) ≡ 2 [a(x)v] − [b(x)v] + c(x)v = 0. dx dx The equation L(u) = 0 is said to be self-adjoint if L(u) ≡ M (u). A linear self-adjoint second-order differential equation defined on [α, β] can always be expressed in the form
d du p(x) + q(x)u = 0, dx dx where p(x) and q(x) are continuous on [α, β] and p(x) > 0. The general equation L(u) = 0 can always be made self-adjoint and written in this form by multiplication by the factor 1 b(x) dx , exp a(x) a(x) when b(x) c(x) b(x) p(x) = exp dx and q(x) = dx . exp a(x) a(x) a(x) In general, if dn u dn−1 u du L(u) = p0 n + p1 n−1 . . . + pn−1 + pn u, dx dx dx then its adjoint is dn dn−1 d [pn−1 v] + pn v. M (v) = (−1)n n [p0 v] + (−1)n−1 n−1 [p1 v] + . . . − HI 391 dx dx dx
16.512 Abel’s identity If p(x) and q(x) are continuous in [α, β] in which p(x) > 0, and u(x) and v(x) are suitably differentiable with
d du p(x) + q(x)u = 0, dx dx then the result
du dv −v p(x) u ≡ const. dx dx is known as Abel’s identity. More generally, if we consider the linear nth -order equation dn u dn−1 u du + pn = 0, p0 n + p1 n−1 + . . . + pn−1 dx dx dx and Δ is the Wronskian of a (fundamental) set of linearly independent solutions u1 , u2 , . . . , un , the Abel identity takes the form
x p1 (x) dx , Δ = Δ0 exp − x0 p0 (x) where Δ0 is the value of Δ at x = x0 . IN 119
Solutions of the Riccati equation
1099
16.513 Lagrange identity If the linear nth -order equation L(u) = 0 is defined by dn u dn−1 u du + pn u, L(u) ≡ p0 n + p1 n−1 + . . . + pn−1 dx dx dx then the expression d vL(u) − uM (v) = {P (u, v)} , dx where M (v) is the adjoint of L(u), is called the Lagrange identity. The expression P (u, v), which is linear and homogeneous in dn−1 u dn−1 v du dv u, , . . . , n−1 and v, , . . . , n−1 , dx dx dx dx is then known as the bilinear concomitant. In the case of the second-order equation d2 u du + c(x)u = 0, L(u) = a(x) 2 + b(x) dx dx with adjoint M (v), the Lagrange identity becomes
d d du vL(u) − uM (v) = − (a(x)v)u + b(x)uv . IN 124 a(x)v dx dx dx
16.514 The Riccati equation The general Riccati equation has the form dz + a(x)z + b(x)z 2 + c(x) = 0, dx and an equation of this form results from the substitution p(x) du dx z= u in the general self-adjoint equation
d du p(x) + q(x)u = 0. dx " xdx The further substitution v = u exp α a(x) dx in the Riccati equation then gives the more convenient form dv + r(x)v 2 + s(x) = 0, dx with x
x
r(x) = b(x) exp − a(x) dx and s(x) = c(x) exp a(x) dx . HI 273 α
α
16.515 Solutions of the Riccati equation If in the Riccati equation dv + r(x)v 2 + s(x) = 0, dx r(x) = 0, while r(x) and s(x) are continuous on the interval [α, β], then every solution v(x) may be expressed in the form 1 Au (x) + Bv (x) , r(x) Au(x) + Bv(x) with A, B arbitrary constants, not both zero, and the prime denoting differentiation, while u and v are linearly independent solutions of
1100
Oscillation and Non-Oscillation Theorems for Second-Order Equations
1 dz d + s(x)z = 0. dx r(x) dx Conversely, if u(x) and v(x) are linearly independent solutions of this last equation and A and B are arbitrary constants, not both zero, the function 1 Au (x) + Bv (x) r(x) Au(x) + Bv(x) is a solution of the Riccati equation wherever Au(x) = Bv(x) = 0. IN 24
16.516 Solution of a second-order linear differential equation A fundamental system of solutions of a homogeneous second-order linear differential equation in the canonical form d2 x dx + b(t)x = 0 + a(t) 2 dt dt is a system of two linearly independent solutions φ1 (t) and φ2 (t). The Wronskian of these solutions is ! ! !φ1 (t) φ2 (t)! ! ! = φ1 (t)φ2 (t) − φ2 (t)φ1 (t), W (t) = ! φ1 (t) φ2 (t)! and the solution to the inhomogeneous equation d2 x dx + b(t)x = f (t), + a(t) dt2 dt subject to the initial conditions x(t0 ) = x0 and x (t0 ) = x1 may be written t φ1 (ξ)φ2 (t) − φ2 (ξ)φ1 (t) f (ξ) dξ, x(t) = c1 φ1 (t) + c2 φ2 (t) + W (ξ) t0 where the constants c1 and c2 are chosen such that x(t) satisfies the initial conditions. The linear combination c1 φ1 (t) + c2 φ2 (t) is known as the complementary function where c1 and c2 are arbitrary constants.
16.61–16.62 Oscillation and Non-Oscillation Theorems for SecondOrder Equations Equations whose solutions possess an infinite number of zeros in the interval (0, ∞) are said to have oscillatory solutions. The following theorems relate to such properties:
16.611 First basic comparison theorem If all solutions of the equation
are oscillatory, and if
d2 u + φ(x)u = 0 dx2 ψ(x) ≥ φ(x),
then all the solutions of d2 v + ψ(x)v = 0 dx2 are oscillatory, and conversely. That is, if ψ(x) ≥ φ(x) and some solutions v are non-oscillatory, then so also must some solutions u be non-oscillatory. BS 119
Szeg¨ o’s comparison theorem
1101
16.622 Second basic comparison theorem If all the solutions of the self-adjoint equation
d du p1 (x) + q1 (x)u = 0 dx dx are oscillatory as x → ∞, and if q2 (x) ≥ q1 (x), p2 (x) ≥ p1 (x) > 0, then all the solutions of the self-adjoint equation
d dv p2 (x) + q2 (x)v = 0 dx dx are oscillatory.
BS 120
16.623 Interlacing of zeros Let y1 (x) and y2 (x) be two linearly independent solutions of d2 y + F (x)y = 0, dx2 and suppose that y1 (x) has at least two zeros in the interval (a, b). Then if x1 and x2 are two consecutive HI 374 zeros of y1 (x), the function y2 (x) has one, and only one, zero in the interval (x1 , x2 ).
16.624 Sturm separation theorem Let u(x) and v(x) be two linearly independent solutions of the self-adjoint equation
d dy p(x) + q(x) = 0, dx dx in which p(x) > 0 and p(x), q(x) are continuous on [a, b]. Then, between any two consecutive zeros of IN 224 u(x) there will be one, and only one, zero of v(x).
16.625 Sturm comparison theorem Let p1 (x) ≥ p2 (x) > 0 and q1 (x) ≥ q2 (x) be continuous functions in the differential equations
d du p1 (x) + q1 (x)u = 0, dx dx
d dv p2 (x) + q2 (x)v = 0. dx dx Then between any two zeros of a non-trivial solution u(x) of the first equation there will be at least one zero of every non-trivial solution v(x) of the second equation. IN 228
16.626 Szeg¨ o’s comparison theorem Suppose, under the conditions of the Sturm comparison theorem, that p1 (x) ≡ p2 (x), q1 (x) ≡ q2 (x), and u(x) > 0, v(x) > 0 for a < x < b, together with
du dv v− u = 0. lim p1 (x) x→a dx dx Then, if u(b) = 0, there is a point ξ in (a, b) such that v(ξ) = 0. HI 379
1102
Oscillation and Non-Oscillation Theorems for Second-Order Equations
16.627 Picone’s identity Consider the equations
d du p1 (x) + q1 (x)u = 0, dx dx
d dv p2 (x) + q2 (x)v = 0, dx dx with p1 , p2 , q1 , and q2 positive and continuous for a < x < b, where q2 (x) > q1 (x) and p1 (x) > p2 (x). Then with a < α < β < b, Picone’s identity is
β β 2
2 β β u du dv du dv p2 du 2 −u = (q2 − q1 ) u ds+ (p1 − p2 ) ds+ ds. v p1 v − p2 u 2 v dx dx ds ds ds α α α v α IN 226
16.628 Sturm-Picone theorem Consider the self-adjoint equations
du p1 (x) + q1 (x)u = 0 dx and
d dv p2 (x) + q2 (x)v = 0. dx dx Let p1 , p2 , q1 , and q2 be positive and continuous for a < x < b, where q2 (x) > q1 (x) and p1 (x) > p2 (x). Then, if x1 and x2 is a pair of consecutive zeros of u(x) in (a, b), v(x) has at least one zero in the open IN 225 interval (a, b). d dx
16.629 Oscillation on the half line Consider the self-adjoint equation d dx
du p(x) + q(x)u = 0. dx
We then have the following results: (i)
Let p(x) > 0 and p, q be continuous on [0, ∞). If the two improper integrals ∞ ∞ dx and q(x) dx 1 p(x) 1 diverge, then every solution u(x) has infinitely many zeros on the interval [1, ∞). Also, if the two integrals 1 1 dx = +∞ and q(x) dx = +∞, 0 p(x) 0 then every solution u(x) has infinitely many zeros on the interval (0, 1).
(ii)
(Moore’s theorem). Every non-trivial solution u(x) has at most a finite number of zeros on the interval [a, ∞) if the improper integral ∞ dx p(x) a converges, and if
Kneser’s non-oscillation theorem
! x ! ! ! ! q(s) ds!! < M !
for
1103
a≤x 0 a finite constant.
16.71 Two Related Comparison Theorems 16.711 Theorem 1 Consider the equations in the Sturm comparison theorem with the same assumptions on p(x) and q(x), and let u(x), v(x) be solutions such that u (x1 ) = v (x1 ) = 0, u (x) = v (x1 ) > 0. Then if u(x) is increasing in [x1 , x2 ] and reaches a maximum at x2 , the function v(x) reaches a maximum HI 376 at some point x3 such that x1 < x3 < x2 .
16.712 Theorem 2 Consider the equation d2 y + F (x)y = 0, dx2 in which F (x) is continuous in (a, b) and such that 0 < m ≤ F (x) ≤ M. Then, if the solution y(x) has two successive zeros x1 , x2 , it follows that πM −1/2 ≤ x2 − x1 ≤ πm−1/2 .
16.81–16.82 Non-Oscillatory Solutions The real solution y(x) of d2 y + F (x)y = 0 dx2 is said to be non-oscillatory in the wide sense in (0, ∞) if there exists a finite number c such that the solution has no zeros in [c, ∞). HI 376
16.811 Kneser’s non-oscillation theorem Consider the equation d2 y + F (x)y = 0, dx2 and let lim sup x2 F (x) = γ ∗ , lim inf x2 F (x) = γ∗ . Then the solution y(x) is non-oscillatory if γ ∗ < 14 , oscillatory if if either γ ∗ or γ∗ equals 14 .
1 4
< γ∗ and no conclusion can be drawn HI 461
1104
Some Growth Estimates for Solutions of Second-Order Equations
16.822 Comparison theorem for non-oscillation Consider the differential equations d2 y + F (x)y = 0, dx2
f (x) = x
∞
F (s) ds,
x ∞
d2 y + G(x)y = 0, g(x) = x G(s) ds, dx2 x where 0 < g(x) < f (x). Then if the first equation is non-oscillatory in the wide sense, so also is the HI 460 second.
16.823 Necessary and sufficient conditions for non-oscillation Consider the equation d2 y + F (x)y = 0. dx2 ∞
lim sup x F (s) ds = F ∗ ,
Then, if
x→∞
lim inf x
x→∞
it follows that:
x ∞
F (s) ds = F∗ ,
x
(i)
a necessary condition that the solution y(x) be non-oscillatory is that F∗ ≤
(ii)
a sufficient condition that the solution y(x) be non-oscillatory is that F ∗
0 be continuous in (−∞, ∞) and such that xG(x) ∈ L(0, ∞). Then the equation d2 y − G(x)y = 0 has one, and only one, solution y+ (x) passing through the point (0, 1), which is positive dx2 and strictly monotonic decreasing for all x, and one and only one solution y− (x) through the point (0, 1), which is positive and strictly increasing for all x. The solution y+ (x) has the property that dy+ (x) 1/2 ∈ L2 (0, ∞). [G(x)] y+ (x) ∈ L2 (0, ∞) and dx 2 2 If, in addition, 0 < α ≤ G(x) ≤ β < ∞, then e−βx ≤ y+ (x) ≤ e−αx
for
x > 0.
HI 359
16.912 General result on dominant and subdominant solutions Consider the equations d2 y d2 Y − g(x)y = 0, − G(x)Y = 0, 2 dx dx2 where g and G are continuous on (0, ∞) with 0 < g(x) < G(x), and xg(x) ∈ L(0, ∞). In addition, let yα and Yα be the solutions of these respective equations corresponding to
A theorem due to Lyapunov
1105
yα (0) = Yα (0) = 1, yα (0) = Yα (0) = α for − ∞ < α < ∞. Let yω and Yω be determined, respectively, by yω (0) = Yω (0) = 0, yω (0) = Yω (0) = 1, and let y+ and Y+ be the subdominant solutions for which y+ (0) = Y+ (0) = 1 2 2 2 2 while y+ (x) , g(x) [y+ (x)] , Y+ (x) , and G(x) Y+ (x) belong to L(0, ∞). Then, if β and γ are such that y−β = y+ and Y−γ = Y+ , it follows that β < γ and yα (x) < Yα (x), yω (x) < Yω (x),
0 < x < ∞,
−γ ≤ α, HI 440
y+ (x) > Y+ (x).
16.913 Estimate of dominant solution Let G(x) be positive and continuous with continuous first- and second-order derivatives satisfying G(x)G (x) < 54 [G (x)] . Then there exists a dominant solution y(x) of the fundamental solutions Y0 (x) and Y1 (x) of d2 y − G(x)y = 0, dx2 determined by the initial conditions 2Y0 (0) = 0, Y1 (0)= 1, Y0 (0) = 1, Y1 (0) = 0, such that x
−1/4 1/2 y(x) < [G(x)] exp [G(ξ)] dξ , 2
0
and a positive constant C such that the normalized subdominant solution y+ (x), for which y+ (0) = 1 2 2 and y+ (x) ∈ L(0, ∞), G(x) [y+ (x)] ∈ L(0, ∞), satisfies x
1/2 y+ (x) > CG(x)−1/4 exp − [G(ξ)] dξ . HI 443 0
16.914 A theorem due to Lyapunov Let y(x) be any solution of d2 y − G(x)y = 0 dx2 with G(x) positive and continuous in (0, ∞) with xG(x) ∈ L(0, ∞). Then x
2 2 exp − [G(ξ) + 1] dξ < [y(x)] + [y (x)] 0 x
< C exp [G(ξ) + 1] dξ , where C = [y(0)]2 + [y (0)]2 .
0
HI 446
1106
Boundedness Theorems
16.92 Boundedness Theorems 16.9216 All solutions of the equation
are bounded, provided that "∞ (i) |φ(x)| dx < ∞, ∞ |ψ(x)| dx < ∞ and (ii)
d2 u + (1 + φ(x) + ψ(x)) u = 0 dx2
ψ(x) → 0 as x → ∞.
BS 112
16.922 If all solutions of the equation d2 u + a(x)u = 0 dx2
are bounded, then all solutions of
are also bounded if
d2 u + (a(x) + b(x))u = 0 dx2 ∞ |b(x)| dx < ∞.
BS 112
16.923 If a(x) → ∞ monotonically as x → ∞, then all solutions of d2 u + a(x)u = 0 dx2
are bounded as x → ∞.
BS 113
16.924 Consider the equation d2 u + a(x)u = 0 dx2
in which
∞
x|a(x)| dx < ∞.
du exists, and the general solution is asymptotic to d0 + d1 x as x → ∞, where d0 and d1 dx may be zero, but not simultaneously. BS 114 Then lim
x→∞
16.9310 Growth of maxima of |y| Sonin’s theorem generalized by P´ olya may be stated as follows: Let y(x) satisfy the differential equation
{k(x)y } + φ(x)y = 0, where k(x) > 0, φ(x) > 0, and both functions k(x), φ(x) have a continuous derivative. Then the relative maxima of |y| form an increasing or decreasing sequence according as k(x)φ(x) is decreasing or increasing. SZ 164
17 Fourier, Laplace, and Mellin Transforms 17.1–17.4 Integral Transforms 17.11 Laplace transform The Laplace transform of the function f (x), denoted by F (s), is defined by the integral ∞ F (s) = f (x)e−sx dx, Re s > 0. 0
The functions f (x) and F (s) are called a Laplace transform pair, and knowledge of either one enables the other to be recovered. If f is summable over all finite intervals, and there is a constant c for which ∞ |f (x)|e−c|x| dx 0
is finite, then the Laplace transform exists when s = σ + iτ is such that σ ≥ c. Setting F (s) = L [f (x); s] to emphasize the nature of the transform, we have the symbolic inverse result f (x) = L−1 [F (s); x] . The inversion of the Laplace transform is accomplished for analytic functions F (s) of order O s−k with k > 1 by means of the inversion integral γ+i∞ 1 f (x) = F (s)esx ds, 2πi γ−i∞ SN 30 where γ is a real constant that exceeds the real part of all the singularities of F (s).
17.12 Basic properties of the Laplace transform 1.8
For a and b arbitrary constants, L [af (x) + bg(x)] = aF (s) + bG(s)
2.
(linearity)
If n > 0 is an integer and lim f (x)e−sx = 0, then for x > 0, x→∞ , + L f (n) (x); s = sn F (s) − sn−1 f (0) − sn−2 f (1) (0) − · · · − f (n−1) (0)
(transform of a derivative) SN 32
1107
1108
3.11
Integral Transforms
"x If lim e−sx 0 f (ζ) dζ = 0, then x→∞
L 0
x
1 f (ξ) dξ; s = F (s) s
(transform of an integral)
L e−ax f (x); s = F (s + a)
4.
SN 37
(shift theorem)
SU 143
The Laplace convolution f ∗ g of two functions f (x) and g(x) is defined by the integral x f ∗ g(x) = f (x − ξ)g(ξ) dξ,
5.
0
and it has the property that f ∗ g = g ∗ f and f ∗ (g ∗ h) = (f ∗ g) ∗ h. In terms of the convolution operation L [f ∗ g(x); s] = F (s)G(s)
(convolution (Faltung) theorem).
SN 30
17.13 Table of Laplace transform pairs f (x)
F (s)
1
1
1/s
2
xn ,
n = 0, 1, 2, . . .
3
xν ,
ν > −1
4
xn− 2
5
x−1/2 (x + a)−1 ,
1
n! , sn+1 Γ(ν + 1) , sν+1 Γ n + 12 1
sn+ 2 |arg a| < π
,
Re s > 0
ET I 133(3)
Re s > 0
ET I 137(1)
Re s > 0
ET I 135(17)
πa−1/2 eas erfc a1/2 s1/2 , Re s ≥ 0
ET I 136(25)
Re s > 0
ET I 142(14)
x for 0 < x < 1 1 for x > 1
1 − e−s , s2
7
e−ax
1 , s+a
Re s > − Re a
ET I 143(1)
8
xe−ax
1 , (s + a)2
Re s > − Re a
ET I 144(2)
9a
e−ax − e−bx b−a
(s + a)−1 (s + b)−1 ,
6
Re s > {− Re a, − Re b}
AS 1022(29.3.12) continued on next page
Table of Laplace transform pairs
1109
continued from previous page
f (x) 9b11
F (s)
αe−ax + βe−bx + γe−cx (a − b)(b − c)(c − a)
(s + a)−1 (s + b)−1 (s + c)−1 , Re s > {− Re a, − Re b, − Re c}
a, b, c distinct , α = c − b, β = a − c, γ = b − a 1011
ae−ax − be−bx b−a
s(s + a)−1 (s + b)−1 , Re s > {− Re a, − Re b}
11 12 13
eax − 1 a eax − ax − 1 a2 ax 1 2 2 e − 2 a x − ax − 1 a3
AS 1022(29.3.13)
s−1 (s − a)−1 ,
Re s > Re a
s−2 (s − a)−1 ,
Re s > Re a
s−3 (s − a)−1 ,
Re s > Re a
14
(1 + ax)eax
s , (s − a)2
Re s > Re a
15
1 + (ax − 1)eax a2
s−1 (s − a)−2 ,
Re s > Re a
16
2 + ax + (ax − 2)eax a3
s−2 (s − a)−2 ,
Re s > Re a
17
xn eax ,
n!(s − a)−(n+1) ,
Re s > Re a
18 19 20 21 22
n = 0, 1, 2, . . .
x + 12 ax2 eax
s , (s − a)3
Re s > Re a
1 + 2ax + 12 a2 x2 eax
s2 , (s − a)3
Re s > Re a
(s − a)−4 ,
Re s > Re a
s , (s − a)4
Re s > Re a
s2 (s − a)−4 ,
Re s > Re a
1 3 ax 6x e
1
2 2x
+ 16 ax3 eax
x + ax2 + 16 a2 x3 eax
continued on next page
1110
Integral Transforms
continued from previous page
f (x) 23
24 25
F (s)
1 + 3ax + 32 a2 x2 + 16 a3 x3 eax
aeax − bebx a−b 1 ax 1 bx − be + ae a−b
26
xν−1 e−ax ,
27
xe−x
2
/(4a)
,
1 b
−
1 a
s3 (s − a)−4 ,
Re s > Re a
s(s − a)−1 (s − b)−1 ,
Re s > {Re a, Re b}
s−1 (s − a)−1 (s − b)−1 ,
Re s > {Re a, Re b}
Re ν > 0
Γ(ν)(s + a)−ν ,
Re a > 0
2 2a − 2π 1/2 a3/2 seas erfc sa1/2
Re s > − Re a
ET I 144(3)
ET I 146(22)
28
exp (−aex ) ,
Re a > 0
as Γ (−s, a)
298
x1/2 e−a/(4x) ,
Re a ≥ 0
1 1/2 −3/2 s 2π
ET I 147(37)
+ , 1 + a1/2 s1/2 exp (−as)1/2 , Re s > 0
308
x−1/2 e−a/(4x) ,
Re a ≥ 0
, + π 1/2 s−1/2 exp (−as)1/2 , Re s > 0
318
x−3/2 e−a/(4x) ,
Re a > 0
ET I 146(26)
ET I 146(27)
+ , 2π 1/2 a−1/2 exp (−as)1/2 , Re s ≥ 0
ET I 146(28)
32
sin(ax)
−1 a s2 + a2 ,
Re s > |Im a|
ET I 150(1)
33
cos(ax)
−1 s s2 + a2 ,
Re s > |Im a|
ET I 154(3)
34
|sin(ax)|,
πs −1 a s2 + a2 coth , 2a Re s > 0
ET I 150(2)
a>0
continued on next page
Table of Laplace transform pairs
1111
continued from previous page
f (x) 3511
36
|cos(ax)|,
a>0
1 − cos(ax) a2
F (s) πs , 2 −1 + s + a cosech , s + a2 2a Re s > 0 −1 s−1 s2 + a2 , Re s > |Im a|
37
38
ax − sin(ax) a3 sin(ax) − ax cos(ax) 2a3
AS 1022(29.3.19)
−1 s−2 s2 + a2 , Re s > |Im a|
AS 1022(29.3.20)
Re s > |Im a|
AS 1022(29.3.21)
2 −2 s + a2 ,
39
x sin(ax) 2a
−2 s s2 + a2 ,
40
sin(ax) + ax cos(ax) 2a
−2 s2 s2 + a2 ,
Re s > |Im a|
Re s > |Im a| 41
x cos(ax)
cos(ax) − cos(bx) b 2 − a2 1
43 44 45
1 − cos(ax) − 12 ax sin(ax) a4 1 1 b sin(bx) − a sin(ax) a2 − b 2
4611
2 2 2 a x − 1 + cos(ax) a4
1 − cos(ax) + 12 ax sin(ax) a2
ET I 152(14)
AS 1023(29.3.23)
2 −2 s − a2 s2 + a2 , Re s > |Im a|
42
ET I 155(44)
ET I 157(57)
−1 2 −1 s + b2 s s2 + a2 , Re s > {|Im a|, |Im b|}
AS 1023(29.3.25)
−1 s−3 s2 + a2 ,
Re s > |Im a|
−2 s−1 s2 + a2 ,
Re s > |Im a|
2 −1 2 −1 s + a2 s + b2 , Re s > {|Im a|, |Im b|} −2 2 2s + a2 , s−1 s2 + a2
Re s > |Im a| continued on next page
1112
Integral Transforms
continued from previous page
f (x) 47
a sin(ax) − b sin(bx) a2 − b 2
F (s) −1 2 −1 s + b2 s2 s2 + a2 , Re s > {|Im a|, |Im b|}
48
sin(a + bx)
−1 (s sin a + b cos a) s2 + b2 ,
Re s > |Im b|
49
cos(a + bx)
−1 (s cos a − b sin a) s2 + b2 ,
Re s > |Im b|
1 50
a
sinh(ax) − 1b sin(bx) a2 + b 2
2 −1 2 −1 s − a2 s + b2 , Re s > {|Re a|, |Im b|}
51
cosh(ax) − cos(bx) a2 + b 2
−1 2 −1 s + b2 s s2 − a2 , Re s > {|Re a|, |Im b|}
52
a sinh(ax) + b sin(bx) a2 + b 2
−1 2 −1 s + b2 s2 s2 − a2 , Re s > {|Re a|, |Im b|}
53
sin(ax) sin(bx)
−1 2 −1 s + (a + b)2 2abs s2 + (a − b)2 , Re s > {|Im a|, |Im b|}
54
cos(ax) cos(bx)
−1 2 −1 s + (a + b)2 s s2 + a2 + b2 s2 + (a − b)2 , Re s > {|Im a|, |Im b|}
55
sin(ax) cos(bx)
−1 2 −1 s + (a + b)2 a s2 + a2 − b2 s2 + (a − b)2 , Re s > {|Im a|, |Im b|}
56
sin2 (ax)
−1 2a2 s−1 s2 + 4a2 ,
Re s > |Im a|
57
cos2 (ax)
2 −1 s + 2a2 s−1 s2 + 4a2 ,
Re s > |Im a|
58
sin(ax) cos(ax)
−1 a s2 + 4a2 ,
Re s > |Im a|
e−ax sin(bx)
−1 b (s + a)2 + b2 ,
59
Re s > {− Re a,
|Im b|}
continued on next page
Table of Laplace transform pairs
1113
continued from previous page
f (x) 60
F (s) −1 (s + a) (s + a)2 + b2 ,
e−ax cos(bx)
Re s > {− Re a, 61
x−1 sin(ax)
arctan(a/s),
62
x−1 [1 − cos(ax)]
1 2
|Im b|}
Re s > |Im a|
ET I 152(16)
Re s > |Im a|
ET I 157(59)
ln 1 + a2 /s2 ,
63
sinh(ax)
−1 a s2 − a2 ,
Re s > |Re a|
ET I 162(1)
64
cosh(ax)
−1 s s2 − a2 ,
Re s > |Re a|
ET I 162(2)
65
xν−1 sinh(ax),
Re ν > −1
1 2
Γ(ν) (s − a)−ν − (s + a)−ν , Re s > |Re a|
66
xν−1 cosh(ax),
Re ν > 0
1 2
ET I 164(18)
Γ(ν) (s − a)−ν + (s + a)−ν , Re s > |Re a|
ET I 164(19)
67
x sinh(ax)
−2 2as s2 − a2 ,
Re s > |Re a|
68
x cosh(ax)
2 −2 s + a2 s2 − a2 ,
Re s > |Re a|
69
sinh(ax) − sin(ax)
−1 2a3 s4 − a4 , Re s > {|Re a|, |Im a|}
70
cosh(ax) − cos(ax)
AS 1023(29.3.31)
−1 2a2 s s4 − a4 , Re s > {|Re a|, |Im a|}
AS 1023(29.3.32)
71
sinh(ax) + ax cosh(ax)
−2 2as2 a2 − s2 ,
Re s > |Re a|
72
ax cosh(ax) − sinh(ax)
−2 2a3 a2 − s2 ,
Re s > |Re a| continued on next page
1114
Integral Transforms
continued from previous page
f (x) 73
x sinh(ax) − cosh(ax) 1
74
a
sinh(ax) − 1b sinh(bx) a2 − b 2
F (s) −2 s a2 + 2a − s2 a2 − s2 ,
Re s > |Re a|
2 −1 2 −1 a − s2 b − s2 , Re s > {|Re a|, |Re b|}
75
cosh(ax) − cosh(bx) a2 − b 2
−1 2 −1 s − b2 s s2 − a2 , Re s > {|Re a|, |Re b|}
76
a sinh(ax) − b sinh(bx) a2 − b 2
−1 2 −1 s − b2 s2 s2 − a2 , Re s > {|Re a|, |Re b|}
77
sinh(a + bx)
−1 (b cosh a + s sinh a) s2 − b2 ,
Re s > |Re b|
78
cosh(a + bx)
−1 (s cosh a + b sinh a) s2 − b2 ,
Re s > |Re b|
79
sinh(ax) sinh(bx)
−1 2 −1 s − (a − b)2 2abs s2 − (a + b)2 , Re s > {|Re a|, |Re b|}
808
cosh(ax) cosh(bx)
−1 2 −1 s − (a − b)2 s s2 − a2 − b2 s2 − (a + b)2 , Re s > {|Re a|, |Re b|}
81
sinh(ax) cosh(bx)
−1 2 −1 s − (a − b)2 a s2 − a2 + b2 s2 − (a + b)2 , Re s > {|Re a|, |Re b|}
82
sinh2 (ax)
−1 2a2 s−1 s2 − 4a2 ,
Re s > |Re a|
83
cosh2 (ax)
2 −1 s − 2a2 s−1 s2 − 4a2 ,
Re s > |Re a|
84
sinh(ax) cosh(ax)
−1 a s2 − 4a2 ,
Re s > |Re a|
85
cosh(ax) − 1 a2
−1 s−1 s2 − a2 ,
Re s > |Re a| continued on next page
Table of Laplace transform pairs
1115
continued from previous page
f (x) 86 87 88 89
F (s)
sinh(ax) − ax a3 cosh(ax) − 12 a2 x2 − 1 a4 1 − cosh(ax) + 12 ax sinh(ax) a4
−1 s−2 s2 − a2 ,
Re s > |Re a|
−1 s−3 s2 − a2 ,
Re s > |Re a|
−2 s−1 s2 − a2 ,
Re s > |Re a|
, + π 1/2 /4 (s − a)3/2 − (s + a)3/2 ,
x1/2 sinh(ax)
Re s > |Re a| −s−1 ln (Cs) ,
Re s > 0
ET I 148(1)
s−1 es/a Ei(−s/a),
Re s > 0
ET I 148(4)
Re s > 0
ET I 148(9)
90
ln x
91
ln(1 + ax),
92
x−1/2 ln x
− (π/s)
0 for x < a H(x − a) = 1 for x > a
s−1 e−as ,
93
|arg a| < π
1/2
ln (4Cs) ,
a≥0
(Heaviside step function) 94
δ(x)
(Dirac delta function)
95
δ(x − a)
e−as ,
a≥0
96
δ (x − a)
se−as ,
a≥0
97
x
Si(x) ≡ 0
98
1 sin ξ dξ ≡ π + si(x) ξ 2
Ci(x) ≡ ci(x) ≡ −
∞ x
998
erf
x 2a
cos ξ dξ ξ
1
s−1 arccot s,
Re s > 0
ET I 177(17)
1 − s−1 ln 1 + s2 , 2
Re s > 0
ET I 178(19)
s−1 ea
2 2
s
erfc(as), Re s > 0, |arg a| < π/4
ET I 176(2)
continued on next page
1116
Integral Transforms
continued from previous page
f (x) 100
F (s)
√ erf a x
−1/2 as−1 s + a2 , Re s > 0, − Re a2
101
ET I 176(4)
, − 1 + 1/2 s−1 s + a2 2 s + a2 −a ,
√ erfc a x
Re s > 0 1028
erfc
a √ x
√ s
s−1 e−2a
, Re s > 0,
1038
104
105
J ν (ax),
x J ν (ax),
Re ν > −1
Re ν > −2
a−ν
ET I 177(9)
Re a > 0
ET I 177(11)
ν −1/2 s2 + a2 s2 + a2 − s ,
Re s > |Im a|, ET I 182(1) , + 1/2 1/2 ,−ν s + s2 + a2 aν s + ν s2 + a2 −3/2 × s2 + a2 , +
Re s > |Im a|, + 1/2 ,−ν aν ν −1 s + s2 + a2 ,
J ν (ax) x
ET I 182(2)
Re s ≥ |Im a| 106
−(n+ 12 ) , 1 · 3 · 5 · · · (2n − 1)an s2 + a2
xn J n (ax)
Re s > |Im a| 107
xν J ν (ax),
Re ν > − 12
xν+1 J ν (ax),
Re ν > −1
I ν (ax),
Re ν > −1
ET I 182(7)
−(ν+ 32 ) , 2ν+1 π −1/2 Γ ν + 32 aν s s2 + a2 Re s > |Im a|
1098
ET I 182(4)
−(ν+ 12 ) , 2ν π −1/2 Γ ν + 12 aν s2 + a2 Re s > |Im a|,
108
ET I 182(5)
ET I 182(8)
+ ,ν −1/2 s2 − a2 a−ν s − s2 − a2 , Re s > |Re a|
ET I 195(1)
continued on next page
Fourier transform
1117
continued from previous page
f (x) 110
xν I ν (ax),
F (s) Re ν > − 12
−(ν+ 12 ) , 2ν π −1/2 Γ ν + 12 aν s2 − a2 Re s > |Re a|
111
xν+1 I ν (ax),
Re ν > −1
−(ν+ 32 ) , 2ν+1 π −1/2 Γ ν + 32 aν s s2 − a2 Re s > |Re a|
112
x−1 I ν (ax),
Re ν > 0
ET I 195(6)
ET I 196(7)
+ 1/2 ,−ν ν −1 aν s + s2 − a2 , Re s > |Re a|
ET I 195(4)
113
sin 2a1/2 x1/2
(πa)1/2 s−3/2 e−a/s ,
Re s > 0
ET I 153(32)
114
x−1/2 cos 2a1/2 x1/2
π 1/2 s−1/2 e−a/s ,
Re s > 0
ET I 158(67)
115
x−1 e−ax I 1 (ax)
+ ,+ ,−1 1/2 , (s + 2a)1/2 − s1/2 (s + 2a) + s1/2 Re s > |Re a|
116
J k (ax) x
k −1 a−k
+
s2 + a2
1/2
,k −s ,
Re s > |Im a|, k > −1 117
x k− 12 J k− 12 (ax) 2a
AS 1024(29.3.52)
AS 1025(29.3.58)
k Γ(k)π −1/2 s2 + a2 , Re s > |Im a|,
k>0
AS 1024(29.3.57)
118
J 0 (ax) − ax J 1 (ax)
−3/2 s2 s2 + a2 ,
Re s > |Im a|
119
I 0 (ax) + ax I 1 (ax)
−3/2 s2 s2 − a2 ,
Re s > |Im a|
17.21 Fourier transform The Fourier transform, also called the exponential or complex Fourier transform, of the function f (x), denoted by F (ξ), is defined by the integral ∞ 1 F (ξ) = √ f (x)eiξx dx. 2π −∞ The functions f (x) and F (ξ) are called a Fourier transform pair, and knowledge of either one enables the other to be recovered. Setting F (ξ) = F [f (x); ξ] , to emphasize the nature of the transform, we have
1118
Integral Transforms
the symbolic inverse result f (x) = F −1 [F (ξ); x] . The inversion of the Fourier transform is accomplished by means of the inversion integral ∞ 1 f (x) = √ F (ξ)e−iξx dξ. 2π −∞
17.22 Basic properties of the Fourier transform 1.
For a and b arbitrary constants, F [af (x) + bg(x)] = aF (ξ) + bG(ξ)
(linearity)
If n > 0 is an integer, and lim f (r) (x) = 0 for r = 0, 1, . . . , n − 1 with f (0) (x) ≡ f (x), then
2.
|x|→∞
+ , F f (n) (x); ξ = (−iξ)n F (ξ)
(transform of a derivative)
SN 27
The Fourier convolution f ∗ g of two functions f (x) and g(x) is defined by the integral ∞ 1 f ∗ g(x) = √ f (x − ξ)g(ξ) dξ, 2π −∞ and it has the property f ∗ g = g ∗ f , and f ∗ (g ∗ h) = (f ∗ g) ∗ h. In terms of the convolution operation,
3.
F [f ∗ g(x); ξ] = F (ξ)G(ξ)
(convolution [Faltung] theorem).
SN 24
17.23 Table of Fourier transform pairs f (x)
F (ξ)
1
1
(2π)1/2 δ(ξ)
27
1 x
(π/2)
3
δ(x)
(2π)−1/2
SU 496
48
δ(ax + b),
a = 0
(2π)−1/2 eibξ/a
SU 517
a>0
(2/π)1/2 ξ −1 sin(aξ)
5 8
6
1/2
a, b ∈ R,
1 |x| < a , 0 |x| > a 0 x0
SU 496
i sign ξ
1 − √ + iξ 2π
π δ(ξ) 2
SU 50
SN 523 continued on next page
Table of Fourier transform pairs
1119
continued from previous page
f (x)
F (ξ) (2/π)1/2 Γ(1 − a) sin 12 aπ
7
1 a, |x|
0 < Re a < 1
8
eiax ,
a∈R
(2π)1/2 δ(ξ + a)
SU 50
9
e−a|x| ,
a>0
a(2/π)1/2 a2 + ξ 2
SU 50
107
xe−a|x| ,
a>0
11
|x|e−a|x| ,
a>0
12
e−a|x| |x|
1/2
,
a>0
,
a>0
13
e−a
14
1 , 2 a + x2
Re a > 0
x , + x2
Re a > 0
157
2
a2
x2
|ξ|
2aiξ(2/π)1/2
2 , (a2 + ξ 2 ) (2/π)1/2 a2 − ξ 2
ξ>0
2
(a2 + ξ 2 ) + 1/2 ,1/2 a + a2 + ξ 2
SU 50
SU 51
1/2 −a|ξ|
(π/2)
e
SU 51
a 1/2 −a|ξ|
i sign ξ (π/2)
e
sin ax2
17
cos ax2
18
e−a|x| cos(bx),
a > 0,
b>0
19
e− 2 ax sin(bx),
a > 0,
b>0
209
sinh(ax) , sinh(bx)
|a| < |b|
(π/2) sin(πa/b) b [cosh (πξ/b) + cos(πa/b)]
219
cosh(ax) , sinh(bx)
|a| < |b|
i (π/2) sinh (πξ/b) b [cosh (πξ/b) + cos(πa/b)]
2
SU 50
SN 523
1/2
x (a2 + ξ 2 ) √ −1 2 2 a 2 e−ξ /4a
169
1
SN 523
1−a
1 cos (2a)1/2
ξ2 π + 4a 4
SN 523
2
ξ π 1 − cos SN 523 4a 4 (2a)1/2 1 1 a(2π)−1/2 2 + a + (b + ξ)2 a2 + (b − ξ)2
1 −1/2 1 (ξ − b)2 ia exp − 2 2 a 1 (ξ + b)2 − exp − 2 a
1/2
SU 123
1/2
SU 123 continued on next page
1120
Integral Transforms
continued from previous page
f (x) 22
sin(ax) x
2311
x sinh x
247
xn sign x,
257
|x| ,
F (ξ)
1/2 |ξ| < a, (π/2) 0 |ξ| > a 3 1/2 πξ 2π e
SN 523
2
SU 123
(2/π)1/2 (−iξ)−(1+n) n!
SU 506
(1 + eπξ ) n = 1, 2, . . .
ν
(2/π)1/2 Γ(ν + 1)|ξ|
−ν−1
cos [π(ν + 1)/2]
−1 < ν < 0, but not integral 267
SU506
i sign ξ(2/π)1/2 sin [(π/2) (ν + 1)] Γ(ν + 1)
ν
|x| sign x,
|ξ|
ν+1
−1 < ν < 0, but not integral ! ! e−ax ln !1 − e−x !,
27
SU 506
π 1/2 cot (πa − iξπ) 2 a − iξ
ET I 121(26)
π 1/2 csc (πa − iξπ) 2 a − iξ
ET I 121 (27)
−1 < Re a < 0 e−ax ln 1 + e−x ,
28
−1 < Re a < 0 In deriving results for the preceding table from ET I, account has been taken of the fact that the normalization factor 1/(2π)1/2 employed in our definition of F has not been used in those tables, and that there is a difference of sign between the exponents used in the definitions of the exponential Fourier transform.
17.24 Table of Fourier transform pairs for spherically symmetric functions
2
e−ar
311
e−ar r
1 E(||k||) = f (||r||)e−ik·r dr (2π)3/2 21 ∞ E(k) = f (r) sin(kr)r dr πk 0 2a 2 π (a2 + k 2 )2 1 2 π (a2 + k 2 )2
411
1
(2π)3/2 δ(k)
1
1 f (||r||) = E(||k||)eik·r dk (2π)3/2 21 ∞ f (r) = E(k) sin(kr)k dk πr 0
Basic properties of the Fourier sine and cosine transforms
1121
17.31 Fourier sine and cosine transforms The Fourier sine and cosine transforms of the function f (x), denoted by Fs (ξ) and Fc (ξ), respectively, are defined by the integrals ∞ ∞ 2 2 f (x) sin(ξx) dx and Fc (ξ) = f (x) cos(ξx) dx. Fs (ξ) = π 0 π 0 The functions f (x) and Fs (ξ) are called a Fourier sine transform pair, and the functions f (x) and Fc (ξ) a Fourier cosine transform pair, and knowledge of either Fs (ξ) or Fc (ξ) enables f (x) to be recovered. Setting and Fc (ξ) = F c [f (x); ξ] , Fs (ξ) = F s [f (x); ξ] to emphasize the nature of the transforms, we have the symbolic inverses f (x) = F s −1 [Fs (ξ); x] and f (x) = F c −1 [Fc (ξ); x] . The inversion of the Fourier sine transform is accomplished by means of the inversion integral ∞ 2 Fs (ξ) sin(ξx) dξ [x ≥ 0] f (x) = π 0 and the inversion of the Fourier cosine transform is accomplished by means of the inversion integral ∞ 2 Fc (ξ) cos(ξx) dξ [x ≥ 0] . SN 17 f (x) = π 0
17.32 Basic properties of the Fourier sine and cosine transforms 1.
For a and b arbitrary constants, F s [af (x) + bg(x)] = aFs (ξ) + bGs (ξ) and
2.
F c [af (x) + bg(x)] = aFc (ξ) + bGc (ξ) (linearity) < If lim f (r−1) (x) = 0 and lim π2 f (r−1) (x) = ar−1 , then denoting the Fourier sine and cosine x→∞
x→∞
transforms of f (r) (x) by Fs (r) and Fc (r) , respectively, (i)
Fc (r) (ξ) = −ar−1 + ξFs (r−1) .
(ii)
Fs (r) (ξ) = −ξFc (r−1) (ξ), r−1 (−1)n a2r−2n−1 ξ 2n + (−1)r ξ 2n Fc (ξ), Fc (2r) (ξ) = −
(iii)
n=0 r−1
(iv)
Fc (2r+1) (ξ) = −
(v)
(ξ) = ξar−2 − ξ 2 Fs (r−2) (ξ), r Fs (2r) (ξ) = − (−1)n ξ 2n−1 a2r−2n + (−1)r ξ 2r Fs (ξ),
(−1)n a2r−2n ξ 2n + (−1)r ξ 2r+1 Fs (ξ),
n=0
(vi)6
Fs
(r)
n=1
(vii)
Fs
(2r+1)
(ξ) = −
r n=1
(−1)n ξ 2n−1 a2r−2n+1 + (−1)r+1 ξ 2r+1 Fc (ξ).
SN 28
1122
Integral Transforms
1 ∞ Fs (ξ)Gs (ξ) cos(ξx) dξ = g(s) [f (s + x) + f (s − x)] ds, 2 0 0 ∞ ∞ 1 Fc (ξ)Gc (ξ) cos(ξx) dξ = g(s) [f (s + x) + f (|x − s|)] ds 2 0 0 (convolution (Faltung) theorem)
3.
(i) (ii)
4.
(i) (ii)
∞
SN 24
If Fs (ξ) is the Fourier sine transform of f (x), then the Fourier sine transform of Fs (x) is f (ξ). If Fc (ξ) is the Fourier cosine transform of f (x), then the Fourier cosine transform of Fc (x) is f (ξ).
(v)
If f (x) is an odd function in (−∞, ∞), then the Fourier sine transform of f (x) in (0, ∞) is −iF (ξ). If f (x) is an even function in (−∞, ∞), then the Fourier cosine transform of f (x) in (0, ∞) is F (ξ). The Fourier sine transform of f (x/a) is aFs (aξ).
(vi) (vii)
The Fourier cosine transform of f (x/a) is aFc (aξ). F s [f (x); ξ] = Fs (|ξ|) sign ξ
(iii) (iv)
SU 45
17.33 Table of Fourier sine transforms f (x) 1
x−1
2
x−ν ,
Fs (ξ) 1/2
(π/2) 0 < Re ν < 2
,
(ξ > 0) ξ>0
ET I 64(3)
(2/π)1/2 ξ ν−1 Γ(1 − ν) cos (νπ/2) , ξ>0
ET I 68(1)
3
x−1/2
ξ −1/2 ,
ξ>0
ET I 64(6)
4
x−3/2
2ξ 1/2 ,
ξ>0
ET I 64(9)
(2/π)1/2 ξ −1 [1 − cos(aξ)] ,
ξ>0
ET I 63(1)
(2/π)1/2 Si(aξ),
ξ>0
ET I 64(4)
5 6 7
1 0<xa x−1 0 < x < a 0 x>a 1 , a−x
a>0
(2/π)1/2 sin(aξ) Ci(aξ) − cos(aξ) 12 π + Si(aξ) , ξ>0
ET I 64(11)
continued on next page
Table of Fourier sine transforms
1123
continued from previous page
f (x) 87
x2
Fs (ξ)
1 , + a2
a>0
(2π)−1/2 a−1 e−aξ Ei(aξ) − eaξ Ei(−aξ) ,
9
−3/2 x x2 + a2 ,
Re a > 0
(2/π)1/2 ξ K 0 (aξ),
10
−1/2 x−1/2 x2 + a2 ,
Re a > 0
ξ 1/2 I 14
117
−ν− 32 x x2 + a2 , Re ν > −1,
12 13
x , 2 a + x2
−1
+
Re a > 0
15
x−1 e−ax ,
16
xν−1 e−ax ,
Re a > 0 Re a > 0
K 14
1
2 aξ
,
ξ>0
ET I 66(27)
ξ>0
ET I 66(28)
π 1/2 2
e−aξ ,
ξ>0
ET I 65(15)
ξ>0
ET I 67(35)
π/2 1 − e−aξ , 2 a
ξ (2/π)1/2 tan−1 , a
ξ>0
ET I 65(20)
ξ>0
ET I 72(2)
−ν/2 ξ (2/π)1/2 Γ(ν) a2 + ξ 2 sin ν tan−1 , a Re ν > −1,
Re a > 0
17
e−ax ,
Re a > 0
18
xe−ax ,
Re a > 0
−ax2
π/8a−1 ξe−aξ ,
2 −1 x + a2 ,
x
2 aξ
ET I 65(14)
Re a > 0
2 x2 )
14
1
ξ>0
ξ ν+1 √ K ν (aξ), 2(2a)ν Γ ν + 32
x (a2
(ξ > 0)
19
xe
,
|arg a| < π/2
20
sin ax , x
a>0
21
sin ax , x2
a>0
2/πξ , 2 a + ξ2 (2/π)1/2 2aξ (a2 + ξ 2 )
,
−ξ 2 ξ exp 4a ! ! !ξ + a! !, ! ln ! ξ − a!
−3/2
(2a)
2
1 (2π)1/2 1/2 ξ π2 1/2 a π2
ξ>0
ET I 72(7)
ξ>0
ET I 72(1)
ξ>0
ET I 72(3)
,
00
a2 x
,
a>0
Fs (ξ) (ξ > 0) π 1/2 1 a ξ −1/2 J 1 2aξ 2 , 2 ξ>0 ET I 83(6) π 1/2 2 1/2 Y 0 2aξ 1/2 + K 0 2aξ 1/2 2 π ET I 83(7)
24
2510
x−2 sin
27 28
29 30
31
2
a x
,
cosech(ax),
26
coth
1 ax − 1, 2
1 − x2
−1
sin2 (ax) , x
2
cos ax
Re a > 0
,
π 1/2 2
a−1 ξ 1/2 J 1 2aξ 1/2 ,
1/2
(π/2)
a−1 tanh
Re a > 0
a>0
a>0
1
2 πa
−1
ξ>0
ET I 83(8)
ξ>0
ET I 88(2)
ξ ,
(2π)1/2 a−1 coth πa−1 ξ − ξ, (2/π)1/2 sin ξ 0
a>0
sin ax2 ,
Re a > 0
sin(πx)
e−ax sin(bx), 2
a>0
ξ>0 0≤ξ≤π π0 ⎧ 1/2 −3/2 ⎪ 0 < ξ < 2a ⎨π 2 1/2 −5/2 ξ = 2a π 2 ⎪ ⎩ 0 2a < ξ , + 2 −1/2 a cos ξ /4a C (2πa)−1/2 ξ , + + sin ξ 2 /4a S (2πa)−1/2 ξ ,
ξ>0 , + 2 −1/2 sin ξ /4a C (2πa) a ξ , + − cos ξ 2 /4a S (2πa)−1/2 ξ ,
ET I 78(7)
ET I 78(8)
ET I 82(1)
−1/2
ξ>0 continued on next page
Table of Fourier sine transforms
1125
continued from previous page
32 337 34 35 367 37
f (x) x arctan , a
2a arctan , x
Fs (ξ) a>0 Re a > 0
1/2 −1 −aξ
(π/2)
ξ
e
(ξ > 0)
,
(2π)−1/2 e−aξ sinh(aξ),
ξ>0
ET I 87(3)
ξ>0
ET I 87(8)
ξ>0
ET I 76(2)
ln x x ! ! !x + a! !, ! ln ! x − a! ln 1 + a2 x2 , x
a>0
(2π)1/2 ξ −1 sin(aξ),
ξ>0
ET I 77(11)
a>0
−(2π)1/2 Ei (−ξ/a) ,
ξ>0
ET I 77(14)
J 0 (ax),
a>0
− (π/2)
1/2
0 1/2
(2/π)
(C + ln ξ) ,
2 −1/2 ξ − a2
0 0
−ν− 12
,
1 Re a > 0, Re ν > − 2 ν a2 − x2 0<xa 0 −ν− 12 2 x − a2
e−ax ,
[Γ(ν)]
ξ>0
2ξ −1/2
Re a > 0
(x2 + a2 )
(2/π)1/2
0a 0 0<xa
11
8
Fc (ξ)
(π/2) 2 (1 + aξ)e−aξ , 2a3 ν √ ξ K ν (aξ) , 2 2a Γ ν + 12
1
2ν Γ(ν + 1)(a/ξ)ν+ 2 J ν+ 12 (aξ), ξ>0
Re ν > −1 0<xa 1 1 − < Re ν < 2 2 Re a > 0
−2−(ν+ 2 ) Γ 1
1 2
ET I 11(8)
ν − ν (ξ/a) Y ν (aξ),
−1 (2/π)1/2 a a2 + ξ 2 ,
ξ>0
ET I 11(9)
ξ>0
ET I 14(1)
continued on next page
Table of Fourier cosine transforms
1127
continued from previous page
f (x) 13
147
Fc (ξ)
xe−ax ,
Re a > 0
ξ>0 ET I 15(7) 2 ξ 1/2 2 −ν/2 −1 (2/π) Γ(ν) a + ξ cos ν tan , a
xν−1 e−ax , Re a > 0,
15
−2 (2/π)1/2 a2 − ξ 2 a2 + ξ 2 ,
x−1/2 e−ax ,
Re ν > a Re a > 0
ξ>0
a2 + ξ 2
−1/2 +
a2 + ξ 2
1/2
ET I 15(7)
,1/2 +a
,
ξ>0 167 17 18
19
e−a
2
x2
,
−1 −x
x
e
2
sin ax
Re a > 0
2−1/2 |a|
−1/2
2 ξ2
a>0
a>0
a>0
217
sin2 (ax) , x2
a>0
227
e−bx sin(ax),
a > 0,
Re b > 0
x2 + a2
+
2 a2 )
−1/2
,
ξ>0
ξ>0 2 2
ξ 1 ξ √ cos + sin , 4a 4a 2 a
ξ>0 a>0
+ 1/2 , , sin b x2 + a2 a>0
ET I 19(7)
ET I 23(1)
ET I 24(7) ξ>0 ⎧ 1/2 ⎪ ξa 1/2 a − 12 ξ ξ < 2a (π/2) ET I 19(8) 0 2a < ξ a+ξ a−ξ (2π)−1/2 2 + , b + (a + ξ)2 b2 + (a − ξ)2
+ 1/2 , sin b x2 + a2 (x2
ET I 15(11)
2 2
ξ 1 ξ √ cos − sin , 4a 4a 2 a
,
tan
ξ>0
sin(ax) , x
−1
,
(2π)
20
24
e
sin x
cos ax2 ,
23
−1 −ξ 2 /4a2
ET I 14(4)
ET I 19(6)
1/2 −aξ
(b/a) (π/2)
e , ξ>0 ET I 26(29) + 1/2 , 0 |Im b|
,
sinh(ax) sinh(bx)
|Re a| < Re b
Fc (ξ) 1/2 (π/2) (a − ξ) ξ < a ET I 20(16) 0 a0 2 2 a +b 2 exp −aξ / 4 a + b2 , + −1 1 − 2 arctan(b/a) , × cos 14 bξ 2 a2 + b2 −1/2
π 1/2 2
2 −1/4
2
ξ>0 sin(πa/b) , b [cosh (πξ/b) + cos(πa/b)] ξ>0
29
cosh(ax) , cosh(bx)
|Re a| < Re b
sech(ax),
Re a > 0
a−1 (π/2)
1/2
32 337
πx x2 + a2 sech , 2a
a2 ln 1 + 2 , x 2
a + x2 ln , b2 + x2
ET I 31(12)
sech (πξ/2a) , ξ>0
31
ET I 31(14)
(2π)1/2 cos(πa/2b) cosh (πξ/2b) , b [cosh (πξ/b) + cos(πa/b)] ξ>0
30
ET I 24(6)
ET I 30(1)
Re a > 0
2(2/π)1/2 a3 sech3 (aξ),
ξ>0
ET I 32(19)
Re a > 0
(2π)1/2 ξ −1 1 − e−aξ ,
ξ>0
ET I 18(10)
(2π)1/2 e−bξ − e−aξ ,
ξ>0
ET I 18(12)
a 0 34
x2 + b2
−1
1/2 −1 −bξ
J 0 (ax), a > 0,
(π/2) Re b > 0
b
e
I 0 (ab),
continued on next page
Mellin transform
1129
continued from previous page
f (x) 35
Fc (ξ)
−1 J 0 (ax), x x2 + b2 a > 0,
(2/π)1/2 cosh(bξ) K 0 (ab), Re b > 0
0 k. Setting f ∗ (s) = M [f (x); s] to denote the Mellin transform, we have the symbolic expression for the inverse result f (x) = M−1 [f ∗ (s); x] .
MS 397(6)
17.42 Basic properties of the Mellin transform 1.
For a and b arbitrary constants, M [af (x) + bg(x)] = af ∗ (s) + bg ∗ (s)
2.
x→0
(ii)
3.
r = 0, 1, . . . , n − 1,
If lim xs−r−1 f (r) (x) = 0, (i)
(linearity)
+ , M f (n) (x); s = (−1)n
Γ(s) f ∗ (s − n) Γ(s − n) (transform of a derivative)
SU 267 (4.2.3)
(transform of a derivative)
SU 267 (4.2.5)
+ , Γ(s + n) ∗ f (s) M xn f (n) (x); s = (−1)n Γ(s)
Denoting the nth repeated integral of f (x) by In [f (x)], where x In [f (x)] = In−1 [f (u)] du, 0
(i)
M [In [f (x)] ; s] = (−1)n
Γ(s) f ∗ (s + n) Γ(n + s) (transform of an integral)
(ii)
M [In∞ [f (x)] ; s] =
Γ(s) f ∗ (s + n), Γ(s + n)
where
In∞ [f (x)] =
∞
∞ In−1 [f (u)] du
(transform of an integral)
SU 269 (4.2.15)
SU 269 (4.2.18)
x
4.
M [f (x)g(x); s] =
1 2πi
c+i∞
f ∗ (u)g ∗ (s − u) du
c−i∞
(Mellin convolution theorem)
SU 275(4.4.1)
Table of Mellin transforms
1131
17.43 Table of Mellin transforms f ∗ (s)
f (x) 1
e−x
Γ(s),
2
e−x
1 2
3
cos x
Γ(s) cos
1
4
sin x
Γ(s) sin
1
5
1 1−x
π cot(πs),
0 < Re s < 1
SU 521(M1)
6
1 1+x
π cosec(πs),
0 < Re s < 1
SU 521(M2)
7
(1 + xa )
2
Γ
Re s > 0
SU 521(M13)
Re s > 0
SU 521(M14)
,
0 < Re s < 1
SU 521(M15)
,
0 < Re s < 1
SU 521(M16)
1 2s , 2 πs
2 πs
Γ(s/a) Γ(b − s/a) , a Γ(b)
−b
0 < Re s < ab SU 521(M3)
8
9
10
11
T n (x) H(1 − x) (1 − x2 )
Tn
Γ
x−1 H(1 − x) (1 − x2 )
P n (x) H(1 − x)
P n x−1 H(1 − x)
1 2
+
1 2s
2−s π Γ(s) , + 12 n Γ 12 + 12 s − 12 n
Re s > 0 2s−2 Γ 12 n + 12 s Γ 12 s − 12 n , Γ(s)
SU 521(M4)
Re s > n 1 1 Γ 2 s Γ 2 s + 12 , 2 Γ 12 s − 12 n + 12 Γ 12 s + 12 n + 1
SU 521(M5)
Re s > 0 2s−1 Γ 12 s + 12 n + 12 Γ 12 s − 12 n √ , π Γ(s + 1)
SU 521(M6)
Re s > n 12
1 + x cos φ 1 − 2x cos φ + x2
13
x sin φ , 1 − 2x cos φ + x2
−π < φ < π
SU 521(M7)
π cos(sφ) , sin(sπ)
0 < Re s < 1
SU 521(M11)
π sin(sφ) , sin(sπ)
0 < Re s < 1
SU 521(M12)
continued on next page
1132
Integral Transforms
continued from previous page
f ∗ (s)
f (x) 14
e−x cos φ cos (x sin φ) , 1 2π
15
− 12 π < φ < 12 π 16
x
J ν (x),
17
Y ν (x),
Re s > 0
SU 522(M17)
Γ(s) sin(sφ),
Re s > −1
SU 522(M18)
< φ < 12 π
e−x sin φ sin (x sin pφ) ,
−ν
Γ(s) cos(sφ),
ν>
− 12
ν∈R
2s−ν−1 Γ 12 s , 0 < Re s < 1 Γ ν − 12 s + 1 −2s−1 π −1 Γ 12 s + 12 ν Γ 12 s − 12 ν × cos 12 s − 12 ν π, |ν| < Re s
ν > 0 2s−1 tan 12 πs + 12 πν Γ 12 s + 12 ν , Γ 12 ν − 12 s + 1 −1 − ν < Re s < min 32 , 1 − ν πn−1 cosec
24
arctan x
πs n
SU 522(M21)
SU 522(M22)
a(s/n)−1 , 0 < Re s < n
n = 1, 2, 3, . . . ,
MS 453
h−1 a−s/h B (s/h, ν − (s/h))
,
h > 0, ln(1 + ax),
2s
SU 522(M20)
+ 12 ν Γ 12 s − 12 ν ,
ν∈R
h > 0, |arg a| < π ν−1 1 − xh for 0 < x < 1 , 0 for x > 1
23
1
3 2
SU 522(M19)
0 < Re s < h Re ν h−1 B (ν, s/h)
MS 454 MS 454
Re ν > 0 |arg a| < π
πs−1 a−s cosec(πs),
−1 < Re s < 0
MS 454
− 12 πs−1 sec(πs/2),
−1 < Re s < 0
MS 454
continued on next page
Table of Mellin transforms
1133
continued from previous page
f ∗ (s)
f (x) −1 1 2 πs
25
arccot x
26
cosech(ax)
Re a > 0
a−s 2 1 − 2−s Γ(s) ζ(s),
27
sech2 (ax),
Re a > 0
4a−s (1 − 22−s ) Γ(s)2−s ζ(s − 1),
28 2911
30
31
32
cosech2 (ax),
x2 + b2
− 12 ν
Re a > 0 + 1/2 , J ν a x2 + b2
+ ⎧ 1 , 2 2 2ν 2 2 1/2 ⎪ a a b − x J − x ⎪ ν ⎨ for 0 < x < a ⎪ ⎪ for x > a ⎩0
0 < Re s < 1 Re s > 1
4a−s Γ(s)2−s ζ(s − 1), 2 2 s−1 a− 2 s b 2 s−ν Γ 1
1
1
1
MS 454
Re s > 2
MS 454
Re s > 2
MS 454
1 2 s J ν−s/2 (ab),
0 < Re s < 2 2 s−1 Γ
MS 454
3 2
+ Re ν
ET I 328
1 − 1 s ν+ 1 s 2 a ,2 J ν+ 1 s (ab), 2s b 2 Re s > 0
MS 455
Re ν > −1 + ⎧ 1 , − ν 2 2 2 2 1/2 2 ⎪ a − x J − x b a ⎪ ν ⎨ 1 −1 1 for 0 < x < a 21−ν [Γ(ν)] a 2 s−ν b−, 2 ν s ν−1+ 12 s, 12 s−ν (ab), ⎪ ⎪ for x > a ⎩0 Re s > 0 MS 455
α−s 2s−2 Γ
K ν (αx) − 1 ν βa2 ++ x2 2 1/2 , × K ν α βa2 + x2
33
sec(πs/2),
Re (α, β) > 0
1
2s
− 12 ν Γ 12 s + 12 ν , Re s > |Re ν|
1 1 1 α− 2 s 2 2 s−1 β 2 s−ν Γ, 12 s K
MS 455
1 (αβ), ν− 2 s
Re s > 0
MS 455
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18 The z-Transform 18.1–18.3 Definition, Bilateral, and Unilateral z-Transforms 18.1 Definitions The z-transform converts a numerical sequence x[n] into a function of the complex variable z, and it takes two different forms. The bilateral or two-sided z-transform, denoted here by Zb {x[n]}, is used mainly in signal and image processing, while the unilateral or one-sided z-transform, denoted here by Zu {x[n]}, is used mainly in the analysis of discrete time systems and the solution of linear difference equations. ∞ The bilateral z-transform, Xb (z) of the sequence x[n] = {xn }n=−∞ is defined as ∞ Zb {x[n]} = xn z −n = Xb (z), n=−∞
∞
and the unilateral z-transform Xu (z) of the sequence x[n] = {xn }n=0 is defined as ∞ xn z −n = Xu (z), Zb {x[n]} = n=0
where each has its own domain of convergence (DOC). The series Xb (z) is a Laurent series, and Xu (z) is the principal part of the Laurent series for Xb (z). When xn = 0 for n < 0, the two z-transforms Xb (z) and Xu (z) are identical. In each case the sequence x[n] and its associated z-transform is a called a z-transform pair. The inverse z-transformation x[n] = Z −1 {X(z)} is given by 1 x[n] = X(z)z n−1 dz, 2πi Γ where X(z) is either Xb (z) or Xu (z), and Γ is a simple closed contour containing the origin and lying entirely within the domain of convergence of X(z). In many practical situations, the z-transform is either found by using a series expansion of X(z) in the inversion integral or, if X(z) = N (z)/D(z) where N (z) and D(z) are polynomials in z, by means of partial fractions and the use of an appropriate table of z-transform pairs. In order for the inverse z-transform to be unique, it is necessary to specify the domain of convergence, as can be seen by comparison of entries 3 and 4 of Table 18.2. Table 18.1 lists general properties of the bilateral z-transform, and Table 18.2 lists some bilateral z-transform pairs. In what 0 for n < 0 follows, use is made of the unit integer function h(n) = , that is, a generalization of 1 for n ≥ 0 1 for n = k the Heaviside step function, and the unit integer pulse function Δ(n − k) = , that is, 0 for n = k a generalization of the delta function. 1135
1136
Definition, Bilateral, and Unilateral z-Transforms
18.2 Bilateral z-transform Table 18.1 General properties of the bilateral z-transform Xb (n) =
∞
xn z −n .
n=−∞
Term in sequence
z-Transform Xb (z)
1
αxn + βyn
αXb (z) + βYb (z)
2
xn−N
z −n Xb (z)
3
nxn
−z
4
z0n xn
dXb (z) dz
z Xb z0
DOC of Xb (z) scaled by |z0 | DOC of Xb (z) scaled by |z0 | to which it may be necessary to add or delete the origin and the point at infinity DOC of radius 1/R, where R is the radius of convergence of DOC of Xb (z) DOC of radius 1/R, where R is the radius of convergence of DOC of Xb (z)
5
nz0n xn
dXb (z/z0 ) −z dz
6
x−n
Xb (1/z)
7
nx−n
−z
8
x ¯n
9
Re xn
10
Im xn
Xb (z) 1 2 Xb (z) + Xb (z) 1 2i Xb (z) − Xb (z)
11
∞
Domain of Convergence Intersection of DOC’s of Xb (z) and Yb (z) with α, β constants DOC of Xb (z), to which it may be necessary to add or delete the origin or the point at infinity DOC of Xb (z), to which it may be necessary to add or delete the origin and the point at infinity
xk yn−k
dXb (1/z) dz
The same DOC as xn DOC contains the DOC of xn DOC contains the DOC of xn
Xb (z)Yb (z)
k=−∞
12
xn yn
1 2πi
Xb (ξ)Yb Γ
z ξ −1 dξ ξ
∞
13
Parseval formula
1 xn y¯n = 2πi n=−∞
14
Initial value theorem for xn h(n)
x0 = lim Xb (z) z→∞
z Xb (ξ)Yb ξ −1 dξ ξ Γ
DOC contains the intersection of the DOCs of Xb (z) and Yb (z) (convolution theorem) DOC contains the DOCs of Xb (z) and Yb (z), with Γ inside the DOC and containing the origin (convolution theorem) DOC contains the intersection of DOCs of Xb (z) and Yb (z), with Γ inside the DOC and containing the origin
Bilateral z-transform
1137
Table 18.2 Basic bilateral z-transforms
1
Term in sequence
z-Transform Xb (z)
Domain of Convergence
Δ(n)
1
Converges for all z
2
Δ(n − N )
z −n
When N > 0 convergence is for all z except at the origin. When N < 0 convergence is for all z except at ∞
3
an h(n)
z z−a
|z| > |a|
4
an h(−n − 1)
z z−a
|z| < |a|
5
nan h(n)
az (z − a)2
|z| > a > 0
6
nan h(−n − 1)
az (z − a)2
|z| < a,
7
n2 an h(n)
az(z + a) (z − a)3
|z| > a > 0
bz az + az − 1 bz − 1 z 1 − (a/z)N z−a
|z| > max
a>0
8
1 1 + n an b
9
an h(n − N )
10
an h(n) sin Ωn
az sin Ω z 2 − 2az cos Ω + a2
|z| > a > 0
11
an h(n) cos Ωn
z(z − a cos Ω) z 2 − 2az cos Ω + a2
|z| > a > 0
12
ean h(n)
z z − ea
|z| > e−a
13
e−an h(n) sin Ωn
zea sin Ω z 2 e2a − 2zea cos Ω + 1
|z| > e−a
14
e−an h(n) cos Ωn
zea (zea − cos Ω) z 2 e2a − 2zea cos Ω + 1
|z| > e−a
h(n)
|z| > 0
1 1 |a| , |b|
1138
Definition, Bilateral, and Unilateral z-Transforms
18.3 Unilateral z-transform The relationship between the Laplace transform of a continuous function x(t) sampled at t = 0, T , 2T , ∞ x(nT )δ(t − nT ) follows from the result . . . and the unilateral z-transform of the function x ˆ(t) =
∞
L{ˆ x(t)} = 0
=
∞
%
n=0 ∞
&
x(kT )δ(t − kT ) e−st dt
k=0
x(kT )e−ksT .
k=0
Setting z = esT , this becomes: L{ˆ x(t)} =
∞
x(kT )z −k = X(z),
k=0
showing that the unilateral z-transform Xu (z) can be considered to be the Laplace transform of a continuous function x(t) for t ≥ 0 sampled at t = 0, T , 2T , . . . . Table 18.3 lists some general properties of the unilateral z-transform, and Table 18.4 lists some unilateral z-transform pairs.
Unilateral z-transform
1139
Table 18.3 General properties of the unilateral z-transform Term in sequence
z-Transform Xu (z)
Domain of Convergence
1
αxn + βyn
αXu (z) + βYu (z)
Intersection of DOC’s of Xu (z) and Yu (z) with α, β constants
2
xn+k
z k Xu (z) − z k x0 − z k−1 x1 −z k−2 x2 − · · · − zxk−1
3
nxn
−z
4
z0n xn
Xu
5
nz0n xn
−z
6
x ¯n
Xu (z)
7
Re xn
1 2
8
∂ xn (α) ∂α
∂ Xu (z, α) ∂α
9
Initial value theorem
x0 = lim Xu (z)
10
Final value theorem
DOC of Xu (z), to which it may be necessary to add or delete the origin and the point at infinity DOC of Xb (z) scaled by |z0 |, to which it may be necessary to add or delete the origin and the point at infinity DOC of Xu (z) scaled by |z0 |, to which it may be necessary to add or delete the origin and the point at infinity
dXu (z) dz
z z0
dXu (z/z0 ) dz
The same DOC as xn
Xu (z) + Xu (z)
DOC contains the DOC of xn Same DOC as xn (α)
z→∞
lim xn = lim
n→∞
z→1
z−1 z
Xu (z)
When Xu (z) = N (z)/D(z) with N (z), D(z) polynomials in z and the zeros of D(z) inside the unit circle |z| = 1 or at z = 1
1140
Definition, Bilateral, and Unilateral z-Transforms
Table 18.4 Basic unilateral z-transforms Term in sequence
z-Transform Xu (z)
Domain of Convergence
1
Δ(n)
1
Converges for all z
2
Δ(n − k)
z −k
Convergence for all z = 0
3
an h(n)
z z−a
|z| > |a|
4
nan h(n)
az (z − az)2
|z| > a > 0
5
n2 an h(n)
az(z + a) (z − a)3
|z| > a > 0
6
nan−1 h(n)
z (z − a)2
|z| > a > 0
7
(n − 1)an h(n)
z(2a − z) (z − a)2
|z| > a > 0
8
e−an h(n)
zea zea − 1
|z| > e−a
9
ne−an h(n)
10
n2 e−an h(n)
11
e−an h(n) sin Ωn
12
zea
|z| > e−a
2
(zea − 1)
zea (1 + zea ) 3
(zea − 1)
|z| > e−a
zea sin Ω − 2zea cos Ω + 1
|z| > e−a
e−an h(n) cos Ωn
zea (zea − cos Ω) z 2 e2a − 2zea cos Ω + 1
|z| > e−a
13
h(n) sinh an
z sinh a z 2 − 2z cosh a + 1
|z| > e−a
14
h(n) cosh an
z(z − cosh a) z 2 − 2z cosh a + 1
|z| > e−a
15
h(n)an−1 e−an sin Ωn
zea sin Ω z 2 e2a − 2zaea cos Ω + a2
|z| > e−a
16
h(n)an e−an cos Ωn
zea (zea − a cos Ω) z 2 − 2zaea cos Ω + a2
|z| > e−a
z 2 e2a
Bibliographic References Used in Preparation of Text (See the introduction for an explanation of the letters preceding each bibliographic reference.) AS AD AK BB BE BI BL BR BS BU
BY CA CE CL CO DW
Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications, New York, 1972. Adams, E. P. and Hippisley, R. L., Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922. Appell, P. and Kamp´e de F´eriet, Fonctions hyperg´eometriques et hypersph´eriques, polynomes d’Hermite, Gauthier Villars, Paris, 1926. Beckenbach, E. F. and Bellman, R., Inequalities, 3rd printing. Springer–Verlag, Berlin, 1971. Bertrand, J., Traite de calcul diff´erentiel et de calcul int´egral, vol. 2, Calcul int´egral, int´egrales d´efinies et ind´efinies, Gauthier-Villars, Paris, 1870. Bierens de Haan, D., Nouvelles tables d’int´egrales d´efinies, Amsterdam, 1867. (Reprint) G. E. Stechert & Co., New York, 1939. Bellman, R., Introduction to Matrix Analysis, McGraw Hill, New York, 1960. Bromwich, T. I’A., An Introduction to the Theory of Infinite Series, Macmillan, London, 1908, 2nd edition, 1926.∗ Bellman, R., Stability Theory of Differential Equations, McGraw-Hill, New York, 1953. Buchholz, H., Die konfluente hypergeometrische Funktion mit besonderer Ber¨ ucksichtigung ihrer Anwendungen, Springer–Verlag, Berlin, 1953. Also an English edition: The confluent Hypergeometric Function, Springer–Verlag, Berlin, 1969. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer–Verlag, Berlin, 1954. Carslaw, H. S., Introduction to the Theory of Fourier’s Series and Integrals, Macmillan, London, 1930. Ces`aro, Z., Elementary Class Book of Algebraic Analysis and the Calculation of Infinite Limits, 1st ed. ONTI, Moscow and Leningrad, 1936. Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Wiley (Interscience), New York, 1953. Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1934.
∗ The Bibliographic Reference BR* refers to the 1908 edition of Bromwich T. I.’A., An Introduction to the Theory of Infinite Series; BR refers to the 1926 edition.
1141
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References
DW61 Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961. Efros, A. M. and Danilevskiy, A. M., Operatsionnoye ischisleniye i konturnyye integraly EF (Operational calculus and contour integrals). GNTIU, Khar’kov, 1937. Erd´elyi, A., et al., Higher Transcendental Functions, vols. I, II, and III. McGraw Hill, New York, EH 1953–1955. Erd´ elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. ET Euler, L., Introductio in Analysin Infinitorum, Bousquet, Lausanne, 1748. EU Fikhtengol’ts, G. M., Kurs differentsial’nogo i integral’nogo ischisleniya (Course in differential FI and integral calculus), vols. I, II, and III. Gostekhizdat, Moscow and Leningrad, 1947–1949. Also a German edition: Differential-und Integralrechnung I–III, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986–1987. Gauss, K. F., Werke, Bd. III. G¨ ottingen, 1876. GA Gel’fond, A. O., Ischisleniye konechnykh raznostey (Calculus of finite differences), part I. ONTI, GE Moscow and Leningrad, 1936. Gr¨ obner, W. and Hofreiter, N., Integraltafel, vol. 2, Bestimmte Integrale, Springer, Wien, 1961. GH2 Giunter, N. M. and Kuz’min, R. O. (eds.), Sbornik zadach po vysshey matematike (Collection of GI problems in higher mathematics), vols. I, II, and III. Gostekhizdat, Moscow and Leningrad, 1947. Gantmacher, F. R., Applications of the Theory of Matrices, translation by J. L. Brenner. Wiley GM (Interscience), New York, 1959. Goursat, E. J. B., Cours d’Analyse, vol. I, Gauthier–Villars, Paris, 1923. GO Gr¨ obner, W. et al., Integraltafel, Teil I, Unbestimmte Integrale, Akad. Verlag, Braunschweig, GU 1944. Gr¨ obner, W. and Hofreiter, N., Integraltafel, Teil II, Bestimmte Integrale, Springer–Verlag, Wien GW and Innsbruck, 1958. Hille, E., Lectures on Ordinary Differential Equations, Addison- Wesley, Reading, HI Massachusetts, 1969. Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities, Cambridge University Press, HL London, 2nd ed., 1952. Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University HO Press, London, 1931. Hurewicz, W., Lectures on Ordinary Differential Equations, MIT Press, Cambridge, HU Massachusetts, 1958. Ince, E. L., Ordinary Differential Equations, Dover, New York, 1944. IN Jahnke, E. and Emde, F., Tables of Functions with Formulas and Curves, Dover, New York, JA 1943. JAC Jackson, J. D., Classical Electrodynamics, Wiley, New York, 1975. James, H. M. et al. (eds.), Theory of Servomechanisms, McGraw Hill, New York, 1947. JE Jolley, L., Summation of Series, Chapman and Hall, London, 1925. JO Kellogg, O. D., Foundations of Potential Theory, Dover, New York, 1958. KE Krechmar, V. A., Zadachnik po algebre (Problem book in algebra), 2nd ed. Gostekhizdat, KR Moscow and Leningrad, 1950. Kuzmin, R. O., Besselevy funktsii (Bessel functions). ONTI, Moscow and Leningrad, 1935. KU Laska, W., Sammlung von Formeln der reinen und angewandten Mathematik, Friedrich Viewig LA und Sohn, Braunschweig, 1888–1894.
References
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Legendre, A. M., Exercises calcul int´egral, Paris, 1811. Lindeman, C. E., Examen des nouvelles tables d’int´egrales d´efinies de M. Bierens de Haan, Amsterdam, 1867, Norstedt, Stockholm, 1891. Lobachevskiy, N. I., Poloye sobraniye sochineniy (Complete works), vols. I, III, and V. LO Gostekhizdat, Moscow and Leningrad, 1946–1951. Luke, Y. L., Mathematical Functions and their Approximations, Academic Press, New York, LUKE 1975. Lawden, D. F., Elliptic Functions and Applications, Springer–Verlag, Berlin, 1989. LW McLachlan, N. W., Theory and Application of Mathieu Functions, Oxford University Press, MA London, 1947. Computation by Mathematica. MC McLachlan, N. W. and Humbert, P., Formulaire pour le calcul symbolique, L’Acad. des Sciences ME de Paris, Fasc. 100, 1950. Morse, M. P. and Feshbach, H., Methods of Theoretical Physics, vol. I, McGraw Hill, New York, MF 1953. Marden, M., Geometry of Polynomials, American Mathematical Society, Mathematical Survey MG 3, Providence, Rhode Island, 1966. McLachlan, N. W. et al., Suppl´ement au formulaire pour le calcul symbolique, L’Acad. des MI Sciences de Paris et al., Fasc. 113, 1950. Mirsky L., An Introduction to Linear Algebra, Oxford University Press, London, 1963. ML MM MacMillan, W. D., The Theory of the Potential, Dover, New York, 1958. Magnus, W. and Oberhettinger, F., Formeln und S¨ atze f¨ ur die speziellen Funktionen der MO mathematischen Physik, Springer–Verlag, Berlin, 1948. Magnus, W., Oberhettinger, F. and Soni, R. P., Formulas and Theorems for the Special MS Functions of Mathematical Physics, 3rd ed. Springer–Verlag, Berlin, 1966. MS2 Mathai, A. M. and Saxens, R. K. , Generalized Hypergeometrics Functions With Applications in Statistics and Physical Science, Springer–Verlag, Berlin, 1973. Mitrinovi´ c, D. S., Analytic Inequalities, Springer–Verlag, Berlin, 1970. MT Milne, E. A., Vectorial Mechanics, Methuen, London, 1948. MV Meyer Zur Capellen, W., Integraltafeln, Sammlung unbestimmer Integrale elementarer MZ Funktionen, Springer–Verlag, Berlin, 1950. Natanson, I. P., Konstruktivnaya teoriya funktsiy (Constructive theory of functions). NA Gostekhizdat, Moscow and Leningrad, 1949. Nielsen, N., Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906. NH Noble, B., Applied Linear Algebra, Prentice Hall, Englewood Cliffs, New Jersey, 1969. NO Nielsen, N., Theorie des Integrallogarithmus und verwandter Transcendenten, Teubner, Leipzig, NT 1906. Novoselov, S. I., Obratnyye trigonometricheskiye funktsii, posobive dlya uchiteley (Inverse NV trigonometric functions, textbook for students), 3rd ed. Uchpedgiz, Moscow and Leningrad, 1950. Oberhettinger, F., Tables of Bessel Transforms, Springer–Verlag, New York: 1972. OB PBM Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Gordan and Breach, New York, vols. I (1986), II (1986), III (1990). Peirce, B. O., A Short Table of Integrals, 3rd ed. Ginn, Boston, 1929. PE Sansone, G., Orthogonal Functions (Revised English Edition), Interscience, New York, 1959. SA
LE LI
1144
SI
SN SM ST STR SU SZ TF TI VA VL WA WH ZH ZY
References
Sikorskiy, Yu. S., Elementy teorii ellipticheskikh funktsiy s prilozheniyama k mekhanike (Elements of theory of elliptic functions with applications to mechanics). ONTI, Moscow and Leningrad, 1936. Sneddon, I. N., Fourier Transforms, 1st ed. McGraw Hill, New York, 1951. Smirnov, V. I., Kurs vysshey matematiki (A course of higher mathematics), vol. III, Part 2, 4th ed. Gostekhizdat, Moscow and Leningrad, 1949. Strutt, M. J. O., Lam´esche, Mathieusche und verwandte Funktionen in Physik and Technik, Springer–Verlag, Berlin, 1932. Stratton, J. C., Phys. Rev A, 43(3), pages 1381–1388, 1991. Sneddon, I. N., The Use of Integral Transforms, McGraw Hill, New York, 1972. Szeg¨o, G., Orthogonal Polynomials, Revised Edition, Colloquium Publications XXIII, American Mathematical Society, New York, 1959. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, 2nd ed. Oxford University Press, London, 1948. Timofeyev, A. F. Integrirovaniye funktsiy (Integration of functions), part I. GTTI, Moscow and Leningrad, 1933. Varga, R. S., Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1963. Vladimirov, V. S., Equations of Mathematical Physics, Dekker, New York, 1971. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, London, 1966. Whittaker, E. T. and Watson, G. N., Modern Analysis, 4th ed. Cambridge University Press, London, 1927, part II, 1934. Zhuravskiy, A. M., Spravochnik po ellipticheskim funktsiyam (Reference book on elliptic functions). Izd. Akad. Nauk. U.S.S.R., Moscow and Leningrad, 1941. Zygmund, A., Trigonometrical Series, 2nd ed. Chelsea, New York, 1952.
Classified Supplementary References (Prepared by Alan Jeffrey for the English language edition.)
General reference books 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Bromwich, T. I’A., An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926 (Reprinted 1942). Carlitz, L., “Generating Functions”, 1969, Fibonacci Quarterly, 7 (4): 359–393. Copson, E. T., An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, London, 1935. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Interscience Publishers, New York, 1953. Davis, H. T., Summation of Series, Trinity University Press, San Antonio, Texas, 1962. Erd´elyi, A. et al. Higher Transcendental Functions, vols. I to III, McGraw Hill, New York 1953–1955. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. Fletcher, A., Miller, J. C. P., and Rosenhead, L., An Index of Mathematical Tables, 2nd ed., Scientific Computing Service, London, 1962. Gr¨ obner, W. and Hofreiter, N., Integraltafel, I, II. Springer–Verlag, Wien and Innsbruck, 1949. Hardy, G. H., Littlewood, J. E., and P´ olya, G., Inequalities, 2nd ed., Cambridge University Press, London, 1952. Hartley, H. O. and Greenwood, J. A., Guide to Tables in Mathematical Statistics, Princeton University Press, Princeton, New Jersey, 1962. Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics, Cambridge University Press, London, 1956. Jolley, L. B. W., Summation of Series, Dover Publications, New York, 1962. Knopp, K., Theory and Application of Infinite Series, Blackie, London, 1946, Hafner, New York, 1948. Lebedev, N. N., Special Functions and their Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1965. Magnus, W. and Oberhettinger, F., Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea, New York, 1949. McBride, E. B., Obtaining Generating Functions, Springer–Verlag, Berlin, 1971. National Bureau of Standards, Handbook of Mathematical Functions, U.S. Government Printing Office, Washington, D.C., 1964. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vols. 1–5, Gordon and Breach, New York, 1986–1992. Truesdell, C. A Unified Theory of Special Functions, Princeton University Press, Princeton, New Jersey, 1948.
1145
1146
21. 22.
Supplemental References
Vein, R. and Dale, P., Determinants and Their Applications in Mathematical Physics, Springer–Verlag, New York, 1999. Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, 4th ed., Cambridge University Press, London, 1940.
Asymptotic expansions 1. 2. 3. 4. 5. 6. 7.
De Bruijn, N. G., Asymptotic Methods in Analysis, North-Holland Publishing Co., Amsterdam, 1958. Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, 3rd ed., Springer, New York, 1971. Copson, E. T., Asymptotic Expansions, Cambridge University Press, London, 1965. Erd´elyi, A., Asymptotic Expansions, Dover Publications, New York, 1956. Ford, W. B., Studies on Divergent Series and Summability, Macmillan, New York, 1916. Hardy, G. H., Divergent Series, Clarendon Press, Oxford, 1949. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.
Bessel functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Bickley, W. G., Bessel Functions and Formulae, Cambridge University Press, London, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vols. I and II. McGraw Hill, New York, 1954. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II. McGraw Hill, New York, 1954. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed., Macmillan, 1922. McLachlan, N. W., Bessel Functions for Engineers, 2nd ed., Oxford University Press, London, 1955. Luke, Y. L., Integrals of Bessel Functions, McGraw Hill, New York, 1962. Petiau, G., La th´eorie des fonctions de Bessel, Centre National de la Recherche Scientifique, Paris, 1955. Relton, F. E., Applied Bessel Functions, Blackie, London, 1946. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958. Wheelon, A. D., Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day, San Francisco, 1968.
Complex analysis 1. 2. 3. 4. 5. 6. 7.
Ahlfors, L. V., Complex Analysis, 3rd ed., McGraw Hill, New York, 1979. Ahlfors, L. V. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1971. Bieberbach, L., Conformal Mapping, Chelsea, New York, 1964. Henrici, P., Applied and Computational Complex Analysis, 3 vols, Wiley, New York, 1988, 1991, 1977. Hille, E., Analytic Function Theory, 2 vols. 2nd ed., Chelsea, New York, 1990, 1987. Kober, H., Dictionary of Conformal Representations, Dover Publications, New York, 1952. Titchmarsh, E. C., The Theory of Functions, 2nd ed., Oxford University Press, London, 1939. (Reprinted 1975).
Supplemental References
1147
Error function and Fresnel integrals 1. 2. 3. 4. 5.
Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Erd´elyi, A. et al., Tables of Integral Transforms, vol. I, McGraw Hill, New York, 1954. Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press, London, 1960. Tricomi, F. G., Funzioni ipergeometriche confluenti, Edizioni Cremonese, Turan, Italy, 1954. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.
Exponential integrals, gamma function and related functions 1. 2. 3. 4. 5. 6. 7. 8. 9.
Artin, E., The Gamma Function, Holt, Rinehart, and Winston, New York, 1964. Busbridge, I. W., The Mathematics of Radiative Transfer, Cambridge University Press, London, 1960. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Erd´elyi, A. et al., Tables of Integral Transforms, vols. I and II, McGraw Hill, New York, 1954. Hastings, Jr., C., Approximations for Digital Computers, Princeton University Press, Princeton, New Jersey, 1955. Kourganoff, V., Basic Methods in Transfer Problems, Oxford University Press, London, 1952. L¨ osch, F. and Schoblik, F., Die Fakult¨ at (Gammafunktion) und verwandte Funktionen, Teubner, Leipzig, 1951. Nielsen, N., Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906. Oberhettinger, F., Tabellen zur Fourier Transformation, Springer–Verlag, Berlin, 1957.
Hypergeometric and confluent hypergeometric functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Appell, P., Sur les Fonctions Hyperg´eometriques de Plusieures Variables, Gauthier-Villars, Paris, 1926. Bailey, W. N., Generalized Hypergeometric Functions, Cambridge University Press, London, 1935. Buchholz, H., Die konfluente hypergeometrische Funktion, Springer–Verlag, Berlin, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. I, McGraw Hill, New York, 1953. Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics, Cambridge University Press, London, 1956. Klein, F., Vorlesungen u ¨ber die hypergeometrische Funktion, Springer–Verlag, Berlin, 1933. N¨ orlund, N. E., Sur les Fonctions Hyperg´eometriques d’Ordre Superior, North–Holland, Copenhagen, 1956. Slater, L. J., Confluent Hypergeometric Functions, Cambridge University Press, London, 1960. Slater, L. J. Generalized Hypergeometric Functions, Cambridge University Press, London, 1966. Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd ed., National Bureau of Standards, Washington, D.C., 1952. Swanson, C. A. and Erd´elyi, A., Asymptotic Forms of Confluent Hypergeometric Functions, Memoir 25, American Mathematical Society, Providence, Rhode Island, 1957. Tricomi, F. G., Lezioni sulla funzioni ipergeometriche confluenti, Gheroni, Torino, 1952.
Integral transforms 1.
Bochner, S., Vorlesungen u ¨ber Fouriersche Integrale, Akad. Verlag, Leipzig, 1932. Reprint Chelsea, New York, 1948.
1148
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
Supplemental References
Bochner, S. and Chandrasekharan, K., Fourier Transforms, Princeton University Press, Princeton, New Jersey, 1949. Campbell, G. and Foster, R., Fourier Integrals for Practical Applications, Van Nostrand, New York, 1948. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Oxford University Press, London, 1948. Doetsch, G., Theorie und Anwendung der Laplace-Transformation, Springer–Verlag, Berlin, 1937. (Reprinted by Dover Publications, New York, 1943) Doetsch, G., Theory and Application of the Laplace-Transform, Chelsea, New York, 1965. Doetsch, G., Handbuch der Physik, Mathematische Methoden II, 1st ed., Springer–Verlag, Berlin, 1955. Doetsch, G., Guide to the Applications of the Laplace and Z-Transforms, 2nd ed., Van Nostrand-Reinhold, London, 1971. Doetsch, G., Handbuch der Laplace-Transformation, Vols. I–IV, Birkh¨ auser Verlag, Basel, 1950–56. Doetsch, G., Kniess, H., and Voelker, D., Tabellen zur Laplace- Transformation, Springer–Verlag, Berlin, 1947. Erd´elyi, A., Operational Calculus and Generalized Functions, Holt, Rinehart and Winston, New York, 1962. Exton, H., Multiple Hypergeometric Functions and Applications, Horwood, Chichester, 1976. Exton, H., Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs, Horwood, Chichester, 1978. Hirschmann, J. J. and Widder, D. V., The Convolution Transformation, Princeton University Press, Princeton, New Jersey, 1955. Marichev, O. I., Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood Ltd., Chichester (1982). Oberhettinger, F., Tabellen zur Fourier Transformation, Springer–Verlag, Berlin (1957). Oberhettinger, F., Tables of Bessel Transforms, Springer–Verlag, New York (1972). Oberhettinger, F., Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973. Oberhettinger, F., Fourier Transforms of Distributions and Their Inverses, Academic Press, New York, 1973. Oberhettinger, F., Tables of Mellin Transforms, Springer–Verlag, Berlin, 1974. Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer–Verlag, Berlin, 1973. Oberhettinger, F. and Higgins, T. P., Tables of Lebedev, Mehler and Generalized Mehler Transforms, Math. Note No. 246, Boeing Scientific Research Laboratories, Seattle, Wash., 1961. Roberts, G. E. and Kaufman, H., Table of Laplace Transforms, McAinsh, Toronto, 1966. Sneddon, I. N., Fourier Transforms, McGraw Hill, New York, 1951. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals, Oxford University Press, London, 1937. Van der Pol, B. and Bremmer, H., Operational Calculus Based on the Two Sided Laplace Transformation, Cambridge University Press, London, 1950. Widder, D. V., The Laplace Transform, Princeton University Press, Princeton, New Jersey, 1941. Wiener, N., The Fourier Integral and Certain of its Applications, Dover Publications, New York, 1951.
Jacobian and Weierstrass elliptic functions and related functions 1. 2. 3. 4.
Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1953. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Physicists, Springer– Verlag, Berlin, 1954. Graeser, E., Einf¨ uhrung in die Theorie der Elliptischen Funktionen und deren Anwendungen, Oldenbourg, Munich, 1950. Hancock, H., Lectures on the Theory of Elliptic Functions, vol. I, Dover Publications, New York, 1958.
Supplemental References
5. 6. 7. 8. 9.
1149
Neville, E. H., Jacobian Elliptic Functions, Oxford University Press, London, 1944 (2nd ed. 1951). Oberhettinger, F. and Magnus, W., Anwendungen der Elliptischen Funktionen in Physik und Technik, Springer–Verlag, Berlin, 1949. Roberts, W. R. W., Elliptic and Hyperelliptic Integrals and Allied Theory, Cambridge University Press, London, 1938. Tannery, J. and Molk, J., El´ements de la Th´eorie des Fonctions Elliptiques, 4 volumes. Gauthier-Villars, Paris, 1893–1902. Tricomi, F. G., Elliptische Funktionen, Akad. Verlag, Leipzig, 1948.
Legendre and related functions 1. 2. 3. 4. 5. 6. 7.
Erd´elyi, A. et al., Higher Transcendental Functions, vol. I, McGraw Hill, New York, 1953. Helfenstein, H., Ueber eine Spezielle Lam´esche Differentialgleichung, Brunner and Bodmer, Zurich, 1950 (Bibliography). Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, London, 1931. Reprinted by Chelsea, New York, 1955. Lense, J., Kugelfunktionen, Geest and Portig, Leipzig, 1950. MacRobert, T. M., Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications, Methuen, England, 1927. (Revised ed. 1947; reprinted Dover Publications, New York, 1948). Snow, C., The Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd ed., National Bureau of Standards, Washington, D.C., 1952. Stratton, J. A., Morse, P. M., Chu, L. J. and Hunter, R. A., Elliptic Cylinder and Spheroidal Wave Functions Including Tables of Separation Constants and Coefficients, Wiley, New York, 1941.
Mathieu functions 1. 2. 3. 4.
Erd´elyi, A., Higher Transcendental Functions, vol. III, McGraw Hill, New York, 1955. McLachlan, N. W., Theory and Application of Mathieu Functions, Oxford University Press, London, 1947. Meixner, J. and Sch¨ afke, F. W., Mathieusche Funktionen und Sph¨ aroidfunktionen mit Anwendungen auf Physikalische und Technische Probleme, Springer–Verlag, Heidelberg, 1954. Strutt, M. J. O., Lam´esche, Mathieusche und verwandte Funktionen in Physik und Technik, Ergeb. Math. Grenzgeb. 1, 199–323 (1932). Reprint Edwards Bros., Ann Arbor, Michigan, 1944.
Orthogonal polynomials and functions 1. 2. 3. 4. 5. 6. 7.
Bibliography on Orthogonal Polynomials, Bulletin of National Research Council No. 103, Washington, D.C., 1940. Courant, R. and Hilbert, D., Methods of Mathematical Physics, vol. I, Interscience, New York, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954. Kaczmarz, St. and Steinhaus, H., Theorie der Orthogonalreihen, Chelsea, New York, 1951. Lorentz, G. G., Bernstein Polynomials, University of Toronto Press, Toronto, 1953. Sansone, G., Orthogonal Functions, Interscience, New York, 1959. Shohat, J. A. and Tamarkin, J. D., The Problem of Moments, American Mathematical Society, Providence, Rhode Island, 1943.
1150
Supplemental References
8.
Szeg¨ o, G., Orthogonal Polynomials, American Mathematical Society Colloquim Pub. No. 23, Providence, Rhode Island, 1959. Titchmarsh, E. C., Eigenfunction Expansions Associated with Second Order Differential Equations, Oxford University Press, London, part I (1946), part II (1958). Tricomi, F. G., Vorlesungen u ¨ber Orthogonalreihen, Springer–Verlag, Berlin, 1955.
9. 10.
Parabolic cylinder functions 1. 2.
Buchholz, H., Die konfluente hypergeometrische Funktion, Springer–Verlag, Berlin, 1953. Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954.
Probability function 1. 2. 3.
Cramer, H., Mathematical Methods of Statistics, Princeton University Press, Princeton, New Jersey, 1951. Erd´elyi, A. et al., Higher Transcendental Functions, vols. I, II, and III. McGraw Hill, New York, 1953– 1955. Kendall, M. G. and Stuart, A., The Advanced Theory of Statistics, vol. I: Distribution Theory, Griffin, London, 1958.
Riemann zeta function 1. 2.
Titchmarsh, E. C., The Zeta Function of Riemann, Cambridge University Press, London, 1930. Titchmarsh, E. C., The Theory of the Riemann Zeta Function, Oxford University Press, London, 1951.
Struve functions 1. 2. 3.
Erd´elyi, A. et al., Higher Transcendental Functions, vol. II, McGraw Hill, New York, 1954. Gray, A., Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions and Their Applications to Physics, 2nd ed., Macmillan, London, 1922. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1958.
Index of Functions and Constants This index shows the occurrence of functions and constants used in the expressions within the text. The numbers refer to pages on which the function or constant appears.
Symbols
arccot function . . . . . . xxxi, xxxii, 51, 56–58, 64, 242, 244, 245, 263, 274, 279, 325, 326, 499, 556, 561, 599, 601–607, 625, 767, 892, 1115, 1133 arccoth function . . . . . . . . . xxxi, xxxii, 56, 60, 62, 75, 131–134, 172, 177, 178, 241, 644, 647 arcosech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 arcsec function . . . . . . . . . . . . . . . . . 61, 66, 99, 242, 244 arcsech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 arcsin function. . . . . . . . . . . . . . .xxxi, xxxii, 27, 56–61, 64, 66, 94, 97, 99, 116, 124–126, 133–138, 173, 179–182, 186–193, 195, 196, 198, 202–213, 225, 241–245, 254, 263, 265, 272, 275, 279, 297, 307, 382, 558, 563, 566–568, 583, 588, 589, 591, 600, 601, 604, 605, 607, 621, 622, 624, 625, 631, 632, 637, 638, 662, 668, 700–702, 713, 717, 718, 727, 728, 743, 744, 748, 755, 767, 768, 793, 815, 860, 914, 989, 1007 arcsinh function . . xxxi, xxxii, 54, 56, 60–62, 64, 94, 97, 126, 133, 135, 139, 240, 241, 371, 382, 386, 448, 588, 624, 637, 638, 1007 arctan function . . . . . . . . . . xxxi, xxxii, 27, 30, 49, 51, 52, 55–61, 63–67, 71–77, 79, 83–85, 87, 90, 97, 103, 104, 106, 114, 116, 117, 119, 126, 128–133, 147, 148, 171–175, 177, 178, 190, 205, 239, 240, 242–245, 254, 263, 272, 274, 279, 294, 295, 307, 317, 324, 329, 346, 363, 372, 373, 381, 393, 409, 453, 493–501, 507, 509, 516–521, 524, 556, 557, 560, 563–565, 593, 599–607, 612, 622–624, 631, 632, 637, 639, 640, 643, 644, 646–649, 748, 763, 860, 884, 885, 890–893, 898, 923, 1007, 1036, 1113, 1125, 1128, 1132 arctanh function . . . . . . . . . . . . . xxxi, xxxii, 56, 60–62, 64, 75, 79, 97, 125–128, 131, 132, 134, 172, 177, 178, 241, 621–623, 977
! and !! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see factorials (m) Sn . . . . . . . . . . . . see Stirling numbers, second kind (m) Zn . . . . . . . . . . . . . . . . . . . . . . . . see Bessel functions, Z ∇ . . . . . . . . . . . . . . . . . xliv, 767, 1050–1053, 1055–1057 β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see beta function δ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see delta function δij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Kronecker delta γ and Γ . . . . . . . . . . . . . . . . . . . . . . see gamma functions λ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1043 μ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1043 ν function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1043 Φ . . . . . . . . . . . see Lerch function and hypergeometric functions, confluent Ψ . . . . . . . . . . . see Euler function and hypergeometric functions, confluent Θ function . . . . . . . . . . . . . . . see Jacobi theta function ℘(x) . . . . . . . . . . . . . . . . . . . . . . see Weierstrass function ξ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 1040 || · || . . . . . . . 1081–1083, 1085, 1086, 1095, 1096, 1120 || · ||1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081–1083 || · ||2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081–1083 || · ||∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081, 1083
A Airy function (Ai) . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii am function . . . . . . . . . . . . . . . . . . . xxxix, 625, 866, 867 Anger function (J) . . . . . xli, 339, 352, 371, 384, 421, 423, 444–446, 670, 671, 946, 948, 949, 992 arccos function . . xxxi, xxxii, 23, 56–60, 64, 99, 135, 139, 173, 179–183, 210, 211, 241–244, 263, 272, 279, 293–296, 307, 312, 313, 318, 393, 454, 511, 558, 562, 589, 600, 601, 607, 610, 624, 695, 730, 742, 767, 768, 861, 890, 936, 993, 994 arccosec function . . . . . . . . . . . . . . . . . . . . . 242, 244, 728 arccosh function . . . . . . xxxi, xxxii, 56, 60–62, 64, 97, 126, 133, 135, 137, 138, 241, 382, 386, 511, 532, 621, 624, 729, 768
1151
1152
INDEX OF FUNCTIONS AND CONSTANTS
associated Legendre functions first kind (P ). . . .xli, 326, 327, 333, 336, 374, 406, 407, 486, 660, 661, 665, 666, 686, 699, 703, 705, 706, 727, 755, 760, 761, 767–789, 792, 793, 797, 806–808, 810, 823, 831, 839, 840, 848, 958–972, 974, 975, 980–983, 992 second kind (Q) . . . . . . xli, 333, 336, 374, 383, 407, 511, 661, 662, 666, 685, 686, 700, 702, 703, 705, 727, 769–781, 783–785, 791, 795, 823, 831, 839, 840, 958–973, 981
B Bn (x) . . . . . . . . . . . . . . . . . . . see Bernoulli polynomials B(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Beta function Bateman function (k) . . . . . . . . . . . . . . . . xli, 349, 1023 bei(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions ber(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions Bernouli number (Bn ) . . xxxii, xxxiii, xxxix, 1–3, 8, 26, 42, 43, 46, 55, 145, 146, 148, 221–224, 353, 356, 376, 379–382, 387, 472, 550–552, 554, 560, 567, 574, 580, 581, 587, 589, 591, 764–766, 899, 906, 936, 1038–1045 Bernouli number (Bn∗ ) . . . . . . . . . . . . . . . . . . . . xxxiii, 62 Bernoulli polynomial (Bn (x)) . . . xxxii, xxxiii, xxxix, 46, 1037, 1041, 1042, 1045 Bessel functions In (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, xli, 13, 320, 339, 340, 345, 347, 350, 351, 368, 382, 385, 419, 435, 441, 444, 445, 470, 477–480, 491, 494, 496, 507, 513–515, 524, 595, 605, 616, 617, 660, 661, 663–681, 684–687, 689, 691, 692, 695–699, 702–716, 719, 720, 722–725, 727, 729, 730, 735, 736, 738, 741, 743, 745–747, 751–760, 762, 779–781, 783–787, 789, 794, 797, 800, 820, 832, 833, 838, 846, 847, 901, 911, 916, 917, 919, 920, 925–933, 943, 954, 1002, 1027, 1028, 1116, 1117, 1123, 1125, 1128 Jn (x) . . . . . . . . . . . xxxvii, xxxviii, xli, 13, 339, 350, 352, 371, 384, 385, 417–421, 423, 435, 440–443, 445, 446, 477–480, 482, 483, 491, 492, 507, 514, 515, 522, 524, 525, 578, 629, 642, 653, 659–694, 696–753, 756–759, 761–763, 767, 768, 777, 779, 780, 782–787, 792–794, 797–799, 802, 803, 808, 811, 812, 818–820, 830–838, 841, 845–848, 854, 855, 900, 910–914, 916, 918–931, 933–950, 954– 957, 963, 964, 972, 992, 1000, 1002–1004, 1017, 1023–1025, 1028, 1034, 1116, 1117, 1124–1129, 1132, 1133
Bessel functions (continued) Kn (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii, xli, 2, 337, 339, 345–348, 350–353, 364–368, 370, 371, 384, 385, 417, 419, 435, 442, 444, 445, 477–482, 490, 491, 504, 505, 507, 514, 515, 529, 573, 575, 576, 578, 595, 638, 645, 648, 653, 654, 657, 660– 682, 684–696, 698–700, 702–716, 718–724, 726, 727, 729–732, 735, 736, 738, 740, 742, 745–753, 756–759, 761, 768, 776–787, 789, 794, 800, 803, 811, 814, 817–820, 828, 832–834, 837, 838, 841, 845, 846, 848, 854, 855, 900, 911, 917–920, 923, 925–933, 939, 942, 945, 955, 957, 1027, 1028, 1035, 1123–1126, 1129, 1132, 1133 Nn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii, 910 Yn (x) . . . . . . . . . . . . . xxxvii, xlii, 338, 339, 345, 346, 351–353, 371, 384, 385, 419, 435, 440, 442, 443, 445, 446, 477–480, 482, 483, 492, 507, 514, 515, 573, 578, 647, 654, 659–664, 666–682, 684–693, 695–700, 705–709, 711, 714–720, 722–724, 726, 728, 729, 732–736, 738, 740–742, 745, 747–752, 756–759, 761, 767, 768, 777, 779, 782, 783, 785, 787, 793, 794, 799, 817–819, 832, 833, 835–837, 847, 848, 854, 855, 910, 911, 914, 918–920, 922, 923, 925–931, 933, 937–939, 941–943, 945, 946, 949, 954–957, 975, 1025, 1034, 1124, 1126, 1132 Zn (x). . . . . . xlii, 483, 629, 630, 767, 910, 911, 926, 931–933, 937, 940, 941, 975 Zn (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii, 629, 630 Hankel . . . . . . . . . . . . . . . . . . . . . . see Hankel function beta function (β) . . . . . . . . . . . 319, 322, 324, 325, 334, 371, 375, 383, 395, 396, 403, 432, 471, 553, 558, 562–564, 573, 586, 602, 904, 906, 907 Beta function (B) . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 6, 129, 175, 315–318, 320, 322–330, 332, 333, 335, 338, 347–349, 351, 359, 360, 364, 368, 370, 372, 374, 375, 382, 383, 395–397, 399–402, 407, 408, 411–413, 440, 442, 444, 460, 469, 472, 485, 486, 490, 512, 539–543, 548, 553, 559, 585, 705, 749, 754, 760, 801, 810, 813, 814, 816, 821, 894, 895, 908–910, 991, 1005, 1023, 1025 Bi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxviii bilateral z transform . . . . . . . . 1135–1137, 1139, 1159 binomial coefficients . . . . . . . . . . . . . . . . . . . . xxxiii, xliii, 1–6, 11, 12, 15, 22, 23, 25, 31, 33, 46, 84, 86, 88, 89, 100–103, 106, 110, 111, 114, 115, 119, 120, 140, 143, 148, 153, 157, 161, 173, 215, 220, 221, 223, 228, 232–238, 241, 316, 326, 329, 354, 357, 361, 362, 386, 393, 394, 397–402, 416, 431, 437, 459–461, 466, 469, 470, 478, 488, 498, 499, 504, 505, 545, 546, 548, 549, 552, 612, 808, 910, 934, 993, 994, 996, 998, 1003, 1023, 1030, 1040, 1041, 1044, 1046, 1047
INDEX OF FUNCTIONS AND CONSTANTS
C Cn (x) . . . . . . . . . . . . . . . . . see Gegenbauer polynomials C(x) . . see Fresnel sine integral and Young function Catalan constant (G) . . . . . . . . . xxxii, xl, 9, 375, 380, 433, 434, 448, 449, 452, 453, 470–472, 530–534, 536–538, 556, 558, 560, 563, 564, 580, 600–603, 632, 633, 1046 cd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Ce function . . . . . . . . . . . xxxviii, xl, 763–766, 953, 954 ce function . . . . . . . . . . . . xxxviii, xl, 763–767, 951–957 Chebyshev polynomials first kind (Tn (x)) . . . . . . . . . xli, 448, 667, 718, 790, 800–803, 983, 988, 993–996, 999, 1131 second kind (Un (x)) . . . . . . . . . xxxvii, xli, 800–802, 994–996 chi function . . . . . . . xxxvi, xl, 142–144, 644, 645, 886 ci function . . . . xxxv, xl, 219–221, 340–344, 423–426, 436, 437, 447, 495, 505, 506, 528, 529, 571, 572, 578, 581, 594, 595, 599, 605, 628, 629, 638–645, 647, 656, 658, 748, 762, 886, 887, 1115 Cin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi cn function . . . . . . . . . . xxxiii, xxxiv, xl, 623–626, 714, 866–873, 875, 879, 880 complex conjugate . . . . . . . . . . . . . . xliii, 293, 341, 342, 421, 422, 424–426, 511, 528, 529, 927, 931, 933, 1060, 1061, 1070, 1071, 1082, 1087, 1136, 1139 confluent hypergeometric functions . . . . . . . . . . . . . see hypergeometric functions, confluent constants Catalan . . . . . . . . . . . . . . . . . . . . see Catalan constant Euler . . . . . . . . . . . . . . . . . . . . . . . . . see Euler constant cos function . . . . . . . . . . xxix, xxxvi, xxxviii, 4, 13, 19, 20, 26–39, 41–52, 54–56, 64, 74–76, 78, 79, 126, 147, 151–237, 249, 250, 253, 254, 317, 318, 320, 322, 323, 326–329, 331–333, 338–345, 353, 358, 372, 373, 377–383, 385, 388–534, 537, 540–545, 550, 551, 561, 563, 565, 567–573, 576, 578, 579, 581–601, 604–608, 610–612, 616, 621–623, 628, 629, 631, 632, 634–644, 647–650, 652, 655, 656, 658, 659, 662–664, 667, 669, 671, 673–675, 677– 681, 686, 688, 689, 691, 692, 695, 703–707, 709, 711, 713, 715, 717–748, 750–752, 754, 756–770, 779–782, 784, 785, 787–789, 792–794, 797–800, 802, 806, 808, 811, 812, 815, 817, 818, 823, 825, 829–831, 833, 836, 837, 844, 845, 847–849, 854, 862–869, 877–880, 882–886, 888–894, 896, 898– 900, 904, 906–910, 912–918, 920, 922, 924, 925, 928, 930, 933–943, 945–951, 953, 954, 958–964, 966–981, 984–993, 997, 998, 1000, 1003, 1005– 1007, 1023, 1025, 1026, 1029, 1030, 1037, 1038, 1041, 1042, 1044, 1053, 1057, 1060, 1061, 1066,
1153
1067, 1090, 1110–1113, 1115, 1117, 1119–1122, 1124–1128, 1131, 1132, 1137, 1140 cosec function . . xxvii, xxix, 36–39, 43, 44, 49, 50, 64, 113–115, 126, 155, 156, 160, 225, 254, 315–323, 325, 327–332, 334, 335, 339, 349, 352, 354, 355, 358, 373, 381–383, 387, 388, 400, 401, 403–408, 410–414, 421, 422, 434, 437–439, 446, 453, 454, 471, 479, 480, 484, 496, 508, 509, 529, 540, 541, 547, 551, 565, 586, 588, 593, 600, 602, 604, 658, 659, 663, 664, 669, 670, 677–682, 689–692, 695, 697, 700, 713, 715, 717, 718, 723, 724, 755, 760, 761, 780, 787, 788, 844, 853, 869, 900, 921, 927, 929, 930, 932, 964, 967, 1131, 1132 cosech function . . . . . . 27, 43, 113–115, 126, 387, 501, 513, 634, 715, 751, 1111, 1124, 1133 cosh function . . . . . . . . . . . . . . xxviii, xxxvi, 27–36, 38, 42, 43, 45, 47, 48, 50–52, 64, 110–151, 231–237, 251, 323, 338, 339, 371–390, 407, 419, 425, 432, 433, 439, 448, 451, 452, 454, 455, 468, 482, 484, 485, 502, 504, 509–527, 568, 570, 573, 578–581, 595, 596, 605, 610, 611, 621, 622, 634, 643–645, 648, 686, 702, 705, 710, 713–717, 722, 723, 729, 735, 747, 750–752, 755, 760, 763–767, 778, 781, 787–789, 791, 792, 802, 806, 843, 844, 886, 888, 896, 906, 908, 909, 912–917, 921, 922, 952–958, 960–963, 967–969, 973, 977, 980–982, 998, 1006, 1043, 1112–1115, 1119, 1128, 1129, 1140 cosine integral (Ci) . . . . xxxv, xxxvi, 886, 930, 1115, 1122, 1126 cot function . . xxvii, xxviii, 28, 36, 37, 39, 42, 44, 46, 49, 52, 56, 64, 147, 157–161, 168, 174, 176–178, 185, 188, 189, 194, 195, 204–207, 213, 222–225, 229, 230, 274, 318–323, 330, 331, 334, 354, 355, 358, 372, 379, 381–384, 388, 395, 396, 400, 401, 403–406, 411–414, 422, 434, 448, 454, 455, 472, 484, 485, 492, 493, 496, 506, 509, 511, 532–534, 540, 542, 543, 546, 558, 559, 565, 567, 568, 587, 588, 590, 594, 595, 604, 606, 611, 623, 631, 632, 637, 663, 664, 669, 670, 676, 678–681, 690–692, 697, 700, 717, 723, 753, 754, 849, 867, 868, 876, 882, 903–905, 912, 914, 915, 922, 927, 929, 930, 932, 954, 967–969, 971, 979, 989, 1013, 1089, 1090, 1120 coth function . . . . . . . 28, 39, 40, 42, 44, 64, 110, 116, 118–120, 124, 130–134, 138, 145–148, 381, 384, 386–388, 485, 489, 502, 509–513, 520, 523, 580, 582, 595, 622, 634, 715, 716, 876, 921, 922, 956, 957, 967, 981, 1029, 1110, 1124 cs function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii curl . . . . . . . . . . . . . . . . . . . . . . . . . 1050–1053, 1057, 1058 cylinder function . . . . see parabolic cylinder function
1154
INDEX OF FUNCTIONS AND CONSTANTS
D dc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263–265 delta function (δ(x)) . . . . . . . . . . 661, 1115, 1118–1120 determinant . . . . . 1070, 1075–1077, 1084–1086, 1096, 1100 dilogarithm (L2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 div . . . . . . . . . . . . . . 1050, 1051, 1053, 1055, 1057, 1058 dn function . . . . . . . . . xxxiii, xxxiv, xl, 623–626, 714, 866–873, 875, 879, 880 double factorials . . . . . . . . . . . . . . . see factorial, double ds function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii
E En (x) . . . . . . . . . . . . . . . . . . . . . . . see Euler polynomials E(x) . . . . . . . . . . . . . . . . . . . . . . see MacRobert function elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 860 complete . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 860, 861 E . . . . . . . . . . . . . . . . xxxi, xl, 60, 135–139, 179–183, 185–195, 197, 198, 200, 202–214, 225, 255–262, 264–274, 280–283, 285–296, 300–315, 410, 604, 606, 621–623, 625, 632, 672, 777, 814, 852, 853, 855, 857, 860, 862–864, 880 complete . . . . . . . . . . xxxiv, xl, 313, 394, 408–410, 472–475, 562, 592, 593, 596–600, 619–622, 632, 633, 671, 696, 704, 714, 729, 860–865, 869, 880, 990 F . . . . . . . xxxi, 60, 61, 134–139, 179–183, 186–195, 197–200, 202–214, 225, 250, 251, 254–315, 407, 408, 410, 411, 470, 486, 563, 568, 602, 604, 606, 621–623, 631, 632, 654, 660, 661, 683, 684, 687, 699, 700, 704, 707, 730, 731, 733, 734, 758, 804, 809–820, 822, 823, 843, 849, 856, 860–864, 889, 918, 959, 963, 964, 968, 970, 971, 974–976, 979, 981, 982, 984, 986, 991, 994 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 complete . . . . . . . . . . . . . . xxxiv, xli, 274, 275, 313, 394, 408–410, 472–475, 538, 539, 562, 567, 585, 588, 592, 593, 596–600, 611, 612, 619, 620, 622, 631–633, 696, 704, 713, 714, 719, 720, 729, 756, 788, 860–870, 875, 879–881, 990 Π . . . . . . . xxxi, xxxiv, xxxix, 51, 52, 135, 137, 138, 180, 181, 183, 197, 198, 200–202, 204, 205, 210, 212–214, 255, 262, 263, 265, 276–279, 284, 285, 293, 295–300, 605, 620, 623, 632, 860, 880 erf . . . . . . . . . . . xxxvi, xl, 107–109, 336, 365, 635, 646, 887–889, 1115, 1116 erfc . . . . . . . xxxvi, xl, 887, 888, 890, 891, 1108, 1110, 1115, 1116 error functions . . . . . . . . . . . . . . . . . . . . . see erf and erfc
Euler constant (C ) . . . . xxxii, xxxv, xxxvi, xl, 3, 15, 321, 323, 330–332, 334, 335, 359, 361, 362, 364, 367, 369–371, 411, 412, 422, 447, 476, 478, 483, 484, 501, 534, 535, 538, 541, 553–555, 558, 559, 570–574, 578–581, 585, 586, 588, 593, 594, 599, 605, 628, 639, 644, 656, 658, 747, 748, 883, 884, 886, 894–896, 898, 903–906, 911, 919, 937–939, 944, 1037, 1038, 1046, 1125 Euler function (ψ) . . . . . . . . . . . . xxxix, 318, 321, 323, 330–332, 334, 335, 356, 359, 360, 364, 369, 382, 387, 388, 400, 403, 405, 411–414, 466, 486, 490, 501, 509, 510, 523, 535–538, 540–543, 553–555, 558, 559, 562, 570–574, 576–579, 585, 586, 588, 594, 595, 607, 617, 658, 659, 747, 769, 770, 820, 842, 902–907, 911, 919, 929, 969, 1011–1013 Euler number (En ) . . . . . . . . . . . . . . . . . . . . xxxii, xxxiii, xl, 8, 43, 145, 146, 221, 223, 376, 379, 380, 533, 550, 580, 1043–1045 Euler polynomial (En (x)) . . xxxii, xxxiii, 1044, 1045 exponential function (exp) . . . . . . . . . . . . . . . . . . . . . . 27, 108, 109, 143, 144, 215, 335–340, 345–349, 352, 364–371, 383–385, 390, 426, 428–430, 439, 440, 480, 482, 484–486, 488, 489, 492–498, 501–509, 513, 516, 521, 523, 524, 526, 574–576, 581, 617, 639, 645–649, 651, 652, 655–658, 693, 697–699, 706–710, 712, 713, 722, 723, 748–753, 759, 760, 768, 778, 781, 785, 791, 805, 810, 811, 815, 826, 828, 829, 831, 834, 837, 841–844, 848–850, 855, 876, 877, 879, 880, 886–888, 890–893, 895, 896, 913, 915–917, 920–923, 928, 933, 935, 967, 997, 1002, 1024–1026, 1029, 1030, 1066, 1090, 1094, 1096, 1098, 1099, 1105, 1110, 1119, 1123, 1124, 1128 exponential integral (En (x)) . . . . . . . . . . . . . . . . . . xxxv exponential integral (Ei(x)) . . . . . . . . . . . . . . . . . . xxxv, xl, 107, 109, 143, 144, 150, 151, 338, 340–344, 361, 370, 375, 386, 421, 422, 424–426, 432, 461, 468, 483, 484, 492, 495, 527–530, 535, 553, 555, 571–573, 577, 578, 594, 595, 605–607, 627, 628, 638–647, 649, 656, 658, 748, 883–887, 900, 902, 931, 1115, 1123, 1125
F
F . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Fourier transform F (x) . . . . . . . . . . . . . . . . . . see hypergeometric function
INDEX OF FUNCTIONS AND CONSTANTS
factorial ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii–xxxiv, xxxvii, xliii, 2–5, 8, 12, 13, 18, 19, 22, 23, 25–27, 33, 34, 42–44, 46, 49–51, 54, 55, 60–62, 66, 68, 77, 79, 85, 90, 92, 94, 106–109, 114, 115, 127, 128, 140–143, 145, 146, 148, 152, 155, 156, 163–167, 174, 215–219, 221–224, 228, 230, 241, 315, 316, 321, 323, 325, 326, 328–330, 332, 336, 340–342, 344, 346, 353, 354, 359, 361, 364, 365, 367, 379, 381, 382, 386, 389, 393, 396–398, 400–402, 405, 408, 417, 419, 430, 431, 436, 437, 441, 444, 466, 469, 470, 472, 486–488, 495–499, 502–505, 508, 510, 512, 517, 522, 528, 530, 531, 533, 535, 550– 552, 555, 559, 560, 567, 572–578, 580, 585–587, 591, 593, 601, 607, 612, 613, 616, 619, 627, 635, 636, 660, 672, 677, 687, 688, 698, 704, 705, 707, 709, 725, 769–771, 789–793, 795–801, 803–812, 840–842, 844, 860–862, 866, 867, 869, 884–886, 889, 892, 893, 895–897, 899–901, 904, 907, 909– 911, 913, 918–921, 923–925, 929, 930, 934–936, 940, 941, 944, 949, 950, 961, 962, 968, 973–975, 977, 979, 982–984, 986, 988–993, 995, 997–1002, 1005, 1010–1013, 1018, 1022, 1023, 1025–1027, 1031, 1034, 1038–1044, 1046, 1047, 1074, 1108, 1109, 1120 double (!!) . . . . . . . . . . . . . . . . . . . . . . . xliii, 23, 77, 79, 94, 110, 111, 113, 114, 127, 128, 146, 152, 155, 156, 174, 222, 226, 245, 250, 316, 319, 324–326, 336, 345, 346, 363, 364, 367, 369, 395–398, 401, 402, 405, 408–410, 420, 430, 435, 459, 460, 466, 467, 469, 470, 472, 478, 488, 531, 538, 539, 543, 551, 573–576, 585, 586, 601, 607, 616, 793, 805, 860–862, 889, 897, 909, 923, 934, 940, 974, 977, 982, 984, 986, 988–990, 992, 994, 997 Fe function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953–955 fe function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953–955 Fek function . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 955, 957 Fey function . . . . . . . . . . . . . . . . . . xl, 765, 767, 955, 956 Fourier transform . . . . . . xliv, 1117, 1118, 1121, 1122, 1129 cosine. . . . . . . . . . . . . . . .xliv, 1121, 1122, 1126–1129 sine. . . . . . . . . . . . . . . . . . . . . . . . xliv, 1121–1125, 1129 Fresnel integral cosine (C) . . . . . xxxvi, xl, 171, 225, 226, 415, 434, 475, 476, 492, 629, 641, 649, 650, 659, 887–890, 935, 1126 sine (S) . . xxxvi, xli, 170, 171, 225, 226, 415, 434, 475, 476, 492, 629, 641, 649, 650, 659, 887–890, 935, 1057, 1124
1155
G
a1 ,... Gpq nm (x | b1 ,... ) . . . . . . . . . . . . . . . see Meijer G function gamma function Γ(x) . . . . . xxxiv, xxxvii–xxxix, xliii, 6, 9, 68, 107– 109, 121–123, 163–167, 264, 296, 317, 318, 321, 322, 324, 326–333, 336–338, 346–355, 358–361, 365–368, 370, 374, 376, 377, 379–384, 386–390, 395, 396, 398–401, 406, 407, 411, 414, 419, 421, 423, 436–445, 459, 460, 462, 466, 472, 479, 486, 491, 492, 497–499, 503, 506, 509, 511, 512, 515, 521–523, 529, 535, 538, 539, 545–548, 550–553, 555, 560, 566–568, 570–574, 576–580, 585, 588, 594, 595, 602, 604, 605, 613–617, 632–640, 645, 646, 648–663, 665–668, 670, 672–688, 690–694, 696–700, 702–712, 715–717, 724–727, 730–734, 736–738, 741, 744–749, 752–761, 769–789, 791– 801, 803–853, 856, 864, 889, 892–902, 904, 909, 910, 912–921, 923, 929, 940–949, 959–964, 966– 975, 978, 979, 981–983, 991–993, 995, 999, 1002, 1003, 1005, 1008, 1009, 1013, 1019–1030, 1032, 1033, 1035–1040, 1043, 1046, 1048, 1056, 1108, 1110, 1113, 1116, 1117, 1119, 1120, 1122, 1123, 1126, 1127, 1130–1133 γ(x) . . 215, 335, 338, 340, 346, 347, 370, 440, 492, 496, 639, 657, 677, 706, 899, 902, 1027 incomplete (Γ(x, y)) . . . . . . . . xxxix, 215, 338, 340, 346–348, 352, 366, 368, 436, 438, 498, 576, 657, 658, 710, 749, 787, 899–902, 1002, 1027, 1110 incomplete (γ(x, y)) . . . . . . . . . xxxix, 439, 899–902 gd(x) . . . . . . . . . . . . . . . . . . see Gudermannian function Ge function . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953–955 ge function . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 953, 955 Gegenbauer polynomial (Cn (x)) . . . . . . . xl, 327, 406, 795–800, 927, 940, 941, 969, 983, 990–993, 995, 997, 999, 1017 Gek function . . . . . . . . . . . . . . . . . . . . . . . . . . . xl, 955, 957 Gey function . . . . . . . . . . . . . . . . . . . . . . xl, 765, 955–957 grad . . . . . . . . . . . . . . . . . . 1050, 1051, 1053, 1055, 1056 Gudermannian (gd) . . . . . . . . . . . . . . . . . xl, 52, 53, 116
H H function . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 879, 880 Hn (x) . . . . . . . . . . . . . . . . . . . . see Hermite polynomials H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . see Struve function H(x) . . . . . . . . . . . . . . . . . . . . . . . . . . see Hankel function H(x) . . . . . . . . . . . . . . . . . . . . . . . . see Heaviside function Hankel function (Hn (x)) . . . . . . . . . . . . . . . . . . . . xxxvii, xli, 339, 350, 351, 368, 370, 385, 492, 653, 663, 688, 691, 693–695, 698, 702, 709, 723, 750, 752, 753, 768, 778, 789, 850, 910, 911, 914–916, 920, 922, 923, 925–928, 931, 940, 944 Hen (x) . . . . . . . . . . . . . . . . . . . see Hermite polynomials
1156
INDEX OF FUNCTIONS AND CONSTANTS
Heaviside Function (H(x)) . . . . . xliv, 642, 750, 1115, 1118, 1131 heiν (z) . . . . . . . . . . . . . . . . . . . . . see Thomson functions herν (z) . . . . . . . . . . . . . . . . . . . . . see Thomson functions Hermite polynomials Hn (x) . . . . . . xxxvi, xxxvii, xli, 365, 503, 803–806, 810–812, 983, 992, 996–998, 1001, 1030 Hen (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi, xxxvii Hermitian . . . . . . . . . . . . . . xliv, 1070, 1071, 1082, 1083 hyperbolic cosine integral . . . . . . . . . . . . . . . . . . . see chi function sine integral . . . . . . . . . . . . . . . . . . . . . see shi function hypergeometric functions F . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, xl, 315–318, 320, 327, 329, 330, 335, 347–349, 351, 368, 370, 374, 375, 394, 398, 436, 438–440, 442, 444, 488, 490, 503, 512, 517, 639, 646, 648, 654, 657, 663, 670, 671, 673, 677–681, 683, 685, 688, 690, 699, 703, 704, 706, 707, 711, 712, 736, 737, 745, 749, 754, 755, 759, 760, 771–776, 779, 780, 784, 791, 792, 794–797, 801, 803, 805, 807–810, 813–818, 821–824, 826–835, 838, 841, 844, 846, 848, 849, 889, 910, 946, 982, 999, 1005–1013, 1015–1023, 1025, 1033, 1035, 1037, 1039 confluent (Φ) . . . . . . xxxix, 1022–1024, 1027, 1028, 1030–1032 confluent (Ψ) . . 816, 1023, 1027, 1028, 1038, 1039
I incomplete beta function I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 910 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix, 910, 1132 incomplete Gamma function . . see gamma function, incomplete inverse functions . . . . . . . . . . . . . . . . . . . . . . . . 1118, 1121
J Jacobi elliptic functions . . . see cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, sn Jacobi polynomial (pn (x)) . . . . . . xli, 998–1000, 1003 Jacobi theta function (Θ) . . . . xxxiv, xxxix, 879, 880 Jacobi zeta function (zn) . . . . . . . . . . . . . . . . . . . . . xxxiv
K kei(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions ker(z) . . . . . . . . . . . . . . . . . . . . . . see Thomson functions K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 k . . . . . . . . . . . . . . . . . . . . . . . . . . xliv, 134, 135, 184–200, 204, 206, 225, 263, 410, 472–475, 562, 567, 568, 585, 588, 592, 593, 596–602, 604–606, 619–626, 631–633, 859–868, 870–873, 875, 879, 881, 1006 Kronecker delta . . . . . . . . xliv, 1046, 1047, 1076, 1088
L
L. . . . . . . . . . . . . . . . . . . . . . . . . . . . see Laplace transform L2 (x) . . . . . . . . . . . . . . . . . . . . . see dilogarithm function Ln (x) or Lα n (x) . . . . . . . . . . . see Laguerre polynomials L(x) . . . . . . . . . . . . . . . . . . . . see Lobachevskiy function L(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . see Struve function Laguerre function (Lα n (x)) . . . . 348, 441, 707, 709, 803–806, 808–812, 840, 901, 983, 1000–1004, 1028 polynomial (Ln (x)) . . xli, 344, 806, 808, 809, 811, 812, 844 Laplace transform . . . . . . xliv, 1107, 1108, 1129, 1138 Legendre functions first kind (Pn (x)) . . . . . xli, 93, 106, 327, 390, 405, 406, 409, 513, 612, 698, 707, 719, 769–772, 774, 776–782, 785, 786, 788–794, 801, 809, 815, 829, 933, 936, 940, 941, 959–961, 963–969, 972–990, 992, 999, 1017, 1131 second kind (Qn (x)) . . . . . . xli, 324, 373, 383, 696, 719, 769–771, 773, 777, 780, 785, 788, 790, 791, 959, 960, 965, 966, 968, 972, 973, 975–981, 986 Lerch function (Φ) . . . . . . . . . . . . . . . . xxxix, 642, 1039 li function . . . xxxv, xli, 238, 340, 527, 553, 636, 637, 883, 884, 887, 902, 1027 limit . . . . . . . xxxii, 6–8, 14, 21, 26, 53, 250–252, 511, 610, 611, 617, 635, 883, 887, 890, 894, 895, 904, 905, 931, 951, 963, 992, 1000, 1003–1006, 1023, 1038–1040, 1067, 1101, 1104, 1106–1108, 1118, 1121, 1130, 1136, 1139 ln function . . . . xxvii–xxix, xxxi, xxxii, xxxiv–xxxvii, 3, 9–11, 23, 26, 27, 43, 44, 46, 47, 49, 51–56, 61–67, 69–85, 87, 90, 94, 97, 99, 103, 104, 106, 113–120, 123–130, 133, 143, 145–148, 150, 155– 161, 167–172, 174–176, 178, 186–197, 199, 200, 204–207, 220–225, 237–245, 250, 316, 321, 324, 326, 330–332, 334, 338–340, 353–364, 369–373, 375, 376, 378–381, 383, 386–390, 395, 402, 410, 431, 433, 434, 438, 447–449, 451–457, 462–466, 470–473, 483–485, 495, 497, 499–502, 517–521, 527–607, 622–628, 631–633, 636–645, 647–649, 656, 658, 659, 661, 668, 671, 672, 695, 702, 718, 719, 728, 747, 748, 755, 763, 861, 862, 868, 880, 882–887, 891–893, 895, 898–900, 902–907, 909, 911, 914, 919, 929, 937–939, 944, 963, 969, 972, 977–979, 981, 982, 990, 1006, 1011–1013, 1026, 1027, 1037–1040, 1046, 1048, 1056, 1113, 1115, 1120, 1123, 1125, 1128, 1132, see log function Lobachevskiy function (L) . . . xli, 147, 225, 375, 380, 381, 530–534, 588, 589, 593, 891 log function . . . . . . . . . . . . . . . . 27, 642, see ln function
INDEX OF FUNCTIONS AND CONSTANTS
Lommel function (S) . . . . . . xxxvi, xli, 339, 346, 352, 371, 384, 386, 417, 670, 674, 676–678, 680, 681, 756, 758, 760, 761, 779, 782, 783, 785, 787, 788, 794, 815, 816, 819, 828, 945–947, 950, 1035 Lommel function (s) . . . . xli, 419–421, 439, 443, 670, 692, 725, 760, 761, 945, 946, 1133 Lommel function (U) . . . . . . . . . . . . . . . . . . . . . . xlii, 947 Lommel function (V) . . . . . . . . . . . . . . . . . xlii, 947, 948
M
M . . . . . . . . . . . . . . . . . . . . . . . . . . . . see Mellin transform Mλ,μ (z) . . . . . . . . . . . . . . . . . . . see Whittaker functions MacRobert function (E) . . . . . . . . . . . . . . . . 1035, 1036 Mathieu functions Se . . . . . . . . . . . . . . . . . xxxviii, xli, 764–766, 953, 954 se . . . . . . . . . . . . . . . . . xxxviii, xli, 763–766, 951–957 max . . . . . 851, 854, 856, 987, 1066, 1081–1086, 1088, 1091 Meijer function (G) . . . . . xl, 351, 444, 654, 690, 691, 704, 711, 758, 776, 778, 817–819, 825–832, 835, 838, 844, 845, 847, 850–856, 1032–1035 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . xliv, 1130 min . . . . . . . . . . . . . . . . . . . . . . 851, 854, 856, 1085, 1091
N Nν (z) . . . . . . . . . . . . . . . . . . . . . . . see Bessel function, Y nc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii nd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Neumann function . . . . . . . . . . . see Bessel function, Y Neumann polynomial (On (x)) . . . . . . xxxvii, xli, 346, 384, 386, 946, 949, 950 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . see || · || and || · ||p ns function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii
O On (x) . . . . . . . . . . . . . . . . . . . see Neumann polynomials orthogonal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798
P Pn (x) . . . . . . . . see Jacobi polynomials and Legendre polynomials Pn (x) . . . . . . . . . . . see Legendre functions (first kind) Pnm (x) . . . . . see Legendre functions (associated, first kind) parabolic cylinder function (D) . . . . . . . . . . xxxviii, xl, 348, 349, 352, 365, 384, 390, 503, 504, 506, 653, 657, 658, 697, 708, 712, 740, 746, 802, 805, 811, 841–850, 1028–1031
1157
Phi function (Φ) . . . . . . . . . . . . . . . . . . . . . . xxxvi, xxxix, xl, 239, 336–338, 344, 345, 353, 354, 358, 364, 367, 371, 376, 379, 381, 384, 390, 489, 503, 504, 526, 574, 604, 629, 640, 645–649, 748, 749, 755, 781, 802, 835, 838, 887–891, 899, 902, 997, 998, 1001, 1050, 1051, 1056, 1057 Pochhammer symbol . . . . . . . . . . . . xliii, 321, 330, 635, 672, 705, 900, 918, 947, 1010–1012, 1018, 1022, 1031, 1048 polynomials . . . . . . . . . . . . . . . . . . . . . . see specific name principal value (PV). . .xliii, 322, 329, 335, 337, 433, 454, 528, 534, 563, 572, 883
Q Qn (x) . . . . . . . . see Legendre functions (second kind) Qm n (x) . . . see Legendre function (associated, second kind)
R root . . . . . . . . . . . . 15, 84, 104, 331, 539, 542, 553, 576 3 . . . . 72, 86–88, 264, 330, 331, 363, 364, 539, 570, 918 4. . . . . . . . .73, 78, 83, 105, 135, 136, 139, 210, 211, 263–265, 272, 295, 296, 312–315, 483, 493–495, 507, 524, 525, 868, 875, 878–881, 997 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894 rot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
S Sn (x) . . . . . . . . . . . . . . . . . . . . . . see Schlafli polynomials (m) Sn . . . . . . . . . . . . . . . . see Stirling numbers, first kind s(x) . . . . . . . . . . . . . . . . . . . . . . . . . . see Lommel function S(x) . . . . . . . see Lommel function and Fresnel cosine integral sc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Schlafli polynomial (Sn (x)) . . . . . . . . . . . . xli, 949, 950 sd function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Se(x) . . . . . . . . . . . . . . . . . . . . . . . . see Mathieu functions se(x) . . . . . . . . . . . . . . . . . . . . . . . . see Mathieu functions sec function . . . . . . . . . . . . . . . . . . 36, 39, 43, 44, 50, 52, 64, 113, 114, 155, 156, 315, 323, 328, 329, 371, 372, 377–379, 389, 395, 396, 400, 401, 403–405, 410–414, 421, 422, 436, 438, 439, 446, 471, 472, 479, 541, 551, 586, 646, 653, 661, 663, 664, 669, 670, 674, 682, 689, 691, 699, 706–708, 710, 713, 716, 718, 719, 726, 728, 734, 740, 749, 757, 761, 783, 806, 820, 825, 845, 846, 864, 921, 922, 932, 1007, 1126, 1132, 1133 sech function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27, 43, 62, 113–115, 323, 387, 509, 634, 715, 750, 751, 787, 800, 802, 841, 883, 1128, 1133
1158
shi function . . xxxvi, xli, 142–144, 495, 644, 645, 886 Si function . . . . . . . . . xxxv, 643, 886, 930, 1115, 1122 si function . . . . . . . . xxxv, xli, 219–221, 340–344, 421, 423–426, 447, 495, 505, 506, 528, 529, 571, 572, 578, 581, 594, 595, 599, 605, 628, 629, 638–644, 647, 649, 650, 656, 658, 748, 762, 886, 887, 992, 1115 sigma function . . . . . . . . . . . . . . . . . . . . . xxxix, 876, 877 sign function . . xlv, 46, 177, 241, 243, 251, 322, 350, 351, 365, 370, 423, 437, 438, 447, 465, 485, 594, 596, 603, 604, 610, 611, 640, 642, 652, 750, 768, 885, 1118–1120, 1122 sin function . . . . . . . . . . . . . . . . xxvii, xxix, xxxi, xxxii, xxxiv–xxxviii, 4, 13, 19, 20, 23, 26–52, 55, 56, 64, 74–76, 79, 147, 151–237, 249, 250, 253, 254, 263–265, 317, 318, 321–323, 325–329, 331–333, 339–345, 348, 354, 355, 358, 359, 371–373, 375, 377–383, 385, 387–534, 537, 539–547, 550, 551, 554, 558, 559, 561–563, 565, 567–573, 576, 578, 579, 581–601, 604–606, 608, 610–612, 616, 621– 623, 628, 629, 631–633, 635–644, 647–651, 655, 656, 658, 659, 662–665, 667, 669, 671–675, 677, 678, 680, 684, 686, 688, 689, 691, 692, 695, 698, 703, 705, 706, 708, 709, 711, 713–715, 717–748, 751, 752, 754–756, 758–760, 762–770, 773, 774, 776, 777, 779–782, 784, 789, 793, 794, 797–800, 802, 806, 808, 810–812, 817, 820, 825, 829, 830, 833, 836, 837, 839, 842, 844–846, 848–850, 853, 859–869, 876–880, 882–893, 896, 898, 900, 904, 906–910, 912–918, 920, 922, 924, 925, 927, 928, 930, 933–940, 942, 943, 945–951, 954, 958, 959, 961–964, 966–981, 984–992, 994, 997, 998, 1000, 1005–1007, 1009, 1013, 1025, 1026, 1029, 1036– 1038, 1042, 1044, 1053, 1057, 1060, 1061, 1066, 1067, 1090, 1110–1113, 1115, 1117–1128, 1131, 1132, 1137, 1140 sinh function . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi, 27–36, 38, 40, 42, 43, 45, 47, 48, 50–52, 64, 110–151, 231–237, 251, 338, 339, 358, 371–390, 407, 419, 425, 432, 433, 438, 439, 448, 451, 452, 454, 455, 461, 466–468, 477, 484, 485, 489, 491, 496, 502, 504, 508–527, 570, 573, 578–582, 595, 596, 606, 610, 611, 621, 622, 634, 643–645, 664, 686, 698, 702, 704, 705, 710, 711, 713–716, 723, 729, 735, 747, 751–753, 755, 760, 763–766, 778, 787–789, 806, 843, 844, 886, 888, 896, 898, 913–915, 917, 943, 953, 955–957, 960–963, 967–969, 980, 981, 997, 1006, 1025, 1029, 1040, 1112–1115, 1119, 1120, 1124, 1125, 1128, 1140 sn function . . . . . . . . . xxxiii, xxxiv, xli, 623–626, 714, 866–873, 875, 879, 880 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859
INDEX OF FUNCTIONS AND CONSTANTS
square root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii, xxxiv, xxxvi, xxxviii, xliv, 2, 9–11, 14, 15, 23, 25, 26, 30, 37, 43, 44, 54–61, 63–67, 71–79, 83– 99, 103–109, 125–139, 158, 170–175, 177–184, 197, 199, 200, 202–214, 225, 226, 230, 239–245, 249, 251, 254–315, 317–319, 321, 324–328, 330, 333, 336, 337, 339, 344–353, 355, 359, 363–376, 380, 382–385, 390, 391, 393–396, 400–402, 404– 411, 414–419, 421, 425, 426, 428–430, 434–436, 440–446, 448, 451–454, 456, 457, 460, 472–479, 481–483, 485, 486, 488–497, 499, 501–507, 511, 513–515, 517, 518, 522–527, 529, 531, 532, 534, 535, 537–539, 542, 543, 545, 549–551, 553, 554, 556–558, 560, 562, 563, 565–568, 570–576, 578– 581, 584, 585, 588, 590–593, 595–606, 608–613, 615–617, 619, 621–623, 629, 631–635, 637–642, 644–651, 653, 657, 659, 661–668, 670, 672–675, 677, 678, 680–683, 685–763, 766–768, 771, 773, 777, 778, 780–782, 785–789, 791–794, 800–808, 810–812, 814, 815, 819, 828, 829, 837, 841, 843– 845, 848, 853, 854, 856, 859–866, 868, 870–876, 879–881, 887–891, 893–902, 905, 908, 909, 913– 915, 917, 918, 920–926, 928, 931–946, 950, 951, 958, 960–974, 976–983, 985, 987, 988, 990–998, 1000, 1002, 1003, 1007, 1009, 1018, 1019, 1023, 1026–1030, 1035, 1038, 1052, 1054, 1055, 1060, 1082, 1116–1121, 1123, 1125–1127, 1129, 1131 step function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 Stirling number first kind (Snm ) . . . . . . . . . . . . . . . . . . . xlv, 1046–1048 second kind (Sm n ) . . . . . . . . . . . . . . . . xlv, 1046–1048 Struve function H(x) . . xli, 345, 351, 421, 435, 442, 443, 573, 647, 659, 660, 663, 664, 669, 675, 677, 679, 680, 692, 694, 708, 722, 725, 735, 753–759, 787, 838, 848, 856, 942, 943, 946, 1035, 1132 modified (L(x)) . . . . . . xli, 345, 350, 351, 435, 441, 515, 595, 605, 663, 664, 669, 671, 675, 676, 678, 679, 692, 722, 736, 753–759, 787, 794, 942, 943
INDEX OF FUNCTIONS AND CONSTANTS
1159
T
U
Tn (x) . . . . . . . . . . . . . . . . . . see Chebyshev polynomials tan function . . . . . . . . . . . . . . xxvii–xxix, 27–30, 35, 36, 39, 40, 42, 44, 46, 50, 52, 53, 55, 56, 60, 64, 126, 151, 155–161, 167–178, 180, 181, 183–185, 188, 189, 194, 195, 199, 200, 202–207, 209, 212, 214, 222–225, 229, 230, 274, 322, 332, 339, 340, 355, 371, 375–379, 381, 382, 384, 388, 389, 393, 395, 396, 400, 403–405, 409–414, 421–423, 433, 434, 451–455, 457–459, 471–475, 483–486, 492, 493, 496, 498, 505, 506, 509, 515, 518, 532–535, 537, 541, 545, 546, 567–570, 579, 582, 586–589, 591– 594, 597–599, 604, 606, 622, 631, 632, 659, 664, 669–671, 682, 686, 699, 716, 717, 725, 728, 744, 745, 754, 766, 849, 862–865, 867–869, 883, 886, 890, 894, 905, 909, 922, 932, 954, 969, 971, 979, 986, 990, 1006, 1007, 1023, 1123, 1127, 1132 tanh function . . . . . . . . . 12, 27–29, 31, 39, 40, 42, 44, 51, 52, 62, 64, 110, 113–120, 123–126, 128–132, 134–137, 139, 145–148, 338, 380–383, 387, 390, 472, 484, 485, 489, 502, 509, 512, 513, 516, 518, 520, 569, 606, 621, 622, 716, 717, 751, 753, 766, 788, 789, 841, 921, 922, 956, 957, 1025, 1124 theta function (θ) . . . . xxxiv, 521, 633, 634, 877–883 Thomson functions bei(x) . . . . . . . . . . . . . . . . . . xxxix, 761–763, 944, 945 ber(x) . . . . . . . . . . . . . . . . . . xxxix, 761–763, 944, 945 hei(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 944 her(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli, 944 kei(x) . . . . . . xli, 641, 663, 672, 674, 748, 762, 763, 944, 945 ker(x) . . . . . . xli, 641, 663, 671, 674, 747, 762, 763, 944, 945 toroidal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see trace trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 transpose . . . . . . . . . . . . . . . xlv, 1069–1073, 1075, 1089
Un (x) . . . . . . . . . . . . . . . . . . see Chebyshev polynomials unilateral z transform . . . . . . . 1135, 1138–1140, 1159
W Wλ,μ (z) . . . . . . . . . . . . . . . . . . . see Whittaker functions Weber function (E) . . . . 338, 339, 346, 353, 371, 384, 421, 423, 670, 671, 751, 943, 946, 948, 949 Weierstrass function (℘) . . . . xxxix, xl, 626, 873–877, 880 Whittaker functions M . . . . . xli, 338, 348, 445, 654, 682, 697, 703, 705, 706, 709, 710, 715–717, 736, 748, 749, 784, 785, 787, 819–841, 1024–1027 W . . . . . . . . . xlii, 338, 346–349, 367, 368, 384, 423, 445, 635, 652, 654, 682, 697, 698, 704, 706, 707, 709, 710, 712, 715, 716, 726, 727, 736, 745, 749, 756, 759, 761, 776–778, 781, 782, 784–788, 803, 814–817, 819–841, 843, 844, 846, 847, 857, 979, 1024–1028, 1035
X Xb . . . . . . . . . . . . . . . . . . . . . . . . see bilateral z transform Xu . . . . . . . . . . . . . . . . . . . . . . see unilateral z transform
Y Y . . . . . . . . . . . . . . . . . . . . . . . . . . . see Bessel function, Y Young function (C) . . . . . . . . . xxxvi, xl, 417, 439, 440
Z zeta function (ζ) . . xxxix, 8, 338, 353, 354, 358, 359, 376, 377, 379–381, 386–389, 433, 434, 449, 471, 509, 540, 542, 543, 550, 552, 560, 567, 569, 576, 577, 580, 587, 591, 593, 607, 626, 633, 634, 658, 659, 802, 876, 877, 880, 894, 898, 903–905, 907, 909, 1036–1041, 1133 zn(x) . . . . . . . . . . . . . . . . . . . . . . see Jacobi zeta function
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Index of Concepts This index refers to concepts appearing in the text.
A
arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 242 arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 244 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 747 argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 of a complex number . . . . . . . . . . . . . . . . . . . . . . . . xliv arithmetic mean theorem . . . . . . . . . . . . . . . . . . . . . 1056 arithmetic progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 arithmetic-geometric inequality . . . . . . . . . . . . . . . 1060 arithmetic-geometric progression . . . . . . . . . . . . . . . . . 1 associated Legendre functions . . . 769, 788, 958, 972, 974 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 789 and Bessel functions . . . . . . . . . . . . . . . . . . . . 782, 787 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 778 and powers . . . . . . . . . . . . . . . . . . . . . . . . 770, 776, 779 and probability integral . . . . . . . . . . . . . . . . . . . . . . 781 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 789 and trigonometric functions . . . . . . . . . . . . . . . . . . 779 associated Mathieu functions . . . . . . . . . . . . . . . . . . . 952 asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 1146 asymptotic result . . . . . . . . . . 21, 356, 895, 1026, 1029 asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . 973, 975 adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 algebraic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059, 1060 algebraic functions . . . . . . . . . . . . . . . . . . . . . . . . . 82, 253 and arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and associated Legendre functions . . . . . . . . . . . 789 and Bessel functions . . . . . . . . . . . . . . . . . . . . 674, 715 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . 344, 363 and hyperbolic functions . . . . . . . . . . . 132, 375, 715 and logarithmic functions . . . . . . . . . . . . . . . . . . . . 538 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 789 and trigonometric functions . . . . . . . . . . . . . . . . . . 434 alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 analytic continuation . . . . . . . . . . . . . . . . . . . . . 970, 1012 Anger functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 angle of parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 anticommutative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 approximate solution . . . . . . . . . . . . . . . . . . . . 1093–1096 approximation by tangents . . . . . . . . . . . . . . . . . . . . . 921 arccosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 242 arccotangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 244 arcsecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
B Ballieu theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 Bateman’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1040, 1045 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 1040, 1041 Bessel functions. . .629, 659, 748, 749, 753, 910, 912, 914, 916–920, 924, 925, 928, 931, 933–937, 940, 941, 954, 1146, see Constant/Function index and algebraic functions . . . . . . . . . . . . . . . . . 674, 715 and arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 and associated Legendre functions . . . . . . 782, 787
1161
1162
INDEX OF CONCEPTS
Bessel functions (continued) and Chebyshev polynomials . . . . . . . . . . . . . . . . . 803 and exponentials . . . . 694, 699, 708, 711, 713, 715, 742, 834 and Gegenbauer functions . . . . . . . . . . . . . . . . . . . 798 and hyperbolic functions . . . . . . . . . . . 713, 715, 747 and hypergeometric functions . . . . . . . . . . . . . . . . 817 confluent . . . . . . . . . . . . . . . . . . . . . . . . . 830, 831, 834 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 794 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 854 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 767 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 854 and parabolic cylinder functions . . . . . . . . . . . . . 845 and powers . . . . . 664, 675, 689, 699, 708, 711, 727, 742, 831, 834 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 670 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 756 and trigonometric functions . . . 717, 727, 742, 747 generating functions . . . . . . . . . . . . . . . . . . . . . . . . . 933 imaginary arguments . . . . . . . . . . . . . . . . . . . . . . . . 911 Bessel inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 bilateral z-transform . . . . . . . . . . . . . . . . . . . . 1135, 1136 bilinear concomitant . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315, 322 Bonnet–Heine formula . . . . . . . . . . . . . . . . . . . . . . . . . 988 bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 boundedness theorems . . . . . . . . . . . . . . . . . . . . . . . . 1106 branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866, 1024 Brauer theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1086, 1088 Buniakowsky inequality . . . . . . . . . . . 1059, 1061, 1064
C Calogero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 Carleman inequality . . . . . . . . . . . . . . . . . . . . 1060, 1066 Catalan constant . . . . . . . . . . . . . . . . . . . xxxii, 1046, see Constant/Function index Cauchy principal value . . . . . . . . . . . . . . . . . . . . . . . . . 528 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . 1093, 1095 Cauchy–Schwarz–Buniakowsky inequality . . . . . 1059, 1061, 1064 Cayley–Hamilton theorem . . . . . . . . . . . . . . . . . . . . 1084 change of variables . . . . . . . . . . . . . . . . . . . 248, 607, 608 characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . 1071 characteristic polynomial. . . . . . . . . . . . . . . . . . . . . .1084 characteristic values . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 Chebyshev inequality . . . . . . . . . . . . . . . . . . . 1059, 1065
Chebyshev polynomials. . . . . . . . . . . . . . . . . . . .988, 993 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 803 and elementary functions . . . . . . . . . . . . . . . . . . . . 802 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 Christoffel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983 Christoffel summation formula . . . . . . . . . . . . . . . . . 986 circle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 circulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 classification system . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi classified references . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145 column norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 comparison of approximate solutions . . . . 1094, 1096 comparison theorem . . . . . . . . . 1100, 1101, 1103, 1104 complementary error function. . .see error functions, complementary complementary modulus . . . . . . . . . . . . . . . . . . . . . . . 859 complete elliptic integrals . . . . . . . . . . . . . 619, 632, 859 complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146 complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii conditional convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 6 conditions, Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 confluent hypergeometric function . . . . . . . . . . . . . . see hypergeometric function, confluent conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980 constant of integration . . . . . . . . . . . . . . . . . . . . . . . . . . 63 constants . . . . . . . . . . . . . see Constant/Function index Catalan . . . . . . . . . . . . . . . . . . . . see Catalan constant Euler . . . . . . . . . . . . . . . . . . . . . . . . . see Euler constant continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902 continuity, Lipschitz . . . . . . . . . . . . . . . . . . . . 1094, 1095 converge absolutely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 conditionally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 uniformly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 convergence circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 19 convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 theorem . . . . . . . . . . . . 1108, 1118, 1122, 1130, 1136 coordinates, curvilinear . . . . . . . . . . . . . . . . . . . . . . . 1052 cosine and rational functions . . . . . . . . . . . . . . . . . . 171, 390 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628, 639, 886 hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 cosine-amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 Cramer’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050
INDEX OF CONCEPTS
1163
curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1052 cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 cylinder function . . . . see parabolic cylinder function
D Darboux–Christoffel formula . . . . . . . . . . . . . . . . . . . 983 de Moivre’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 decreasing solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 definite integrals . . . . . . . . . 247, see integrals, definite delta amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 derivative of a composite function . . . . . . . . . . . . . . . 22 determinants . . . . . . . . . . . . . . . . . . . . . . 1075, 1076, 1078 Gram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Vandermonde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 differential equations . . . . . . . . . . . . . . . . . . . . . 873, 874, 910, 931, 944, 947–950, 952, 958, 974, 975, 980, 981, 983, 993, 995, 998, 1000, 1003, 1011, 1013, 1015, 1024, 1031, 1034, 1093 adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 exact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 homogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 hypergeometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018, 1031 Riccati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014, 1022 second-order . . . . . . . . . . . . . . 1017, 1098, 1100, 1104 self-adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 special types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 variables separable . . . . . . . . . . . . . . . . . . . . . . . . . 1097 differentiation of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 1064 of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 dilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 diophantine relations. . . . . . . . . . . . . . . . . . . . . . . . . . 1089 directional derivative. . . . . . . . . . . . . . . . . . . . . . . . . . 1051 Dirichlet conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 DOC (domain of convergence) . . . . . . . . . . . . . . . . 1135 domain of convergence (DOC) . . . . . . . . . . . . . . . . 1135 dominant solutions . . . . . . . . . . . . . . . . . . . . . . 1104, 1105 double factorial symbol . . . . . . . . . . . . . . . . . . . . . . . . xliii double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 610, 1021 doubling formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 doubly-periodic function . . . . . . . . . . . . . . . . . . 866, 874
E eigenvalues . . . . . . . . . . . . . . . . . . . 951, 1071, 1084, 1087 eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 elementary functions . . . . . . . . . . . . . . . . . 25, 247, 1006 and Chebyshev polynomials . . . . . . . . . . . . . . . . . 802 and Gegenbauer polynomials . . . . . . . . . . . . . . . . 797 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 792 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 850 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 850 indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 elliptic functions . . . . . . . . . . . . . . . . 619, 631, 865, 1148 Jacobian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866, 870 order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 Weierstrass . . . . . . . . . . 626, see Weierstrass elliptic functions elliptic integrals . . . 104, 184, 619, 621, 631, 632, 859 complete . . . . . . . . . . . . . . . . . 394, 472–474, 632, 859 derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 394, 863, 865 functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . 863 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 equations differential . . . . . . . . . . . . . see differential equations first-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093, 1096 linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 special types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094, 1095 error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 887, 1147 complementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 essential singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 Euclidean norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 Euler constant . . . . . xxxii, see Constant/Function index dilogarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892, 908 numbers . . . . . . . . . . . . . . . . . . . . . . . . 1040, 1043, 1045 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044 substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 exact differential equations . . . . . . . . . . . . . . . . . . . . 1097 expansion of determinants. . . . . . . . . . . . . . .1075, 1076 expansions, asymptotic . . . . . . . . . . . . . . . . . . . . . . . 1146 expansions, Weierstrass . . . . . . . . . . . . . . . . . . . . . . . . 869 exponential integrals . . 627, 636, 638, 883, 885, 1147 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 106, 334 and algebraic functions . . . . . . . . . . . . . . . . . 344, 363 and associated Legendre functions . . . . . . . . . . . 776 and Bessel functions . . . . . 694, 699, 708, 711, 713, 715, 742, 834 and complicated arguments . . . . . . . . . . . . . . . . . . 336
1164
INDEX OF CONCEPTS
exponentials (continued) and exponential integrals . . . . . . . . . . . . . . . . . . . . 628 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 652 and hyperbolic functions . . . . . . 148, 338, 382, 386, 522, 525, 713, 715 and hypergeometric functions . . . . . . . . . . . . . . . . 814 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822, 834 and inverse trigonometric functions . . . . . . . . . . 605 and logarithmic functions . . . . . 339, 571, 573, 599 and parabolic cylinder functions . . . . . . . . . . . . . 842 and powers . . . . . 148, 346, 353, 363, 364, 386, 497, 525, 573, 699, 708, 711, 742, 754, 776, 834, 842 and rational functions . . . . . . . . . . . . . . 106, 340, 353 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 754 and trigonometric functions . . . 227, 339, 485, 493, 495, 497, 522, 525, 599, 742 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
F factorial symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 figures . . . . . . . . . . . . 608–610, 892, 913, 915, 916, 1036 final value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139 finite sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 first mean value theorem . . . . . . . . . . . . . . . . . . . . . . 1063 first-order equations . . . . . . . . . . . . . . . . . . . . 1093, 1096 first-order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix, xxxi, 82, 132, 247, 248, 274, 397, 410, 547, 656, 859, 867, 908, 920, 931, 981, 991, 1039, 1141 Fourier series . . . . . . . . . . . . . . . . . . . . 19, 46, 1066, 1067 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 1107, 1117 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121, 1129 properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126 exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129 sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121, 1129 properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118, 1120 fourth roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 fractional transformation . . . . . . . . . . . . . . . . . . . . . . 1014 Fresnel integrals . . . . . . . . . . . . . . . . 629, 649, 887, 1147 Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 functional series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
functions . . . . . . . . . . . . . see Constant/Function index inner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 outer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii fundamental inequalities . . . . . . . . . . . . . . . . . . . . . . 1094 fundamental system. . . . . . . . . . . . . . . . . . . . . . . . . . . 1100
G gamma functions . . . . . . . . . . 650, 892, 894, 895, 1147 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 and trigonometric functions . . . . . . . . . . . . . . . . . . 655 incomplete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657, 899 Gauss divergence theorem. . . . . . . . . . . . . . . . . . . . . 1055 Gegenbauer functions and Bessel functions . . . . . 798 Gegenbauer polynomials . . . . . . . . . . . . . . . . . . . . . . . 990 and elementary functions . . . . . . . . . . . . . . . . . . . . 797 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 general formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65, 249 generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . 635 generalized Fourier series . . . . . . . . . . . . . . . . 1067, 1068 generalized Legendre polynomials . . . . . . . . . . . . . . 990 generating functions Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1040 Bernoulli polynomials . . . . . . . . . . . . . . . . xxxii, 1041 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933 Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . 995 Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 Euler polynomials . . . . . . . . . . . . . . . . . . . . xxxii, 1044 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . 997 Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 1000 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . 988 Neumann polynomials . . . . . . . . . . . . . . . . xxxvii, 950 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . 1046, 1047 geometric progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Gerschgorin theorem . . . . . . . . . . . . . . . . . . . . 1083, 1088 grad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 Gram determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 Gram inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 Gram–Kowalewski theorem . . . . . . . . . . . . . . . . . . . 1080 Green theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1055, 1056 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 growth estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1104 growth of maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Gudermannian (gd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
INDEX OF CONCEPTS
1165
H Hadamard’s inequality . . . . . . . . . . . . . . . . . . . . . . . . 1077 Hadamard’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 Hankel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 910, 925 Heaviside step function . . . . . . . . . . . . . . . . . . . . . . . . xliv Heine formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . 767, 1052 Hermite method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Hermite polynomials . . . . . . . . . . . . . . . . . 803, 996, 997 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . 1077, 1089 Hessian determinant . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 H¨ older inequality . . . . . . . . . . . . . . . . . 1059, 1061, 1064 homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 homogeneous differential equations . . . . . . . . . . . . 1097 hyperbolic amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 cosine integral . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 sine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 hyperbolic functions . . . . . . . . . . . . . . . . . . . 28, 110, 371 and algebraic functions . . . . . . . . . . . . . 132, 375, 715 and associated Legendre functions . . . . . . . . . . . 778 and Bessel functions . . . . . . . . . . . . . . . 713, 715, 747 and exponentials . . . . 148, 338, 382, 386, 522, 525, 713, 715 and inverse trigonometric functions . . . . . . . . . . 605 and linear functions . . . . . . . . . . . . . . . . . . . . . . . . . 120 and logarithmic functions . . . . . . . . . . . . . . . . . . . . 578 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 763 and parabolic cylinder functions . . . . . . . . . . . . . 843 and powers . . . . . . . . . . . . . . . 139, 148, 386, 516, 525 and rational functions . . . . . . . . . . . . . . . . . . . . . . . 125 and trigonometric functions . . . 231, 509, 516, 522, 525, 747, 763 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 240 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110, 120 hypergeometric differential equation . . . . . . . . . . . . . . . . . . . . . . . . 1010 series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005, 1008 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 hypergeometric functions . . . . . . . 812, 841, 946, 1005, 1006, 1039, 1147 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 817 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812 and trigonometric functions . . . . . . . . . . . . . . . . . . 817
hypergeometric functions (continued) confluent . . . . . . . . . . . . . . 820, 841, 1022, 1023, 1147 and Bessel functions . . . . . . . . . . . . . . 830, 831, 834 and exponentials . . . . . . . . . . . . . . . . . . . . . . 822, 834 and Legendre functions . . . . . . . . . . . . . . . . . . . . . 839 and parabolic cylinder functions . . . . . . . . . . . . 849 and polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 840 and powers . . . . . . . . . . . . . . . . . . . . . . . 820, 831, 834 and special functions . . . . . . . . . . . . . . . . . . . . . . . 839 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . 838 and trigonometric functions . . . . . . . . . . . . . . . . 829 several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022 two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018
I identities Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098 Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Picone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . 251, 252 incomplete beta functions . . . . . . . . . . . . . . . . . . . . . . 910 incomplete gamma function . . . . . . . . . . . . . . . . . . . . 657 increasing solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 indefinite integrals elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . 63 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 induced norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 inequalities . . . . . 950, 963, 979, 987, 997, 1041, 1061, 1083–1085, 1094 algebraic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059, 1060 Carleman . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060, 1066 for sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 Hadamard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063–1066 Schur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 infinite products . . . . . . . . . . . . . . . . . . . . . . . . . 6, 14, 862 initial value theorem . . . . . . . . . . . . . . . . . . . . 1136, 1139 inner function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi integer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 integer pulse function . . . . . . . . . . . . . . . . . . . . . . . . . 1135 integral differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 1064 formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063–1066 inversion . . . . . . . . . . . . . . . . . 1107, 1118, 1121, 1129 part (symbol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii representations . . . . . . 887, 888, 892, 898, 900, 902, 906, 908, 912, 914, 916, 942, 946, 950, 960, 974, 976, 980, 981, 985, 991, 996, 1005, 1021, 1023, 1025, 1028, 1035, 1036, 1039, 1040, 1067
1166
INDEX OF CONCEPTS
integral (continued) theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107, 1147 relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129 integrals definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 double. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610, 1021 elliptic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104, 859 Fresnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 improper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251, 252 indefinite . . . . . . . . . . . . . . . . . see indefinite integrals Mellin–Barnes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607, 612 pseudo-elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 integration constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 termwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 inverse z-transformation . . . . . . . . . . . . . . . . . . . . . . 1135 inverse hyperbolic functions . . . . . . . . . see hyperbolic functions inverse trigonometric functions . . . . . . . . . . . . . 599, see trigonometric functions and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 605 and logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 and powers . . . . . . . . . . . . . . . . . . . . . . . . 600, 601, 607 and trigonometric functions . . . . . . . . . . . . . 605, 607 inversion integral . . . . . . . . . . . . 1107, 1118, 1121, 1129
J Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . 806, 998 Jacobi theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 Jacobian determinant . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Jacobian elliptic functions . . . . . . 866, 870, 879, 1148 Jacobian elliptic integrals . . . . . . . . . . . . . . . . . . . . . . 623 Jensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066
K Kneser’s non-oscillation theorem . . . . . . . . . . . . . . 1103 Kowalewski theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1080
L L2 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . 1059, 1099 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . 808, 1000 Laplace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Laplace integral formula . . . . . . . . . . . . . . . . . . . . . . . 985
Laplace transform . . . . . . . . . . . . . . . . . . . . . . 1107, 1129 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767, 1051 latent roots values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 least common factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 least common multiple . . . . . . . . . . . . . . . . . . . . . . . . . 798 Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . 975, 1149 and hypergeometric functions confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 associated . . . . . see associated Legendre functions special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 Legendre normal form . . . . . . . . . . . . . . . . . . . . . . . . . 859 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . 983, 988 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 794 and elementary functions . . . . . . . . . . . . . . . . . . . . 792 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791 lemmas Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Gronwall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 Riemann–Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . 1067 letters, conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 linear dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 L∞ norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . 1094, 1095 Lobachevskiy’s “angle of parallelism” . . . . . . . . . . . . 51 Lobachevskiy’s function . . . . . . . . . . . . . . . . . . . . . . . . 891 logarithm integrals . . . . . . . . . . . . . . . . . . . . . . . . 636, 887 logarithms . . . . . . . . . . . . . . . . . . . . . . . . 53, 237, 527, 529 and algebraic functions . . . . . . . . . . . . . . . . . 238, 538 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 747 and exponentials . . . . . . . . . . . . . . 339, 571, 573, 599 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 656 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 578 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 and inverse trigonometric functions . . . . . . . . . . 607 and powers . . . . . . . . . . 540, 542, 553, 555, 573, 594 and rational functions . . . . . . . . . . . . . . . . . . 535, 553 and trigonometric functions . . . 339, 581, 594, 599 gamma functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 Lommel functions . . . . . . . . . . . . . . . . . . . . . . . . . 760, 945 two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947 Lyapunov theorem . . . . . . . . . . . . . . . . . . . . . . 1089, 1105
M MacRobert functions . . . . . . . . . . . . . . . . . . . . . 850, 1035 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 854 and elementary functions . . . . . . . . . . . . . . . . . . . . 850 and special functions . . . . . . . . . . . . . . . . . . . . . . . . 856
INDEX OF CONCEPTS
Mathieu functions . . . . . 763, 950, 951, 953, 954, 1149 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 767 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 763 and trigonometric functions . . . . . . . . . . . . . . . . . . 763 imaginary argument . . . . . . . . . . . . . . . . . . . . . . . . . 952 matrix adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 cofactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 determinants . . . . . . . . . . . . . . . . . . . see determinants diagonal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . 1071 differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074 Hermitian . . . . . . . . . . . . . . . . . . . . . . 1070, 1077, 1089 idempotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 irreducible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 principal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 nilpotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 non-negative definite . . . . . . . . . . . . . . . . . . . . . . . 1071 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082, 1083 null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 orthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 positive definite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 reducible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 skew-symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 special . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069, 1070 triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 unitary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 mean value theorems . . . . . . . . . . . . . . . 247, 1063, 1064 Meijer functions. . . . . . . . . . . . . . . . . . . . . . . . . . 850, 1032 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 854 and elementary functions . . . . . . . . . . . . . . . . . . . . 850 and special functions . . . . . . . . . . . . . . . . . . . . . . . . 856 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . 1107, 1129 basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131 Mellin–Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . 1021 metric coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 metrical coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054 Minkowski inequality . . . . . . . . . . . . . . 1059, 1061, 1065 modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632, 859, 860 multiple angle expansion . . . . . . . . . . . . . . . . . . . . . . . . 31 multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . 607, 612
1167
N named theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 natural norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 Neumann functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910 Neumann polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 949 nome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 non-oscillation . . . . . . . . . . . . . . . . . . . . 1100, 1103, 1104 normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 compatible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 Euclidean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 induced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082, 1083 natural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083 spectral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii
O one-sided z-transform . . . . . . . . . . . . . . . . . . . . . . . . . 1135 order of presentation . . . . . . . . . . . . . . . . . . . . . . . . . xxvii ordinary differential equations . . . . . . . . . . . . . . . . . 1093 orthogonal curvilinear coordinates . . . . . . . . . . . . . 1052 orthogonal polynomials . . . . . . . . . . . . . . 795, 982, 1149 oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100, 1102 Ostrogradskiy–Hermite method . . . . . . . . . . . . . . . . . 67 Ostrowski inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 Ostrowski theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 outer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
P parabolic cylinder functions. . . .841, 849, 1028, 1150 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 845 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 843 and hypergeometric functions . . . . . . . . . . . . . . . . 849 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 848 and trigonometric functions . . . . . . . . . . . . . . . . . . 844 parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 parameter of the integral . . . . . . . . . . . . . . . . . . . . . . . 859 Parodi theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 Parseval formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 Parseval theorem . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 partial sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Perelomov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
1168
periodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 951 periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865, 870 permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Perron theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . . 1088 Picone identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Pochhammer symbol . . . . . . . . . . . . . . . . . . . . . . . . . . xliii Poincare’s separation theorem . . . . . . . . . . . . . . . . . 1087 points, singular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958 Poisson integral . . . . . . . . . . . . . . . . . . . . . . . . . 1056, 1057 poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865, 870, 874, 892 polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 254, 313, 322 and hypergeometric functions confluent . . . . . . 840 characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 Chebyshev . . . . . . . . . . . see Chebyshev polynomials degree 3 or 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 Gegenbauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 Hermite . . . . . . . . . . . . . . . . see Hermite polynomials Jacobi . . . . . . . . . . . . . . . . . . . see Jacobi polynomials Laguerre . . . . . . . . . . . . . . . see Laguerre polynomials Legendre . . . . . . . . . . . . . . see Legendre polynomials orthogonal . . . . . . . . . . . . . . . . . . . . . . . . 795, 982, 1149 positive definite . . . . . . . . . . . . . . . . . . . . . . . . . 1071, 1072 positive semidefinite . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–18, 25 expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 363 and arccosecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and arcsecant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 and associated Legendre functions . . 770, 776, 779 and Bessel functions . . . . . 664, 675, 689, 699, 708, 711, 727, 742, 831, 834 and binomials . . . . . . . . . . . . . . . . . . . . . . . . . . 315, 322 and Chebyshev polynomials . . . . . . . . . . . . . . . . . 800 and exponential integrals . . . . . . . . . . . . . . . . . . . . 627 and exponentials . . . . 148, 346, 353, 363, 364, 386, 497, 525, 573, 699, 708, 711, 742, 754, 776, 834, 842 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 652 and Gegenbauer polynomials . . . . . . . . . . . . . . . . 795 and hyperbolic functions . . 139, 148, 386, 516, 525 and hypergeometric functions . . . . . . . . . . . . . . . . 812 confluent . . . . . . . . . . . . . . . . . . . . . . . . . 820, 831, 834 and inverse trigonometric functions . . . . . 600, 601, 607 and Legendre polynomials . . . . . . . . . . . . . . . . . . . 791 and logarithmic functions . . . . . 540, 542, 553, 555, 573, 594 and parabolic cylinder functions . . . . . . . . . . . . . 842
INDEX OF CONCEPTS
powers (continued) and rational functions . . . . . . . . . . . . . . 353, 401, 553 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 754 and trigonometric functions . . . 214, 397, 401, 405, 411, 436, 459, 475, 497, 516, 525, 594, 607, 727, 742, 779 binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 hyperbolic functions . . . . . . . . . . . . . . . . . . . . 110, 120 trigonometric functions . . . . . . . . . . . . . . . . . 151, 395 principal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi natural norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 252, 528 vector norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 probability function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 probability integrals . . . . . . . . . . . . . . . . . . 629, 645, 887 and associated Legendre functions . . . . . . . . . . . 781 problem, Cauchy . . . . . . . . . . . . . . . . . . . . . . . . 1093, 1095 product finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 infinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 14, 45 of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 8 pseudo-elliptic integrals . . . . . . . . . . . . . . . . . . . 105, 184 pulse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135
Q q-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880 quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 quasiperiodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878
R radius of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 rate of change theorems . . . . . . . . . . . . . . . . . . . . . . . 1057 rational functions . . . . . . . . . . . . . . . . . . . . . . 66, 253, 254 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 789 and associated Legendre functions . . . . . . . . . . . 789 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 670 and cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171, 390 and exponentials . . . . . . . . . . . . . . . . . . . 106, 340, 353 and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . 125 and logarithmic functions . . . . . . . . . . . . . . . 535, 553 and powers . . . . . . . . . . . . . . . . . . . . . . . . 353, 401, 553 and sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171, 390 and trigonometric functions . . . . . . . . 401, 423, 447 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv reciprocal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 reciprocals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 12
INDEX OF CONCEPTS
1169
references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141 supplementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145 remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 representation theorem. . . . . . . . . . . . . . . . . . . . . . . .1056 residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870 Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Riemann differential equation . . . . . . . . . . . . . . . . . 1014 Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 1038 Riemann zeta functions . . . . . . . . . . . . . . . . . 1036, 1150 Riemann–Lebesgue lemma . . . . . . . . . . . . . . . . . . . . 1067 Rodrigues’ formula . . . . . . . . . . . . . 993, 995, 998, 1000 roots . . . . . . see square roots and Constant/Function index fourth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Routh–Hurwitz theorem . . . . . . . . . . . . . . . . . . . . . . 1086 row norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083
S saltus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 Schl¨ afli integral formula . . . . . . . . . . . . . . . . . . . . . . . . 985 Schl¨ afli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949 Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Schwarz inequality . . . . . . . . . . . . . . . . 1059, 1061, 1064 second mean value theorem . . . . . . . . . . . . . 1063, 1064 second-order equations . . . . . . 1017, 1098, 1100, 1104 self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098 semiconvergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 separation theorem . . . . . . . . . . . . . . . . . . . . . 1087, 1101 series . . . . . . . . . . . . . . . . . . . . . . . . . 860, see specific type alternating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 asymptotic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 diverge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Fourier . . . . . . . . . . . . . . . . . . . . . . . . 19, 46, 1066–1068 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 51 hypergeometric . . . . . . . . . . . . . . . . . . . . . . . 1005, 1008 generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 of exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 of logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–18 rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 semiconvergent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 trigonometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46, 862 sign function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 signum function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv
sine and rational functions . . . . . . . . . . . . . . . . . . 171, 390 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628, 639, 886 hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 886 multiple angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 sine-amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . 958, 1038 solenoidal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 Sonin theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix and hypergeometric functions confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839 and MacRobert functions . . . . . . . . . . . . . . . . . . . . 856 and Meijer functions . . . . . . . . . . . . . . . . . . . . . . . . . 856 indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 spectral radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083 spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974 square roots . . . 84, 88, 92, 94, 99, 103, 179, 184, 254 and cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 and sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 408 Steffensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv Stieltjes’ theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . 1046, 1048 table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047, 1048 Stokes phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 920 Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Struve functions . . . . . . . . . . . . . . . . . . . . . 753, 942, 1150 and Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 756 and exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 and hypergeometric functions confluent . . . . . . 838 and parabolic cylinder functions . . . . . . . . . . . . . 848 and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 and trigonometric functions . . . . . . . . . . . . . . . . . . 755 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . 1101 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . 1101 Sturm–Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . 1102 Sturmian separation theorem . . . . . . . . . . . . . . . . . 1087 subdominant solutions . . . . . . . . . . . . . . . . . . . . . . . . 1104 subordinate norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082 substitutions, Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986 summation theorems . . . . . . 940, 986, 992, 998, 1002, 1030, 1042
1170
INDEX OF CONCEPTS
sums binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 powers of trigonometric functions . . . . . . . . . . . . . 37 products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 products of trigonometric functions . . . . . . . . . . . 38 reciprocals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 12 tangents of multiple angles . . . . . . . . . . . . . . . . . . . . 39 trigonometric and hyperbolic functions . . . . . . . . 36 supplementary references . . . . . . . . . . . . . . . . . . . . . 1145 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . 1072 symbol binomial coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . xliii factorial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii integral part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii Pochhammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii synonyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii system of equations . . . . . . . . . . . . . . . . . . . . . 1094, 1095 linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 Szeg¨ o comparison theorem . . . . . . . . . . . . . . . . . . . . 1101
T table usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi tangent approximation . . . . . . . . . . . . . . . . . . . . . . . . . 921 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 termwise integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 tests, convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 19 theorems addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973, 975 arithmetic mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 Ballieu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Brauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086, 1088 Cayley–Hamilton . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 comparison . . . . . . . . . . . . . . . 1100, 1101, 1103, 1104 convolution . . . . . . . . . 1108, 1118, 1122, 1130, 1136 de Moivre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 final value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139 Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 general nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Gerschgorin . . . . . . . . . . . . . . . . . . . . . . . . . . 1083, 1088 Gram–Kowalewski . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 Green . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055, 1056 Hadamard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077 initial value . . . . . . . . . . . . . . . . . . . . . . . . . . 1136, 1139 integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 Jacobi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076
theorems (continued) Kneser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103 Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089, 1105 mean value . . . . . . . . . . . . . . . . . . . . . . 247, 1063, 1064 named . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 non-oscillation . . . . . . . . . . . . . . . . . . 1100, 1103, 1104 oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100 Ostrowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 Parodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 Parseval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067, 1068 Perron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Perron–Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Poincare’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072 rate of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 reciprocal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056 Routh–Hurwitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086 second-order equations . . . . . . . . . . . . . . . . . . . . . 1100 separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Sonin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Stieltjes’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Stokes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Sturm comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 Sturm separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 Sturm–Picone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102 Sturmian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 summation . . 940, 986, 992, 998, 1002, 1030, 1042 Szeg¨ o comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 vector integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 Wielandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633, 877 Thomson functions . . . . . . . . . . . . . . . . . . . . . . . . 761, 944 total variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084 transformation formulas. . . . . . . . . . . . . . . . . . . . . . .1008 transforms Fourier . . . . . . . . . . . . . . . . . . . . see Fourier transform fractional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 Hankel. . . . . . . . . . . . . . . . . . . . . see Hankel transform integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 Laplace . . . . . . . . . . . . . . . . . . . see Laplace transform Mellin . . . . . . . . . . . . . . . . . . . . . . see Mellin transform of a derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 triangle inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 trigonometric functions . . . . . . . . . . . 28, 151, 390, 415 and algebraic functions . . . . . . . . . . . . . . . . . . . . . . 434 and associated Legendre functions . . . . . . . . . . . 779 and Bessel functions . . . . . . . . . . . 717, 727, 742, 747
INDEX OF CONCEPTS
1171
trigonometric functions (continued) and exponentials . . . . 227, 339, 485, 493, 495, 497, 522, 525, 599, 742 and gamma functions . . . . . . . . . . . . . . . . . . . . . . . . 655 and hyperbolic functions . . . . . . 231, 509, 516, 522, 525, 747, 763 and hypergeometric functions . . . . . . . . . . . . . . . . 817 confluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 and inverse trigonometric functions . . . . . 605, 607 and logarithmic functions . . . . . 339, 581, 594, 599 and Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 763 and parabolic cylinder functions . . . . . . . . . . . . . 844 and powers . . . . . 214, 395, 397, 401, 405, 411, 436, 459, 475, 497, 516, 525, 594, 607, 727, 742, 779 and rational functions . . . . . . . . . . . . . . 401, 423, 447 and square roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . .408 and Struve functions . . . . . . . . . . . . . . . . . . . . . . . . 755 inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56, 241 powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151, 459 trigonometric series . . . . . . . . . . . . . . . . . . . . . . . . 46, 862 triple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 triple vector product . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 two-sided z-transform . . . . . . . . . . . . . . . . . . . . . . . . . 1135
V Vandermonde determinant . . . . . . . . . . . . . . . . . . . . 1078 variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097 variational principles . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 vector differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049 product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049
W Weber functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 Weierstrass elliptic functions . . . . 626, 873, 880, 1148 Weierstrass expansions . . . . . . . . . . . . . . . . . . . . . . . . . 869 weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982 Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 Wielandt theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 Wronskian determinant . . . . . . . . . . . . . . . . . . . . . . . 1079
Y Young inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065
U
Z
uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 unilateral z-transform . . . . . . . . . . . . . . . . . . . 1135, 1138 unit integer function . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 unit integer pulse function . . . . . . . . . . . . . . . . . . . . 1135 use of the tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi
zeros . . . . . . . . . . . . . . . . . . . . . . 865, 870, 879, 972, 1038 interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 simple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1000 zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 z-transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135
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