AN ATLAS OF FUNCTIONS Jerome Spanier The Claremont Graduate School
Claremont. California. U.S.A.
Keith B. Oldham Trent...
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AN ATLAS OF FUNCTIONS Jerome Spanier The Claremont Graduate School
Claremont. California. U.S.A.
Keith B. Oldham Trent University Peterborough. Ontario. Canada
O HE1VHSPHERE PUBLISHING CORPORATION A subsidiary of Harper & Row. Publishers, Inc.
Washington
New York
London
Distribution Outside North America
SPRINGER-VERLAG Berlin
Heidelberg
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London
Paris
Tokyo
AN ATLAS OF FUNCTIONS Copyright G 198' bs Hemisphere Publishing Corporation. All rights reserved Printed in the Urated States of America. Except as permitted under the United States Copyright Act of 1976. no pan of this publication may be reproduced or distributed in any form or by any means. or stored jr. a data base or retneNai system. without the prior written permission of the publisher.
1234567890 EBEB 89876 This book was set in Times Roman by Edwards Brothers. Inc. The editor was Sandra Tamburrino. Edwards Brothers. Inc.. was printer and binder.
Libran of Cow Cataloging in Publication Data Spanter. Jerome. date
An atlas of functions.
Bibliography: p. Includes indexes 1. Oldham. Keith B. D. Title. QA331.S695
1997,
ISBN 0.89116-573-8
515
86.18294
Hemisphere Publishing Corporation
DISTRIBUTION O( WE NORTH AMERICA: ISBN 3-540-17393-1
Berlin
CONTENTS
Preface
lx
0
General Considerations
1
The Constant Function c
11
2
The Factorial Function n! and Its Reciprocal
19
3
The Zeta Numbers and Related Functions
25
I
4 The Bernoulli Numbers, B,
35
5
The Euler Numbers, En
39
6
The Binomial Coefficients (m )
43
7
The Linear Function bx - c and Its Reciprocal
53
8
The Unit-Step u(x - a) and Related Functions
63
9
The Integer-Value Int(x) and Fractional-Value frac(x) Functions
71
10
The Dirac Delta Function S(x - a)
79
11
The Integer Powers (bx + c)" and x"
83
12
The Square-Root Function -.'bx + c and Its Reciprocal
91
13
The Noninteger Powers x'
99
14
The b.ia= - x: Function and Its Reciprocal
107
15
The bv'.t= + a Function and Its Reciprocal
115
CONTENTS
vi
16
The Quadratic Function ax' + bx + c and Its Reciprocal
123
17
The Cubic Function x' + ar= + bx + c and Higher Polynomials
131
18
The Pochhammer Polynomials (x).
149
19
The Bernoulli Polynomials B,(x)
167
20
The Euler Polynomials E,(x)
175
21
The Legendre Polynomials P,(x)
183
22
The Chebyshev Polynomials T,(x) and U,(x)
193
23
The Laguerrre Polynomials L,(x)
209
24
The Hermite Polynomials H,(x)
217
25
The Logarithmic Function ln(x)
225
26 The Exponential Function exp(bx + c)
233
27 Exponentials of Powers exp(- arxD
253
28 The Hyperbolic Sine sinh(x) and Cosine cosh(x) Functions
263
29 The Hyperbolic Secant sech(x) and Cosecant csch(x) Functions
273
30 The Hyperbolic Tangent tanh(x) and Cotangent coth(x) Functions
279
31
The Inverse Hyperbolic Functions
285
32
The Sine sin(x) and Cosine cos(x) Functions
295
33
The Secant sec(x) and Cosecant csc(x) Functions
311
34
The Tangent tan(x) and Cotangent cot(x) Functions
319
35
The Inverse Trigonometric Functions
331
36
Periodic Functions
343
37 The Exponential Integral Ei(x) and Related Functions 38
Sine and Cosine Integrals
351
361
39 The Fresnel Integrals S(x) and C(x)
373
40 The Error Function erf(x) and Its Complement erfc(x)
385
41
The exp(x) erfc(f) and Related Functions
395
42
Dawson's Integral
405
vU
CONTENTS
43 The Gamma Function r(x)
411
44 The Digamma Function $(x)
423
45
435
The Incomplete Gamma y(v;x) and Related Functions
46 The Parabolic Cylinder Function D,(r)
445
47 The Kummer Function M(a;c;x)
459
48 The Tricomi Function U(a;c;x)
471
49 The Hyperbolic Bessel Functions L(x) and I#)
479
50 The General Hyperbolic Bessel Function 1,(x)
489
51
The Basset Function K,(x)
52 The Bessel Coefficients J,(x) and J,(x)
499 509
53
The Bessel Function J,(x)
521
54
The Neumann Function Y,(x)
533
55
The Kelvin Functions
543
56
The Airy Functions Ai(x) and Bi(x)
555
57
The Struve Function
563
58
The Incomplete Beta Function B(v;µ;x)
573
59 The Legendre Functions P,(x) and Q(x)
581
60 The Gauss Function F(a,b;c;x)
599
61
The Complete Elliptic Integrals K(p) and E(p)
609
62
The Incomplete Elliptic Integrals F(p;b) and E(p;4)
621
63
The Jacobian Elliptic Functions
635
64
The Hurwitz Function ((v;u)
653
Appendix A Utility Algorithms
665
Appendix B Some Useful Data
673
References and Bibliography
679
Subject Index
681
Symbol Index
691
PREFACE
The majority of engineers and physical scientists must consult reference books containing information on a variety of mathematical functions. This reflects the fact that all but the most trivial quantitative work involves relationships that are best described by functions of various complexities. Of course, the need will depend on the user, but all will require information about the general behavior of the function in question and its values at a number of arguments. Historically. this latter need has been met primarily by tables of function values. However, the ubiquity of computers and programmable calculators presents an opportunity to provide reliable, fast and accurate function values without the need to interpolate. Computer technology also enables graphical presentations of information to be made with digital accuracy. An Atlas of Functions exploits these opportunities by presenting algorithms for the calculation of most functions to more than seven-digit precision and computer-generated maps that may be read to two or three figures. Of course, the need continues for ready access to the many formulas and properties that characterize a specific function. This need is met in the Atlas through the display of the most important definitions, relationships, expansions and other properties of the 400 functions covered in this book. The Atlas is organized into 64 chapters, each of which is devoted to one function or to a number of closely related functions; these appear roughly in order of increasing complexity. A standardized format has been adopted for each chapter to minimize the time required to locate a sought item of information. A description of how the chapters are sectioned is included in Chapter 0. Two appendixes, a references/bibliography section and two indexes complete the volume. It is a pleasure to acknowledge our gratitude to Diane Kaiser and Charlotte Oldham who prepared the typescript and
to Jan Myland who created the figures. Their skill and unfailing good humor in these exacting tasks have exceeded all expectations. We would also like to record our indebtedness to our families for their support and patience during the many years that the preparation of the Atlas took time that was rightfully theirs. To them, and especially to Bunny and Charlotte, we dedicate this book. Jerome Spanier Keith B. Oldham
ix
CHAPTER
T GENERAL CONSIDERATIONS
In this chapter are collected some considerations that relate to all, or most, functions. The general organization of the Atlas is also explained here. Thus, this could be a good starting point for the reader. However, the intent of the authors is that the information in the Atlas be immediately available to an unprepared reader. There are no special codes that must be mastered in order to use the book, and the only conventions that we adopt are those that are customary in scientific writing. Each chapter in the Atlas is devoted to a single function or to a small number of intimately related functions. The preamble to the chapter exposes any such relationships and introduces special features of the subject function.
0:1 NOTATION The nomenclature and symbolism of mathematical functions are bedevilled by ambiguities and inconsistencies. Several names may be attached to a single function, and one symbol may be used to denote several functions. In the first section of each chapter the reader is alerted to such sources of confusion. For the sake of standardization, we have imposed certain conventions relating to symbols. Thus, we have eschewed boldface and similar typographical niceties on the grounds that they are difficult to reproduce by pencil on paper, or with office machines. We reserve the use of italics to represent numbers (such as constants. c: coefficients, a a,, a3, ... and function arguments. x) and avoid their use in symbolizing functions. Another instance of a convention intended to add clarity is our distinction in using commas and semicolons as separators, as in F(a.b;c;x). Elements in a string may be interchanged when they are separated by commas but not where a semicolon serves as separator. The symbol z(=x + iy) is reserved to denote a complex variable. All other variables are implicitly real.
0:2 BEHAVIOR Some functions are defined for all values of their variable(s). For other functions there are restrictions, such as - I
< x s I or n = 1, 2, 3...., on these values that specify the range of each variable and thereby the domain of the function. Likewise, the function itself may be restricted in range and may be real valued, complex valued or each of these in different domains. Such considerations are discussed in the second section of each chapter. This section also reveals how the function changes in value as its variables change throughout their ranges. thereby exposing the general "shape" of the function. This information is conveyed by a verbal description, sup1
2
GENERAL CONSIDERATIONS
0:3
plemented by a graphical "map." All diagrams have been computer generated and are precisely drawn. Each graph has been scaled so that the graticule spacing is either ten or fifteen millimetres (2.00 or 3.00 mm between adjacent dots), which permits interpolation to an accuracy of two or three significant digits. Much greater precision is available from the algorithms of Section 8. A univariate function (i.e., a function depending only on a single variable, its argument) is generally illustrated by a simple graph of that function versus its argument. Bivariate functions of two continuous variables are often diagrammed by contour maps whose curves correspond to specified values of the function and whose axes are the argument and the second variable. In discussing the behavior of some functions it is convenient to adopt the terminology quadrant. There are
four quadrants defined as illustrated in Figure 0-1.
FIG 0-1 x>0
x 0
f (x) >0
second
first
quadrant
quadrant
........................................
third
fourth
quadrant
quadrant
0
f(x)>
f=n!
1
v>I
v>0
THE ZETA NUMBERS AND RELATED FUNCTIONS
27
3:5
and
30) =
3:3:4
2r(v)
v>0
t"-' sech(r) dt
[o
involving functions discussed in Chapters 13, 26,29 1 and 43.
The most commonly encountered definitions of zeta, lambda, eta and beta numbers are via the infinite series 3:6:1-3:6:4. These series may be reformulated as limits, for example: I
I
Y(v)=lim
3:3:5
J-=
ll =limYj" '
I
3
2"
J -=
and apply generally for v > I in the cases of i(v) and A(v), or for v > 0 in the q(v) and p(v) cases. However, when sufficient additional terms are included in series 3:3:5, to give 3:3:6
;(v) = lim J-=
I+
I
2"
+
l
+
+
3`
1
(J - 1)"
1
+
1
(v - 1)J"
+ '
2J"
+
v
v(v + 1)(v + 2)
121"''
720J""
the limit provides a definition of the zeta function for any order v: positive, negative or zero. The general expression
for the kth appended term in 3:3:6 involves functions from Chapters 2, 4 and 18 and is (v)j_1B5/(k!Jr*'-'). It is necessary to include only those appended terms for which v + k - I is negative or zero, but a few extra will speed the approach to the limit. The corresponding definition of the beta function is
+-+ p(v)=lim I--+--...J- = 5" (J - 2)' 2J" 2J'-' 3" 1
3:3:7
1
I
l
v
v(v + I)(v + 2)
6J"3
+...1
the kth appended term being (v)t_,2'(2' - I)BR/(2k!J'-"-') in this case, J being a number equal to 4n + I where n is a natural number.
3:4 SPECIAL CASES The zeta, lambda, eta and beta numbers may be regarded as special cases of the corresponding functions. No further specialization is fruitful.
3:5 INTRARELATIONSHIPS The zeta and beta functions satisfy the reflection formulas
(1 - v) =
3:5:1
2r(v)4(v) (2a)"
V'Tr
cost 2
p(I - v) = I -J r(v)p(v) sine
3:5:2
involving the gamma function [Chapter 43) and the functions of Chapter 32.
With f(n) representing any one of the four numbers I,(n), Mn). 11(n) or (3(n). one may sum the infinite
series E(-1)"f(n)/n, as well as the series of complements E[I - f(n)], E(-1)"(1 - f(n)]. E[I - f(n)]/n and F(-1)"[1 - f(n)]/n. With the lower summation limit taken as n = 2, these sums are as shown in Table 3.5.1. These sums involve the logarithmic function [Chapter 25] of various constants. Archimedes number it (Section 1:7), Euler's constant y (Section 1:7) and the ubiquitous constant U = rr(.'-)/V ar = 0.8472130848 [see Section 1:7). Also listed in Table 3.5.1 are the sums E(- I)"f(n). Strictly, these particular series do not converge, the tabulated entries being the limits 3:5:3
lim ff(2)
- f(3) + f(4) -
1
= f(J - I) ± - f(J)
which do converge and whose values may be associated with E(-1)"f(n).
2
THE ZETA NUMBERS AND RELATED FUNCTIONS
3:6
28
Table 3.5.1 f(n)
(
f=;
(-1)' n
1), fin)
f=A
In(2)
7 2
r,
4
-In(V2)
+ In (-)
Y- 1
2
-
In(2) - I
\'n.
Y
- In(2)
2
In (4
In(4) - I
f-d
2
V_` -In(-I 4 U n
2
3
In(4) - 1
z - In(4)
- In(\n) - I I
4
2
I-In4)-Y
\\n
2
8
n n -+In(-)-I .\'8U
I
IT
I - fin)
1- IM2) - Y
_
In(n) -
-+In(\2)--n4
-+In(\2)-I 4
CC
n
-
-1
Y
1
1 - f(n)
)-1)'[I - f(n))
11 - f(n)]
I - In(-) n
1-In(V2U)--4
3:6 EXPANSIONS The series
C(v)+2 +3 1
3:6:1
1
1
1
3"
5"
v>I
v>1
3:6:2
TI(v)=1_' 1
3:6.3
3:6:4
_
1
]
1
3''
5'
v>0
v>0
are the most useful representations of the four functions and serve as definitions of the zeta and lambda numbers,
l(n) and X(n), for n = 2, 3, 4, ... as well as for the eta and beta numbers, q(n) and [3(n), for n = 1. 2. 3, .... Zeta and lambda functions are expansible as the infinite products 1
3:6:5
6(v)
1 -2 I
3:6:6
1
1
I
I
1
I
1 -3 "I -5-" 1 - 7-" 1
I -3-''1 -5-"1 -7-"l - 11-
ri 1
7r 1
v>I
v>1
where a; is the jth prime number.
3:7 PARTICULAR VALUES In Table 3.7.1 Z has the value
3:7:1
Z = 4(3) = 1.202056903
and G is Catalan's constant [see Section 1:7].
For n = 2, 4, 6, ... all zeta, lambda and eta numbers equal e' multiplied by a proper fraction [the fraction is related by equation 3:13:1 to the Bernoulli number B. of Chapter 4]. Similarly, for n = 1, 3, 5, ..., P(n) is proportional to n", the proportionality constant being a proper fraction related to the Euler number E,_, [see Chapter
29
THE ZETA NUMBERS AND RELATED FUNCTIONS
3:8
Table 3.7.1
v=-5
v=-4
-I
'=-2
v=-3 I
1
252
120
12
31
-7
1
0 252
120
I
-1
0 1
1
4
2
0
6
-
0
v=4
v=s
a'
Z 90
u'
a'
72
8
8
%
n'
32
7a'
4
720
In(2)
-1
5
v-3
a'
1
2
0
8
0
v-2
v= I
12
0 4
v= 0
0
0
WY)
v=-1
12
1
rr
2
d
G
0
-
rr' 32
A(4)
51 by equation 3:13:2. For negative integer orders, the zeta and beta functions are related much more simply to the Bernoulli and Euler numbers; thus: 1,(-n) =
3:7:2
" -B+ I
n=1,2,3,...
and
n=0, 1,2....
3:7:3
3:8 NUMERICAL VALUES We present two algorithms. The first is designed to calculate zeta, lambda and eta numbers for n >- 5. so that, in conjunction with the table in Section 3:7. one may evaluate 4(n), x(n) or rl(n) for all positive integer orders. This algorithm is based on expansion 3:6:6, using primes as large as 43, and on relation 3:0:1. The precision exceeds 24 bits.
Input n >-,
Storage needed: n, k and f Set k = 7 + 2 Int(9/n) =
Input c >
Set f=(c-2")/(1-2) k
_
(1) If frac(k )fracO = 0 go to (3)
(2) Replace f by f/(t - k-") (3) Replace k by k - 2
Input restrictions: n >- 5; code c = 0. 1 or 2 function code, c ;(n) 0 1
x(n)
2
r((n)
Ifk>8goto(1)
fo = ;(n) f.
J
Mn)
rl(n)
If k > 2 go to (2)
Test values:
Output f
;(6) = 1.01734306 A(6) = 1.00144708
K> If v = I go to (7) Set f = (3HcH-2 - c)/4
Use degree mode or change
Ifv=0goto(7)
90 to rr/2.
Set f = (1cl - 2")/(I - 2") 1
If v
- go to (3)
Replace v by I - v Set w = v
ifc I
3:10:1
and 3:10:2
( 1
0(t)dr = v +
0
(-I)' In(2j + 1)
[1 - (2j + 1)-']
v -> 0
but not as established functions. At v = 0 the derivative of the zeta function equals -InV 22rr.
3:11 COMPLEX ARGUMENT The representation of 4(v + iµ) in terms of its real and imaginary parts is given by 3:11:1
cos{µ In(k)}
sin{µ ln(k)}
k'
k"
> 1
l;(v + iµ) = ,_,
3:12 GENERALIZATIONS The four functions of this chapter are special cases of the Hurwitz function of Chapter 64. Thus: 3:12:1
{(v) = (v:I)
THE ZETA NUMBERS AND RELATED FUNCTIONS
3:13
M(v) = 2-% v,
3:12:2
32
1) 2
1
3:12:3
*v) = 2 " v: 2) - ;(v:I) = J(v:1)
S(v) = 4 "LL1
3:12:4
v:4) - C v: 4)] = 2
v: 2)
The last pair of equations shows that the bivariate eta fu nction [see Section 64:13] is also a generalization of the eta and beta functions.
3:13 COGNATE FUNCTIONS When n is even t(n) is related to the Bernoulli number B. [Chapter 4] by
I(n)=
3:13:1
(2a)"IB"I
n=2,4,6,...
2n!
For odd n the zeta number is related to the Bernoulli polynomial [Chapter 191 via the integral {(n) _
3:13:2
(2a )"
n = 1. 3, 5, ...
B"(r)cot('nt)dr
2n!
The beta number of odd argument is related to the Euler number E"_1 [Chapter 5) by
n l IE.-,I 2/ 2(n - 1)!
[3(n)=
3:13:3
n=1,3,5....
while for even n the relationship is to the integral of an Euler polynomial [Chapter 201 3:13:4
n=2.4,6,
E,(t)scc(at)dr
[i(n)
= 4(n - 1)!
..
0
3:14 RELATED TOPICS The four number families occur as coefficients in power series expansions of trigonometric and hyperbolic functions of argument ax or arx/2: 2 cot(ax)_---jlQn)x7rx
-1<x>
Setf=0 p»» If frac(n/2) + 0 go to (1) Set f = (2/7r)"+' n!/5
Replace f by -1 - 10 Int(f)
1
Input restriction: n must be a nonnegative integer
If frac(n/4) + 0 go to (1) Replace f by 4 - f
Test values:
Setf= 1
ES=0
Ifn+0goto(1)
f=E"
0.
c
-x < - < -X b
c -x0 x
Set n = 0 Input x >> (3) If Int(x) * 0 go to (4) Replace x by 32r
Intervention to halt output is needed if x is recurrent in base 32.
Replace it by it - 1 Go to (3) (4) Replace x by x/32 If Int(x) = 0 go to (5)
Replace n by n + 1
Go to (4)
Test values:
(5) Output n
x= a,02=2
(6) Replace x by 32x
Output: n = 1 followed by the digits
n n. The numerical values of all other coefficients may be calculated from the recursions Yo' _ y('-0/2, y(') = (,y('-')/2) + yo-" and for m ? 2 y." = (y."_-." + Expansions similar to 11:6:5, but involving the Chebyshev polynomials of the second kind U"(x), also hold. In fact, such expansions exist with any set of orthogonal polynomials [Chapters 21-241.
11:7 PARTICULAR VALUES For b * 0 one has
X --= Ibx + cf. n = -2. -4 -6....
(bx+c)".a=-1,-3.-5..-. (bx+c)' (bx+c)".n- 1.3.5....
x
c-1 b
0
l
0
-I
I
x =
Ibx + c)", n = 2. 4. 6, ...
X
_-cb-
x -I -b-c
x . x
1
0
1
0
=
1
1
0 0
-I 1
1
1
x =
11:8 NUMERICAL VALUES These arc readily calculated using the "y'" key present on most programmable calculators or using the corresponding computer instruction. Where such a facility is absent, the simple algorithm below may be employed.
Setf Input n >> Input x >>
Storage needed: n, x and f 1
Ifn=0goto(2)
Ifn>0goto(I)
Input restrictions: n must be a nonnegative integer but x is unrestricted.
Replace x by I /x
Replace n by -n (1) Replace fby xf Replace n by n - 1
Ifn*0goto(I)
f=X,
88
12 'v n(
Icy
ibi
11:10 OPERATIONS OF THE CALCULUS Differentiation and integration of the integer powers are easily accomplished: d
11:10:1
(bx+c"= (bx + c)"-'
11:10:2
r Or + c)'dt =
11:10:3
n=0.1.2....
b(n + 1)
(bt + c)"dt =
J
x
n=-1.-2.-3.
x
n = -1, 0. 1. 2.
(bx + c)
n-2.-3.-4,
b(-n - 1) (bx, + c)"'' - (bxo + c)"
11:10:4
J"
n = 0, 1, ±2,±3....
b(n + 1)
(bt + c )'dt =
bx, + c
1
b
n= -1
bxo+c)
Differintegrals of the simple powers x' are given by the formula
nom-'
11:10:5
n=0,1,2....
f(n-v+l)
dx'
where r is the gamma function of Chapter 43.
11:11 COMPLEX ARGUMENT
Forn= 1,2,3,... n(n - 1)
(x + iy) = x" -
n(n - 1)(n - 2)(n - 3)
x"-'y2
+
2!
n(n - 1)(n - 2)
r + i nx"''y -
5!
X. 'y' +
n(n - I)(n - 2)(n - 3)(n - 4)
3!
(n)()2)k(n )(-?)
x" `y' 4!
where each upper limit in the summation is an integer chosen to make the final binomial coefficient (:) or Some examples are 11:11:2
(x + iy)2 = (x2 - y2) + i(2xy)
11:11:3
(x + iy)' = (x' - 3x 2) + i(3x2y - y')
11:11:4
(x + iy)4 _ (x' - 6x2y2 + y') + i(4x'y - 4xy3)
89
THE INTEGER POWERS (bas + c)' AND x
11:14
The corresponding formulas for negative n involve infinite sums
i
11:11:5
- 1J ) r ? 1 _
(2k-
(x+iy)"=x'
2k
-,y j (1k
J
L
k-0
[?]* + 1)
x
An example is
/2
1 3y' 5v' (x + iy)-x = (X-2 - x +
11:11:6
xr
xb
- ss +
jr,
- ... I
11:12 GENERALIZATIONS In the present chapter the power is restricted to be an integer. This restriction is removed in Chapter 13.
11:13 COGNATE FUNCTIONS Any polynomial, including all the functions discussed in Chapters 16 through 24. consists of a finite sum of the functions of this chapter.
11:14 RELATED TOPICS Many functions of x can be expanded as an infinite series: _ i a,x'
f(x) = ao + a,x + axxx +
1 1 : 1 4 : 1
I-o
of terms of which ax' is typical, j being a nonnegative integer and a, a constant. Such sums are known as power series and examples will be found in Section 6 of most chapters of this Atlas. In the notation of Chapter 17, power series are polynomial functions of infinite degree, and could be symbolized p.(x). The somewhat more complicated series aox° + a,x°'B + a,X'-' +
11:14:2
a1.t
-o
may also be treated as a power series because a redefinition of the argument and isolation of the factor x' relates 11:14:2 to the 11:14:1 function
Za,x°la=xf(xe)
11:14:3
i-o
Depending on the values of the coefficients a,, power series may converge for all x, only over a certain range of argument values. or may represent an asymptotic expansion near some x = xo value [see Section 0:6]. Provided that a power series converges for argument x, it may be differentiated term by term: d 11:14:4
1)a;.,x,
dx,-o
or integrated:
dt=i-o j
Jo i-a
11:14:5
i-i
j
to yield another power series.
Power series may also be raised to a power where
11:14:6 o
o
(jn-kn-k)a,_,c,
co=ao,c,= jao t-o
j= 1,2,3,.,.
THE INTEGER POWERS (bx + c)' AND x"
11:14
90
Similarly, two power series may be multiplied: where
11:14:7
Y, akbj_2
c,
or divided:
a1x
x,
11:14:8
'-a
"X'
where
ci =
bi
1'
a
bi-icA.
bo k-0
bo
1-0
If f is given by the power series Ea,x', then the inverse function (see Section 0:3] is another power series of
argument If - ao)/a namely so c1(f
11:14:9
a2
where c, = a,, c2 = -a,a,, c, = 2ala22 - alla,, c. = Sa a2a, - Sa,a; - a3la., c, = 14a,a2 + 3aja3 + 21 a;a';a, - a;a, and ce = 84a;a, + 7o;a,a. + 7a;a a, - 42a,a2 - 28a3,a - 28a3a2'a, - a;ab. The procedure of converting a power series for f(x) into a power series for x is known as reversion of series. Any function that can be repeatedly differentiated may be expanded as a power series by utilizing the Maclaurin series (the special y = 0 case of the Taylor series 0:5:1]: df 11:14:10
f(x) = f(0) + x
dx
d2f (0) +
(0) + -
2
T2
=
x'
dJf (0)
1_o j! dx'
Whether or not such a series is convergent, it provides an asymptotic representation of f(x) as x -a 0. Functions expansible as power series may also be represented as series of Bessel functions [see Section 53:14] or as continued fractions [Section 0:61.
CHAPTER
12 THE SQUARE-ROOT FUNCTION bx + c AND ITS RECIPROCAL
Functions involving noninteger powers are known as algebraic functions. The square-root function V and the reciprocal square root 1/Vx are the simplest algebraic functions. In this chapter, as in the previous one, we generalize the argument of these simplest algebraic functions and consider mainly the bx + c and 1/vi"" + c functions.
A graph of 'V bx + c versus x generates a curve known as a parabola and the adjective parabolic is therefore appropriately applied to the (bx + c)'r function. Some geometric properties of the parabola are noted in Section 12:14.
12:1 NOTATION Especially in computer applications V' is sometimes denoted SQRT(x) or SQR(x). The notation lx is also occasionally encountered. The symbols x"' and V xoften interpreted as defining equivalent functions but in this Atlas we make a distinction between the two. If x is positive x 12 has two values, one positive and one negative. The square-root function f, however, is single valued and equal to the positive of the two x"= values. Hence the relation between the two functions is 12:1:1
bx+c=I(bx+c)"'1
or
12:1:2
(bx + c)"2 = *_
bx +
Si m ilarl y
12:1:3
I
bx+c
= Ox + c)-'"I
and c)-1"2
12:1:4
(bx +
=
91
12:2
THE SQUARE-ROOT FUNCTION
bx --c AND ITS RECIPROCAL
92
12:2 BEHAVIOR Figure 12-I is a map of the functions V box + c and I /V bT under standard conditions, that is, when b and c are both positive. The orientation of
bx + cchanges to those shown in Figure 12-2 when b and/or care negative.
The Vbx + c function is not defined for values of x more negative than -c/b. The function itself takes all nonnegative values. As Figure 12-I shows, V b x+ is zero at x - -c/b, at which point it has an infinite slope. Likewise, 1/\'bx + c is defined only for bx > -c and takes all positive values. Figure 12-1 reveals that the reciprocal square-root function steadily declines from an infinite value at -c/b toward zero as x - x.
FIG 12-1
12:3 DEFINITIONS The square-root function is defined as the inverse of the square function (Chapter I11. Thus, V'bx + c is the
nonnegative number whose square equals bx + c; that is: 12:3:1
bx + c = Ill
where
f2 = bx + c
A parabola is defined geometrically as constituting all points P whose distance PF from a fixed point F (called the focus of the parabola) equals the shortest distance from P to a straight line DD (the directrix of the parabola). If the x-axis of a Cartesian coordinate system is placed along DF. the shortest line joining the directrix to the focus bx + c (see Figure 12-3), and if d is the length of that shortest line, then the equation of the parabola is f =
where b = 2k and c = k2 + lay, y being the distance from the focus to the coordinate origin.
THE SQUARE-ROOT FUNCTION
93
bx T -c AND ITS RECIPROCAL
12:5
12:4 SPECIAL CASES When b = 0 both
bx + c and I/V bx c reduce to constants.
12:5 INTRARELATIONSHIPS The multiplication and division of square-root functions generate root-quadratic functions [see Section 15:121: 12:5:1
b,x + c,
byr + r. _
b,b x1- + (bic: + b.ci)x + c,c,
(b,.,, - c,)(b,x + c,) =
and
12:5:2
bix+ci/ b.r+c,_
(b,x
+c,)/(b,x+c,)_
%lb b x ,
,
' + (b c. ,
+ b . c )x + c c , ,
,
bx+c,
while the rule 12:5:3
6.r+c)
j(bx+c)"" J(br+c)":I
n=0,2.4.... n= 1.3.5.-..
governs the raising of V bx + c to an integer power.
If f(x) = V'bx + c, then the reflected function f(-x) coexists with f(x) only if c is positive and then only in the range -c/jbI 0
The function d' in 12:10:7 is given by
b>0
x > -c/b
b 0 v
nfm =1,2.3_] n2,4,6.... J
0
(-1)"
undef
undef
undef
undef
=
I
0
_1
0
1
undef
0
I
1
I
undef
undef
0
1
under
undef
0
±1
(-1)"s
(-1)"
0
1
Al
n1 . 2, 3. ... v=-m n-1,3,5,... n
13:8 NUMERICAL VALUES permit values of x" to be found quickly and easily, as do Computer language instructions such as x ' v or keys on most programmable calculators. Such operations often accept only positive x values and yield only firstquadrant [see Section 0:21 values of x". Care is therefore required to interpret the result correctly when v is rational. Of course, the reflection formula 13:5:1 may be used to calculate x' when x is negative and v is a suitable rational number.
THE NONINTEGER POWERS x'
103
13:10
Alternatively, the following simple algorithm, based on the formula x' = exp(v In(x))
13:8:1
[see Chapters 25 and 26 for the In and exp functions], may be used.
Input x »»
Storage needed: v, x and f I
Setf = ln(x)
Input restrictions: The argument x must be positive but v is unrestricted. Output values are positive.
Input v »» Replace f by exp(vf)
f=x'««
- l
x >0
and
13:10:3
1
(-t)'dt =
-( -.t)"
i
v+1
v< - I
x -1; otherwise we have
J`t'dt=1n(x)
13:10:4
v=-I
and
t'dt = - v < - I
13:10:5 I
v+1
A general expression for the indefinite integral of a power function multiplied by a linear function raised to any power involves the incomplete beta function [see Chapter 58]:
THE NONINTEGER POWERS x"
13:10
c-'.' 13:10:6
b""
t"(bt + c)"dt =
B v+ l;-µ-v+1;
104
bx
b>0
bx+c
01
blal
FIG 14-1 b,e/ l-E
The second geometric definition of an ellipse is as the locus of points P such that the sum PF + PF' of the distances from P to two points F and F' (the foci of the ellipse) obeys the relationship 14:3:3
PF + PF' = a constant = o
Of course, a must exceed the interfocal distance FF' = 4. If a cartesian coordinate system is erected with its
THE 6V.' -72 FUNCTION AND ITS RECIPROCAL
109
14:7
origin 0 at the midpoint of the line FF', which is chosen as the x-axis, then the equation of the ellipse is again ±b a- - x- where a = a/2 and b = I - ((b/a)'.
14:4 SPECIAL CASES The semielliptic function bV a -i becomes the semicircular function V-. when b = I (see Figure 14-21. The parameter jal is then known as the radius. The geometrical definition of a circle is as the locus of all points P lying at a constant distance (equal to the radius p) from a fixed point called the center of the circle. If the origin of a cartesian coordinate system is placed
at the center, then the equation of the circle is f = x a- - r- where a = ±p. Other geometrical properties of the circle are discussed in Section 14:14.
14:5 INTRARELATIONSHIPS The function f(x) = W a - x is an even function, that is, the reflection relationship
f(-x) = f(x)
14:5:1
is satisfied.
The multiplication formula 14:5:2
f(vx) = by
(a/v)' - x'
shows that multiplication of the argument of a semielliptic function by a constant generates another semiellipse, one semiaxis being unchanged. Choosing v = 1/b generates a semicircle of radius jabl.
14:6 EXPANSIONS Two useful infinite expansions are
x a'-x==la1b2a'1 -----
14:6:1
X2
'16
8a'
16a°
x - 128x°
\x'1 and I
14:6:2
ba' - xZ
=
I jajb
1+-+-+-+-+ = l x'
3x'
5x°
35x°
2a'
8a'
16a°
128x"
1
-1/2
Ialb J=u \
J
lI ax1
-a<x -x. These asymptotes
asymptote of the bV +x function. Similarly, -bx is an asymptote of b are shown in Figures 15-I and 15-2.
15:3 DEFINITIONS Algebraic and arithmetic operations define b x + a and its reciprocal in a straightforward fashion. A hyperbola may be defined geometrically as the locus of all points P such that the length of the line PF connecting P to a fixed point F (a focus of the hyperbola) obeys the relationship 15:3: I
PF
_ the eccentricity _ a constant PD = greater than unity of the hyperbola - e
1<E<x
with respect to the length PD of the shortest line from P to the straight line DD (called a directrix of the hyperbola). +x is The eccentricity of the hyperbola ±bV 1 + b-. An illustration of relationship 15:3:1 is shown in Figure
15-3. Note that the hyperbola of Figure 15-3 has two branches and that there exists a second directrix D'D' and a second focus F' symmetrically positioned with respect to a line parallel to both directrices and midway between them.
An alternative geometric definition of a hyperbola is as the locus of all points P such that 15:3:2
IPF - PF'I = constant
where F and F' are the foci.
15:4 SPECIAL CASES When a - 0, the bV7x 'T-aand I /(bx' -+a) functions reduce to the linear bx and reciprocal linear 1/(bx) functions, respectively. When b = 1, the function tbV +x becomes tV +x and is termed a rectangular hyperbola or an equilateral hyperbola and has the interesting rotational properties now to be described.
THE b
117
x2 + a FUNCTION AND ITS RECIPROCAL
15:4
If a function g(x), interpreted as a cartesian graph of g(x) versus x, is rotated through a positive angle 0 in a
counterclockwise direction about the point (X,G) then a new function f = f(x) is created that is related to g(x) by the general rotation formula
15:4:1
fcos(O) _ [x - Xjsin(B) - [1 - cos(e)]G + g(y)
y = X + [x - X]cos(h) + If - Gjsin(O)
When rotation occurs about the origin, so that X = G = 0, the simpler relationship f cos(h) _ x sin(O) + g(x cos(O) + f sin(g))
15:4:2
holds. This formula will be used to rotate the function g(x) = ±Vx +a about the origin through angles of
-ir/4 and -n/2. The equation f/V 2 = -x/V2- _ -n/4 in 15:4:2, simplifies to
[(x/V - f/V)[' + a, obtained by setting g(x) = ± x + a and 0 = a
f=-
15:4:3
2.x
Similarly, the equation 0 = -x ± V(f)2 a, which arises from setting g(x) to ±V
+x
a and 0 to -e/2 in
equation 15:4:2. reduces to
f= ± x'"-a
15:4:4
Thus, the three functions x V +x a, a/2r and ±\/x a all have exactly the same shape and differ only in their orientations, as illustrated in Figure 154. The function -a/2r is also a member of the quartet of identically shaped rectangular hyperbolas.
It ma be apparent from Figure 15-4 that the curve f(x) = ±Vlx' - a can be obtained from g(x) _ ± x' + a not only by rotation but also by reflection in their common asymptote. A very general result states that the reflection of the function g(x) in the straight line bx + c generates the function f = f(x) where 15.4.5
11-b2[f=2bx+2c-II+b2[g(v)
y=
2bf + x(1 - b2] - 2bc
1+62
When c = 0 and b = 1. this general reflection formula collapses to .x = g(f(x))
15:4:6
showing that the reflection of a function g(x) in the straight line x generates the inverse function [see Section 0:31
of g(x). Hence, the rectangular hyperbola ±V a is the inverse function of the rectangular hyperbola and conversely.
THE W x* * a FUNCTION AND ITS RECIPROCAL
15:5
118
15:5 INTRARELATI ONSIQPS The f(x) = b x ++ a function is even and therefore satisfies the reflection formula
f(-x) = f(x)
15:5:1
The multiplication formula f(vx) = bpvl
15:5:2
+ a/(v'')
shows that multiplication of the argument x of the semih perbolic function by a constant generates another semihyperbolic function. The inverse function of ±b x + a is also another hyperbola
f=
15:5:3
±1
x2-ab'
b
where
x= tbVj'+a
The two hyperbolas tb -x + a and ±bV? - a are said to be conjugate to each other. As shown in Section 15:4, they share the same asymptotes.
15:6 EXPANSIONS The semihyperbolic function may be expanded binomially as 15:6:1
b x=+a=bVa I +
x`
x6
8a'+16a'
(112) X '
5x8
28a`
Usx'sa
when x is small, and as /I
a
a
a3
(2) /b gx'+l1+..)-bXj
lx' )
1r!'
when x is large. It is this latter equation that describes the approach of the semihyperbola to its linear asymptote.
THE b
119
x=
a FUNCTION AND ITS RECIPROCAL
15:9
Similarly, the reciprocal function is expansible in the two following ways:
x'
1
15:6:3 b
(1
x +a
3x'
a
b
_
35x"
5x6
'_
l6a '+ 128a
+ 8a'
b
x'
1/2 )
1
a
J
a
0 0
x+
2Va
1
15:9:2
15:9:3
=
b x2+
2a - x=
2bVa
b
x2 +
b
x= + a
1
15:9:4
bx +
2x=-a 2bWI
8-bit precision
xj s 0.3 V a
a>0
8-bit precision
W a 2.5Vj
8-bit precision
x a 3.3 V lad
THE bV x a FUNCTION AND ITS RECIPROCAL
15:10
120
15:10 OPERATIONS OF THE CALCULUS Differentiation gives d -b x'+a dx
15:10:1
bx
x'+-
and
-dxW;'+ . d
15:10:2
I
-x (x'+a)'
_ b
The simplest formulas for indefinite integration must employ differing limits according to the sign of a. Thus: ab
bx
t2+adt= 2 z2+a+ 2 arsinh
b
15:10:3
1. 15:10:4
J
_b t'+adt=bx2 dt
15:10:5
dr
15:10:6
J .-e b
f
r'+a
dt
/x\
_ b arsinhl
a
b
= I arcosh(' x
I
//
'arcsch(
`
-,V,' +a
x
I
aO 1
a< O
x>0
a>0
dt
a
a>0
-a
V-a///
b
1
t'+a I
15:10:8
2
I
t -+a
Jo b
15:10:7
+a+abarcoshl
a
x V;
-a arcsecl V-a/
a0 b x + 2c
)
xV-O
and
_ -1
16:13:4
got
arcoshrbx - 2c
at+bt+c \
c, > 0
p 3b. (i.e, if P, given in 17:1:2, 131
17:3
THE CUBIC FUNCTION x' - ax= + bx - c AND HIGHER POLYNOMIALS
132
is positive). The cubic function always acquires the value zero at least once: that is, there is at least one real root to the equation p3(x) = 0. Section 17:7 discusses the circumstances under which a second or third mat root exists. The behavior of a polynomial of higher degree is determined in a complicated fashion by its coefficients and by its degree. Generally, a graph of a polynomial p (x) exhibits a number of inflections (at the zeros of the polynomial of degree n - 2 that is the second derivative of p (x)-see Section 0:7), and local maxima and minima (at the zeros of the polynomial dp,(x)/dx of degree n - 1). Some general rules concerning the number of zeros of p,(x) (i.e., the number of real roots of the equation 0) are given in Table 17.2.1.
17:3 DEFINITIONS The operations of addition, multiplication and raising to a power fully define a polynomial. Written as a concatenation
133
THE CUBIC FUNCTION x' + ax, - bx + c AND HIGHER POLYNOMIALS
17:5
Table 17.2.1
n=2,4,6....
n-2,
a, > 0
a, < 0
rt=3,5.7....
I S.N,sn-2
Number N, of inflections
Number N. of minima
OSN,S 2 IsN,a -
Number N. of maxima
NM
Number N, of zeros
1aN,sn
n
N.
N,
NM-l n
N.-N.-1
1 0
and
J-0
J-0
-I
-
(j + 1)a). I.r!,r = 0
17:7:3
(j + 1)(j + 2)aJ,:xM < 0
and
i-0
p.(x.,) = minimum p,(XM) = maximum
J-0
Equation 17:7:1 is equivalent to the condition dzp.(x,)/dx2 = 0, while equations 17:7:2 and 17:7:3 state that dp.(x,,,)/
dx = 0, d2p.(x.)/dx2 > 0 and dp.(xM)/dx = 0, d'p.(x. )/d.c2 < 0, respectively [see Section 0:71. Zeros of p,(x) correspond to the roots of the equation p.(x) = 0, that is. r is a zero if 17:7:4
7- a,r'=0
p.(r)=0
J-0
The numbers of inflections, minima, maxima and zeros depend not only on the degree of the polynomial but on the values of the coefficients; see Table 17.2.1. For the cubic function x3 + axe + bx + c, an inflection occurs at x = -a/3 [see Figure 17-I]. Provided a2 exceeds 3b. a maximum and a minimum occur at arguments (- a - 3b - a)/3 and ( a- - 3b - a)/3, respectively. This cubic function has the single zero 17:7:5
r=(Q+1r)"'+(Q-Vt1)1J3-3
D>0
17:7
THE CUBIC FUNCTION x' + ar' + bx - c AND HIGHER POLYNOMIALS
136
if the discriminant D [defined in equation 17:1:2] is positive, and three distinct zeros
r, = 2VPI cos(4/) 3
17:7:6
)
r_ = -2V IPI cosl d' +
where
3
rr = 2V cos
0-
3)
¢ = 3 autos I - I
D 0.6 will give r2.
Output r
r K go to (7) Replace bk by bk(2k + 1)
s
°
Setj=O (6) Replace j by j + I Replace bk by bk/(J + j)
Ifk<jgoto(5)
Replace bk by bk(J - k + j) Goto(6)
----------------------------
(7) Set ao - k = 1 (8) Set at = 0
Replace k by k + 1
If k:Kgo to (8)
Setj=k=0
Go to (10) (9) Replace j by j + 1
s
Ifj>Kgoto(ll) Setk=j
Set ale = at-,(2k - l)(J/k)/(J - k + 1)
u0
Replace bk by bleak (10) Replace k by k + 2
If k > K go to (9) Set at = ak_. - (at-2 - ak_,)(2k - 1)(J/k)/(J - k + 1)
J= 8,xa=0,x1=80,
Replace b, by b; +
K = 3, and
bleak
Test values:
----------------------------
Go to (10)
x
(11) Setk=K+ 1
I
0
(12) Replace k by k - 1
10
14
Set at = k!bk
20 30 40 50 60 70 80
20 26 22 26 32
Set) = k
(13) Replace j by j + 1 If j > K go to (14)
Replace at by at + bj!(u/h)'-k/(j - k)! Go to (13) (14) Replace at. by ale/[k!(-h)'] Output at
52
Output: a3 = 0.0003754, a2 =
ax,a5-1, ..., as < <
K, the hypergeometric series necessarily converges for all finite values of x. If L = K, convergence is generally limited to the argument range (xl < 1. If L < K the series diverges (unless it terminates) for all nonzero arguments, but it may nevertheless usefully represent a function asymptotically [see equation 37:6:5 for an example]. The name "hypergeometric" arises because the 18:14:1 series can be regarded as an extension of the geometric series [see equation 1:6:5J to which it reduces when K = L = 0. Other small values of K and L give rise to certain well-known families of functions, discussed elsewhere in this Atlas, as well as a number of generic classes of functions. Table 18. 14.1 summarizes these types of functions. Notice that the so-called generalized hypergeomerric function is not, in fact, fully general because one of its denominatorial parameters is constrained to be unity. Hypergeometric functions are important because most of the so-called special functions of mathematical physics (i.e., functions that arise as solutions to differential equations of practical importance) are instances of this class. The cylinder functions of Chapters 49-56 provide examples, all being K at 0, L = 2 hypergeometric functions. Cases with L = 4 and K = 0, 2 or 4 occur in the theory of elasticity and are encountered in Sections 47:11, 53:6 and 55:6 of this Atlas. Functions that are expressible as hypergeometric series may be evaluated by exploiting convenient features of expansion 18:14:1. If the abbreviation 18:14:2
G=
(a, + j)(a2 +j)(a3 + j) ... (aK + j) (Cl + j)(cr + j)(cs + j) ... (cr. + j)
is adopted, then a hypergeometric function of argument X may be represented as 18:14:3
where R, is the remainder when the hypergeometric series 18:14:1 is truncated after the j = J - I term. If J is chosen to be larger in magnitude than any of the a or c parameters, and also to exceed IXIIIIL-10. then the dominant term in G, is JK-L. Unless L < K, the R, remainder will become steadily smaller in magnitude as J increases. For large enough J, the R, term may be well approximated by a geometric series 18:14:4
+G,X+GIG, IX2+G,G,.,G,.2X'+ = GoG,GI ... fJ,-,X'[1 + jK-LX + j2,1-21X2 + JM-XX) + ...] .
Go
I - JK-LX
THE POCHHAMMER POLYNOMIALS (x),
18:14
156
With approximation 18:14:4 incorporated when appropriate, expression 18:14:3 forms the basis of a universal hypergeometric algorithm that may be used to calculate numerical values of a wide variety of functions. The algorithm, which is detailed below, requires the following inputs: (a) K, the number of numeratorial parameters; (b) L, the number of denominatorial parameters; (c) a a2, a3, ..., aK, C1, c2, c3, ..., cL, the values of the parameters (these are stored serially in the algorithm as b1, b,, b3... , bK.L); and (d) X, either the value of the argument x or some simple variant of it, such as I - x or x2/4.
Input K >>
Seti=0
Input L >>
la»»
Storage needed: 7 + K + L registers are required to store the following quantities: K, L, i, br, b2, b3, ..., bK+L+ X, t, f and j.
(1) Replace i by i + 1 If i > K + L go to (2)
Input b, > parameter
Go to (1)
(2) Set j = 0
Input X >>
Y»» Setr=f= I
I Intervention to halt output is necessary.
Go to (4)
(3) Replace t by t/(b; + J) (4) Replace i by i - I
Ifi>Kgoto(3) Ifi=0goto(5)
Replace t by t(b; + J) Go to (4) (5) Replace t by tX Replace f by f + t
Replace jbyj+l
input: K = 1, L = 1; parameters: 1, Z; X = 4 output: 1.57229437, 1.57081694,
1.57079671,1.57079633,
1.57079633(=Tr/2), ...
input: K = 0, L = 2;
If frac(j/5) * 0 go to (7)
parameters: 21, 1; X = 1 output: 3.76219871, 3.76219569.
If I >- j L-K/X go to (6)
3.76219569(=cosh(2)), ...
Set i = r/(1 - (jL_x/X)1 (6) Output f - i Approximate f(x) value
Test values:
input: K= 1,L=0; parameter: 1; X = 0.04
(7) Seti=K+L+I Go to (4)
output: 1.021340580, 1.021340743
1.021340745,1.021340744
1.021340744(=10 daw(5)), ...
Because the performance of the algorithm depends on many factors, it is not possible generally to predict the value of J required to achieve a sought precision in f(X). Accordingly. the universal hypergeometric algorithm sets
J = 5. 10. 15, ... and generates an output for each of these J values. leaving it to the operator to observe when convergence has been achieved, or to judge when the output value has sufficient accuracy for his purpose. In some cases, convergence may occur but be tediously slow. Note that when L < K, the algorithm may fail to produce an acceptable value unless X is sufficiently small; in this case the output values may initially tend toward the correct answer but later diverge. Table 18.14.2 lists some of the functions that may be evaluated using the universal hypergeometric algorithm.
For many of these functions alternative algorithms will be found in the appropriate chapter of this Atlas. Such "custom" algorithms are generally more satisfactory than the universal algorithm, but the latter has the supreme advantage of versatility.
1
THE POCHHAMMER POLYNOMIALS W.
1S7
18:14
Table 18.14.2 Chapter
No. of
Puamnen
or
secti on
Parameter values
K
L
b.. b,. b,.... b,,.,.
X
sought run"" ftxi
2
2
Restrictions
3 1:7
1:
I
1
2
1
2
1
3 3 I
22 2 2 I.
.
.
..
I : 2...
2
22
'g.)
n - 1. 2. 3....
3
fi(n)
n - I. 2. 3....
3
-
(I m x)°
-1<x k/bl
II
1: I
-
V'bx + Ic/bl
12
141 < ION
12
1>k/M
12
5IO :0 .1 .2 ,
28:13
THE POCHHAMMER POLYNOMIALS W.
159
18:14
Table 18.14.2 (Continued) No. 0(
Chapter
pvatnetm
oe
Paranveter values
K
L
I
2
2
section X
.-. b,_,
b,. b, b,.
! I.-2
22
1. 2 ;
2
-
2. 2
3
2
2
I. 2: 2. 2
2
2
1. 2: 2. 2 I
2 2 1
x
0
2
x
0
1.
2
4[l12/x) - arseclex)1/x'
0<x I
31
ananbul
-Io
x'b'(sl
1
43:13 44
- (Mx) - a(>9t
I. t + u
2vI; :
v> 0
r B(x.yl
1+v
v: I.
42
V.v
A
Mta:c:x)
-x _I
exp(-x)M(ar;x) r"Uta:r:xl
a,
45 45
large x
45
large v
45
v * I. 3, s.
46
v*0.2.4....
46
* 1. 3.5....
46
+0.2.4....
46
large x
46 47
47 large x
48
x
+I+2µ.I
2 I
0
2
-l
I
I,
46:13
I
r
s"'exDl 2)W.,,(x) Ivfx)
large x
48:13
49
4 I,Lv/
I, 2
4
49
THE POCHHAMMER POLYNOMIALS (x),
18:14
162
Table 18.14.2 (Continued) No. of Parameters Parameter values
K
L
0
2
X
b1. bl. b,, ..., br.t
1,
+R
1
Sough function f(xl
_I
3-:
2
'_
0
2
I
2
2
1
2v
-2 -
-
- + v,
I
v 2 3
x7
2
4
1
1
2
2
-1
- - v. - - v: 1 -R. 1 + R: 1
1
V 2 = exp-x)1 jx)
Imp x
49
V'2trx exp(-x')1i(x)
latgt x
49
r(t + vN2/ 19,(x)
r = -l. -2. -3, ...
50
v + -I. -2. -3....
50
V2wxexp(-x)1,(x)
Urge
30
(2R+I)!!x-"itxl
m1.0.1.2....
50:4
large x
51
Al
IK2/x)*tap(x)ldx1
2s
1.-+a
1
2
a
- + v: 1. 1 + 2.
2
49
n
2x
1+
1,
1
0
I I
1
0. 1. 2....
R421x 1"1,1x1
4
2
2
Resu.ctions
+
txplx)ILIx)
s
-x'
0
2
1.
0
2
1. 2
w = 0. 1. 2....
exp(a IkJxl
lyfx)
1
51:4
52
4
-x: 52
1,(x1
4
0
2
I. I + n
0
2
1.
1
-r 4
-x'
1+
+ v; 1. 1 + v. I + 2.
3
n - 0, 1. 2, ...
n!(2/x)"1,(x)
r(1 + vK2/x)9,Ix)
v
52
-1. -2, -3....
53
-Jr
rYl + vg4/x°YY;1xI
v + -I. -2. -3....
53
-x:
avcvcfav)1,(xv-W
v * s1, x2. x3, ...
53
(2n + I )!!x""J,(x)
R - -1. 0. 1. 2....
53:4
v a -1. -2. -3....
53:6
large x
53:6
2 1
-: 1 - r. 1, 1 + v
3
-x
3
0
2
0
2
2
-4R
1.
1+v 3
1
I - a. - - a; -.
I
4
1.
a
3 3 a.a+-.--a.2-a:1.22 1
2
0
0
=. 1.
4
-1
1
Ptn:x1 with 2xQ(v:x)
4
v
-2 3
v
4
2
wither= --bes(xl
I
va: -. larger
53:6
55
236
33
-i
2
256
I. I. -.
4
!, 1.
4
,
-
1
2
0
3
-x'
I2, I
0
r0 + vIC,(xl
-1 I
2
4
-x
-
r bei(xl
55
\'e [bee(x) + bei:fx)]
55
16
1+v 4 2
4
2
t+2
-x
256
r(I + vl(
x
Fe,(xl
55:6
THE POCHHAMMER POLYNOMIALS (x).
163
18:14
Table 18.14.2 (Continued) No. o(
Chapter
/lr
parameters
Parameter values
K
L
0
4
J
v3-v
2
2
Sough (union r(s)
X
bi, b;. b,..., bx.t
r12 + v)t.-)
OeJsl
x
2
3"'r(*) Btu)
0
2
1
9
3
_
2
I
3
9
45
-9
-33-
1
4
2
5
-
II
1
7
I
11
I
13 17
-9
2
4s'
I. -
2 2
-.r
2 2
4
-2 2
2
-
-xt
v
4
13
2
0
--
2
0
--
I
-4 5-
2
0
-r(-+,11-) ,. x 3
2
2
2
x'
Va
4
4
xl
- (hdx) -
v
r
v' 2
l
v; 1+ v
x
I
u * µ: 1 + v
x
I
r+ µ; I+ r
I- x
I-x
2
2
3
-.1+-: I. -2 +r 2 2
large positive x
56
large positive s
56
I
57
Iarge x
57
large x
57
' 4x11
large x
57
(1i.1x1 - YJx)I
large x
$7
s>0
57.13
(hi(s) - Y{.rll
33
1*
56
>0
- (Y .4x1 - A.,(x)l
Y+r r(2 *
-
large positive x
2
-4
2
56
57
/
3
r(2+oil-I 1
0
large positive x
57
I- I 2 2
2
56
57
A
1
*
a1
a
*Vx (sBN-s) - cAif-x11. c - cm
2,2
0
2
IJAI(-xl + cBif -x11.:
-4
-1
0
Bilsl
l Q
4
2 2
I
VrV r expl
large positive x
4
JS 33
0
11
-r -
2
0
56:13
4
3
3
2
-r
Z 2 0
36
>> I>>>>>
Fs-t,,age needed: n. x. f, g and j
Setf=g=0
Input x >>
Input restrictions: The degree n must be an integer not less than 5. The argument x is unre-
(1) If x_ 2.3(n - 1)
follows from 19:6:2.
19:10 OPERATIONS OF THE CALCULUS Additional to the general formulas of Section 17:10 are the following special results for the operations of differentiation and integration of Bernoulli polynomials d" 19:10:1
dx" d
19:10:2
dx
B"() = n'
B,(x) = nB"_,(x)
B"(t)dt =
19:10:3
B,_1(x) - B,_1
n+l
ro
nx*0
19:10:4
19:10:5
'
I
n=0
B"(t)dr = fo1
n= 1,2,3,...
0
19:10:6
f B"(t)B.(t)dt = Ju
m!W !
(m + n)!
B.."
m,n= 1,2,3,...
19:11 COMPLEX ARGUMENT Bernoulli polynomials are defined for real arguments only.
19:12 GENERALIZATIONS A class of functions defined by the generating function
t' exp(xt) 19:12:1
2,...
is known as Bernoulli pohnomials of order m [see Korn and Kom, page 824]. They represent a generalization of the Bernoulli polynomials discussed in this chapter, which are of first order. A still more general set of higher order Bernoulli polynomials is discussed by Erdelyi, Magnus, Oberhettinger and Tricomi (Higher Transcendental Functions, Volume 1. pages 39 and 40].
173
THE BERNOULLI POLYNOMIALS B.(x)
19:14
19:13 COGNATE FUNCTIONS Bernoulli functions are closely related to the Euler polynomials that are the subject of Chapter 20. Equations 20:3:3 and 20:3:4 make this connection explicit.
19:14 RELATED TOPICS From integrals 19:10:4 and 19:10:3 it follows that x"
19:14:1
+(1+x)+(2+x) ' +...+(m-1+x)= B". (m+x)-B"+,(x) n+1
The most important applications of this general formula arise by setting x equal to unity; B"+ (m + 1) 1" + 2" + 3" +
19:14:2
+ m" =
+ (- 1)'B"+1 n = 0, 1, 2, ...
p + 1
or a moiety: 19:14:3
1" + 3" t 5" + - + (Zm - 1)"
1)
=
2"B"+,(m) n + 1
(2"
1)B".,
n=0.1,2,...
Further simplification occurs if is = 2. 4. 6. ... because B"_ r is then zero. Some examples of these finite summations are presented in Section 1:14. See Section 20:14 for cases of alternating signs.
CHAPTER
20 THE EULER POLYNOMIALS En(x)
Certain series involving the natural numbers can be conveniently expressed using Euler polynomials, as explained in Section 20:14.
20:1 NOTATION Euler polynomials of degree n and argument x are generally denoted E (x), although the symbol E,(x) is occasionally encountered. The E (x) symbolism is also used to denote Schldmilch functions, which are quite unrelated to Euler polynomials. The Schldmilch functions are briefly discussed in Section 37:14, but they are not used elsewhere in this Atlas.
20:2 BEHAVIOR The Euler polynomials are defined for all nonnegative integer n and for all real values of x. Figure 20-1 shows the behavior of the first few Euler polynomials in the range 0 < x t 1, which is the most important range for these functions. Over this range the inequality
E
E , (x )
20:2:1
0
n
2
0 5 x s 1
n= 0, 2, 4, ...
is valid for even n, E. being the nth Euler number [Chapter 51. As is evident from the diagram, all Euler polynomials of even degree (except Eo(.r)) are zero at x = 0 and x = I and exhibit a local extremum [see Section 0:7) at x = 1. In complementary fashion, all Euler polynomials of odd degree are zero at x = 1 and (except for E,(x)) display a local maximum or minimum at x = 0 and x = I. Figure 20-2 demonstrates the behavior of typical Euler polynomials outside the 0 rt x -S I range.
20:3 DEFINITIONS Euler polynomials are defined by the generating function [Section 0:31
20:3:1
l2+xexp(t)
E.(x) -o
n! 175
p, 44
44
44
44
0144 a44O*44 O44O44O*44O44a 44 00 44
4r
.
.
.
.
.
.
.
.
.
.
44
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
'
1. 1
1.0 :....:....:.............................
FIG 20-1
:
........:...
:....:....:...
..
.............:. 0.9
..
:
:
:..............
...
..
..
.......... :.... ................. ........................ .
'
..;....: ...
....
:
................
.
0.7
.................... ...
....:... :....:....:....:....:.............:
....
0.8
0.5 SCx
: :
j .............
0. 3
THE EULER POLYNOMIALS E.(x)
177
An alternative definition in terms of Euler numbers E; [Chapter 51 is
E.(.x)=2
2"'' E.(x)n+1 _-
(x
B.,1 (
(n)(2x-1)"-'E,
'I) - B.,,
(-x)
n = 1. 2. 3, ...
2
and
20:3:4
n+1
1B,-,(x) - 2`1313.,,
(2)J
n = 1, 2. 3, ...
relating Euler polynomials to Bernoulli polynomials (Chapter 19] also serve as definitions.
20:4 SPECIAL CASES The first eight Euler polynomials are 20:4:1
Eo(x) = 1
20:4:2
E1(x)=x-2
20:4:3
E2(x) = xz - x
20:4:4
E3(x)-x3-
3x2
1
+4
2 20:4:5
20:4:6
20:4:7
E4(x) = x4 - 2x3 + x
E,(x)=x5-
5x4
5x2
1
+2 2 E6(x)=x6-3x5+5x3-3x 2
20:4
THE EULER POLYNOMIALS E"(x)
20:5
no
and
ET(x) = x' -
20:4:8
7x6
+
2
35x'
21x2
4
2
17
+
8
Coefficients of all terms in E"(x) for n 5 15 are listed by Abramowitz and Stegun [Table 23.11.
20:5 INTRARELATIONSHIPS The Euler polynomials possess even or odd symmetry about x = Z: 20:5:1
and obey the reflection formula
(-1)"[2x" - E"(x)]
20:5:2 The argument-addition formula
E"(x + y) =
20:5:3
I " JE,(x)ti "
Ito
has the special cases
\nlE,(x)=2x"-E"(x)_(-I)"E"(-x).
E"(1+x)_
20:5:4 and
1
E"(I2+xl
20:5:5
!) n
-x"
E,
(2x)'
Because all Euler numbers of odd index are zero, about half of the terms in the last summation [which is equivalent to 20:3:21 vanish. The argument-multiplication formula takes different forms:
E"(mx)=Z(-1)'B".,Ix+l m// n 1 ;_(1 `
20:5:6
m=2,4,6,...
(
,
20:5:7
m= 1,3,5,...
l
m
according to whether the multiplier is even or odd. Definition 20:3:3 is a special case of formula 20:5:6.
20:6 EXPANSIONS Definition 20:3:2 leads immediately to the polynomial (
20:6:1
E"(x) = 1 x +
1
n(n - 1)
8
n(n - 1) E_2 2
x - 2!"
lx --
2"-' \
5n(n - 1)(n - 2)(n - 3)
1
I
2
384
(x - 2
+-lx--l nE"_, 2"-'
I
\
2/
+
E. 2"
in x - Z; all terms containing odd-indexed Euler numbers are zero. Polynomials in x: 20:6:2
E"(x)=x._2x.
,+4(3)x" ,_2(5)x" `+
171x" z+ 8
179
THE EULER POLYNOMIALS E,.(x)
20:8
or x - I : 20:6:3
E"(x)=(x- I)"+Z(x- 1)'-' -4(3)(x- 1)"'+2(5)(x- I)" s- 1717)(x- I)"-'+... g \7
may also be written; these contain (n + 2)/2 terms if n is even, or (n + 3)/2 terms if n is odd; see Section 20:4 for the coefficients in expansion 20:6:2 when n adopts specific small values. The Fourier expansion (see Section 36:61 20:6:4
4n!
(2j +
E.(x) = -
sin (2j + 1)ax -
reduces to
E"O= x
20:6:5
2
i
(- I)"1-4n! r.-I
nir
sin[(2j + 1)inx]
2j+I".I
]
n = 1 , 2, 3 , ...
n=2,4,6,...
x
0
1
05xsl
when the degree is even, and to
(-1)4a`11n4n! . cos[(2j + I)irx] 20:6:6
E"(x) =
"
I
'=o
n = 1, 3, 5, ...
1
(2j + t)
0 > Input x > >
Storage needed: n, x, s, f, g and j
Sets=f= 1 Ifn=0goto(5)
Setf=g=0
1
Input restrictions: n must be an integer not less than 5. x is unrestricted if the green commands are included.
(I) If x I go to (3) Replace x by x - 1 Replace g by g + 2sr" Replace s by -s
Goto(1)
Use degree mode or change 90 to n/2. 1
THE EULER POLYNOMIALS E"(x)
20:9
180
(2) Replace g by g + 2sx" Replace s by -s Replace x by x - 1
(3) Ifx0goto(4)
E6(0.4) = -0.906624
Replace f by [4sn!/(2wrr)"'1]f
E13(1) = 2730.5
Replace f by f - g
E1(-1.5) = 56.66015625
(5) Output f
E9(rr) = 1902.63483
f=E"(x)< For arguments in the range 0 s x s 1, the portions of the algorithm shown in green may be omitted. The algorithm is based on the Fourier expansion 20:6:4. truncated at j = 1 + Int(30/n). This choice ensures that the absolute error in E"(x) does not exceed 6 x 10' for n > 5, although the relative error may be greater than this, particularly near a zerb of E"(x). When the green portions of the algorithm are included, arguments of any magnitude can be accommodated. Recursion 20:5:4, or its equivalent:
E"(x - 1) = 2(x - 1)" - E"(x)
20:8:1
is employed to shift the argument into the range covered by the Fourier expansion.
20:9 APPROXIMATIONS For small n and arguments between zero and unity, Figure 20-1 can provide approximate values of E"(x). For large n and arguments between zero and unity, the leading term of expansion 20:6:4 provides the approximation E"(x) =
20:9:1
4n.r
sin i r x -
nn 2
n-. x
05xs1
Approximation formulas may be derived from 20:6:2, 20:6:1 or 20:6:3 for x values close to 0. i or 1, respectively. For x values of large magnitude, the approximation 1
E"(x) _ I x - Z I
20:9:2
Ix
8-bit precision
(
-
12
> 4(n - 1)
follows from 20:6:1.
20:10 OPERATIONS OF THE CALCULUS In addition to results that follow from the general formulas of Section 17:10, there are several special results from applying the differentiation and integration operators to the Euler polynomial. These include d"
20:10:1
E"(x)
n!
dx"
d E"(x) = nE"-1(x)
20:10:2
dx
20:10:3
r
E"-3(x)
n+1
n+1
n1,3,5, in 0,2,4,...
xa
=0
x0 =2
THE EULER POLYNOMIALS E"(x)
181
20:14
and 20.10:4
f
, E"(t)E",(t)dt =
J0
4m!n! 1)
(m+n+2)!B° .*_
20:11 COMPLEX ARGUMENT Euler polynomials are defined for real argument only.
20:12 GENERALIZATIONS Euler polynomials may be generalized in the manner described in Erdelyi, Magnus, Oberhettinger and Tricomi [Higher Transcendental Functions, Volume 1, page 43J.
20:13 COGNATE FUNCTIONS Bernoulli polynomials [Chapter 19] are closely related to Euler polynomials. Equations 20:3:3 and 20:3:4 establish a direct link.
20:14 RELATED TOPICS Iteration of recurrence formula 20:5:4 leads to
=
20:14:1
E,(.r) ± E"(m + x)
where the upper/lower signs correspond to odd/even m. Setting x = I in this relationship yields the useful result 20:14:2
1
while setting x = ¢ generates 20:14:3
1)'=
E. + 2
The first of these results is presented in Chapter 1 as equation 1:14:9. and equations 1:14:6-1:14:8 are special cases.
CHAPTER
21 THE LEGENDRE POLYNOMIALS Pn(x)
The Legendre polynomials arise in several branches of applied mathematics (as, for example, in describing electrostatic or other fields), especially in problems involving spherical symmetry. The Legendre polynomials are among the simplest of the families of orthogonal polynomials [see Section 21:14].
21:1 NOTATION The symbol P,(x) is standard for the Legendre polynomial of degree n and argument x, although the P is often italicized. The name spherical polynomial is also encountered. Sometimes normalized Legendre polynomials are of this chapter is normalized through multiplication by (2n + 1)/2 [see Section specified: the polynomial 21:141. Yet another name is zonal surface harmonic [see Section 59:141. The symbol P,(x), where v is not necessarily an integer, denotes a member of the function family known as Legendre functions of the first kind, which are a subject of Chapter 59. When v takes any integer value, positive, negative or zero, P,(x) reduces to a Legendre polynomial 21:1:1
21:1:2
Po(x) = P,(x)
P,(x) =
v = n = 0, 1. 2...
v = -n = -1, -2, -3, ...
The polynomials of the present chapter are thus special instances of Legendre functions of the first kind and sometimes are so identified. The reader is referred to Sections 21:12 and 21:13 for explanations of the related notations P:(x), P,'(x). P.*,"(x) and P"(x).
For -1 s x s I the argument x is often replaced by the cosine of a subsidiary variable 0: 21:1:3
0 = arccos(x)
as a reflection of the fact that Legendre polynomials often arise naturally in scientific and engineering applications with arguments that are cosines of salient angles.
21:2 BEHAVIOR In this chapter we restrict the degree n to be a nonnegative integer; n = 0, 1, 2..... Although the argument x of the Legendre polynomial P (x) may take any real value, the most important range is -1 s x 5 1; this is the only range accessible when x is replaced by cos(9). Within this range the polynomials never exceed unity in magnitude 183
THE LEGENDRE POLYNOMIALS P,(x)
21:2
sl
21:2:1
as illustrated in Figure 21-1. The Legendre polynomial
-1<x1
0
the latter being known as Laplace's integral representation. The Legendre polynomial function f = P,(x) satisfies Legendre's differential equation
(1 - x2)
21:3:7
-- - 2x
+ n(n + l)f = 0
The most general solution of this equation is c,P,(x) + c2Q,(x). where c, and c2 are arbitrary constants and Q,(x) is the function discussed in Section 21:13. Figure 21-3 illustrates a geometric definition that explains how Legendre polynomials arise in certain applications. The triangle shown has two of its sides, one of unity length and one of length r, enclosing the angle e. 1 - 2r cos(9) + r-. In some physical By the cosine law [see Section 34:14], the length z of the third side equals problems it is necessary to expand z-' as a power series in r-1. Formula 21:3:1 shows this series to be 21:3:8
1
Po(cos(8))
z
r
P,(cos(8))
+
+
P2(cos(6))
r'
r=
if r > 1, or a similar series if r < 1.
1
21:4 SPECIAL CASES The first eight polynomials are 21:4:1
Po(x) = I = Pa(cos(9))
21:4:2
P,(x) = x = cos(8)
3x' - I 21:4:3
P2(x) =
2
1 * 3 cos(29)
=
4
+
THE LEGENDRE POLYNOMIALS P.(x)
187
21:4:4
P3(x) =
5x3 - 3x
3 cos(8) + 5 cos(36)
=
2
35x4 - 30x' + 3
8 9 + 20 cos(29) + 35 cos(49)
=
P.(x) =
21:4:5
64
8
21:4:6
P5(x) =
63x5 - 70x3 + 15x _ 30 cos(O) + 35 cos(30) + 63 cos(50) 8
21:4:7
214:8
P6(x) =
PT(x) =
21:5
128
231x6 - 315x' + 1051' - 5
50 + 105 cos(20) + 126 cos(49) + 231 cos(60) =
16
429x' - 693x' + 315x' - 35x
=
512
175 cos(8) + 189 cos(30) + 231 cos(58) + 429 cos(78) 1024
16
The formulas given above for P.(x) apply for all values of x, whereas those for P"(cos(h)) are restricted to - I S
cos(h) S 1.
21:5 INTRARELATIONSIIPS Legendre polynomials are even or odd
P.(-.r) _ (-1)"P.(x)
21:5:1
according to the parity of n. The recurrence formula
P.(x)=
21:5:2
2n-1 is
n-1
XP._1(x)--P._2(x) n
n=2.3,4,...
relates three polynomials of consecutive degrees. Positive integer powers can be written as finite series of Legendre polynomials. For example: x2 =
21:5:3
1
3 Po() W +
2 P2(x) 3
x' = 5 Po(x) + 5 P3(x)
21:5:4
X4
21:5:5
=
l 5
8
Po(x) + 4 P2(x) + 7
35
P.(-V)
Other well-bchavcd functions of x can be expressed as infinite series of Legendre polynomials P.(x), and a collection
of such formulas is presented by Gradshteyn and Ryzhik [Section 8.92J. Many such series expansions may be obtained by the procedure explained in Section 21:14. Among summation formulas for Legendre polynomials arc 21:5:6
(2n - 1)P._1(x) + (2n - 3)P._2(x) + (Zn - 5)P"_3(x) + =
n(P.-i(x) - P.(x)]
+ 3P,(x) + Po(x)
x*I
I - x
and 21:5:7
(2n - 1)P._;(x) - (2n - 3)P._2(x) + (2n - 5)PP_3(x) _ n[P._,(x) + P.(x)] 1+x
x + -1
± 3P,(x) + Po(x)
THE LEGENDRE POLYNOMIALS P"(x)
21:6
188
21:6 EXPANSIONS Definition 21:3:3 leads to the finite series
(n - 1)!! r
n(n + 1)
(n/2)! 21:6:1
x2+
(n - 2)n(n + 1)(n + 3) X4-
n-1
1
n=0,2,4,...
4!
(n - 1)(n + 2)zt
n!!
_U12
P"(x)
2!
3!
+
(n - 3)(n - 1)(n + 2)(n + 4) 5!
xa
- ...]
2
n= 1,3,5. that are valid for all arguments x. The infinite Fourier series P"(cos(8)) = ?
G (+ 1)" sin[(n + I - 2j)01 it j_0 (j + Y2)"., involves coefficients that are ratios of Pochhammer polynomials (Chapter 18]. A third expansion is 21:6:2
P"(x)=1E
21:6:3
(n+j)! 2
2
11
Other expansions can be found in Gradshteyn and Ryzhik [Section 8.9111 and yet others may be derived from the
formulas in Section 59:6 with v = n, or by combining equations 21:12:1-21:12:4 with the expansions given in Chapter 60.
21:7 PARTICULAR VALUES Pa-z) n - 0
P.(-1)
P,(0)
P,(I)
P.(.) I
I
1
1
I
-x
1
0
1
(- 1)*"(" - 1)^
21:8 NUMERICAL VALUES For small n, values of P"(x) are easily found from the expressions in Section 21:4. The algorithm Input n >>
Input x >>
>>>>
Setf=g=x If n = I go to (2) Setf=h=j= 1 If n = 0 go to (2) (1) Replace j by j + I
Replace f by [(2j - 1)xg - (j - 1)h]/j
Ifj=ngoto(2) Set h = g
Setg=f Go to (1) (2) Output f
f = P"(x)
8V
which follows from expansion 21:3:3.
21:10 OPERATIONS OF THE CALCULUS Differentiation of a Legendre polynomial gives d
21:10:1
dx
P"(x =
n
1 - x,
[P.-c(x) - xP.(x)l
n = 1, 2. 3, ...
while integration produces J
21:10:2
Pjr)dt =
P.-1(X)
-+ P1.-i(x)
n = 1. 2. 3....
i
From this last result it follows that
P.(t)dr-0
J
21:10:3
in view of 21:7:4. Other definite integrals involving Legendre polynomials include
P.(t)dt _
TI
21:10:4
T P.(r)dr
21:10:5
-
I
=
tz
n
N/"8-
2n+I
,
[(n - 1)! ' J-
n!!
0
n=0.2.4.... I, 3. 5, ...
m=n - 1,n-2,n-3,... m!
21:10:6
I
t'P.(t)dt =
J
n 2
!(m + n + 1)!!
m=n,n+2,n+4,...
m=n+ I,n+3,n+5,...
and
f (µ-n+2)(µ-n+4)(µ-n+6).. tµ- 2± 1
21:10:7
1
t"P.(t)dt =
12)
\
/ (µ+n+1)(µ+n-1)(µ+n-3)Iµ+2+ZI
µ> -2 ± 2
THE LEGENDRE POLYNOMIALS P"(x)
21:11
190
where the upper/lower signs apply according as n is even/odd. Gradshteyn and Ryzhik devote several pages [Sections 7.22-7.251 to a listing of yet other definite integrals involving Legendre polynomials. The integral of a product of P,(x) with some other function f(x) is frequently evaluable by utilizing Rodrigues' formula 21:3:2 and parts integration ([Section 0:10]. Whenf the integration limits are ±1. the simple formula 21:10:8
J
d"f(t) (1
1
f(t)P,(t)dt =
'
2'n!
- tz)"dt
dt'
emerges. The integral
f
21:10:9
mn
0 P,(t)P"(t)dr =
2
in
2n + 1
It
establishes the orthogonalirv of Legendre polynomials over the interval - I n or if j and n have unlike parities. The simplest coefficient yo' equals unity, and others can be calculated by means of the recursions y;"' _ 2 y;=;" + 2 y;"-i" except -y"' = Yo 1) + 2 yz and Yo where yi"-11
There are formulas relating the product of two Chebyshev polynomials to (usually the sum of two) other Chebyshcv polynomials: 1
1
22:5:8
T.(x)T,.(x) = 2 T".,.() + - T._ (x)
22:5:9
U,(z)U,.(x) =
T_(x) -
nsm
2(1 - x2)
1
I
2
2
n L, m
- U..,,(x) + - U_(x)
n:5 m
1
22:5:10
T,(x)U,.(x) =
2 U_() 1
2 U..,.() as well as the so-called
n = m + I 1
2 U._._2(x)
n >- m+ 2
formulas for sums of products of Chebyshev polynomials of different
arguments: 1
22:5:11
22:5:12
T; (x)T; (y) _
2 I
-
+7
U,(x) U ,(y)
-
T..j(x)T,(y) - T,(x)T,.dy)
x-y U,.,(x)U,(y) - U.(x)U,.,(y)
-
THE CHEBYSHEV POLYNOMIALST.0) AND U.(x)
22:6
198
Series of Chebyshev polynomials of even degree have the sums 22:5:13
To(x) + T2(x) + T.(x) + ... + T.(x) =
22:5:14
U(x) + U2(x + U4(x) +
1
+
1
-2 -2
n = 0, 2, 4, -
U.(x)
.
1 - T,.2(x) 2(1 - x2)
. + U, x
while the corresponding sums for odd degree are 1
+ T,(x) _ - U,(x)
T,(x) + T)(x) + T3(x) +
22:5:15
n = 1, 3, 5, .. .
2
+ U.(x) =
U1(x) + U3(x) + U5(x) +
22:5:16
x - T..2(x)
n = 1. 3. 5....
2(1 - x2)
22:6 EXPANSIONS Explicit expressions for the two Chebyshev polynomials are 22:6:1
T.(x) = 2 Y n ' . (n
j j) (2x)'-2'
n = 1, 2, 3, .. .
U.(x)=I(-I)' n
j)(2x)w2J
n=0.1.2....
and
22:6:2
`
J=o
where J = Int(n/2). but these expansions may also be written/ T,(x) = X. - 1 2 1 x"-2(I - x2) + 14 1 x"-4(I - x2)2 -
22:6:3
\J
111////
x"''-(I
and
22:6:4
U"(x) =
- (n
rn + I)
`
I)
- x 2) + 1 n + I I x"-4(l - x')= - .. .
3
X.
If t;," represents the coefficient of x1 in the expansion of T.(x) so that
T.(x) = i t."x,
22:6:5
J=o
j((n+-J
then tw1' = 0 when j and n are of unlike parity. All nonzero values of t," are integers given by (-1)4.-))/2n(
22:6:6
2J-1/
1
\n
2
j = 1, 2. 3..... n
)!J!
except that t`,° equals unity.
22:7 PARTICULAR VALUES T.(0)
Tj1)
1
I
1
0
1
1
f-1Y''
1
T,(-1)
n=0
-I
1 ,3 . 5 ,
n=2.4.6....
i
U,(-1) 1
-n- 1
n+l
U,(0)
U,(1)
I
n-I
0
(-I)
I
n+I
THE CHEBYSHEV POLYNOMIALS T,(x) AND U,(x)
199
22:10
The zeros of the Chebyshev polynomials occur at arguments given by T,(r1) = 0
22:7:1
r1 = cost
(2j - 1)a
i
2n
J
j= 1, 2, 3,...,n
and
U,(r,)=0
22:7:2
rj =cos
a n+1
j=1,2,3,...,n
22:8 NUMERICAL VALUES For small degree, values of the Chebyshev polynomials are readily calculated from the expressions in Section 22:4. The following algorithm employs recurrence formula 22:5:2 and provides exact values of the Chebyshev polynomials. T,(x) is generated if the command in black is executed; U,(x) is produced by the green alternative (the two algorithms differ only by a factor of 2 in the command that sets up the value For - I < x < 3, values of the Chebyshev polynomials may also be found using the universal hypergeometric algorithm [Section 18:14). Input it >>
Input x >>
31»»
Storage needed: n, x, f, g, h and j
»»
1
Setf=g={2x If n = l go to (2)
Input restrictions: n must be a nonnegative integer. x may adopt any value.
Setf=h=j= 1 If n = 0 go to (2) (1) Replace j by j + I Replace f by 2xg - h If j = it go to (2)
Test values: To(any x) = 1
U1(x)=2x
Replace h by g Replace g by f Go to (1) (2) Output f
f=
T12(0.2) = -0.748302037 U,(0.5) = 0
1.239131013
l U,(x))
22:9 APPROXIMATIONS Figures 22-1 and 22-2 can provide approximate values of T,(x) and U,(x) for degrees up to 5. For large values of
the argument, the following approximations are valid:
T,(x) - 2 (2x
22:9:1
22:9:2
- I) n
l
U,(x) = 2x1 2x - 2x)
8-bit precision
x > (18n)'14
8-bit precision
x > 2(n - 1)'I4
22:10 OPERATIONS OF THE CALCULUS Differentiation gives
_
d
22:10:1
dx
T,(x)
it
T,+i(x)) = nU,-,(X) 1 - xr [ ',(x) -
it -- 2
n?2
AND U.(x)
THE CHEBYSHEV POLYNOMIALS
22:11
d
22:10:2
dx
1
U.(x) =
7- i [(n + 1)U.-i(x) - nxU.(x)]
The Chebyshev polynomial of the second kind may be integrated indefinitely r'
U , (r)dt =
22:10:3
1 -
T.- I (x)
n+1
J
but we know of no corresponding integral for T.W. The definite integral
m*n
f0 f' Tj:)T,(t)
22:10:4
m=n=0
dr= J ir IT
m=n=1,2,3,...
2
establishes that the polynomials T0(x), T,(x), T,(x), ... are orthogonal (see Section 21:14] on the interval -1 S x S I with respect to the weight function I / 1 - x2. Likewise, the integrals 0
1 - t2 dt
22:10:5
mm #=n
= aa
n=1 ,2,
2
1
or m3,=..n.=0
2
establish the orthogonality of the polynomials U,(x). U2(x), U3(x).... on the same interval with respect to the weight function V 1- x2. Many other definite integrals are listed by Gradshteyn and Ryzhik [Sections 7.34 to 7.36]. including
f' t T.(t) dr = 0 1-r2
22:10:7
T"(r)dr =
T
4n2-22:10:6 4n2 - 1
trr(v + 1) 2
,rlv (-1)./2a
22:10:8
v>-1
n=0,2,4,...
b>0
2
J. 0
2
\
n+llr(vn}I
1 - rr
and
f
sin(bt)T (22:10:9
1-t2
dr=
b>0
2
n=1,3,5,
2
where r denotes the gamma function [Chapter 43] and J. the Bessel coefficient [Chapter 52] of order it. On Hilbert transformation (Section 7:10] the product of one kind of Chebyshev polynomial and its weight function gives the other kind: T,(t)dr
1
22:10:10
a
J 1
r
22:10:11
r
77
12 (tS)
+_
J i
-1 <s< I
= U.-I(s) dr
t - s)
_
-T.- ds)
22:11 COMPLEX ARGUMENTS Applications of Chebyshev polynomials are usually restricted to real arguments.
THE CHEBYSHEV POLYNOMIALS T"(x) AND U"(x)
201
22:12
22:12 GENERALIZATIONS Chebyshev polynomials may be generalized to Gegenbauer polynomials, which are themselves special cases of Jacobi polynomials. This section briefly discusses both of these generalizations. is a quadrivariate function, its value being deThe Jacobi polynomial or hypergeometric polynomial pendent on: the argument x, which may take any real value with interest being concentrated on the -1 s x >>>>> Setx=x0 Input x, > >>>>> Set h = (x, - x)/99
Setj=k=E=0 Input n >> »» (1) Output j j(as cue) < «« Input A, > >>>>>
Replace j by j + I
If j5ngo to (I)
(2) Set f = 0 Replace j by n + 1
(3) Replace j by j - 1
Replaccfbyfx+A,
Ifj*0goto(3)
If E> If go to (4)
a,
j =0, 1,2,...,In j = m + 1,m+2,m+3,...,n Storage needed: n + 8 registers are required for Ao,
A,. A2, ..., A,,, x, h, j. k, n, E and f.
THE CHEBYSHEV POLYNOMIALS T,(x) AND U,(x)
207
Replace E by Ill (4) Replace x by x + h Replace k by k + I
E
N/:);;, that is, when the Hermite polynomial leaves its oscillatory region [see discussion in Section 24:2J.
+
44
440' i4
i4
44
44
44
44
C4
%.
441 0 44 0 44ti
441:Iro+4 ej
\n14 x\. x:.. a 12
.. a ..,t
*n=10
.....:....:. *:
*\:
..nab
*. *....
FIG 24-2
n-2
:.........:.... :....:....:....:.... .... .
.
.
.
.
..' n=0
THE HERMITE POLYNOMIALS H"(x)
24:10
222
From 24:9:2 it follows that the n zeros of H"(x) are approximated by
1=±1,t3,±5,...,±(n-1) 2n+1 J=0,±2,±4,...,±(n-1)
neven
j7r
r=
24:9:3
2
nodd
The lines drawn on Figure 24-2 connect the zeros predicted by this approximation, blue for even n and green for odd n. Evidently, the approximation is good for the inner zeros but worsens the further the zero is from x = 0.
24:10 OPERATIONS OF THE CALCULUS We have the following simple results for differentiation and indefinite integration: d
d 24:10:2
n = 1, 2, 3... .
H"(x) = 2nH"_,(x)
24:10:1
dx
[exp(-x')H"(x)] = -exp(-x')H"-,(x) ( H".,(x)
rr
24:10:3
n=0.2,4,...
2(n + 1)
H It'-it =
H" ,(x) - (-2)". "n
J1
n= 1.3,5,
2(n+ 1) and
24:10:4
J exp(-t')H"(t I = i o
n = 2, 4, 6, .. .
exp(-x2)H".1(x)
(-2)'"-'11'(n - 2)!! - exp(-s-)H"_,(x)
n = I, 3. 5, .. .
Three important definite integrals are 24:10:5
J= r
V; n!(b' - 1)"2/(n/2)! 0 0
t exp(-t2 )H"(bt)dt =
24:10:6
J
exp(-t')H"(bt)dt =
=
\cn"b(2b' -
2)"-" I'
n = 0, 2, 4, ...
n= 1,3,5,... n=0,2,4,... n=1,3,5,...
and
f. r" exp(-t')H"(bt)dt = \n!P"(b)
24:10:7
where P. is the Legendre polynomial of degree n (Chapter 21], and many others are listed by Gradshteyn and Ryzhik [Sections 7.37 and 7.38]. The orthogonality of Hermite polynomials is established by
r
24:10:8
0
J = exp(-t2)H"(t)H,"(t)dt = jV'n 2"n!
m*n m=R
24:11 COMPLEX ARGUMENT As for other orthogonal polynomials, the argument of the Hermite polynomials is generally encountered as a real variable.
THE HERMITE POLYNOMIALS H"(x)
223
24:14
24:12 GENERALIZATIONS Hermite polynomials are special cases of the parabolic cylinder function [Chapter 46J in which the order is a nonnegative integer: 24:12:1
2"12 expl 2
n = 0, 1, 2, ...
I
11
Because the parabolic cylinder function itself generalizes to the Tricomi function [Chapter 481 or the Kummer function [Chapter 471 these latter functions may also be regarded as generalizations of Hermite polynomials H"(x) = 2"U
24:12:2
I-n3 x` 2
/-n 24:12:3
-(-2)`" 1)
2 1
.\
3 x' n!! M/1-n 2 ; 2;
n = 1. 3. 5....
24:13 COGNATE FUNCTIONS The alternative Hermite polynomial He"(x) is related to H"(x) by a simple variable change 24:13:1
He"(x) = 2-"1=H"(x/\)
24:14 RELATED TOPICS Equation 24:3:6 arises in the application of Schradinger's equation to simple harmonic oscillators. The solution 24:14:1
f=expl 2'IH"(.t)=f(x)
i
is known as a Hermire function. In addition to sat sfying a typical orthogonality relationship 24:14:2
1 " f(t)f"(t)dt =IV'0'nn!2"
m*n
m=n
Hermite functions also satisfy such related expressions as 24:14:3
f(t) f (tt -
0
m* n t 1
m=n-!
CHAPTER
25 THE LOGARITHMIC FUNCTION ln(x)
The logarithmic function may be regarded as the simplest transcendental function, that is, ln(x) is the simplest function of x that cannot be expressed as a finite combination of algebraic terms. Prior to the advent of electronic calculators, logarithms [especially decadic logarithms. see Section 25:14] were extensively used as aids to arithmetic computation, both in the form of tables and in the design of devices such as slide rules. In addition to its treatment of the logarithmic function, this chapter includes brief discussions of the logarithmic integral li(x) and the dilogarithm diln(x).
25:1 NOTATION The name logarithm of x is used synonymously with logarithmic function of x to describe ln(x). Alternative notations are log(x) and log,(x). although the former is also used to describe a decadic logarithm- The initial letter is sometimes written in script type to avoid possible confusion with the numeral "one." See Section 25:11 for the significance of Ln(:). To emphasize the distinction from logarithms to other bases [Section 25:141, In(s) is variously referred to as the natural logarithm, the Naperian logarithm, the hyperbolic logarithm or the logarithm to base e.
25:2 BEHAVIOR The logarithm In(x) is not defined as a real-valued function for negative argument [but see Section 25:11] and takes
the value -x when x is zero. As shown in Figure 25-1, In(x) is negative for 0 s x < I and positive for x > 1. The slope dln(x)/dx continuously decreases as x increases so that, even though ln(x) increases without limit as x tends to infinity, its rate of increase becomes ever more leisurely. In fact, In(s) increases more slowly than any positive power of x: In(x)
25:2:1
x''
-.0
x-,x
v>0
25:3 DEFINITIONS As illustrated in Figure 25-2, the logarithmic function is defined through the integral 1
25:3:1
In(s)
-dt
x>0 us
THE LOGARITHMIC FUNCTION ln(x)
25:4
4
4
4
4
4
4
4
4
4
4
4
226
4
4
4
4
4
:....:....:....:....:....:....:....:....:....:....:....:.... ....:..-
5
.:....:....:........:........:....:....:....:1...:....:....:....:.. 4
It may also be defined as the limit 25:3:2
ln(x) = lim[v(xU' - I)]
x>0
v>0
or by the rule 25:3:3
ln(x)=f
if
x=el
where e is the number 2.1718281828 defined in Chapter 1.
25:4 SPECIAL CASES There are none.
25:5 INTRARELATIONSHIPS The logarithmic function of any argument in the range zero to unity may be related to ln(x) where I < x < x by the reciprocation formula
THE LOGARITHMIC FUNCTION In(x)
227
In(-' = -ln(x)
25:5:1
X
25:8
X>0
The logarithmic functions of products, quotients and powers may be expressed by the formulas 25:5:2
ln(xv) = ln(x) + ln(y)
25:5:3
ln(x/y) = ln(x) - In(y)
25:5:4
In(x) = vin(x)
X>0
y > 0
x>0
y>0
X>0
If a function f(x) is expressible as the finite (or infinite) product IIf,(x), its logarithm is the finite (or infinite) sum
ln(f(x)) _
25:5:5
all f;(x) > 0
In(f1(x))
provided, in the infinite case, that the series in question converges.
25:6 EXPANSIONS The logarithmic function may be expanded as a power series in a variety of ways, of which the following are representative: x2
25:6:1
ln(x) x_-- + 1
25:6:2
X2
25:6:3
(x - 1)'
x
-1
ln()= x + I
+
27
(x' - W 3(x + 1)'
+
(-x)' I
x3
+
(x - 1)' 3.r'
_
+
(x - 1)'
1
ra-
jx'
2
(x2 - 1)S
I
r' - 1)7j"
5(x' + 1)5
2j + I
l
i=o
x>0
As well, the logarithm is expansible through the continued fractions x
I
25:6:4
4x
x
9x
16x
In(1 + x) = 1 - -
I+ I-x+ 2-x+ 3-2x+ 4-3x+ 5-4x+
and
In(l +x) 25:6:5
x
_
I
x
x 4x 4x 9x 9x 16x
1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+
25:7 PARTICULAR VALUES
In(O)
In(I)
In(e)
We')
0
1
n
25:8 NUMERICAL VALUES Computer languages invariably permit logarithms to be calculated by a single command. Likewise, scientific calculators mostly incorporate a key that generates the value of In(x) from an x value in the calculator's register. Sometimes a "log" key must be struck and the result multiplied by 2.302585093 (see Section 25:14]. Because of the widespread availability of such devices, we include no algorithm for the logarithmic function, although the universal hypergeometric algorithm [Section 18:14] does permit logarithms to be evaluated to any sought precision.
THE LOGARITHMIC FUNCTION In(s)
25:9
228
25:9 APPROXIMATIONS A number of approximations, including
ln(x) =
25:9:1
x-I
3
4
4
3
- s x :S -
8-bit precision
x and
25:9:2
(x - I)
ln(x)
(6)3f5
-sx52
8-bit precision
1 + 5x
2
are available when x is close to unity. Based on the limiting expression
n-' x
In(n) -. -y + E 7,
25:9:3
y = 0.5772156649
/=1 1
where y is Euler's constant [Chapter I), we have the approximation 25:9:4
In(x) _
frac(x)
`n"'
- 0.58 +
X
1
8-bit precision
j.1 J
x z 31
valid for large arguments.
25:10 OPERATIONS OF THE CALCULUS Single and multiple differentiation give b
d
25:10:1
25:10:2
rIn(bx+c)=bx+c
I:/ - ln(bx + c) = -(n - I)! dx"
/ -b 1
n= 1,2,3,...
bx + c
while indefinite integration yields the following results: 25:10:3
In(bt + cdr = x +
C
[ln(bx + c) - 1)
)/b
25:10:4
b) [-In(x)1'
In"(t)dt = (- I)"n!x
fl "
J=r)
n = 1.2.3....
j!
+
25:10:5
f
r' ln(r)dr =
v
l1
I
In(x) -
25:10:7
v * -1
+
In'(x)
V = -I
2
dt 25:10:6
V.
_ li(bx + c)
J-/bin(bt+c)
b
t''dr ,0 In(t) - {In(ln(x))
v = - 1
The Ii function in the last two formulas is the logarithmic integral function [see Section 25:131. In addition to definite integrals that can be evaluated as special cases of the last five formulas, the following are of interest:
THE LOGARITHMIC FUNCTION ln(x)
229
25:12
In(-In(t))dr = -y
25:10:8
J0 (-mj/12
ln(r)dt
25:10:9
'- r
( In(t)dr _
25:10:10
J o t ± tIn(t)ln(1 t r)dt =
25:10:11
l -a'/6 (-G 1
2 - (ir'/12) - In(4)
f2 - (a'/6)
Others are given by Gradshteyn and Ryzhik [Section 4.2-4.41. The constants appearing in these expressions are defined in Section 1:7. Semidifferentiation and semiintegration, with a lower limit of zero, yield
d'r
25:10:12
in
ln(4bx) In(bx)=-
and
d1/`
25:10:13
x
T7, ln(bx) = 2
(ln(4bx) - 21
25:11 COMPLEX ARGUMENT When the argument of the logarithmic function is replaced by the complex number x + iy, the result is a function that adopts infinitely many complex values for each pair of x.y values. To emphasize this many-valued property. the first letter of the function's symbol is capitalized in writing: 25:11:1
-tr>
If x0
THE EXPONENTIAL FUNCTION exp(bx + c)
26:7
236
x x x x x exp(x)1- = - - - - - --1+ 2- 3+ 2- 5+ 1
26:6:3 and
x x x exp(x)=1- 2+ Ix +-----... 3- 2+ 5X
26:6:4
26:7 PARTICULAR VALUES exp(-z)
exp(-I)
exp(o)
exp(I)
0
t/e
1
e
exp(-)
26:8 NUMERICAL VALUES Most calculators and computers incorporate means by which exp(x) can be calculated directly. Therefore, no algorithm for the exponential function is included in this chapter although the universal hypergeometric algorithm (Section 18:141 may be employed to calculate exp(x) for any finite value of the argument x.
26:9 APPROXIMATIONS Relaxing the n -p x condition in definition 26:3:3 leads to 130
26:9:1
exp(x) = 1+
130
-I 5 X:5 1
8-bit precision
See the test values for an 'economized polynomial" approximation eo + e,x + e2x2 to exp(-x) in the range 0 5 x 5 1. This approximation has a precision superior to 8 bit, and is presented in Section 22:14. Some rational function approximations to exp(x) are given in Section 17:13.
26:10 OPERATIONS OF THE CALCULUS Single differentiation, multiple differentiation and indefmite integration of the function exp(bx + c) give d 26:10:1
dx
exp(bx + c) = b exp(bx + c)
d-
26:10:2
a;
exp(bx + c) = b" exp(bx + c)
and exp(bx + c) 26:10:3
f exp(bt + c)dt =
b>0
b
These three results may be generalized to the rule d" exp(bx + c) = b"exp(bx + c)
26:10:4
b > 0
[d(x + °C)]"
for differintegration (Section 0:101 with a lower limit of -x. The generalized differintegral with a lower limit of
THE EXPONENTIAL FUNCTION exp(bx + c)
237
26:10
zero is a function containing an incomplete gamma function [Chapter 45] 26:10:5
d"
Y(-v bx)
exp(bx + c) = b" exp(bx + c)
dr"
T(-v)
= x 'exp(bx + c)y*(-v;bx)
Differentiation of the self-exponential function gives d 26:10:6
x' = x`[1 + ln(x)J
dx
For positive b, indefinite integrals of t"exp(br + c) are generally evaluable as Kummer functions [Chapter 471
J r` exp(bt + c)dt = x
26:10:7
v
e
p(c)
v>-l
M(v + l;v + 2;bx)
and the following special cases apply:
exp(bx + c) - exp(c)
26:10:8
exp(bt + c)dt =
b
Jo
26:10:9
r" exp(bt + c)dt =
n! exp(bx + c)
c)dt =
exp(bt 26:10:10
-exp(bx+c)dawV
exp(b t + c)
J
26:10:11
n=0.1.2....
[exp(-bx)-a"(-bz)J
b>0
b>0
dt = exp(c) Ei(bx)
=
exp(br + c ) 26:10:12
dz
_ (nb"
I) exp(bx + c)Lexp(-bx) Ei(bx)
0!
1!
(n - 2)!
bx
(bx)r
(bx)"-'
n=2,3,4,
.
b>0
the a", daw and Ei functions being explained in Section 26:13, Chapter 42, and Chapter 37. See Table 37.14.1
for a compilation of indefinite integrals of .r"exp(x) for commonly encountered values of v. Equations 26:10:7-26:10:9 apply also when b is negative, but the general expressions 26:10:13
exp(c) I;
L I
26:10:14
J
exp(c)
t"exp(bt+c)dt
1
r(v+ I;-bx)
v > -1
b0
of Laplace transformation is easily implemented but the linear shift property 26:14:4
L{f(bi + c)} = b expl
I
[()
J f(t) expl
b Idt]
b>0
is generally less useful than the corresponding rule [equation 26:14:2\3] for Laplace inversion. Of course. if f is a function (such as sin or exp) for which an argument-addition formula [Section 0:51 exists, transformation of f(bt + c) may be accomplished by that route.
26:14
THE EXPONENTIAL FUNCTION exp(bx + c)
242
With a > 0 and u representing the unit-step function [Chapter 81. the function u(t - a)f(t - a) may be obtained by nullifying the t < 0 portion of f(t) and then translating the residue along the t-axis as portrayed in Figure 264. Such a function may be Laplace transformed by the delay property 26:14:5
L{u(t - a)f(t - a)} = exp(-as)fL(s)
a>0
For certain simple functions g(t), the composite function f{g(t)} may be Laplace transformed as the definite integral 26:14:6
L{f{g(t)}}
G(s.t)far)dt 0
where G(v) is a bivariate function, some examples of which are shown in Table 26.14.1. The most general chain rule for Laplace transformation is 26:14:7
L{f{g(t)}} = J
G(s,t)fL(1)dt
tIo
where G(u;t) is the function resulting from Laplace inversion of exp{-uG(s)}dG/ds, G being the inverse function of g [that is, G{g(x)} = x: see Section 0:3]. There is no simple product rule for Laplace transformation: that is. the transform of f(t)g(t) cannot be simply written in terms of 1L(s) and gas). For certain instances of f(t), however, simple relationships do exist. Thus: 26:14:8
L{exp(bt + c)g(t)} = exp(c)gL(s - b)
Table 26.14.1 g(r)
G(sx)
1 (2\ sr)
exP(_
4r /
exp(r)- I sinh(r)
V exp(-r)
I'(I+s) J,(r)
THE EXPONENTIAL FUNCTION exp(bx + c)
243
26:14
L{sinh(bt)g(t)} = I (gL(s - b) - gi(s + b)]
26:14:9
and similarly the Laplace transform of cosh(bt)g(t) is [gL(s - b) + gL(s + b)]/2. When f(t) is a power of r, Laplace transformation of the product f(t)g(t) corresponds to performing operations of the calculus on gi(s). The simplest cases d
L{tg(t)} = -
26:14:10
ds
gc(s)
and
r =J
1 g(r) L7 )
26:14:11
gc(s)ds
iterate to
L{t"g(t)} _ (- 1)"
26:14:12
n = 0, 1, 2, .. .
gr.(s)
ds'
and (
l
LS gr")} =
26:14:13
r
n = 2,3,4, .. .
JL(s)(dsr
J
and all these rules are subsumed in the general power multiplication property
d'
L{t'g(t)} = [d(-s + x)]' AL(S)
26:14:14
which corresponds to differintegration [Section 0:10] with respect to -s. the lower limit being -x. If f(t) is the derivative of a function F(t), then L{f(t)} = sFL(s) - F(0) and if f(t) is an indefinite integral of ti(t), then L{f(t)} = [IL(s) + f(0)J/s, Both these rules of Laplace transformation are special cases of the general differintegral transformation property [see Oldham and Spanier, Section 8.11 i
Lif (t) } = s fc(s) - Z s' (
26:14:15
l
If
J = -Int(-v)
tit, (0)
'
This rule applies for all is, but the summation is empty if v s 0. The periodicity property permits the Laplace transform of a periodic function [Chapter 361 to be replaced by an integral over a single period L{per(t)}
I-exp(-Ps)
I
I2 - I2 cothl Ps 1 / `26:14:16
per(t) exp(-st)dt =
per(t) exp(-st)dt
2 ///
Many important periodic functions are symmetrical in the sense that translation by a half-period changes the sign:
that is, if ger(t) is such a symmetrical periodic function of period Q, qer[t + (Q/2)J = - Ber(t). In such cases Laplace transformation may be accomplished by the formula 26:14:17
L{per(t)} = gerL(s) =
ger(t) exp(-st)dt
1 + exp
Qs
o I
2
= [.!.+
qer(t) exp(-st)dt
tanhl Qs JJ J
\ /
o
in which integration is over a half-period. The half- and fully rectified analogs [Section 36:131 of such a symmetrical periodic function have transforms equal to [Z - coth(Qs/4)] gerL(s) and coth(Qs/4) gerL(s). respectively, where gerL(s) is given by 26:14:17. The rules above, in conjunction with Table 26.14.2, permit a very large number of Laplace transforms to be
THE EXPONENTIAL FUNCTION exp(bx - c)
26:14
244
Table 26.14.2 Chapter or
Chapter or
section
section no
fm
no
7 10
r8( t)
1:13
p(c:h:t - a)
2c
hsj
exp(-as)sinh1 `1
28
i
8
steepening staircase
j - 0. 1. 2....
£u4t - Wit)
- {(sl
cs+b
bt + c
7
I
s
I
I
s'
-1
7
b
cs
cxp b) Ei
cs .
b
a >0
u(t - a)
j = 0, I. 2, ...
2£(-1)'uU - b - 2jb1
2£u(r -(j +
1 + 2!u)r - j:w')
7 7
29
840:s)
lengthening
2b
s
2Ps staircase
2
P/
LLL
10
6(r - a)
10:12
611 - a)
11
(br+cY
11
bc'-'(br + c)-
30
)
1
30
J
n - 2. 3.4. ..
(n!)'s
ta P-// J ,nh\l
30
4
exp(-as)
26 26
n'/r"'
II
b )'
j=0.1.2.....n
An -j)' ( CS, exp(rs\
b
29
s exp(-as)
._
^=2.3.4. -.
t"' exp( b
2
Ps'
a>0 a>0
n = 0. I. 2.
r"
(
+ I - coth Ps s
2
wave
2
1J
?t csch(-s) triangular
+ (-1)1-,, frac(2i) - J
9
-
12b.
sawtooth
P/
26
s'
b
1
Int(bt - 1)
9
-
27:13
-c exp(-) + cs - b
b Mhr s)
staircase
r
fret
9
27:13
g,(Os)
treads
b>0
Int(bt)
- sech(bs)
pulses
2s
16
37
26
with
j = 1, 2. 3....
c - bt
9
26
function
- exp(-asl
Staircases
j = 0. 1. 2....
)'7r-)
8
11
rs,
bs,c
8
8
Euler's
cs
b
26
8
(b
bs +c
10:12
8
/
I
E.
(cs\
Schl6milch functions
b/1
II 37:13
1)
at'+bi+c
(^ - 1)!b'(bs + rY'
n = 2. 3. 4, ...
c
b
2a
S
S'
7
1I
245
THE EXPONENTIAL FUNCTION exp(bx + c)
26:14
Table 26.14.2 (Continued) Chapter or
Chapter
or section
section
n = 0. 1. 2.
23
11
12
12
Vr 12
13
- + -2 16 ex -
12
12
1°C
cE
exp(Cs) erfcl
1 b1+c
b.
7 (1 - Wt - b)j
12
bJ
bsexp(b) erfcl
b1+ 12
b)
41
40
YY J
12
40
R erfclV bsl
Rb
26
12
bs+c 41.42
exp(-bf) erf(V -br)
or
VR
daw(V br)
bg0
12
a cxp(drl erfc(a V 1)
41
12
V Rt 41
exp(a-r) erfc(aVr)
40
erf(Vbf)
1b
Ib1+c)"
b""f' bt\\+ c
13
13
v>-1
lt-aY-'u(t-a)
13
13
12
s-b rl o /s,
13
13
12
V,I VJ +a) s
13
41
b
`.
,J-erf(VbJ)
Vr
41
(rbbl`ulr-b)
r(I-u(f-b)j b'P-'01 +cr-'
exp(b) r( I + v: b rl vl
v>0
v>-1 v>-1
v>-1
v>0 -1
cos(2
r
(Z
46
27:13
sinh(2\)
or
exp[-2V
27
J-' exp(-
)i/ y t
0v:t)
sin12\ b,)
El.
s
vk s
I
Vs O,(v:r)
27
b20
27
b 1E 0
27
(2s)", CXP(- V'2-b;)
27
S.
sech(V S) stnh(2v y s)
28. 29
sech(\/-S) sinh(Vs - 2vVs)
28. 29
csch(V) cosh(Vs - 2vVst
28. 29
2
27:13
27:13
c
S)
E1,11 exp(b)
rt
f xP(Q
46
27
exp
S
28. 32
46
b 3: 0
exp\s/
n = -1. 0, 1. 2...
or
v > -1
(bl
exp(br + c) erfc(V bt +
Icosh(2V )
J
r(v)exp(4b) D "\b)
2Vb-t 28. 32
a
\ 3s
51
0Kvs1
Vs
THE EXPONENTIAL FUNCTION exp(bx + c)
26:14
248
Table 26.14.2 (Continued) Chapter
Chapter
or
or
section no.
section no.
f(r)
IvI s 2
B.(v;r)
27:13
28. 29
e.(o;s)
square wave
/bt
27:13
y..(b)
21x-br)"exPl 2) csch(b)
29
csch(V s) cosh1206)
lengthening
sgnjsin(V ;))
32
Vs
44:12
n0.I.2...
22
29
1 G(s
sech(br)
+ bl l 2b
2b
t
v>0
(2hr1' csch(br)
29
44:13
fll+v) I 1 + v: s+h) b
64
2b
29, 30
b'? I I + csch(br) - coth(br))
64
29. 30
b"r"-'II - csch(br) - cothlbr))
64:13
30
tanh(br)
l G12 I_1 b
2b 32
sgn sin
ar
1
squarewave
(26
tanh(bs)
(-2b)
h
b
Zh.
[h.(!) b.``
31
2 arstnh(br)
31
arcosh(l + br) K (bi)
iY,(br)
54. 57
32
32
y'r
sinr
rh(br)
or
l
or
l2rl
Vt+b 2r)
0 * v > -I
r" sin(bi)
bc02V br)
bei(2Vbr1
or
32:13
)-J
b sinc w
35
1
= I sin(bi)
larcoshll\\
31
b
bs,
or or arsinh
expl-
,
)
31
32
r( sin v (s + b')"r
arccot(-J)
35
b
in erf(V b%4s )
I
S
I
arsinh
SI
K()
s
Sin(V'h)
32
55
Vs'-h=
57
lb. J
exp(b/ I
44
s
-Y,(s)J
S
51
30
1n(!-ly(s -I
coth(bi) - br
30:13
44:13
s
/6 \
I
s
S
40
i-
cos - or-sin I g
32 S
-
b arccat - I = arcten b s. s
35
rl
2 arctan(V)
exp\6/ e1fc1 5
1
bl
41
THE EXPONENTIAL FUNCTION exp(bx + c)
249
26:14
Table 26.14.2 (Continued) Chapter
Chapter
or
or
section no.
f(r)
35
arctan(br)
section no.
US)
MsCirbl u/b) l1 /J
I- Si(-)
cos(S
IlL
38
I
57:13
srC,lbt)
38
Si(br)
2
s
arccot(/b)
2 daw(2
45
2"yl v:
b=r'1
v>0
I
2/
35
2b
s- `
41
/ s
expl - I D_,.l - I
20s)'
)
46
b
4b
2s b
37
Ei(b' )
4b'
f(2v)
f 2v;-+
45
arctanlsI
I exp(s _) erfcl s s
/
s
s'
b
erf(bt)
40
35
\\\
1
br
42
s/ s)
V7 \b'
K_.(2Vbs)
51
///
56
47
50
48
l
}}
2
r
r(3/ (2)"'Jexpr \b/ r( 3: c > -1
r MIa;2c br)
r(c)'vbr"-"'
00'
1._ ,(2V'
r
f(I
+ c) FI a, I + c:2c: b1 I
I b\ -s"M ax; f(a)
br)
l
45
60
47
$
t'"'exp(-Br)U(c - ba + a - 6:811
B''" F a,b:c:
s
60
B
2b sinc(vtb - s)'-'tb + s)-" 48:4
k
I - v; I + v:
B(
49
bin(br)l,lbr)
49
sl
2b
s K(-)
61
I - II E(26)
6I
LLL
50 r 2 51
51
KdV 'b,)
K,;.."(br)
-I S v t 0
a
b 1f exp(8s/ K.(±)
51
a csc(va)P.(b)
$9
exp(b) K. :.,(bl
51
YYYar
59
\ P,11 + b,)
THE EXPONENTIAL FUNCTION exp(bx + c)
26:14
250
determined. The table lists pairs f(t), fL(s) of functions and their Laplace transforms. To use this table for transformation, first locate the chapter or section of the Atlas in which the f function appears- Then scan the numerical index that constitutes the first column of the table to find entries containing the f function. Of course, there exist many Laplace transform pairs that are not listed in Table 26.14.2 and that cannot be conveniently deduced via rules 26:14:2-26:14:18. We recommend the comprehensive tabulation by Roberts and Kaufman as a source of several thousand varied transform pairs. A useful check on the accuracy of an f(t), fL(s) pair is provided by their limiting forms as the arguments approach zero and infinity. These limiting properties are 26:14:18
and
L {lim{f(t)}} = lim{sfL(s)} ,-.o
L f lim{f(r)}} = lim{sir(s)}
,-u
An even simpler check is provided by the requirement that the inverse Laplace transform of any continuous function
f(t) must tend to zero as s -. X. Section 18:14 demonstrates that the majority of the functions in this Atlas may be expressed as hypergeometric functions. Let f(t) be such a function and let us adopt (a,_x), as an abbreviation for the product of K Pochhammer polynomials
(a )j(w),(a3); ... (ax),
f(t) -
26:14:19
(a;-.k); P
=o (c1);(c.) (c3); ... (CL),
,_o (ci-L),
Then fL(s) is also a hypergeometric function, namely that given by expression 18:14:7. Moreover, many functions closely related to f(t) may also be Laplace transformed to hypergeometric functions with one or more extra numeratorial parameters. Thus: 26:14:20
L{t"f(br)} =
r(1 + v)
(1
i
+ v) (ai_x)
'
(cl.L)j
,=u
S
-v > -1 (b)'
'
s
and
1
26:14:21
L{t`f(sb r)} =
1
r(l + v) s
2 of
I I+ 2/
(ai-.x),
+ 4b'
'
` s'
(ci-.L);
j=0
(-
while the transform of ?f(±\/) is the sum 26:14:22
r(1 + v)
(I + v)j (;a _x) (. + ;ai .x) \4x-ib
r(# + V)a,.K
sin"cI-L
. o ± (4 + v)3 (} + }ai_K)j(l + ial-x),
4x_Lb
s J (i + ;ci_L),(1 + 1c'-L), of two hypergeometric functions, each with 2K + 1 numeratorial and 2L denominatorial parameters. The table of transform pairs 26:14:2 is also valuable for Laplace inversion. To use it for that purpose, first ;=o
scan the right-most column to locate the chapter or section number of the Atlas where the function fL is discussed. Then the second column lists the inverse transform of the function in the third column. As for transformation, there are a number of general properties of Laplace inversion that permit the table to be used to invert many more than the 120 entries that are displayed. Some of these general rules are presented in the following paragraphs. The linearity property of Laplace inversion follows immediately from equation 26:14:2. The linear shift property (
26:14:23
/
/\l
fL(bs + c) = LS b expl b tt)fl L I } =1110
of Laplace inversion incorporates the scaling property,, which`` is the c instance of 26:14:23. The most general chain rule for Laplace inversion gives the inverse transform of the composite function fL{g(s)}
as the definite integral of the product F(ru)f(u), where f(t) and F(ru) are respectively the inverse transforms of fL(s) and exp{-ug(s)}. That is:
251
26:14:24
(THE EXPONENTIAL FUNCTION exp(bx + c)
fc{g(s)} = Li
J0
exp(-ug(s)} = L{F(ru)}
where
mF(t;u)f(u)du}
26:14
Unfortunately, there exist rather few g(s), F(t:u) pairs that can be used to exploit relationship 26:14:24. Table 26.14.3 lists three. Whereas there is no such simple rule for Laplace transformation, there does exist a general product rule for Laplace inversion. This is the so-called convolution property 26:14:25
- u)g(u)du } = L{f(t)*g(t)}
fL(s)gc(s) = L{ J ,f(t a
111
Two straightforward applications of this property lead to the inversion rules
26:14:26 and
26:14:27
b>0
exp(-bs)gL(s) = L{u(t - b)g(t - b)}
//bsr `111
I
l
J.1
1
cothl -2) gi(s) = Li g(t) + 21 u(t - jb)g(r - jb) }
/t
J = intl -) \b
To Laplace invert the product s"gL(s) involves operations of the calculus applied to the inverse transform. The simplest cases l 26:14:28
s
(r
k(s) = L j
J0
g(t)dt
l
and
sgc(s) = L
26:14:29
d
(t) +9(0)8(1)
ddtt
iterate to 26:14:30
AL(s)
(
= L{
r
lJ
l
... J0g(t)(dt)" }
-'
;
n = 2. 3.4....
JJJ
and
26:14:31
s"gt(s) = L {dtg (t) +
(0)8 2-o g d
n = 1, 2, 3, ...
)(t)}
Here 8'`(t) is the n" derivative of the Dirac delta function (Section 10:131 at t = 0. away from the immediate vicinity of t = 0 all such functions are zero. Ignoring the contributions from these delta derivatives, the inverse Laplace transform of S' L(s) is the differintegral d'g/dt' for all values of v. A number of useful rules exist for the Laplace inversion of the quotients ft{g(s)}/g(s) and fL{g(s)}/h(s) where g(s) and h(s) are such simple functions as f, I/s, s- . o-. etc. Inversion usually leads to definite or indefinite Table 26.14.3 g(s)
F(ru)
In(s)
rim)
exp
s
(3r'')"'
AiIIII
Q
\(3:t'
THE EXPONENTIAL FUNCTION cxp(bx + c)
26:14
252
integrals of the product of f(t), the inverse transform of 0s), with other functions. Examples include
fL\s/
(
= Li
26:14:32
s'+a)_
fL(
26:14:33
s+a
=
cj
(
l tu)f(u)du }
cos(2
1
LS i Jo(
ar - aue)f(u)du i
l
a?0
1
with Io( au2 - ate) replacing Jo( at - au2) if a is negative; Roberts and Kaufman [pages 171-1741 may be consulted for others. If fL(s) is the hypergeometric function
k(s) _
26:14:34
I j (a)/(a2), ... (aK)/
I
s ,-o (e1,(c., ... (CL),
s
/
1
(a .r );
s
(c,_L)/ (S')
then inverse transformation of fL(s), and a number of related functions, give hypergeometric functions of t. The results 26:14:35
1." fL s
=L
b
t`
/t e
S. fLl bs I = L
26:14:36
1
26:14:37
(± L\
f
s
b/
T'(1 + v) ,_o (1 + v), (c1-L),
_
L
C(1
v> -1
(a1~x),(±b't /4)' + V),( V\'
+ v)
v > -1
J-0 (I
(lair)/(i + 'a,-x), (br14X"L)' t" f(1 + v),_o (1 + . + ;c 1-L), 1112 "ai-x LS
I'( + v) l b ci_c
(# + #a1-x)/(1 + lahl )/(bt/4K L (j + v),(! + ICI-J,
are seen to be strict counterparts of the transforms 26:14:20-26.14:22. We conclude this section by drawing to the attention of the reader the Heaviside expansion theorem that is where discussed in Section 17:10. This theorem provides a procedure for the Laplace inversion of is a polynomial function of degree n. A similar procedure may be employed to invert the rational function p,,(s)/ provided m < n, via the decomposition detailed in Section 17:13. The Heaviside theorem may even be applied for infinite values of m and n, provided that the denominatorial degree exceeds that of the numerator, and hence may be employed to invert the quotient of two transcendental functions that may be written, for example. as (Aa + A1s + A2s2 + - )/(a1s + a2s2 + as' + - ).
CHAPTER
27 EXPONENTIALS OF POWERS exp(- ax" )
Many of the results of this chapter follow by combining the properties of the functions discussed in Chapters 12, 13 and 26. It is appropriate to devote a chapter to the functions exp(-ax"), however, because they are of such widespread importance, particularly when a is positive. For example, the temperature dependence of many physical properties obeys this functionality with v = - 1. Random events often involve the v = 2 instance and lead to important distributions, as discussed in Section 27:14.
27:1 NOTATION No special notation or symbolism need be considered beyond that addressed in Sections 12:1 and 26:1.
27:2 BEHAVIOR As with the functions discussed in Chapter 13, the range of exp(-ax") depends in a rather detailed way on the characteristics of the number v. In this chapter, however, attention will be confined to the range x >- 0. except when v is an integer. Figures 27-1 and 27-2 are maps of the functions exp(x') and exp(-.r") for assorted values of v. The important graph of exp(-x2) versus x, known as a Gauss curve, displays a maximum value of unity at x = 0 and points of inflection [see Section 0:71 at x = ±1/V2. Similarly, exp(-1/x) has an inflection at x = #.
27:3 DEFINITIONS With x replaced by -ax', any of the definitions in Section 26:3 can serve to define exp(-ax').
27:4 SPECIAL CASES When v = 0 or 1, the exp(-ax') function reduces respectively to a constant [Chapter I] or a simple exponential [Chapter 26].
254
EXPONENTIALS OF POWERS exp(-ax")
27:5
0
O
i
0
i.
4
4
R
ti
'r0' p
4
4
+NO
+p 00 4ti b+pN 4 4 4 ,4
0
iX10 4
A
11.
CO
4~
4
4
:exp(-1/x7
exp U /x)
O
v
a
Co
if if
ti
a
i4
tiCj
06 O.0
4f
if if
4f .
.
ti
ti .
a
v
4b
if if .
.
1.0
27:5 INTRARELATIONSHIPS The relationships of Section 26:5 hold when x is replaced by -ax". Summation formulas for series of exponentials of I/x include
/
X1
exp(
27:5:2
expl X1 I - expl
sI
\
/
and
\
9)
4
I + expl x + exp( z
27:5:1
///
+ expl
+
9J -
az
rtx - 1
)
= 2 93(0;x) - 2 = =
Z
-z
z 03(0:x)
2
Z
+ \ exp(_ r2x)
- N/Trx expl
4xI
x>0
x >0
EXPONENTIALS OF POWERS exp(-ax')
255
27:5:3
-+
e.p(ex - + P( xl) + ex p\ x / p(x5)
_ -4 B. (0;-
4/
27:9
- 2exp
4
4=x/
X>0
in addition to variants of equations 26:5:5-26:5:12. The B functions are explained in Section 27:13. The final approximations improve as x increases but are valid to better than I part in 109 for x z 1.
27:6 EXPANSIONS The series expansion
exp(-ax')
27:6:1
=1--- ax"
+
= i (-ax')'
a'x''
I!
2!
j,
i-s
holds for all a, for all v and for all x for which x' is defined. The continued fraction expansion 27:6:2
exp(V)=1+
2V
--
x/3 x/15 x/35 x/(4j' - 1)
2 - + 2+
2+
2+
2+
is rapidly convergent.
27:7 PARTICULAR VALUES Choosing a = I or a = - I and taking care that x' is defined we find
X = 0
exp(x')
exp(-x')
Jx
v < 0`
v>0
I
{
x = x
x = t
{x1
0 v0
1
{0
v0
v0
27:8 NUMERICAL VALUES Values of exp(-ax') are calculable by first evaluating -ax', followed by exponentiation.
27:9 APPROXIMATIONS Though crude, the approximation 27:9:1
exp(-x')-
I0
-
-
IrI ;!: V-1r
is never in error by more than 0.09. This triangular approximation to the Gauss curve has an area of V, equal to that under the curve.
EXPONENTIALS OF POWERS exp(-(xx')
27:10
256
27:10 OPERATIONS OF THE CALCULUS Differentiation gives d
d exp(-ax') =
27:10:1
-avx"_i
exp(-ax")
Indefinite integration of the general expressions cxp(-at') and t" exp(-at') can be accomplished by making the substitution -at' _y and utilizing the general formulas contained in equations 26:10:7. 26:10:13 and 26:10:14. Below are listed some important indefinite integrals: exp(t')dt = exp(x2) daw(x)
27:I0:2 Jo
J exp(V)dt = 2(f - 1) exp(
27:10:3
J
27:10:4
)
Idt = (x + f) expQ10 - Ei(VIx)
j exp(
x>0
o
f
27:10:5
exp(-dt t
Ei11f
= x exp(1
x>0
x
x
J exp(-r)dt = 2 erf(x)
27:10:6
27:10:7
f
27:10:8
f
exp(-V)dt= 2(f + 1) exp(-Vx)
exp(:'
dt=(x-V)exp(-:1
71
N/_X
-I I exp(-) dt=xexpl-)+Ei J! /B2- 4ay 1
27:10:9
t
1
1
x///
x/
1
6 exp(-at' - At - y)dt = 2
27:10:10
-Ei1Vzl
expl
4a
J
erfc l
x>0
x>0 tax + 2Va
a
>0
Chapters 42, 37 and 40 are devoted to the daw, Ei and eerfe functions. Some important definite integrals include
ll v
j:exP(_w)dt =
27:10:11
-(D+
J
1> 0
a> 0
and
27:10:12 o
t'
dt=111 exp(-2V)
a>0
0
but for a more comprehensive listing see Roberts and Kaufman, in which the notation for theta functions is almost identical with that adopted in this Atlas. The identity of expressions 27:13:5-27:13:8 with definitions 27:13:1-27:13:4 permits a reformulation of series of exponential functions, as follows: 27:13:15
I I
1 + 21 (-1)' cxp\(
L
ll
x/J
= 92(0;x) = 27J=o
exp-
(2j 41)'n'x
(
x>0
2
27:13:16
1
I + 2i expl
27:13:17
eXP((24x =o
x IJ = 93(0;x) = I + 21 exp(-j'a'x)
1)') =
94(0:x) = I + 27_ (-1)' exp(-j-ar2x)
x>0 x>0
,-I
Theta functions with v = 0 are important in their own right. Several of the representations coalesce when x leading to the particular values 27:13:18
2'/'e2(0; !) = 93(0; !) = 2'"9.(0; It
i) = 1.086434811 zr
IT
where U is the ubiquitous constant discussed in Section 1:7.
Theta functions play an important role in the theory of elliptic functions [Chapter 63]. As well as the theta functions discussed above, there are other versions known as Neville's theta functions and Jacobi's theta functions [see Sections 63:8 and 63:131.
27:14 RELATED TOPICS When a measurement or observation is repeated a large number N of times, it frequently happens that the values found are not identical. We say that the measured quantity x has a distribution. Sometimes (as in rolling dice) only a finite set of values is available for x. Here, however, we consider the continuous case (exemplified by sizes of raindrops) in which possible x values are limited in proximity only by the discrimination of the measuring device.
Let the measurements be arranged in order of size along the line -x < x < X. Then, as N -' x. it often becomes possible to delineate a density function or frequency function f(x) with the property that 2f(x)dx gives the approximate probability that any single measurement of x will lie in the range x - dx to x + dx. Associated with each distribution is a mean defined by 27:14:1
and a variance defined by
µ = J jt f(t)dt
27:14:2
f
EXPONENTIALS OF POWERS exp(-ax')
259
v= =
j (I - p.)'f(t)dt = -12 + Jf
27:14
t2f(t)dt ? 0
The limits, xo to x,, on the integrals demarcate the range that is accessible to x, often -x to x or 0 to x. The mean of a distribution provides a measure of the average value of the property x, while the variance describes the dispersion of the distribution about the mean. If the distribution has too great a dispersion, no finite variance exists. The Lorentz distribution [see Table 27.14.11 is a case in point: for this particular distribution, depicted in Figure 27-3 for a = 8, the integral in 27:14:2 diverges. Even integral 27:14:1 diverges for the Lorentz distribution, but the so-called Cauchy principal value of the integral, namely c
lim f rf(r)di
27:14:3
=L O
h
g.
4)
O
xa
g
(z3
:....: 0.8
....: X. ....:........... 0.8 :....:....:
0.4
FIG 27-3 is as tabled.
Accompanying each distribution is a cumulative function or distribution function, given by 27:14:4
Kr) = f x f(t)dt
where, by definition of f(x), F(x,) = 1. The value of F(x) necessarily lies in the range 0 s F(x) s 1 and expresses the fraction of the measurements that (for N -. x) will lie in the interval between ro and x. Statisticians speak of percentiles or percentage points: the p'" percentile is the value x,1100 of x such that 27:14:5
F(x,i,oo) = 100
The fiftieth percentile .r1j2 is also called the median of the distribution and satisfies the condition 27:14:6
f(t)dt =
f
f(t)dt = 2
Though with symmetrical distributions they are the same, the mean and the median are not, in general, identical, nor does either necessarily correspond to the peak that often exists in a graph of f(x) versus x. Where such a peak exists, its s-coordinate is known as the mode or most probable value of the distribution. Distributions occur widely, especially in statistics and physics. Examples from both of these fields have been assembled into Table 27.14.1, and a selection is displayed graphically. The archetypal distribution is the normal distribution shown in Figure 27-4 for a = 3. The Gauss distribution is the p. = 0 instance of this. Notwithstanding
EXPONENTIALS OF POWERS exp(-axe)
27:14
260
Table 27.14.1 D01nMron
U. l*m nnanpularl
Llmm ,n In al
Mph
Dense/ luncuon
Yarulwc
Cunwlwn< fuacu0n
0
n+r
Fn
an c,l
dl\¢al
-
Gws. 2v
`!scap{- I.r-411
.1.,110 Gems!
tap-normal Raylcl circular nonnel
01o=
`'! 0 exp -0 ln'(
va
.
Ii
µ!
l
(. Cl17\ain-µ1I 7
µ tap(
- 11
160 1
r
010
0
a.
np-a l
` eap -o r 1
rr, \
0 n=
I - ecpl
Iwhlvl, - wl
weh'Ivu - r11 1
spa.. - yl
tapl-0p - µll Bcllxmann 1¢
a\e
I -cap -.5 1
tar cap - an' I
0 1,,
µ
L
1 - eap-nn
o tap-0r!
010=
II - c.M-=1 - rlll
e%p0ntnltall
Fenru Dlrac
Burt-Eimmn
ae w
n0
dan,J -Br
IM1 - BXB - enpvnl
01a1 01
-v0
dlln! l -BI
MI I -
n lafl - B1
B)
- IMB' cap,,')
o lnl a - B,
In. I
nhwI -0.
'Bl
- Inkapvnl - 01
Inl1 -Bl
r
In! i - BI
010 = ,rln/21
0.-
tw(-v µ
r101,
v0
0.-
B01a
v tMl - Bl
Blv.B!
Sntdaa s+ w>11
ow. a1rU. all1.0agh
IfeQel
-C .. by,
µ - Cawk
a rc s
*0 LMt\ ul, - on 2
pnwipal
A
valve
fkll. la CwMll
-0)
-a10%
In-II'
µ 1..
S[ 12//=11!l!'
f
2\rn-2l"`
4, 6 6.
I
SIu
m,-r
n=3.5.7...
-sa:
la - II!!
r\iln - 21!'
I1
na i°-1.: 1
0
]
1
_
-aacuw,/\ nl 121 f' 121
W
EXPONENTIALS OF POWERS exp(-ax")
261
,, 6
,'L6
x6
ig 4q 44
4
4
,6
xti6 ig.t
27:14
x
iqt
:....:....:...0.4 0. 2
x6
,
rL6
It
q'}
x10
4
:....:....:....:..............:....:....:.1.0
.:....:....:....:.0.6
:....;. /.:.....:....:.
...:....: 0.4
0. 2
: FIG 27-5 :
:.:....:.............. :....:....:-.. 0
6
,6
it
4 .........
:
ig
it
......:....:....:.0.e
.:....\ .. Z ....: ....................0.6
FIG 27-6 :....:....: ...................0.2 r
.f Cx) .
27:14
EXPONENTIALS OF POWERS exp(-ar")
262
its name, the normal distribution is encountered in practice rather seldom. The asymmetrical log-normal distribution, in contrast, finds more frequent application. An example of this latter distribution in which µ = 2v is illustrated in Figure 27-5. Even less symmetrical is the Boltzmann distribution depicted in Figure 27-6: such a distribution describes, for example, the variation of the numbers of molecules with height in the earth's atmosphere. The shape of the Weibull distribution is a very strong function of a (which must be positive) and this distribution therefore finds widespread empirical application in failure analysis.
CHAPTER
28 THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
This chapter and the next two chapters address the six so-called hyperbolic functions. The present chapter deals with the two most important of the six: the hyperbolic sine and the hyperbolic cosine. These two functions are interrelated by 28:0:1
cosh''(x) - sinh'(x) = I
and by each being the derivative of the other [see equations 28: 10:1 and 28:10:2].
28:1 NOTATION The names of these functions arise because of their complex algebraic relationship [see Section 32:111 to the sine and cosine functions. Their association with the hyperbola is explained in Section 28:3. The notations sh(x) and ch(x) sometimes replace sinh(x) and cosh(x). Although they cause confusion, the symbolisms Sin(x) and Cos(x) are occasionally encountered.
28:2 BEHAVIOR Both functions are defined for all arguments but, whereas the hyperbolic sine adopts all values, the hyperbolic cosine is restricted in range to cosh(x) ? 1. Figure 28-1 shows the behavior of the functions for rather small arguments. For arguments of large absolute magnitude, both functions tend exponentially towards infinite values.
28:3 DEFINITIONS The hyperbolic sine and cosine functions are defined in terms of the exponential function of Chapter 26 by 28:3:1
sinh(x) =
28:3:2
cosh(x) =
exp(x) - exp(-x) 2
exp(x) + exp(-x) 2
263
28:4
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
ti
440
264
44
To provide a geometric definition of the hyperbolic functions, consider the positive branch of the rectangular x - 1 depicted in Figure 28-2. The green area is bounded by the hyperbola and by a pair of straight lines OP and OP' through the origin with slopes that are equal in magnitude but opposite in sign. Let a denote this shaded area: it can take values between zero (corresponding to points P and P' coinciding with A) and infinity (corresponding to lines OP and OP' having slopes of +I and -1 and constituting the asymptotes of the hyperbola). The lengths PQ and OQ may then be regarded as functions of a and are, in fact, the hyperbolic sine and cosine of a hyperbola [Section 15:41
28:3:3
PQ = sinh(a) = sinh (green area)
28:3:4
OQ = cosh(a) = cosh (green area)
The second-order differential equation d2f 28:3:5
dx2
= b2x
has the general solution f = cl sinh(bx) + c2 cosh(bx), where b, cl and c2 are constants.
28:4 SPECIAL CASES There are none.
28:5 INTRARELATIONSHIPS The hyperbolic cosine function is even 28:5:1
cosh(-x) = cosh(x)
265
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
whereas the hyperbolic sine is an odd function
sinh(-x)
28:5:2
-sinh(x)
The duplication and triplication formulas 28:5:3 28:5:4
cosh(2x) = cosh 2(X) + sinh2(x) = 2 cosh2(x) - 1 = 1 + 2 sinh((x) sinh(2x) = 2 sinh(x) cosh(x) = 2 sinh(x)
1 + sinh2(x)
28:5:5
cosh(3x) = 4 cosh((x) - 3 cosh(x)
28:5:6
sinh(3x) = 4 sinh((x) + 3 sinh(x) = sinh(x)[4 cosh2(x) - 1]
generalize to
28:5:7
cosh(nx) = TA(cosh(x)) = i t' cosh(nx) !-a
and (tp'cosh(x) - tt_',] cosh((x)
28:5:8
sinh(nx) = sinh(x)Un_,(cosh(x)) = i sinh(x)
j_a
where the T. and U. Chebyshev polynomials are discussed in Chapter 22, as are the Chebyshev coefficients De Moivre's theorem 28:5:9
cosh(nx) ± sinh(nx) _ [cosh(x) ± sinh(x)]" = exp(±nx)
is also useful. Equations 28:5:3 and 28:5:4 may be regarded as special cases of the argument-addition formulas 28:5:10
cosh(x ± y) = cosh(x) cosh(y) ± sinh(x) sinh(y)
28:5:11
sinh(x ± y) = sinh(x) cosh(y) ± cosh(x) sinh(y)
From 28:5:3 one may derive the expressions x) 28:5:12
=
/cosh(x) + 1
cosh)
2
2
x2-1
28:5
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
28:5
266
and
cosh(2) - I
sinhl 2 I = sgn(x)
28:5:13
for the hyperbolic functions of half argument`, as well as the formulas cosh(2x) + 1 28:5:14 coshz(x) = 2 and cosh(2x) - 1 28:5:15
sinh'(x) =
2
for the squares. These latter may be generalized to the expressions
a-nr ( \ nlcoshl(n-2j)x]
1
2" 28:5:16
1
}
j
,=o
n = 1,3,5,...
1 (J) cosh[(n
cosh"(x) =
21
1
n=2,4,6,... and
1
2",=o 28:5:17
sinh'(x) =
2"--1 1 (-I)' I n I sinh[(n - 2j)x]
(-1N I n J sinh[(n - 2&1 =
_o
n = 1,3.5,... (n)
I
2"
cosh[(n - 2j)x1 =
(-l)"/=(n n!!
a/A-1
+
n = 2, 4, 6, ...
I n I cosh[(n - 2j)r1
1)
2"-1
j)
,-o
for any positive integer power of the hyperbolic cosine or sine. The function-addition formulas cosh(x) i- sinh(x)//= exp(±x)
28:5:18
28:5:19
cosh(x) + cosh(y) = 2 cosh( x 2 ,_ I cosh (x
sink( Z ) sinh \\(x
28:5:20
cosh(x) - cosh(y) = 2
28:5:21
sinh(x) ± sinh(y) = 2 sinhl Z cosh lx
Z
,J
2 y)
Y)
and the function-multiplication formulas
2
)
\\ 1
28:5:22
sinh(x)sinh(y)
=
cosh(x + y) 2
28:5:23
sinh(x) cosh(y) =
1
2 1
28:5:24
cosh(x - y)
2
sinh(x + y) + 2 sinh(x - y) 1
cosh(x) cosh(y) = - cosh(x + y) + - cosh(x - y)
2
2
complete our listing of intrarelationships between these most maleable functions.
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
267
28:9
28:6 EXPANSIONS The hyperbolic sine and cosine functions may be expanded as infinite series x2/*r
x5
x3 sinh(x)=x+-+-+ 5! 3!
28:6:1
x2 x4 cosh(x) = 1 + - +
28:6:2
4!
2!
,=o (2j + 1)! x2j
_ i -
+
(2j)!
/-o
or as infinite products
_) x2
sinh(x) = x I +
28:6:3
zr
28:6:4
cosh(x) = 1 1 + \\\\\\
//
1+
I
\\
Z) 4n x2
=xnI+ ;_1
x2
jn
22
4x22) ... _ I+ 4x2)(I + 4x) 9 rr (j +x2 0) zrr z '+7 1\ \\I + 25.r j.1
28:7 PARTICULAR VALUES x =-1
x= -m
I
sinh(x)
e2
x=0 0
x=I e-1
x=m
2e
2e
I+ e' coah(x)
1
2e
2e
28:8 NUMERICAL VALUES These are easily calculated via equations 28:3:1 and 28:3:2. There is an algorithm in Section 29:8 that enables any one of the six hyperbolic functions, including sinh(x) or cosh(x), to be evaluated. As well, the universal hypergeometric algorithm [Section 18:14] permits values of sinh(x)/x and of cosh(x) to be found.
28:9 APPROXIMATIONS The hyperbolic sine and cosine may be approximated by polynomials; for example, x3 28:9:1
sinh(x) = x +
8-bit precision
jxj < 0.84
6 28:9:2
/ cosh(x) = 1 I+
z
4)
jxl < 0.70
8-bit precision
at small arguments and by exponential functions 28:9:3
28 : 9 : 4
when the argument is large.
sinh(x) = sgn(x) cosh(x)
=
2
exp2lxi)
jxj > 2.78
8-bit precision 8 -bi t prec i s i on
l
xi > 2 . 78
28:10
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
268
28:10 OPERATIONS OF THE CALCULUS Differentiation and indefinite integration of sinh(bx) and cosh(bx) give d 28:10:1 sinh(bx) = b cosh(bx) dx d
28:10:2
cosh(bx) = b sinh(bx)
I
28:10:3
sinh(bt)dt =
cosh(bx) - I b
0
sinh(bx)
. cosh(bt)dr =
28:10:4
Jo
b
The general formulas
-id: n!!
"
cosh (t)dt =
28:10:5 o
(2j)!!
Lx + sinh(x)
(-1)u-1)/2(n
.o (2j + 1)!!
sink"(t)dt =
, ...
n = 2, 4, 6, .. .
cosh'-'"(x)
r"-1W2
- 1)!r E (- Iy (2j(2')r .r
- 1)!! cosh(x)
nn n!! 28:10:6
3. 5
(2j)!! w12 _ r
(n - I)u rr
n,
n=1.
cosh2'(xl
_o
r"(2j)!!
(-1)"'2(n - 1)!!
[x+cosh(x)
1
(2j + 1)!!
;_o
n = 1, 3, 5, ...
sinh2'(x)
.
'_o
(-1)'sinh2'+'(x)n!! n = 2, 4, 6, ...
permit the indefinite integration of integer powers of the hyperbolic sine and cosine functions. Alternative expressions may be derived by integration of equations 28:5:16 and 28:5:17. Noninteger powers are treated in Section 58:14. Other important classes of indefinite integral include
-+
n!sinh(bx)
b"' -
(bx)"
(b.)" -2
M!
(n - 2)!)
n! cosh(bx) r (bx)"-'
n! sinh(bx) r(bx)" t° cosh(br)dt =
+ bx
l
- n!cosh(bx) [(bxr+ (bx)s (n --I)!
+n!
(n --3)!
n!
(n - 2)!
+.+1
- n! cosh(bx) r (bx)"-'
j;77- L (n - I)!
Jt" sinh(bt)dr =
28:10:8
n!
- b-+,
J
n=1.3,5,...
a!cosh(bx) 1(bx)" + (bx)" 2 b0+1
n=2.4.6,...
(bx)"'2
I - + + L n! (n - 2)!
b1
1
(bx)"-3
+
IL
28:10:7
1
+...+1J
(bx)"-,
] + ... i. bx
+ (n - 3)!
n = 2, 4, 6. ...
0
n! cosh(bx)
(bx)°
(bx)r'2
+
-
L n! n!sinh(bx)
+ bx
]
(n - 2)! (bx)"''
- 1)!
+
(bx)°''
(n - 3)!
n=1,3,5,I( ...
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
269
28:13
The bracketed series in the above integrals may be expressed as [e (bx) ± e.(-bx)J/2, where the e function is discussed in Section 26:13. Similar indefinite integrals for is = -1, -2, -3, ... are listed by Gradshteyn and Ryzhik [Section 2.4751; they involve the chi and shi functions defined in Chapter 38. A large number of indefinite and definite integrals involving the hyperbolic sine and cosine functions exist. The reader is referred to Chapters 2.4 and 3.5 of Gradshteyn and Ryzhik.
28:11 COMPLEX ARGUMENT When the argument x of sinh(x) or cosh(x) is replaced by x + iy, we have 28:11:1
sinh(x + iy) = sinh(x) cos(y) + i cosh(x) sin(y)
28:11:2
cosh(x + y) = cosh(x) cos(y) + i sinh(x) sin(y)
For a purely imaginary argument 28:11:3
sinh(iy) = i sin(y)
28:11:4
cosh(iy) = cos(y)
28:12 GENERALIZATIONS The Jacobian elliptic functions nc(x;p) and nd(x;p) may be regarded as generalizations of cosh(x), to which they reduce when p = 1. Likewise, sc(x;p) and sd(x;p) reduce to sinh(x) when p = I and therefore generalize the hyperbolic sine. See Chapter 63 for all these Jacobian elliptic functions.
28:13 COGNATE FUNCTIONS The expressions 1 - sech2(x)
sgn(x)
28:13:1
sinh(x) = sgn(x)
1
cosh(x)
sech(x)
cosh(x) =
'1+ sinh2(s) =
) + tanh2(x)
1 + csch(x)
)
28:13:2
sgn(x)
tanh(x)
I - cosh2(x) _
sech(x)
coth2(x) - 1
Icoth(x)I
Icsch(x)l
1 - tanh2(x)
Vcoth2(x)
-1
relate the hyperbolic sine and cosine to the other hyperbolic functions (see Chapters 29 and 301. The hyperbolic sine and cosine functions are closely related to those hyperbolic Bessel functions I.(x) in which v is an odd multiple of ± . Examples are 28:13:3
sinh(x)
lt 2(x) =
Y
YYYY Irx
2 28:13:4
cosh(x)
1_112(x) =
Trx sinh(x)1
28:13:5
I cosh(x) -
I,,,(x) =
V VVVVTrxL
X
and others may be constructed by use of the recursion formula 28:13:6
2v
L. A) + - L(x) - I.-AX) = 0 X
These functions, some of which are graphed in Figure 28-3. share the properties of all hyperbolic Bessel functions as discussed in Chapter 50. The name modified spherical Besse! function and the symbol i,(x) is sometimes given to the function V/Tr/2r I,,,i,:(x).
28:13
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
270
28:14 RELATED TOPICS If a heavy rope or flexible chain of length 2L is freely suspended from two points, separated by a horizontal distance
2h, but at the same level, then the rope adopts a characteristic shape known as a catenary is cosh(bx) - cosh(bh) 28:14:1
f(x) =
b
and its shape is illustrated in Figure 28-4. The coefficient b is related to the lengths L and h by the implicit definition 28:14:2
bL = sinh(bh)
271
THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS
titer ..............:..............1.p
FIG 28-4
28:14
CHAPTER
29 THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS
Of the six hyperbolic functions, the two treated in this chapter are perhaps the least frequently encountered. The property 29:0:1
(1 - sech'(x))(t + csch2(x)) =
interrelates the two. Two features of this chapter-sec Sections 29:3 and 29:8-deal with all six hyperbolic functions.
29:1 NOTATION The notation cosech(x) is sometimes used for the hyperbolic cosecant. Some authors admit only four hyperbolic functions, using 1/cosh(x) and 1/sinh(x) to represent the secant and cosecant.
29:2 BEHAVIOR Figure 29-1 shows the behavior of the two functions. Both approach zero as their arguments tend to ±x. Both functions accept any argument but, whereas csch(.r) adopts all values, the hyperbolic secant is restricted in range
to 0 : sech(x) s 1. 29:3 DEFINITIONS The relationships 2 29:3:1
sech(x) _
exp(x) - exp(-.x)
cosh(x)
and 2 29:3:2
csch(x) =
exp(x) - exp(-x)
_
sinh(x)
are the usual definitions of the hyperbolic secant and cosecant. 273
THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS
29:3
274
h
O
if if if if i?
41
cech (x>
....:.. 1.5
sech(x)
.
.
...
.
.
...
...
...
...
...
.
0
:.........:..............:....:.-0.5
: FIG 29-1
cacti (x)
:....:.......:.............. :....:.-1.0
/ /
X
each (x) of
FIG 29-2
Figure 29-2 depicts three similar right-angled triangles. If the sides that are drawn as dashed lines are of unit length, then each of the other six sides equals one of the hyperbolic functions as labeled. Pythagorean and similarity properties thus enable the interrelationship between any two hyperbolic functions to be deduced and may serve as the definition of any one function in terms of any other. Note that the argument x is shown in the diagram, but it is not related to the triangles by any simple construction. It is, in fact, related to the angles in the triangles via the gudcrmannian function (see Section 33:14). The differential equations 29:3:3
d
dx
+
oz-_j'-0
are satisfied respectively by f = a csch(ax) and f = a sech(ax).
THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS
275
29:6
29:4 SPECIAL CASES There are none.
29:5 INTRARELATIONSHIPS The hyperbolic secant and cosecant obey the reflection formulas 29:5:1
sech(-x) = sech(x)
29:5:2
csch(-x) _ -csch(x)
and the duplication formulas
sech(2x) =
29:5:3
sech2(x)
2 - scch2(x) sech(x) csch(x)
29:5:4
2
csch(2x) =
Other relationships may be derived via the equations of Section 28:5. but these are generally more complicated and less useful than are the intrarclationships of the hyperbolic sine or cosine.
29:6 EXPANSIONS The functions sech(x) and x csch(x) may be expanded as power series 29:6:1
x'
Sx`
6lx°
2
24
720
/
4x2 \i
,=o
it --<x>»»
Storage needed: x, f, g and c
Set f = exp(x) Set g = 1 /f
Input restriction: x 0 code, c function
Replace f by If - g)/2
Input code c
>>> If cs1go to (2)
0
sinh csch cosh sech
Replace g by f + g If c > 3 go to (1) Replace f by g
tanh
Go to (2)
coth
(1) Replace f by f/g fo = sinh(x)
Test valucx=a
(2) If frac(c/2) = 0 go to (3)
f, = cosh(x) (3) Output f
fo = 11.54873936 f. = 11.59195328
f3 = sech(x)
0
coth(r)dt = ln(sinh(.r))
xo = ln(1 + V2) = 0.88137 ...
coth(t) - - Jdt =
30:10:5 10
30:10:6
m/sinh(x))
t
I
1`
x
/J1
[coth(t) - l]dt = In[I - exp(-2r)]
x > 0
J
30:10:8
30:10:9
30:10 10
cxp(-2x)]
[I - tanh(i)Jdt = In[I
30:10:7
r.
f
tanh(t)dt
r
tanh(bx)
0
b x0 = 1.19967864
coth2(t)dt = x - coth(x)
n = I. 3. 5...cr
In(cosh(x)) - o
fx - v
tank"'(x)
n-2j+1
The indefinite integrals of tanh"(x) and coth"(x), where X is an arbitrary power, are discussed in Section 58:14. The semiderivative, with lower limit -x, of [I + tanh(x/2)1/2 is an important function, known as the RandlesSevcik function in electrochemistry d 112
30:10:11
where X =
rsf(x) =
(d
I+tanh
(x + x )] 1/2
2
(-I
''//;
=-
(-1) VJexp(jx)
rn
V X- AX + 2r) X
(21 - 1)2a- + x2. It is related to Lerch's function [Section 64:121 by the identity rsf{In(.r)} = r(D(-.r:
-1/2;1). 30:11 COMPLEX ARGUMENT When the argument of the hyperbolic tangent and cotangent functions becomes x + iy, we have 30:11:1
tanh(x + iy) =
30:11:2
coth(x + iv) =
sinh(2x) + i sin(2y)
cosh(2r) + cos(2v) sinh(2r) cosh(2x)
- i sin(2v) -
cos(2v)
THE HYPERBOLIC TANGENT tanh(x) AND COTANGENT coth(x) FUNCTIONS
30:12
284
For purely imaginary argument 30:11:3
tanh(ii') = i tan(v)
30:11:4
coth(iy) _ -i cot(v)
30:12 GENERALIZATIONS The Jacobian elliptic functions sn(xp) and ns(xp) ]see Chapter 63] may be regarded as generalizations of tanh(x) and coth(x), respectively. Asp - 1, sn(x;p) -> tanh(x) and ns(x;p) -> coth(x).
30:13 COGNATE FUNCTIONS The expressions cosh(x) - I
sinh(x)
30:13:1
sgn(x) sgn(x)
tanh(x)
1 - sech2(x)
sgn(x)cosh(x)
717 sinh22(x)
1 + csch2(x)
coth(x)
and
30:13:2
coth(x) =
I + sinh2(x) sinh(x)
-
sgn(x)cosh(x)
-
sgn(x)
cosh2(x) - I
= gs n(x)
1 + csc hx)= 2(
1 - sech2(x)
tanh(x)
relate the tangent and cotangent to other members of the hyperbolic family. Figure 29-2 is useful in expressing these relationships.
The function coth(x) - (1/x), which occurs in the theory of dielectrics, is known as the Langevin function. It is mapped in Figure 30-1 and its integral is given in 30:10:5. The Langevin function can be expanded via 30:6:6 and its reciprocal as 1
30:13:3 coth(x) - (I/x)
3 =-+2x x
-
1
2
x' + r,'(1)
where r,(1) denotes the j ° positive root of the equation tan(y) _ y 1see Section 34:7].
CHAPTER
31 THE INVERSE HYPERBOLIC FUNCTIONS
The six functions of this chapter are interrelated by the following permutations of argument: 31:0:1
arsinh(x) = sgn(x) arcosh(
1
1
1 + x2) = sgn(x) arsech
= aresc i - = artanh x
l -+X1 = arcothl
z + x2
l + x2 x
31:0:2
arcosh(x) = arsinh(
x2 -
(1 ( ) = arsechl xJ = - areschl
x
= arcoth
1/x2 - 1
I
xZ
) =artanh)
x
x >_ 1
x2- 1 X
WI
7,I
=arcoth
x2 I = artanh(
1 - x')
05xs I
x2
x1/
31:0:4
1 + 1/x2) = sgn(x) arscch(
aresch(x) = arsinh - 1 = sgn(x) arcosh(
I
= sgn(x) artanh
I
= sgn(x) arcoth( V1 + x2)
x2/(1 + x2))
x*0
+ x2 31:0:5
/
artanh(x) = arsinhl
= areschl
x
1
1
1 - x2 I = sgn(x) arcosh
1-x2 x
1
I = arcothl - I
1 - x2 = sgn(x) arsech(
1
--X 2)
-1 s x < 1 285
THE INVERSE HYPERBOLIC FUNCTIONS
31:1
31:0:6
arcoth(x) = sgn(x) arsinh
= sgn(x) arcssh(
+
.
x-- 1
)
I
x=
x2/(x2 - 1)) = sgn(x) arsech(
= sgn(x) arcosh(
1) = artanh
286
1 - 1/x2)
` Gxi > 1
I
x/
31:1 NOTATION The prefix `ar" means "area," and its pertinence can be appreciated by reference to Figure 28-2. The inverse hyperbolic sine of x, for example, denotes that area which. in the Figure 28-2 construct, has a hyperbolic sine of a. That is: if x = sinh(a) = PQ arsinh(x) = a = shaded area 31:1:1 The symbolisms arcsinh(x), agsinh(x), arsh(x) and sinh-'(x) are all used in place of arsinh(x). Corresponding variants are encountered for the other inverse hyperbolic functions. The same symbolisms with a capitalized initial
letter-Arcsinh(x), Arsh(x), Sinh-'(x), etc.-are sometimes used synonymously with arsinh(x). but more often they denote the multiple-valued functions that we discuss in Section 31:12.
31:2 BEHAVIOR The behaviors of the six inverse hyperbolic functions are shown in Figure 31-1. Notice that all functions exist in the first and third quadrants (see Section 0:2) only. The inverse hyperbolic sine has a behavior that is simpler than the other five. It accepts all arguments and adopts the sign of its argument. The inverse hyperbolic cosine is normally defined only for arguments x ? I although some authors extend its domain of definition to W a I via arcosh(-x) = arcosh(x). It adopts only positive values. The inverse hyperbolic secant is likewise normally defined only for 0 s x s 1 although its definition may be extended to 0 s i s I by means of arsech(-x) = arsech(x). The values of arsech(x) are always positive. The inverse hyperbolic cosecant has two branches, as mapped in Figure 31-1. It adopts the sign of its argument and exhibits a discontinuity at x = 0. The inverse hyperbolic tangent is defined only for -1 5 x 5 1 and approaches ±x as x -+ ± 1. The inverse hyperbolic cotangent has two branches. For I s x I
Test values:
K««
arsinh(-w) _ -1.862295743 arcosh(ir) = 1.811526272 arcoth(ir) = 0.3297653150
31:9 APPROXIMATIONS For large x we can use equations 31:6:4 and 31:6:3 to approximate
31:9:1
arsinh(x) = sgn(x)I ln(2k1) + L
31.9:2
arcosh(x) = In(2x) - 4s=
1
z2
8-bit precision
4,2 J
8-bit precision
2.2
For small x the approximation 8-bit precision
artanh(x)
is useful.
31:10 OPERATIONS OF THE CALCULUS Differentiation of the six inverse hyperbolic functions gives d 31:10:1
&
1
arsinh(x) =
d 31:10:2
1+x 1
x >- I
a rcosh(x) =
dx
- 2I < x S ` I
x
31:9:3
x'-1
THE INVERSE HYPERBOLIC FUNCTIONS d
- arsech(x)
0Sx
»»>
Setk=N=210
Setj=0
Storage needed: 2N + 9 registers are required to store:
p, k. N, j. to, io, r,. is. r2...., iN-2, rx-1, iN-i, m, K, M, ! and R.
(1) Output j j(as cue)
Input ri > Input it >
»»>
I »»> Replace j by j + I Ifj*kgoto(1)
(2) Replace k by k/2
Set m = -2k
FUse degree mode or replace 360 by 2w. 1
THE SINE sin(x) AND COSINE cos(r) FUNCTIONS
309
(3) Replace m by m + 2k
Input restriction: The parameter p must be an integer.
SetK=m/k Set M = 0 Set j = P1
(4) Replace j by j - I Replace .M by M + K2' Replace K by Int(K/2) Replace M by M - 2K21
1fj*0goto(4) Set ! = 360M/N Set R = cos(t)
Replace I by (p/Ipl) sin(1)
Setj=k+m - 1 (5) Replace j by j + I
SetK=r,R+i,! SetM=i,R-r,l
Replace r,_,, by r,_, + K
Set r,=r,-4-2K Replace i,-4 by i,-4 + M
Set i,=y-t-2M If)+ I I go to (2)
Set m = -1 (6) Replace m by m + I
Set Km Set,M = 0
Set) = pl i7) Replace j by j - l Replace M by ,M - K2' Replace K by Int(K/2) Replace M by M - 2K21
Ifj*0goto(7)
JIM Sm go to(8)
SetR=r,
Setr,,,rM Set ', R Set ! = Im
Set i = i,w Set i,w = I
(8) If m < N - I go to (6) Set k = -1
Ifp>0goto(9) Set p = 0 Set N= 1
(9) Replace k by k + I
k«« R=R,
l
x,so 2j+ I
(2j - l)!! x" (2J)!! 2j+I
I X 3 x° I x 3 x 5.e aresin(x)=x+--++ 23 2x45 2x4x67 1 x3
but the multifarious interrelationships between the six functions allow easy adaptation of these series to other inverse
trigonometric functions. For example, in tight of equation 35:0:7, replacement of x on the right side of 35:6:3 by (I --x)/2 provides an expansion for 2 arccos(x). Continued fraction expansions of the inverse trigonometric functions include x jr' 4x' 9xr 16x' arctan(x) = - - - - - .--
35:6:4
1+ 3+5+ 7+ 9+
and, for -1 < x < 1 35:6:5
arcsin(x)
_
x V ` - x= (I x 2)x (1 x 2)x (3 x 4)x (3 X 4)x (5 x 6)x
3-
1-
5-
7-
II-
9-
35:7 PARTICULAR VALUES The entry "undef" in Table 35.7.1 indicates that the function is undefined at the argument in question.
35:8 NUMERICAL VALUES Calculators usually incorporate keys for evaluating the arctan, arcsin and arccos functions, but computer languages rarely incorporate any inverse trigonometric function other than arctan. Accordingly, we present an algorithm that
Table 35.7.1
x=0
x=V2
x=1
x=V2
x=s
-0.615...
0
0.615._.
n/4
0.955...
-n/4
0
n/4
n/2
undcf
n/2 undef
-n/2
undef
undef
undef
2.186...
n/2 n/2
0.955...
3n/4 undef
undef
undef
n/2 n/4 0 0
-I
x=-x
x=-V2
-1/2 undef
-0.955...
-n/4
aresin(.)
-n/2
amcsc(x)
0
undef -n/4
arccot(x)
it
2.526...
amtan(x)
V2
arwos(x)
undef
undef
3n/4 n
aresec(x)
n/2
3n/4
n
1
n/4
n/4
0
0.615...
0
undef
undef
n/4
n/2
THE INVERSE TRIGONOMETRIC FUNCTIONS
35:9
338
computes any one of the six inverse trigonometric functions by utilizing a built-in arctan function. The algorithm, which is exact, uses relationships 35:0:1 and 35:0:2. -ts-well, the universal hypergeometric algorithm (Section 18:141 may be used to calculate inverse trigonometric functions.
Input x »4»»
Storage needed:
Setf = I
x, c (the code) and f
If c 1 - c
b
6
Indefinite integrals of the form ftarctrig(bt + c)dt include 35:10:10
r J t/bt arctan(bt + Odt =
::
I
+
z
-
arctaa(bx + c) -
1 +62x2-C2 35:10:11
Jtarccot(b:+c)dJ=
arccot(bx + c) +
t arcsin(bt + c)dt =
BYxt _ 2c' 2b 4b
- 1 arcsin(bx + c) +
bx - 3c
I - (bx + c)2
462
22_
I c 111
t arccos(bt + c)dt =
x
/b
x
-
4b
bx - 3c
-
1 + (bx + c)2
l0
c - bile
2
I + (bx + c)
c bt
+
r
t
462
r/b
35:10:13
c+
2b,
2bx+2c-ir
262
35:10:12
b
I
-c - I s bx s I - c
arccos(bx + c) +
I-(bx+c)2
462
-
-c-Isbx51-c
but none of ft-'arctrig(bt + c)dt has been evaluated as a finite number of terms. See Spiegel [pages 82-841 for a long list of indefinite integrals of the form ft="arctrig(t)dt for integer n. Gradshteyn and Ryzhik [Section 2.81 list similar integrals and include additional entries such as farctrig"(t)dt and f(bt + c)-'arctrig(t)dt. Gradshteyn and Ryzhik [Section 4.5] also present over 100 definite integrals involving inverse trigonometric
functions; acctan(t) dt
35:10:14
J
- f arccot(t) dt = G
and r' aresin(t)
35:10:15
I
r
a dt = 2 ln(2)
0
are typical examples. G is Catalan's constant (see Section 1:7). The inverse trigonometric functions play a role in the fractional calculus [see Oldham and Spanierl. For example:
THE INVERSE TRIGONOMETRIC FUNCTIONS
35:11
d1/2
35 : 10: 16
aretan(1x)
1
dx'n
V
dr 'n atrcot(V x)
Vx
1 + x)
'V;
d'R aresin(V x-) 35:10:18
340
2(1 - x)
dx'/2
35:11 COMPLEX ARGUMENT A variety of ways may be used to extend the inverse trigonometric functions to complex argument. One might couple the definitions 35:3:1-35:3:6 with integration in the complex plane or, alternatively, permit power series such as 35:6:1 and 35:6:3 to accept a complex argument. Another route makes use of the relationships 35:13:135:13:6, which are valid for complex argument as well. The results, however one proceeds, are rather formidable looking. The most useful of the resulting multivalued functions are Arcsin(x + iy) = kir + (-1)4arcsin(Y) + (-1)ri In[X + X' --I] 35:11:1 Arccos(x + iy) = 2k7r ± {arccos(Y) - i In[X +
35:11:2
35:11:3
Arctan(x + ry) = kir + 1 arctan 2
XZ - ]}
2 + t In x2 + (y + 1)2 4 x2+(y-1)
2x
1-x2-y-
x2 + (V - 1)2 * 0
where k is any integer and 35:11:4
X=1
2
(x+l)2+y2+
2
y=!2 (x+1)2+y^-2
(x-1)2+yr
(x-1)Z+y2
The logarithmic function In occurring in the above equation is the single-valued logarithm defined in Chapter 25.
35:12 GENERALIZATIONS Because the trigonometric functions we periodic, their unrestricted inverses have infinitely many values for each acceptable argument. These multivalued inverse functions are distinguished from the single-valued functions treated elsewhere in this chapter by having their initial letter capitalized. The definitions are encompassed by 35:12:1
Arctrig(x) = y
trig(y) = x
when
With k = 0, ±1, ±2, ..., the relationships 35:12:2 35:12:3
Aretrig(x) = arctrig(x) + 2kir
arctrig = aresin, arccos, aresec, arcsec
Arctrig(x) = arctrig(x) + for
arctrig = arctan, arccot
hold. The inverse tangent and inverse sine functions are each special cases of the Gauss F function [see Chapter 60): 35:12:4
arctan(x) = xF(2,1; 2;-x2
35:12:5
arcsin(x) = F1 2, 2; 2;x2)
I
-1 < x < 1 -1 <x < 1
Also, the inverse sine and cosine are special cases` of the incomplete beta function of Chapter 58:
341
THE INVERSE TRIGONOMETRIC FUNCTIONS
35:12:6
r l 12;x' aresin(x)= 1 2B 2; !1
35:12:7
35:13
1
B 2, 2;1 - xr
arccos(x) =
2
35:13 COGNATE FUNCTIONS The inverse trigonometric functions are closely related to the inverse hyperbolic Chapter 30. One has 35:13:1
35:13:2
arctan(x) _ -i artanh(ix) arccot(x) = i arcoth(ix)
35:13:3
arcsin(x)
-i arsinh(ix)
35:13:4
arccos(x)
±i arcosh(x)
35:13:5
arccsc(x) = i arcsch(ix)
35:13:6
aresec(x) _ ±i arsech(x)
functions that are considered in
CHAPTER
36 PERIODIC FUNCTIONS
Periodic functions play an important role in the solutions of many difficult problems in applied mathematics. Also, periodic functions, or nearly periodic functions, form the medium by which a good many information transfers take place. For example, beams of light, sound waves, and a variety of telecommunications signals are instances of periodic functions, often "modulated" in some way.
36:1 NOTATION We shall use per(x) to represent any periodic function, and occasionally qer(x) to represent a second periodic function. Throughout this chapter P will denote the period of per(.r). The quantity 27r/P is sometimes known as the frequency of the periodic function and is often denoted by w.
36:2 BEHAVIOR Apart from their characteristic of indefinitely repeating, periodic functions share no common behavior. Periodic functions may be continuous or discontinuous, simple or complicated. Examples of periodic functions are shown in Figures 36-1 through 36-7, exhibited later in this chapter.
36:3 DEFINITIONS A function of argument x that satisfies the condition 36:3:1
f(x) = f(x + kP)
k=0 .±1 .± 2 ,
for all x is a periodic function of period equal to the smallest positive value of P that satisfies equality 36:3:1. A periodic function may be 'naturally" periodic, as are each of the functions cited in Section 36:4, or it may be "synthesized" from an aperiodic function. An example of a "synthetic" periodic function, in this case created f r o m the square function [Chapter I I ], is 36:3:2
per(x)=(x-2k)2
2ksx - l .
2jax`
PERIODIC FUNCTIONS
347
36:14
36:11 COMPLEX ARGUMENT For constant y, the periodic function per(x + iy) remains periodic in x. The Fourier coefficients of such functions are then complex numbers that can be evaluated with the help of equations 32:11:1 and 32:11:2. Certain functions, such as In and sinh, which are aperiodic for real argument, become periodic when the argument is imaginary or complex. The Jacobian elliptic functions [Chapter 63] are doubly periodic when their argument is complex. That is, they satisfy the recurrence relations 36:11:1
per(x + iy) - per(x + P + iy) = per(x + iv + IQ) = per(x + P + iy + iQ)
where P and Q are the real and imaginary periods.
36:12 GENERALIZATIONS Apart from the double periodicity cited in Section 36:11. no generalization of the concept of a periodic function has been made.
36:13 COGNATE FUNCTIONS The terms full-rectification and half-rectification, which have their origin in the technology of alternating electrical currents, are sometimes encountered in connection with periodic functions. Using the notation of the absolute-value function [see Chapter 8), full- and half-rectification converts the function per(x) into 36:13:1
lper(x)I
full-rectification
and 1
Z {Iper(x)I + per(x)}
36:13:2
half-rectification
Figure 36.2 shows an example of each. Full- or half-rectification of a periodic function maintains its periodicity and, generally, the period remains unchanged, Sometimes, as in the full-rectification example shown in Figure 362, rectification decreases the period (increases the frequency) by a factor of 2 (or more).
sin(2nx/P)
FIG 36-2
7
half rectification
36:14 RELATED TOPICS Table 36.14.1 lists some frequently encountered periodic functions, together with their Fourier coefficients (see equations 36:6:1-36:6:31. For further examples of Fourier coefficients, see equations 32:5:29, 32:5:30, 20:6:420:6:6 and 19:6:5-19:6;7.
348
PERIODIC FUNCTIONS
36:14
Table 36.14.1 C
Name of (unction S q uare wave (odd)
Square wave
(even)
Sawtoot6 wave
Triangular wave
Figure
4
363
/4x 4 P
36-4
(-I)' n = Intl
36.6
Full-rectified sine wave
36-2
t-,)
iw _I
0 1
r I ' 2rs'frac( 2+ P
_l]
-puha---P JP)
_4 2
:
h
P
-
!2 zin l P) + !2
r
-k - sint -- j
ch
/ xx `p
sinl
Ixsr.
36.2
0
0
2P
fracI 2 xP
36-5
367
sine wave
0
1-1)b"iSie
Jn
Pulse train
Half-rectified
St
(O
pert.)
P
rlrrh - sinll-
0
0
0
0
0
4c
P.
Is
-4 0
zi ng/
2a r P
l
1
(j: _ I M
ut2-J)
0
r
:....;....:FIG
36-3:....:..p
:....:....:FIG
36-4;......p
;Q{s,
0
44Q
;...
4+Q
4 0. 5
FIG
365;......0
0
PERIODIC FUNCTIONS
349
36:14
;4
ti A?
y4Q
ti
44,Q,
y4p
+4e
y440
+4q?
:FIG 36-7 :
CHAPTER
37 THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
Indefinite integrals of the form ft-'exp(±t)dt cannot be expressed in terms of elementary functions for n = I. 2, 3..... The exponential integral function Ei(x) fills this deficiency. A closely related function, the entire exponential integral, is also discussed in this chapter, it is related to Ei(x) by an identity 37:0:1
Ein(x) = y + ln(kl) - Ei(-x)
y = 0.5772156649
involving Euler's constant y [Section 1:71 and the logarithm of the absolute value of the argument x. The logarithmic integral 11(x) is discussed in Chapter 25 but, because of its simple relationship 37:0:2
li(x) = Ei(ln(x))
to the exponential integral, many of its properties are given in this chapter.
37:1 NOTATION Notations abound for the exponential integral function: Ei*(x), Es(x), Ei(x), i(x), E`(x) and E_(x) have all been used to denote Ei(x) or some very similar function. When one of these symbols is encountered, care is needed to ascertain whether its definition differs from that which is employed for Ei(x) in this Atlas. When x is negative, the expression -E,(-x) is often used to replace Ei(x) in view of relationship 37:13:7. The E,(x) symbol here represents a Schl6milch function [see Section 37:13] and should not be confused with the identical symbol used for Euler polynomials [Chapter 20]. To add to the confusion, the notation ei(x) is sometimes encountered for the Schli milch E,(x) function.
37:2 BEHAVIOR Figure 37-1 maps the behaviors of the exponential integral Ei, entire exponential integral Ein and logarithmic integral li functions. All three functions increase without limit as x - x but, following an inflection [Section 0:7] at x = 1, the increase is dramatic for the exponential integral function (e.g., Ei(10) - 2 x 10' and Ei(100) - 3 X 10"). Note also that Ei(x) rapidly approaches zero as x -. - x and exhibits a discontinuity at x = 0. Observe also that li(x) is discontinuous at x = 1 and is defined only for x ? 0. 351
FIG 37-1 Black
37:3
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
352
37:3 DEFINITIONS The exponential integral function is defined by the indefinite integral 1
exp(t)
Ei(x) = J
37:3:1
dt
I
a
for all x, as illustrated in Figure 37-2. At zero argument the integrand in 37:3:1 encounters an infinity so that for x > 0 the integral is to be interpreted as the Cauchv limit 37:3:2
limSEi(-e)+
j
ezp(r) r
l
x>0
dt}
a>0
The exponential integral may also bee expressed as a definite integral in a number of ways, including 37:3:3
Ei(±x) = exp(±x)
-I
dt
Jo
ln(t) ± x
x>0
Gradshteyn and Ryzhik (Section 8.212) list many other integral representations of Ei(x). The definition of the entire exponential integral 37:3:4
1 - exp(-t)
Em()
dt o
t
353
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
37:4
is illustrated in Figure 37-3. Its relationship, equation 37:0:1, to the exponential integral Ei permits several alternative definitions of the Ein function. Definitions of the logarithmic integral function were presented in Section 25:13. Some authorities regard the logarithmic integral as defined only for arguments exceeding unity, but in this Atlas li(x) exists for 0 s x < 1 with a Cauchy limit interpretation, similar to 37:3:2. serving to extend the definition to .t > 1.
37:4 SPECIAL CASES There are none.
1A
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
37:5
354
37:5 INTRARELATIONSHIPS Relating the exponential integral and its entire analog is the identity
Ei(x) + Ein(-x) = Ei(-x) + Ein(x)
37:5:1
that follows from equation 37:0:1. or through definitions 37:3:1 and 37:3:4. We know of no reflection, recurrence, addition or multiplication formulas for the exponential integral functions that involve a finite number of elementary functions. However the reflection formulas
Ei(-x) = Ei(x) - 2 Shi(x)
37:5:2 and
Ein(-x) = Ein(x) - 2 Shi(x) = -Ein(x) - 2 Chin(x)
37:5:3
may be written in terms of the functions of Chapter 38. Also, the argument-addition formula
j! [exp(y)e/-y)-1 ]
Ei(x + y) = Ei(x) + exp(x)
37:5:4
1x1 > Lv
X ',I
i-0
for the exponential integral exists and utilizes the exponential polynomial [see Section 26:13].
37:6 EXPANSIONS Two alternative power series: xr
x3
Ein(x)=x-+18 4
37:6:1
(-X)'
X +
i- j!j
96
and
37:6:2
3x2
Ilx'
25x'
I
I
4
36
288
2
3
ex p
1
x'
j/ j!
['y+yU+1)]X! are available to express the entire exponential integral function. In 37:6:2, 0 is the digamma function described in Chapter 44. The exponential integral function is expansible in terms of Lagucrre polynomials [Chapter 23J:
Ei(x) = -exp(x)
37:6:3
L;(-x) (-x)
_ 22, the algorithm employs the continued fraction 37:6:4 terminated at (J - 2)/((1 - (J - 1)/(x J/(1 - J/x))) where J = Int(5 + The algorithm will generate li(x) when the additional instruction shown in green is included. Because Ei(0) = Ii(l) = -x. the algorithm returns -1099 at these arguments. The universal hypcrgcomctric algorithm [Section 18:141 also permits Ei(x), Ein(x) and li(x) to be determined for suitable values of the argument x.
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
37:9
356
37:9 APPROXIMATION Based on the identity 37:0:1 and expansion 37:6:1, the approximation 1.05 s x < 0
36x + x=
Ei(x) ^
37:9:1
+ ln(1.78[xl)
0 < x s 0.36
8-bit precision
36 - 8x
0. 38
5 x s 1.25
is valid for most small values of x, while the approximation exp(x)
Ei(x) =
37:9:2
X:5
8-bit precision
-9
X? 15
based on expansion 37:6:5, is useful for arguments of large magnitude. The very simple approximation to the entire exponential integral Ein(x) =
37:9:3
36x - xr
-1.1 :5x:5 1.4
8-bit precision
36 + 8x
may be used near the origin.
37:10 OPERATIONS OF THE CALCULUS The following rules apply for differentiation and indefinite integration:
exp(hx + c)
d
()
d Ei(bx + c) =
37:10:1
x + c/b I - exp(-bx - c)
d
37:10:2
c)
- Ein(bx dx
x + (c/b) x"-'
vf0 -li(x")=dx In(x) d
37:10:3
1 Ei(bx + c) +
Ei(br + Odr = (X +
37:10:4
I - cxpbbx + c)
b
Ein(br+c)dt=
37:10:5
J
37:10:6
li(bi + c)dt h
Ix+b) [Ein(bx+c)- 11+
x + b l li(bx + c) - I li(bY + 2bcx + c')
\
/
exp1`C -
BC Bc
B+b
exp(Br + C) Ei(br + c)dt = J
c
Iexp(Bx+ b)Ei(bx+c)- Ei1 (B+b)Ix+b)
b 37:10:7
1 - exb bx-c)
B+b*0
b
/b
ex C+c b
)
/ lnlx+ c) +y-exp(-bx-c)Ei(bx+c) \\\
\l
1l
l
J
//
B+b=0
When n = -1, the indefinite integral fI"Ei(bl + c)dl cannot be evaluated in a finite number of terms, but for n
= 0, 1, 2, 3, ... we have
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
357
Ei(bx + c)
37:10:8
t
Ei(bt + e)dt
(l c
x"+' -
n+l
i
l
\b/
n!
+
(-b)"+'
j
c"-'[exp(bx + c)e,(-bx - c)- l)
_o
(j + l)(n - j)!
J J-pie I while the n = -2, -3. -4, ... cases can be evaluated via the lengthy expression 37:10:9
J
`is
Ei(bt + c) Ei(bx + c) 1-1 dt = n(-c/b)" +
exp(c)e"_,(-c
n(-c/b)"
r
37:11
cl l [I - (bx/ J
Ei(-c) - Ei(bx) +
j!
[exp(bx)
- exp(-c) 1J [ I
(-c)'*'
(bx)j*'
)_o
e,(-c) 11 Jf
in which e,(x) denotes the exponential polynomial [Section 26:13]. Included among the definite integrals tabulated by Gradshteyn and Ryzhik [Section 6.21-6.231 are
J
37:10:10
f
37:10: l l
=
Ei(bt)dt = b1
=
t' Ei(-bt)dt =
I
37:10:13
,10
37:10:14
z
(
bB
I 0ln(Bt) Ei(-bt)dt
37:10:12
)6+s`
(b
Ei(bt) Et(B:)dt = - In
t"-'exp(-Bt) Ei(-bt)dt = '
t" li(t)dt =
f
f li(t)
J
t"
dt =
B
- In(2 + v)
v> 0
b+B>0
I
/
v> -2
I+v
o
37:10:16
B
B>0
v>-I
bz0
BI 0.v; b \
J
b>0
b
-fY1 v) + (1 + v)b'-"
B>0
b>0
)
b"Ba
I + y + ln(b/B)
Io
37:10:15
b>0
-ln(v - 2)
v>2
v- I
The r and B functions occurring in integrals 37:10:13 and 37:10:14 are, respectively, the gamma function [Chapter 43] and the incomplete beta function (Chapter 58].
37:11 COMPLEX ARGUMENT Even with a complex argument, the Ein function is single valued and continuous. For small complex arguments, the series 37:11:1
Ein(x +
x-- y x3-3xy3 x'-6x'y'+v
[
iv) = x -
+
2!2
+
i[y
2xy
2'2 +
3!3
-
3xy- - y' 3'3
-
+
4!4
l
4.x'y - 4xy' + .. . 4!4
converges rapidly and can be used to find values of Ein(x + iy). For purely imaginary argument, one has 37:11:2
Ein(iv) = Cin(v) + iSi(y)
where the Cin and Si functions are discussed in Chapter 38.
The exponential integral Ei is a many-valued function, when its argument is complex, because equation 37:0:1 generalizes to 37:11:3
Ein(x + iy) + Ei(-x - iv) = y + Ln(x + iv)
37:12
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
where Ln is the multivalued logarithmic function discussed in Section 25:11. However, as explained in that section, one usually selects a principal value of the Ln function, which leads to the single value 1
37:11:4
Ei(x+iy)=y+2ln(x'+y)-Ein(-x-iy)+iO
where 0 is the angle defined in Section 25:11. Notice that, according to 37:11:4, the exponential integral may have a complex value even when its argument is real. In fact: 37:11:5
Ei(x + Oi) = y + In(x) - Ein(-x) + in
x>0
The imaginary term in this expression is ignored in the rest of the chapter. Abramowitz and Stegun [Tables 5 and 5.7] provide tables from which the real and imaginary parts of Ei(x + iy) may be determined. For purely imaginary argument, one has
Ei(iy) = Ci(y) + i Si(y) -
37:11:6
n
sgn(y)
2
in terms of the functions of Chapter 38.
37:12 GENERALIZATIONS The exponential integral function is a special case:
Ei(x) = -f(0;-x)
37:12:1
of the complementary incomplete gamma function [Chapter 45J and of the Tricomi function [Chapter 481
Ei(x) _ -exp(x) U(l;l;-x)
37:12:2
The entire exponential integral function is a hypergeometric function [Section 18:14]:
(I)i(-x) '
Ein(x) = x
37:12:3
-o
(2),(2),
37:13 COGNATE FUNCTIONS In addition to the Schl6milch functions (discussed later in this section], two other function families, related to exponential integrals, are sometimes encountered. The alpha exponential integral function of order n is defined by
t"exp(-xt)dt n = 0, 1, 2, ... J The first member ao(x) equals [exp(-x)]/x and because of the recurrence relation 37:13:1
n
37:13:2
a"(x) _ - a"-,(x) + ao(x) x
n = 1, 2, 3, .. .
all of these functions can be reduced to the elementary functions 37:13:3
a"(x) =
n! exp(-x) x'.
ejx)
n = 0, 1, 2, .. .
where e, is the exponential polynomial (Section 26:13].
The beta exponential integral function family is defined by a similar integral, but with changed limits: 37:13:4
Nx) = J t" exp(-xt)dt
n = 0, 1, 2, .. .
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
359
37:14
The first member po(x) = 2[sinh(x)]/x and all members are reducible, via the expression n!
37:13:5
n = 0, 1, 2. ...
[exp(x)e"(-x) - exp(-x)e"(x)]
[3"(x)
to more elementary functions. A family of functions defined by
E"(x) = I
37:13:6
exp(--rt)
t.
n = 0, 1, 2, ...
dt
was introduced by Schldmilch. The first member E0(x) = [exp(-x)]/x is elementary, while the second
E,(x) = -Ei(-x)
37:13:7
is related in a simple way to the exponential integral function. These identities, coupled with the recurrence relationship .r
E"(x) _
37:13:8
n = 2, 3, 4, .. .
[Eo(x) - E,-,(x)]
n - I
enable the properties of E"(x) to be deduced from those of Ei(-x). The general relationship is 37:13:9
E"(x)=
-(-x)" ' rEi( (n - 1).
-x) +
exp(-x) I
--+ + I
I
X
x
2!
(n - 2)!1 ) 1
X2
(-x)
I
n=2,3,4,...
Some familial properties of Schldmilch functions are
E"(0)=n-1 1
37:13:10 d
37:13:11
dx
E. (x)
E"(x)
E"(x) = x"-'1'(1 - n;x)
37:13:12
n=2,3,4,... n = 1. 2, 3.... n = 0, 1, 2....
and the asymptotic expansion 37:13:13
E"(.r)
I
exp(-x) Is -
n
z'
+
n(n + 1)
(n),
x)I,i
Jr,
The function 37:13:14
xl exp(/ l Ei( Xl l - 1 - x + 2x= - 6r' + ... + j!(-.r)1 + ...
r/
has been named Euler's function and symbolized E(x). It should not be confused with the complete elliptic integral of the second kind [Chapter 61 ] for which the same notation is adopted. The avoid an unnecessary proliferation of functions, none of the functions discussed in this section is used
elsewhere in the Atlas.
37:14 RELATED TOPICS Very general expressions for indefinite integrals of the form f t`exp(bt)dt were presented in Section 26:10. Nevertheless. because of the very widespread occurrence of the integrals 1
t"exp(tt)dt
37:14:1
3
v = 0, t 2, t I, t 2, ...
J
in applied mathematics, Table 37.14.1, containing such indefinite integrals, may be useful. Integrals of the forms
THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS
37:14
u"cxp
37:14:2
-+Iu du
360
3 v=0,±-2 ,+I,a- - 2, 1
and
n = 0, ±1, 22, ±3, ...
J v" exp(±v2)dv
37:14:3
are similarly ubiquitous. The substitutions u = 1/1 or v = vt convert these into the form of 37:14:1 so that Table 37.14.1 may also be used to evaluate integrals of the 37:14:2 and 37:14:3 families. The erfc and daw functions occurring in Table 37.14.1 are the error function complement [Chapter 401 and Dawson's integral [Chapter 421. The lower limits xM and x; are, respectively, the argument values corresponding to the maximum and the point of inflection of a graph of daw(V') (see Section 42:71. Table 37.14.1
V
xu
J
-3
t
r' exp(odr
Ei(x)
(x + 1) exp(x)
(1 - x)exp(-x)
2
2x'
2x lI
cxp(x)Ir d.-(N/;) 3
-2
2 exp(-x) rr 1
3Vx
L
-x
e exp(-10
J
Ei(x) -
exp(x I
Ei(-x) 2
11
4V,
Lx-21 3 erfNl`l
L
Ei(-x) +
exp(-x)
X
X.
A
2 exp(-x)
2 exp(x) 2 daw(V) -
Vx
Vx
-x
B(x) )
0
2 exp(x) daw(Vx)
0
exp(x)UV -daw(Vx)l
exp(x)
- 2 V a erfc(Vl)
v' - erfc(V/X) + V. exp(-x) 2
rr
(x - I) exp(x) 3Vr 3
0 2
2
(x2 - 2x + 2) exp(x)
\r
(x 1 1) exp(-x)
- erfc(v;) + VT x + - cxp(-x) 34A
(
2)
(x= + 2x + 2) exp(-x)
CHAPTER
38 SINE AND COSINE INTEGRALS
This chapter concerns functions defined in terms of indefinite integrals of sin(s)/x. cos(x)/x and their hyperbolic analogs. Whereas the definitions of the hyperbolic sine integral Shi(x) and the sine integral Si(x) present no difficulty, the corresponding integrals of cosines diverge at zero argument. Accordingly, in addition to the hyperbolic cosine integral Chi(x) and the cosine integral Ci(x), it is useful also to define an entire hyperbolic cosine integral Chin(x) and an entire cosine integral Cin(x). These entire integrals are related by 38:0:1
Chin(x) = Chi(x) - 1n(lxl) - y = Chi(x) - In(ix;) - 0.5772156649
and
38:0:2
Cin(x) = y + ln(Ixl) - Ci(x) = ln(I.781072418kxi) - Ci(x)
to the Chi and Ci functions.
For large arguments. certain auxiliary functions discussed in Section 38:13 are more convenient than Ci and Si.
Because of their wider applicability, this chapter places more emphasis on the Si and Ci functions than on their hyperbolic counterparts.
38:1 NOTATION The initial letter of Shi and Chi is not always capitalized. The "h" that is used to identify the hyperbolic integrals may occur elsewhere in the function's symbol. For example, Sih(x) sometimes replaces Shi(x). and the anagram Cinh(x) is often used instead of Chin(x). Some authors use ci(x) synonymously with Ci(x), but others employ it to denote -Ci(x). The notation si(x) is usually encountered with the meaning 38:1:1
n si(x) = Si(.r) - -
Neither ci nor si is utilized in this Atlas. 361
38:2
362
SINE AND COSINE INTEGRALS
38:2 BEHAVIOR Figure 38-1 maps the Shi, Chi and Chin functions. Note that for large positive arguments Shi(x) and Chi(x) converge and approach the value ; Ei(x), where Ei is the exponential integral [Chapter 37].
The damped oscillatory behavior of the Si and Ci functions is evident in Figure 38-2. As x -+ ± x. Si(x) approaches ±0r/2) and Ci(x) approaches zero, whereas Cin(x) approaches infinity logarithmically via a series of plateaus.
FIG 38-1 : ..:.... :.... :.... :.... :.... :.... :.... :
J.:.. 3
2
38:3 DEFINITIONS In addition to their definitions as integrals, the hyperbolic sine and cosine integrals may be defined in terms of the
functions of Chapter 37:
SINE AND COSINE INTEGRALS
363 -.1
4
47
4
0
ZEi(-x) =
Ei(x)
` sinh(t)
t
0
38:3:2
38:3:3
Chi(x) =
ti0
4
Shi(x) =
38:3:1
6
-A
'L
ti
38:3
Ei(.r) + Ei(-x) 2
Chin(x) _ -
(
cosh(:)
=J
Ein(x) + Ein(-x) 2
dt =
dt
xa = 0.52382257
f cosh(:) - 1 t
o
dt
The sine integral, cosine integral and entire cosine integral are defined by f0.
Si(x) =sin(t) - dt = m -o 1
38:3:4
1
J
+
sinc(t)dt =
- - J sin(:) -t dt It
2
1'L
SINE AND COSINE INTEGRALS
38:4
364
cos(t) Ci(x) _ -
38:3:5
and
Cin(x)
38:3:6
r 1=I
COW)
o
dt
t
Some other definitions may be constructed by employing relationships 38:0:1 and 38:0:2.
38:4 SPECIAL CASES There are none.
38:5 INTRARELATIONSHIPS All cosine integrals, like the cosine itself, are even functions:
f(-x) = f(x)
38:5:1
f = Chi, Chin, Ci, Cin
whereas the sine integrals are odd:
f(-x) _ -f(x)
38:5:2
f = Shi. Si
as is the sine function.
Expressions for the Ci and Si functions in terms of the auxiliary functions of Section 38:13 arc
Ci(x) = sin(x) fi(x) - cos(x) gi(x)
38:5:3
Si(x) =
38:5:4
2
- cos(x) fi(x) - sin(x) gi(x)
38:6 EXPANSIONS The sine integral may be expanded as the power series 38:6:1
x3 x3 x7 Si(x)=x--+-600 35280
=
(-x Z
,_o (2j + 1)(2j + 1)!
18
The similar series without alternating signs, namely E.r2J"/(2j + 1)(2j + I)!. represents Shi(x). Likewise, replacement of all the - signs by + in the expansion x4
38:6:2
96
(-x2)'
x6
Cin(x)X24 = - - - 4320 +x6 - - 322560
!_0 23(2j)!
of the entire cosine integral produces the series F.xz'/(2j)(2j)!, which represents Chin(x). The sine integral is also expansible in terms of spherical Bessel functions of the first kind [Section 32:13J: 38:6:3
Si(x) = it
J .1/2
/ Il (2/l = 4-sins\2! (l + 16[ - sin1 2/ - Z cost _)]
+5s6L\1-121sin12J-2cos(2) Jz+...
llz
SINE AND COSINE INTEGRALS
365
38:7
More rapidly convergent even than 38:6:1 and 38:6:2 are the composite power series
SO) -
38:6:4
2 - cos(x)
sin(x) +
x
=
x 3
x2
-
(-xY
x5
x3
180 + 12600
-a (2j + 1)(2j + 3)!
and
sin(x)
38:6:5
Cin(x) +
+
- ;' (-x')'
I - cos(x)
3
x'
xx
xb
x2
2
24
1440
120960
x
)-1j(2j + 2)!
Asymptotic expansions of Ci(x) and of Si(x) - (a/2) may be constructed by combining expressions 38:5:3 and 38:5:4 with the expansions 38:13:5 and 38:13:6.
38:7 PARTICULAR VALUES With n = 1, 2, 3...., Table 38.7.1 lists particular values and features of the six functions. Included in Table
38.7.1 are the arguments at which Si(x) and Ci(x) acquire values that are locally maximal or minimal. These extrema also correspond to special values of the auxiliary integrals discussed in Section 38:13:
n 2 + fi(2na)
Si(±2na) _
38:7:1
minimum maximum of Si(.t)
Si(±(2n - l)ir) _ ±
38:7:2
it
± fi((2n - 1)a)
maximum minimum
n= 1.2,3.... 38:7:3
Cii±l 2n - 1)- I = W . gil 1 2n - D
38:7:4
CiI±(2n-2)a1
2
2
= +gtl l,2»
-)
2 1rI
minimum maximum
of Ci(x) maximum minimum.
At x = :2a, ±4a,
a horizontal inflection is displayed by Cin(x), that is, d Cin(x)/dx and d' Cin(x)/ d 2 are both zero at these values of the argument [see Section 0:71. The zeros of the Chi(x) function occur at x = ±0.52382257. The first zeros of the Ci(x) function are found at =0.61650549, and others occur close to the points of inflection of the function, which are given by d2
ar Ci(x) = 0
38:7:5
.r = ±p,(-l)
j = 1, 2. 3, ...
Similarly, the sine integral inflects at Table 38.7.1 x - -2n.
11 - 2n)4
(1 - 2,,),
- 2n)n
SM.)
x -0
12n -
)a
2n - i).
2n - 1)n
0
CMtx)
Chin(.)
0
mu
Si(x)
min
max
0
2
0
min
max
max
min honz.
innc
366
SINE AND COSINE INTEGRALS
38:8
ad, Si(x)=0
38:7:6
x
x=0.±r,(I)
j= 1,2,3, ..
Here p,(-I) and r,(1) are, respectively, the roots of the equations cot(x) = -x and tan(x) = x (see Section 34:7].
38:8 NUMERICAL VALUES The hyperbolic integrals Shi(x) and Chi(x) arc conveniently evaluated by using the algorithm in Section 37:8 to generate values of Ei(±x) and then employing the identities given in equations 38:3:1 and 38:3:2. I Storage needed:
Input x >>
x, J. f and g
If 14<jxj go to (3)
If(xj>
Storage needed: x, f, g and J
1f0.1 <x2goto(1) Set f = (2x'13) exp(-x'/14) Set g = 2x exp(-x4/10) Go to (5)
(1) If x2 > 15 go to (3) Set J = 2 Int(5 + 1.2x2)
Use radian mode. 1
Set f = 11(2J - 1)
Setg - 1/(2J-3)
Xf
1
(2) Replace f by
2J - 5 I
Replace g by
2J - 7
J(J + 1) Xg
(J - 1)J
Replace J by J - 2
1fJ*0goto(2) Replace f by [-
st2) - cos(x2) - Z 1 /x J
Replace g by (2x2 sin(x2) - cos(x2) - 3gJ/2x3 Go to (5)
(3) Set J = 2 Int(4 + 60/x2)
Setf=g= 1
(4) Replace f by I - f I J 4\)\
Replace g by I - (i2
- /xs
Replace J by J - 2 If J * 0 go to (4) Set J = f sin(x2) Replace f by
f f cos(x2) - g sin(x) + 2x2=
I
2
2J /x
THE FRESNEL INTEGRALS S(x) AND C(x)
379
39:10
g cos(x2)
+ J +
Replace g by
Test values:
(5) Replace f by f/V 2 Output f
S(0.2) = 0.00212744901 C(0.2) = 0.159551382
Replace g by g/V 2
S(-n) = -0.616486872 C(-,r) = -0.451358120
f = S(x) < Output g
S(5) = 0.421217048 C(5) = 0.487879892
g - C(x)
- 0
o
or as the nonperiodic component of the semiintegral and semiderivative, respectively, of the sine function
d-v2 39:13:3
sin(x) = sinx 4d
39:13:4
&T5 sin(x) =sin l\x
v2
f
+ \ Fres(f) +4-
with zero lower limit. Their relationships to the ordinary Fresnel integrals
- S(x)1 - sin(x-)I 2-
Fres(x) = cos(x'-)I
39:13:5
C(x)j
x?0
2 1
1
1
Gres(x) = sin(x2)12 - S(W )J + cos(x2)12 - C(x)
39:13:6
x?0
may also serve as definitions.
The auxiliary Fresnel integrals may be expanded as the convergent power series 39:13:7
39:13:8
Fres(x)
1r
-
32x"
2
cos(x') - sin(x2) 2
3
sin(x2) + cos(x2)
1
rr
- Gres(x)
Va
2
8
105 + 10395 4.r'
16x°
15
+ 945
ILx
-
((4j-+4x`)
_o
V
3)!!
(-4,e)' a,=o (4j + 1)!!
or as asymptotic inverse-power series
x 1
39:13:9
39:13:10
Fres(x) --
Gres(x)-
1
3 + 105
+ (4j - 1)!! +
16x°
(-4x')'
15
945
(4j + I)!!
4x`
16x'
-+
+
(-4x`)1
the latter being valid for large x.
39:14 RELATED TOPICS Fresnel integrals play a paramount role in the theory of the diffraction of light. Indeed, it was his studies in physical optics that led Augustin Fresnel to construct the integrals that now bear his name. A very useful construct in diffraction theory is Cornu's spiral, also known as the clotoid curve. This double spiral is defined parametrically by 39:14:1
f = S(w)
where
C(w) = x
and is shown, in part, in Figure 39-2. The two points x = ±_, f = t (marked A and B on the figure) are approached asymptotically by the ever-tightening spirals. Cornu's spiral has the interesting property that its curvature, that is, the quantity
THE FRESNEL INTEGRALS S(x) AND C(x)
383
1 IO0
0
h
a
a
ti
'b
10,
ti
%
39:14
h
a
'b
m
0
A
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
FIG 39-2:
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
:...
..
...:....:...
..
..
..
..
..
..
...:....:
.
.
.
........................................................... ............
.
...... 0.2
..... ....... 0.1
S(r). x-C(w)
...... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :...... -0.3
:.... :.... :.... :.... ..... ..... .....
....:...
:....:....:...
.... ....
..
-0.5
...:....:....:....:....:....:.-0.7
dzf dx2
T/2
39:14:2 I
39:14:3 J0
l + (dx/
1+
dt) dt
between the origin 0 and point P. These two quantities are, in fact, w y tar and wv 2%a, respectively, where w is the paramctcr dcfincd in 39:14:1.
CHAPTER
40 THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x)
The functions of this chapter are interrelated by 40:0:1
erf(x) + erfc(x) = I
and occur widely in problems of heat conduction and similar instances of the diffusion of matter or energy. The name of the function arises from its importance in probability theory [see Section 40:141.
40:1 NOTATION The function erf(x) is also known as the probability integral and is sometimes denoted H(x) or 4)(x). The related notations (b1(x), 4);(x), etc., then denote successive derivatives of the error function 40:1:1
4)"(x) =
erf(x) d
Sometimes the initial letter of the erf and erfc notation is capitalized without change of meaning, but Erf(x)
and Erfc(x) may also denote ('/2) erf(x) and ('c/2) erfc(x), respectively. Changed arguments are common, the name "probability integral" or the Gauss probability integral often being given to erf(x/'v 2) or [erf(x/V 2)]/2.
40:2 BEHAVIOR Figure 40-1 includes maps of erf(x) and erfc(x). both of which are sigmoidal functions that rapidly approach limits
as x -, _w. These limits are ± I for erf(x) and 0 or 2 for crfc(x).
40:3 DEFINITIONS The most useful definitions of these functions are as the indefinite integrals 40:3:1
sgn(x) f ezp(-t) dt J Vr o
erf(.r) _ - J exp(-t')dt = Vtr ao
395
40:4
THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x)
386
...............................................
and
40:3:2
erfc(x) =
J m exp(-r2)dt = I - erf(x)
but definitions as the definite integrals 40:3:3
2 rcxp(-r2) sin(2xt) erf(x) = dt Ju
40:3:4
2x
erf(x) _ - J exp(-x-t2)dt o
as well as others. also apply.
40:4 SPECIAL CASES There are none.
1.2
387
THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x)
40:8
40:5 INTRARELATIONSIIIPS The error function is an odd function
erf(-x) = -erf(x)
40:5:1
while its complement obeys the reflection formula 40:5:2
erfc(-x) = 2 - erfc(x)
40:6 EXPANSIONS The error function may be expanded in many ways, including
40:6:2
s
erf(x)?
40:6:1
elf(x)
x
-x+ x 10
3
2
- exp(-xz) x + - +4xs -+ 2x3 3
40:6:3
_x (-x)zj v"Goj!(j+i) xzi+1
= exp(-x')
15
erf(x) = V-2[11 12(x') - l312(xz) - I...(xz) + 1712(x2) + ...) _
l'(j+ 2+ l=lntr4
, (-1)'li,i/z(x2) 1-0
while its complement is expansible asymptotically as 40:6:4
erfc(x)-
z exp(-x)
1
3
15
(2j - 1)!!
X V'ir
2xZ
4X4
ix-;
(-2x2)
x- x
The functions 1 and l,, ,z are discussed in Chapter 43 and Section 28:13, respectively. See Section 41:6 for other related expansions.
40:7 PARTICULAR VALUES Included in Table 40.7.1 are values that have a relevance to probability theory [Section 40:14].
40:8 NUMERICAL VALUES The following algorithm calculates numerical values of the error function or its complement with 24-bit precision
(i.e., the relative error in erf(x) or erfc(x) never exceeds 6 x 10-1) for any value of the argument. The erf(x) function is computed if the code c is set equal to 0; erfc(x) is evaluated when c = I is input. For -1.5 s x < 1.5, the algorithm uses a concatenated form of expansion 40:6:1, truncated at j = 3 + lnt(91xl). Note that this truncation actually generates more precision in erf(.r) than is necessary but that this gratuitous accuracy is needed to preserve sufficient precision on complementation to give erfc(x) when x is close to 1.5. For x > 1.5, the algorithm utilizes the continued fraction expression 41:6:3 in the truncated form
Table 40.7.1 x e 0
erf(x) erfc(x)
-1 2
x a 0.476936276
0
0.682689492
1
1
0.317310508
0
THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x)
40:9
erfc(x) _
40:8:1
2/a exp(-x)
2
1
x
.v-2+
2
3
388
2 + Int(32/x)
x\'2
xV2+ xa%2+
The same continued fraction is employed for x < -1.5, with -x replacing x. and the reflection formula 40:5:2 is then utilized.
Some care is needed in programming this algorithm. For instance. the final parentheses in the penultimate command should not be omitted. If they are, the altered order of operation may cause insufficient precision in an output value of the error function complement. For example, erfc(3.5) = 7.43098372 x 10-7 but, on a computing device that retains only 10 significant digits, the result of the calculation erfc(3.5) + I - 1 is 7.43000000 x 10-'. Numerical values of [ n erf(x)1/2x are also available via the universal hypergeotnetric algorithm of Section 18:14.
Storage needed: c (the code),
Input code c
Input x »:
x..iandf 1f1.5o
are useful for large x. The truncation error is less in absolute value than the first neglected term and of the same sign.
It follows f r o m 41:6:2 that the function (l / V nx) + exp(x) erfc(- Vx) is expansible as the series rV-ulr I I fz
-
11 F
j+Il 2
= exp() +
(2x)
1
V rcx =u (2j-1)!!
with uniformly positive terms.
41:7 PARTICULAR VALUES
x--x
x - 0
x - 0.204053939
x=x 0
ezp(x) erfc(V x)
undef
1
0.641304446
exp(x) erfc(- Vx)
undef
I
1.81142415 = inflection
V i exp(x) erfc V )
undef
0
0.513465985
I
3.06039578 = minimum
x
0.805767647
0
V nx
+ exp(x) erfc(-f)
undef
ezp(xr) erfc(r)
I
41:S NUMERICAL VALUES The algorithm presented below generates values of exp(x) erfc(Vx) to 24-bit precision (i.e., the relative error in the output does not exceed 6 x 10-') for any input x ? 0. Because exp(x) erfc(V) does not exist as a real number for x < 0, the input of a negative number -x is treated by the algorithm as an instruction to calculate the alternative function exp(x) erfc(- Vx), using identity 41:0:2. If the facility to calculate this alternative function is not needed, the portion of the algorithm shown in green may be omitted. The algorithm resembles that in Section 40:8. For x < 2.25, it uses the concatenation
J) +1} x(i - J) erf(V)=((( r/(1-J)+Il (J -x(?(J - 2)(J - j) 1)(J - 1)
41:8:1
.
+...+II -l.r+II -dx+112 2x
Ix
VVVVrt
which is based on expansion 40:6:1, truncated with J = 3 + 1nt(9V). For x > 2.25 a continued fraction expansion analogous to 40:8:1 is utilized.
Self= g = 0 Input ± .s >>> I
1f .s 0goto(l)
Storage needed: j', g, x and j
THE exp(x) erfc(U'x) AND RELATED FUNCTIONS
41:9
400
Replace x by -x Set g = 2 ex[(-v)
(I) If 1.50 b>0
s>0 s>0
o
42:10:10
1 exp(-at') daw(bt)dt J0
42:10:11
4(b'b
a)
a>0
f sin(Bt) daw(bt)dt = b expl 4bB > 0
b>0
DAWSON'S INTEGRAL
42:11
410
With lower limit zero, the semiderivative and semiintegral of daw(Vx) are due
2 exp(-x) -jr , daw(\) = V-
42:10:12 and
d-'I2 42:10:13
ue/2
d aw(
Vx ) =
2 1 1 - exp( - x)1
42:11 COMPLEX ARGUMENT The relationship of Dawson's integral to the error function of complex argument W(x + iv) is discussed in Section 41:11. With a purely imaginary argument one finds
iVn daw(iv) =
42:11:1
2
exp(y) erf(y)
42:12 GENERALIZATIONS Dawson's integral is a special case of the incomplete gamma function [Chapter 45] 42:12:1
daw(x) =
2
x exp(-x2)
and of the Kummer function (Chapter 47) daw(x) = x M(1;;;-x2)
42:12:2
42:13 COGNATE FUNCTIONS Dawson's integral is the n = 2 instance of functions defined by 42:13:1
r
exp(r" - x")dr
n = 2, 3, 4, ...
The other members also have some importance. All display a peak in the vicinity of x = 0.7.
CHAPTER
43 THE GAMMA FUNCTION r(x)
The gamma function is unusual in the simplicity of its recurrence properties. It is because of this that the gamma function (and its special case, the factorial) plays such an important role in the theory of other functions. The reciprocal 1/r(x) and the logarithm ht(r(x)) are also important and are discussed in this chapter, as is the related complete beta function B(x.y), which is addressed in Section 43:13. Formulas involving the gamma function often become simpler when written for argument 1 + x rather than x, and we have sometimes taken advantage of this fact. Because r(x) = r o + x)/x, a change of argument is readily achieved.
43:1 NOTATION The gamma function is also known as Euler's integral of the second kind. r(1 + x) is sometimes symbolized x! or n(x) and termed the factorial function or pi function, respectively. To avoid possible confusion with the functions of Chapter 45, r is distinguished as the complete gamma function.
43:2 BEHAVIOR The behavior of r(x) for -5 < x < 6 is shown on the accompanying map, Figure 43-1; it is complicated. For positive argument, the gamma function passes through a shallow minimum between x = I and x = 2 and increases
steeply as x = 0 or x = x is approached. On the negative side, r(x) is segmented: it has positive values for -2
<x0
More comprehensive are the Gauss limit definition 43:3:2
r (x) = lim
...+n1/\l+n/ lr T(l+x)I+2(I /I
and the infinite product definition of Wcierstrass
x exp(W) !
43:3:3 C(x)
where y is Euler's constant [Chapter 1].
l l+ 7 expl -x I `J !
;-1 \
THE GAMMA FUNCTION r(x)
413
43:4
The gamma function may be expressed as a definite integral in many ways apart from the one given above. Gradshteyn and Ryzhik [Section 8.311 give a long list of which the following are representative: (!
x>-1
1
r(1 +x)= J In'I-Idt
43:3:4
t
o
t'-' exp(-st)dt
r(x) = s`
43:3:5
s> 0
x> O
Ja
43:3:6
r(x) = s' sec
(2 Jt'"' cos(st)dt I
0<x0
Alternatively, in 43:3:6 sec and cos may be/replaced by csc and sin, respectively. Likewise, there are several ways of representing the logarithm of the gamma function by means of integrals. One is
In(r(l+x))=I
43:3:7
( ` -
0 I
-II
t
-xJ
x>-1
In(t)
and others may be found in Gradshteyn and Ryzhik [Section 8.34].
43:4 SPECIAL CASES The gamma function reduces to the factorial function [Chapter 2] when its argument is a positive integer: 43:4:1
n = 1, 2, 3....
r(n) _ (n - 1)!
The gamma function of = is V n, and the gamma function of an odd multiple of i involves a double factorial function [Section 2:131. Comprehensive formulas are 43:4:2
r(n + 1) _ (2n - I)!! r(,)/2" = (2n - 1)!!\/2"
0, 1, 2, ...
= 1.772453851
and
43:4:3
r(, - n) _ (-2)" r(j)/(2n - 1)!! _ (-2)"Vr/(2n - 1)!!
it = 0, 1, 2, ...
Similar to 43:4:2 and 43:4:3 are the formulas 43:4:4
r(n + }) = (3n - 2)!!! r(1)/3"
43:4:5
r(; - n) _ (-3)" r(J)1(3n - 1)!!!
43:4:6
n = 0, 1, 2, ... n = 0, 1, 2, ...
r(n + 3) = On - I)!!! r(32)/3" = 2zr(3n -
T(3)
it = 0, 1. 2,...
and
43:4:7
r(, - n) _ (-3)" r(5)/(3n - 2)!!! = 2a(-3)"/Vi(3n - 2)!!! r(#)
n = 0, 1, 2....
involving the triple factorial function [Section 2:131 and where 43:4:8
r(1) = 2.678938535
and
r(;) =
2a
73 r(ti)
= 1.354117939
Likewise, for arguments that are odd multiples of ,, the gamma function takes the values 43:4:9
43:4:10 43:4:11
r(; + n) _ (4n - 3)!!!! r(14)/4" it = 0, 1, 2.... r(; + n) _ (4n - 1)!!!!r(i)/4" = V 2-.(4n - 1)!!v/4" r(,'-!) it = 0. 1. 2.
r(} - n) _ (-4)" r(1)1(4n - 1)!!!!
n = 0, 1, 2, ...
or
43:4:12
r(j - n) = (-4)"(})/(4n - 3)!!!! = \rr(-4)"/(4n - 3)!!!! r(,)
it = 0, 1, 2.
THE GAMMA FUNCTION r(x)
43:5
414
involving the quadruple factorial function [Section 2:131 and where
r(}) = a'/' , 1- = 3.625609908
43:4:13 or
r(:) = - = a'I V U = 1.225416702 r(:)
43:4:14
U being the ubiquitous constant [see Section 1:7].
43:5 INTRARELATIONSHIPS The gamma function obeys the reflection formulas
C(-x) =
43:5:1
-iresc(ax) _ lresc(-rrx) xr(x) r(1 + x)
and
r(1-x)=
43:5:2
asec(trx)
1(}
+x)
The recurrence formulas 43:5:3
C(1 + x) = xr(x)
and
r(x - I)x-=-I r(x)
43:5:4 generalize to 43:5:5
1 +x)r(x)=(x)"r(x)
r(n+x)=x(1
n=0. 1,2,..
and 43:5:6
r(x - n)
(- I)" r(x)
r(x)
(x-
(1 -x)"
n=0.1.2....
in terms of the Pochhammer polynomial [Chapter 18]. The duplication and triplication formulas
r(2x) =
43:5:7
4
2V
r(x) r(# + x)
and (27).
r(3x) =
43:5:8
r(x) r(# + x)r(i + x)
2rm V3 are the n = 2 and 3 cases of the general Gauss-Legendre formula
43:5:9
r(nx) _
2a
n°
n
H r(n + i.n
which applies for any positive integer multiplier n. From 43:5:5 and 43:5:6, one may derive the expressions
zI
n = 2. 3. 4... .
THE GAMMA FUNCTION r(x)
415
r(n + x)
43:5:10
r(x)
43:6
n=1,2,3,...
=(x)
and
r(x - n)
(-1)"
_
43:5:11
n = 1, 2, 3, ...
(1 - x)
r(x)
for the ratio of the gamma functions of two arguments that differ by an integer. These formulas may be used even
when the individual gamma functions are infinite (e.g., r(x - 3)/r(x) -. -6 as x -. 0). Because of the frequent occurrence of the reciprocal gamma function in power series expansions of transcendental functions, particular values of the latter functions often serve as sums of infinite series of reciprocal gamma functions. For example, on account of expansion 41:6:5, we have 1
43:5:12
r(s)
+-++
1
1
1
1
r(1)
r(3)
r( J/2)
17
+eerfc(-l)=5.573169664
while, from the expansions in Section 45:6 1
_y(x- 1:-I)
1
43:5:13
r(x + I) + r(x +T)
Fix)
e
,zo r(x + j)
43:6 EXPANSIONS The power series expansions for the gamma function and its reciprocal are 43:6:1
[r(x)J" = x'' j( l
where ao = 1 and 43:6:2
a_,
ya.;_, + (=1), J
1
Tr2\
Y2
Yx + ( 2 - 12/ x7 ..
I
x
a_1x'='
k )..k
j = 1, 2, 3, ...
k-o
-1, -2. -3, .
Y=
0.5772156649
Here y denotes Euler's constant [Chapter I I and ;(n) is the n'" zeta number [Chapter 3). Numerical values of ao, a-,, a_2, .... a_2s are listed by Abramowitz and Stegun [page 256). The power series expansion of the logarithm of the gamma function is less complicated: 43:6:3
-1<xr= l
,.2 J
12
More rapidly convergent is the similar series 43:6:4
!n(r(1 + x)) = ( 1 - y)x + I In 2
'(J) - l
arx(l - x) (1 + .r) sin(irx)
j
,
,
j = 3, 5, 7....
The gamma function may also be expanded as the infinite product //
I
1\
I + -1 I
\\
43:6:5
r(1 + x) _ fl ,_1
1+
X
J An asymptotic expansion of the gamma function is provided by Stirling's formula: 43:6:6
r(x)
2,rr X
exp(-x)x,(1 +
\
I
12x
+
1
288x'
-
139
51840x
1-
-I 0
_ }tJo(2V-[)
(1)i
X250
XI exp(X)
43:14:2
I_o (1), I
43:14:3 f-o
I-X
-1<X>
)a»»
(1) If x = 0 go to (2) Replace g by g + x
Replace x by x + I
lfx0
F
Many definite integrals, including 44:10:4 and
rY
y(I + r>
dr = -a csc(vvn);(v)
I>
(I) Ifx=0goto(4) Seth = -l/x Setj = n
(2) Replace h by -hj/x Replace j by j - 1
1fj*0go to (2) Replace g by g + h Replace x by x + I
Input restriction: n must be a positive integer.
44:13
THE DIGAMMA FUNCTION 4(x)
432
Ifx0
o
open up other ranges of the parameter and argument. The more complicated integral representations
y(v;x) = x'
45:3:5
453:6
rvx) =
}
t"-''r exp(-t) J,(2V)dt
J0 2x'/' exp(-x)
r(1 - v)
(' J
t-.12'
v>0
exp(-t) K,(2V)dt
v 2. The infinite series in 45:8:1 is evaluated via the truncated concatenation i 45:8:3
'
r(r
0(n+v),, (((
r
\ n+v+J-x +1lIn+v+J - l
+1
n +v+J-2
THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONS
441
x
x
n+v+2
n+v+l
45:10
+1n+vl -
By use of the empirical assignment J = Int(5 + (3 + kxl)/2) one ensures that approximation 45:8:3 has 24-bit precision (the relative error is less than 6 X l0-"). The final segment of the algorithm is concerned with calculating the multiplier exp(-x)/r(n + v), using a procedure similar to that employed in Section 43:8.
Input v >>
Setg=p= I
Storage needed: v, g. p. x, j and f
Input x >> (1) Replace v by I + v
Ifv>2goto(2)
Replace g by gx Replace p by pv + g Go to (1) (2) Set j = lnt(5(3 + !x1)/2)
Set f = 1 /(j + v - x) (3) Replace j by j - I Replace f by (fx + 1)/(j + v)
Ifj * 0 go to (3) Replace p by p + fgx (1 g= 1- 2 _ 2 I] /30v2 7v'3v'
\1
Set
Replace g by g
12v
- v[In(v) - I]
Set f = p exp(g - V)v/2a Output f
Test values: y*(#;-0.49) = 1.34326134
y*(-3.9;0) _ -0.521258841 y*(I;a) = 0.304554469
f = y*(v;x)
Of course, numerical values of y(v;x) and r(v:x) may be computed from y*(v;x) by using equations 45:0:1.
Alternatively, the universal hypergeometric algorithm of Section 18:14 may be used in several versions.
45:9 APPROXIMATIONS Whereas it is exact only if the parameter is a nonpositive integer, the expression
y*(v;x) - X
45:9:1
is a useful approximation for all v if x is positive and sufficiently large.
45:10 OPERATIONS OF THE CALCULUS Differentiation of the incomplete gamma function and its complementary analog give d
d 45:10:1
F(v;x) = x°'exp(-x) = y(v;x) - (v - 1)-y(v - I;x)
y(v;x)
dx
dx
The differintegration [Section 0:10] formula for the entire incomplete gamma function d" 45:10:2
If exp(x)y*(v;x)] = x"-" exp(x)y*(v - µ;x)
dx"
holds for any value of p., positive or negative, integer or noninteger.
THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONS
45:11
442
Two definite integrals are
/
J
45:10:3
1
b s\
v> 0
y(v;bt) exp(-st)dt = ysv) 1 b
a
and
('
J
45:10:4
r"-'r(v;bt)di=
r(v + µ)
0
µbµ
v+ IL >0
p.>0
and many others are listed by Gradshteyn and Ryzhik [Section 6.45).
45:11 COMPLEX ARGUMENT Here we shall cite the form acquired by the complementary incomplete gamma function when its argument is imaginary and omit the more general case of /complex argument- The result: 5:11:1
r(v;iy) = sinl 2 I S(y;v) + cost
2/
C(v;v) + it sing 2 L
C(y;v) - cos(!_.) t S(v;v)]
\\\
)
involves Boehmer integrals (seee Section 39:12`].
45:12 GENERALIZATIONS The three incomplete gamma functions are special cases either of the Kummer function [Chapter 471 45:12:1
x"
x"
y(v;x) _ - exp(-x) M(1;l + v:x) _ - M(v;l + v;-x) V
y*(v'x)
45:12:2
V
exp(-x) M(v;I + v;-x) M(1;1 + v;x) = r(1 + v) r(1 + v)
or of the Tricomi function (Chapter 481 45:12:3
r(v;x) = x"exp(-x) U(1;1 + v;x) = exp(-x) U(I - v;l - v;x)
45:13 COGNATE FUNCTIONS Just as the incomplete gamma function derives from Euler's integral of the second kind [the complete gamma function, Chapter 431 by allowing the upper limit to become indefinite, so the incomplete beta function (Chapter 58] derives from Euler's integral of the first kind (the complete beta function, Section 43:131 by similarly making the upper limit indefinite.
45:14 RELATED TOPICS Several functions are arranged below in a hierarchical chart. The higher placed functions are the more general: arrows represent the effect of restricting one variable to a specific value. Numbers in brackets refer to chapters or sections of this Atlas. This chart is not, of course, exhaustive; thus, the Basset function [K,(x), Chapter 51] is another special case of the Tricomi function, while both y*(v,x) and K,(x) include the exponential function (Chapter 26] among their special cases.
443
THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONS
.
45:14
General hypergeometric function [18:14] with K + L = 3, IK - LI = l
\
Kummer function [47J
Tricomi function 1481
Incomplete gamma function, etc. [451
Parabolic cylinder function (46]
I
Dawson's integral [421
Error function, etc. [40, 411
Constants [11
I Exponential integral (391
CHAPTER
46 THE PARABOLIC CYLINDER FUNCTION
This function arises in the solution to many practical problems that are conveniently expressed in parabolic cylindrical coordinates. This coordinate system, and others. are discussed in Section 46:14.
46:1 NOTATION The parabolic cylinder function is also known as the Weber function or the Weber-Hermite function. The name "Whittaker's function' is also encountered, but confusion should be avoided with the identically named functions of Section 48:13. An alternative to the usual D,(.r) notation is U(-v - k,x). The variables v and x are known, respectively, as the order of the parabolic cylinder function and its argument. The origin of the name "parabolic cylinder" function is made evident in Section 46:14.
46:2 BEHAVIOR The D,(x) function is defined for all real values of v and x. Figure 46-1 is a contour map showing some values of the parabolic cylinder function. For v less than about -0.20494, D,(x) is a monotonically decreasing positive function of x. For larger orders,
the parabolic cylinder function displays a number of zeros, maxima and minima in the -x < x < x range, as shown in Table 46.2.1.
Table 46.2.1 Number
Number of maxima
Number of minima
0
1
0
1
1
0
of zeros -0.20494 < v s 0
0 -1
\\\
and
-xt)e,
42
D.(x)=I(1exp
46:3:2
Jexpl
/
0
2
v0
1
4
and a
Dv2(x )- [KI/4(X42)
46:4:6
x>0
+K3/44
and others may be evaluated via recurrence 46:5:1.
46:5 INTRARELATIONSHIPS The parabolic cylinder function satisfies the recursion formula
D,.,(x) = xD,(x) - YD,-,(x)
46:5:1
and the argument-addition formulas 46:5:2
D,(x + y) = exp
/2xy 4+y2I =Q
j!
2xy - y2
D,.j(x) = expl
4
\
)
v.,y;D._,(x) 11 f=o \
The sum or difference D,(x) t Dr(-x) can be expressed in terms of Kummer functions [Chapter 47] 46:5:3
46:5:4
D,(x) + D,(-x) =
D,(x) - D,.(-x) =
2° 7-'n
r(12v\
x'\ -v1 x'\ exp(I Ml - - - I = 2 2 2/ 4/
/ x2 /1-v 3x2 l r(_v xexpl 4 ) MI 2 .2. 2 ;/'eJ
2)
\
/xz\
2`+2 R
r112v\
-
_
/1 + V; 1 _ x2 \
expl - I MI
\4/ \
2
x2 r(-v 3n xezp\4/ M\1 2)
`
2yJ
2
2
v 3_
x2
+ -; 2; 2
46:6 EXPANSIONS A number of power series expansions exist for exp(±x2/4) D.(x). Thus, one has
r- zv f (-xV2)' P('
46:6:1
2a
'
p(x)
D,(x) =
46:6:2
2r(/ v)
(-x V [)J 2
\ 2.
46:6:3
D,(x) = V
exp(x I i cos( 4 / ro
+7 1 2
/
j!
/
Alternatively, one may add equations 46:5:3 and 46:5:4 and then use expansion 47:6:1 so as to express the parabolic cylinder function as the sum of two power series. Expansion in terms of Hetmite polynomials [Chapter 24] is possible in two distinct ways:
THE PARABOLIC CYLINDER FUNCTION D,(x)
449
2i2
D,(x) =
46:6:4
/2)
r(
exp (
) \ 4 '/
and
D,(x) =
46:6:5
expl \\\
-
H2,(x/ V 2) (I ' 4 } j, (j
x>0
(__!)
x>0
j!
!
2
46:8
2
The asymptotic expansion of the parabolic cylinder function
D,(x) - ? exp
46:6:6
p(-x2)[ 4
I-
(-v)(1 - v) 2x2
+
(-v)(1 - v)(2 - v)(3 - v) 2!(2x)2
(- v),,
j!(-2x')J valid for large x, is a consequence of relationship 46:12:2 or 46:12:3, together with 48:6:1.
46:7 PARTICULAR VALUES D.(0)
D,(-)
0>»»
Set G = -b.\/ Set J = 3 + 61x1 + (I(x + 2)(2 - v)I/5) + (x2/2) Replace x by x 2/2 //
Setw= explvln\2-x-x Replace G by``Gw
//
SetfF=aw Set g=G
Set d = (F + G) exp(x)
Set w = (v + 1)/2
Setj=0 (5) Replace j by j + I //
1
Replace F by Ij- ! - N. Fx/j
1
fx``/j
Replace f by
Replace g by gx/j
Replace G by (j - Nw) Gx/j j +
Replace dbyd+(F-f)+(G-g) If)SJgoto(5) Output d
d = D,(x) ««
0 x?0
46:13 COGNATE FUNCTIONS The function
C(-v)
46:13:1
[D,.(-x) - cos(Tr0D,(x)1
IT
is sometimes encountered and is symbolized V(-v - ?,x).
46:14 RELATED TOPICS In many contexts there arises a need to map space using an origin and three coordinates. In this section we discuss a number of orthogonal coordinate systems, which constitute the most useful ways of performing such mappings. If we use r, q and p to denote three general coordinates, specifying the triplet (r,q.p) of numbers locates a unique point in space. Specifying two of the coordinates, say q and p, but allowing the third to adopt any permissible value, defines a line (generally a space curve) that we can denote (q,p). Specifying only one coordinate, say r, defines a surface (r). It is a characteristic of an orthogonal coordinate system that, at any point (r,q,p). the three surfaces (r), (q) and (p) are mutually perpendicular. Likewise, the three lines (q,p), (r,p) and (r,q) are mutually perpendicular at (r,q,p). In simple physical applications, each (r) surface, defined by allocating a specific value to the r coordinate, may correspond to a particular value of some scalar quantity F (temperature, energy, concentration, electric potential, etc.). Such surfaces are sometimes called equipotentials. Some physical body, known as the generator of the coordinate system, may occupy the r = 0 surface. The line (p,q) corresponding to specified values of the p
and q coordinates is known by a variety of names such as "line of force," "flux line," "field vector," "line of steepest descent," "streamline," etc., depending on the field of application. Here we use the name streamline. Of course, the most familiar set of orthogonal coordinates is the cartesian coordinate system, (x,y.z). We may think of this as arising from a generator corresponding to the infinite plane y = 0 with uniformly spaced equipotentials y = ±1, y = t2, y = t3, .... The streamlines are straight lines. A cartesian coordinate system is depicted in Figure 46-2. As in the other diagrams of this section, streamlines arc shown in red, equipotentials in green and the generator in blue; the z-coordinatc is perpendicular to the plane of the paper. The rectangular coordinate system may equally well be represented by Figure 46-2; it is the two-dimensional equivalent of the cartesian system. The cylindrical coordinate system is illustrated in Figure 46-3. The coordinates consist of two lengths, r and z, and one angle, 0. The relationship to cartesian coordinates is 46:14:1
x=rcos(8)
y=rsin(8)
05r
2Ial + Icd + I and IT,I < IT,_,I are met. If not, j is incremented until all three conditions are satisfied.
Setf=t
I
Storage needed: a, f, t. c, j, g, x and p
Setj=g=0 Ifx0
c*1
C> 0
and we also cite the important definite integral
r
t'-'M(a;c;-r)dt -
47:10:5 J
r(v) r(c) r(a - v) r(a) r(c - v)
00
which is defined by the integral 48:4:10
k,(x) =
2
J * cos(x tan(s) - vt)dt 0
With the c parameter equal to 1 or ?, the Tricomi function is a parabolic cylinder function (Chapter 46]: 48:4: 11
U(a !;x) = 2a exp 2 D_,(2V) (X)
48412
U(a;];x) = 2a exp 2
Di_y,(2)
THE TRICOMI FUNCTION U(a;c;x)
48:5
474
Clearly, these last two expressions can be used in conjunction with recursion 48:5:3 to express U(a;n + 2;x) where n is any integer. The limit operation
{r(1 + a - c)UI a;c; I } = 2x!2K,. (2 V x) ` a/11
lint
48:4:13
produces a Basset function ]Chapter 51] of order c - 1. In this section we have addressed the effect of specializing one of the two parameters. As will be clear from Sections 4 of Chapters 13, 23, 45, 46 and 51, still simpler functions are generated when both a and c are specialized.
48:5 INTRARELATIONSHIPS The important transformation
U(a;c;x) = x'-` U(I + a - c;2 - c;x)
48:5:1
relates two Tricomi functions of common argument. Recurrence formulas may be written interrelating three Tricomi functions whose parameters differ by unity. Examples are
2a-c+x I I U(a;c;x) a(I+a-c) U(a - I;c;x) _ -a U(a;c;x) - -a U(a;c - 1;x) a(l+a-c) 1
48:5:2
U(a + I;c;x) =
48:5:3
U(a;c + l;x) =
l+a-c
c-l+x U(a;c;x) +
c-a
1
U(a:c;x) + - U(a - 1; c;x)
U(a;c - 1;x) _
x
x
x
X
and
all + a - c) U(a:c;x) =
48:5:4
x
U(a + I:c;x) + -
a+x
a+x I
a - I+x
U(a - l;c;x) -
U(a;c + l;x)
I+a-c U(a;c - l;x)
a-I+x
Analogous to equation 47:5:5 is the argument-addition formula
j
U(a;c;x + Y) = Z (a)'( y) U(a + j;c + j;x)
48:5:5
LvI
J
=u
>
R, + 2i x
p»» Set s = 1
Ifx-0goto(1) Sets= -I
Replace x by -x
(1)Setr=f=0
Set j = 10 + lnt(x) (2) Replace f by I + fr Replace r by x/(xr + 2,p
j=J,J-1,...,3,2,1
Storage needed: x. s. r. f and j
THE HYPERBOLIC BESSEL FUNC71ONS lo(x) AND 1,(x)
485
49:10
Replace j by j - 1
Ifj * 0 go to (2)
f-10(x)
-
(I - r')"-'t' exp(±xt)dt
I
i
Substituting t = cos(h) in 50:3:1 gives a second definition and some others are listed by Gradshteyn and Ryzhik [Section 8.4311. A Kummer function [Chapter 471 whose denominatorial and numeratorial parameters are in a two-to-one ratio
is related to a hyperbolic Bessel function by 50:3:2
(x/2)" exp(-x)
1,(x) =
r(l + v)
M( + v;l + 2v;2x)
The operations discussed in Section 43:14 can generate any hyperbolic Bessel function from any other hyperbolic Bessel function and hence from any of the functions discussed in Chapter 49. For example, in the notation of Section 43:14
lox)
50:3:3
-v r(+ v)
(x/2), Mx)
0
21`
x=
A solution of the modified Bessel equation
d'f df .r-+x--(x'+ir)f=0 dx' d.r
50:3:4
is c,I,(x) + c2unless v is an integer [see Section 51:3 for that ease]. The terms c, and c2 arc arbitrary constants. Other differential equations that are satisfied by hyperbolic Bessel functions are x
50:3:5
a + (2v + 1) of - xf = 0
f = c,x `I,(x) +
and
50:3:6
x
dZf
dx-
+ (v + 1)
df dx
-f = 0
f = c,(2\)-" 1.(2V) + c.,(2V)'I_.(2V )
unless v is an integer.
50:4 SPECIAL CASES The hyperbolic Bessel functions of integer order are the subject of Chapter 49. When the order of the hyperbolic Bessel function is an odd multiple of =;, reduction occurs to the simpler functions discussed in Section 28:13. The simplest cases are cosh(z)1
2 50:4:1
sinh(x) -
1_312(x) =
x
ax
11,(x) _
50:4:3
50:4:4
I2 J1(x) =
ax
(x + 1
exp(x)
1 s
- cosh(x)
ax
sinh(x) = ? ax
sinh(x) I cosh(x) x [
!2!L +
`
tax
exp(-x)
2x
2az
cxp(x) P(z)
tax
x-!
exp(-x)
- 1
2az
1_112(x) _
50:4:2
/x - 1
2ax
exp(x) tax
+
s
(x+l) exp(-x) 2ax
THE GENERAL HYPERBOLIC BESSEL FUNCTION 11(x)
50:5
492
and others may be constructed via recursion formula 50:5:1. Some of these functions are mapped in Figure 28-3. The s} mbol i (x) and the name modified spherical Bessel function of the first kind are sometimes applied to the Ni it/2x I., 112W function.
Hyperbolic Bessel functions of order ± are closely related to Airy functions [Chapter 561. One has
X= (2x
4X [Bi(X) = V /Ai(X)I
50:4:5
n
Hyperbolic Bessel functions of order ±14 are expressible as parabolic cylinder functions [Chapter 46] of order -j. The relationships are
13,14-2N6) + D_112(2\) I-114(x) _
50:4:6
x1/4
Similarly. 1:314(x) may be expressed in terms of parabolic cylinder functions by 50:4:7
1,1/4(x)
D,,2(2vx) - D12(-2V)
-VxD_,,2(2Vx) - VxD_,R(-2Vx) 2.x7/4
and, hence, with the help of recursion formula 50:5:1, so may 45/4(x), etc.
50:5 INTRARELATIONSHIPS Hyperbolic Bessel functions satisfy the recursion relationship 2v
I,. (x) = I,_,(x) - -11(x)
50:5:1
X
and the argument-multiplication formula b''
11.(bx) =
50:5:2
r- - ) 2
j!
11.;(x)
Setting b = I + (y/x) converts 50:5:2 into an argument-addition formula, while setting b = i =
generates
the summation formula _
50:5:3
I,() - x1,.1(x) + 21
= 1..2(x) - ... _ ` (-x) I,.1(x) = i "1,(ix) = J,(x) l
j=o
%
The hyperbolic Bessel functions 1,(x) and I_,(x) are identical if v is an integer; otherwise: 2
50:5:4
I_,(x) = 1,(x) + - sin(vir) K,(x)
it
where K, is the Basset function [Chapter 51]. This equation may be regarded as an order-reflection formula, interrelating I_,(x) and I,(x). There are similar order-reflection formulas for the product of two hyperbolic Bessel functions whose orders sum to ± 1. namely 2 50:5:5
1_,(x) 1,_1(x) = I,(x) 1_,1,(x) + - sin(vtr)
ax and
50:5:6
2
1-,-112(x)1,-112(x) = 1112..(x) 112_,(x) + - cos(va) 'nx
THE GENERAL HYPERBOLIC BESSEL FUNCTION I,(x)
493
50:8
50:6 EXPANSIONS The hyperbolic Bessel functions may be expanded in the convergent series (x/2)4.. (x/2Y + + + 50:6:1 I,(x) = r(1 + v) l!r(2 + v) 2!r(3 + v) ;_,j!r(j+v+1)
=i
That this expansion involves a hypergeometric series [Section 18:141 becomes more evident when it is rewritten in the form 50:6:2
I,(x)
+
r(1 + v)
(x/2)" .
x2
4(1 + v)
+
x4
32(1 + v)(2 + v)
x6
+
384(l + v)(2 + v)(3 + v)
+
(x2/4)'
r(1 + o, (l),11 + v)i where (I + v); denotes a Pochhammer polynomial [Chapter 181. The expansion
I,(x) = 1,(x) + xi,.1(x) +
50:6:3
2
xij,
_
J,+2(x) +
Jj.,(x)
j=0
in terms of Bessel functions [Chapter 531 is also convergent. An asymptotic expansion is provided by the series 50:6:4
I,(z) -
exp(z) (
27
(2 - y2)(1 - v)
- v= 1+
2x
+
(;-v))(;+v);
8x2
+
(114_ v2)(/
+
- v2)(i - v2)
48x3
xc
1
j!(2x)' which is valid if x, but not jvi, is large. This series terminates when v = ±; ±? ±i
.. but the expansion is
not exact even under those circumstances.
50:7 PARTICULAR VALUES
v>0
44c)
b(x) I,(x)
-I >>>>
X »»>
Storage needed: Y. f, x and j 1
Setf= I If x a 8 + (v2/12) go to (4) If v + Intflvj) * 0 go to (1) Replace v by -v (1) Set j = Int(x + 3) 3x\
Input restrictions: Any value of v may be input, but x must exceed zero.
If(v-5) v+10+/2 1>0goto(2) Replace j by j + lntl 4 /- 3vl +/JI
(2) Replace f by I + fx2`/I4j(j Replace j by j -
v)]
Ifj*0goto(2) Ifv=0goto(6)
Replace f by f/v (3) Replace f by 2fv/x Replace v by v + I
Ifvs3goto(3) 2 2 1--1 Setj=L1+ L 3v2 1
Replace j by L:11 121
Replace f by Go to (6)
/30v'
7y'
+ v[ I - ln(2v/x)]
f exp(j)
(4) Set j = lnt(5 +
v/2a
-- + ll I
If x < 100 go to (5)r
///
/
rLL\
(5) Replace f by I + f
Replace j by j -
(i
\ Vx 4 IvIJ 1
Replace j by) - IntLl 2 2
V1/ v2J /2jx
Test values: Id(5) = 27.2398718
1112(n) = 5.19875924 L,(l) = 0.0221684248
495
THE GENERAL HYPERBOLIC BESSEL FUNCTION 1,(x)
If j*0go to(5)
50:10
1-r,2(2) = -0.628009049
(6) Replace f by f exp(x)/ Output f
12(10) = 2281.51897
2srx
1,00000) = 4.64153494 x 1021 1_,(125) = 6.88584377 x I0s2
f - 1,(x) -1
/ z\
l
o
Replacing the argument x by 2N/-x in the first of the two Rayleigh formulas 50:10:8
1d {x dx} [x"I.(x)) = x"-"I.-.(x)
50:10:9
f x dxf [x_"I"(x)J = x-"-"1...(x)
for multiple differentiation leads to a relationship that generalizes to the very simple expression d"
50:10:10
dx"
[xvn-I.(2f )J = xr"-" r_4
Here the operator d"/dx" signifies differintegration with lower limit zero to an arbitrary (positive or negative, integer or noninteger) order [see Section 0:10).
50:11 COMPLEX ARGUMENT Replacing x in expansion 50:6:1 by x + iy leads to
.
50:11:1
1.(x + iy) _ Z ,=a
j
cos((b) + i sin(do) (z' + yzl'""t2)
t
4
j!r(j + I + v)
where 4 = (2j + v) (8 + 2ka), 0 and k having retained their significances from Section 13:11. When the argument is purely imaginary, the hyperbolic Bessel function becomes a Bessel function [Chapter 531 of real argument:
r
/ \
/
l
2 )j My) (2 ) ] when its argument is negative. We Unless its order is an integer, the hyperbolic Bessel function is complex
50:11:2
I.(iy) - i"J"(y) = I cosy 2 I + i
have
50:11:3
I.(-x) = (- 1)" I"(x) _ [cos(vw) + i sin(var)J 1.(x)
50:12 GENERALIZATIONS The function (x "," .
(x2/4)'
50:12:1 \2)
0 r(1 + µ + j)r(1 + v + j)
497
THE GENERAL HYPERBOLIC BESSEL FUNCTION l,tr)
50:13
closely related to the Kummer function [Chapter 47], is a generalization of the hyperbolic Bessel function, which is the µ = 0 instance of 50:12:1. The hyperbolic Struve C function [Section 57:131 is also a particular instance of the general function 50:12:1.
50:13 COGNATE FUNCTIONS In the next chapter the cognate Basset function is discussed. Also closely related to hyperbolic Bessel functions arc the Kelvin functions (Chapter 55] and, of course, the Bessel functions [Chapter 531 themselves.
CHAPTER
51 THE BASSET FUNCTION
Because the Basset function of noninteger order is related so simply [see 51:3:51 to the hyperbolic Bessel function of the previous chapter, this chapter concentrates on the Basset functions Ko(x), K,(x). K2(x), ... of integer order.
51:1 NOTATION Alternative names for the Basset function are the modified Bessel function of the third kind, Bessel's function of the second kind of imaginary argument, Macdonald's function and the modified Hankel function. We use v generally to represent the order of a Basset function but replace this symbol by n to specify integer order.
51:2 BEHAVIOR The Basset function K,(x) is infinite for x = 0 and complex for x < 0 [see equation 51:11:31. Accordingly, we restrict attention to x > 0 here and generally throughout this chapter. For all v, K,(x) is a positive and monotonically decreasing function of its argument x, approaching zero as x - %, in accordance with expression 51:9:6, in a manner increasingly independent of the order v. However, the approach to infinity as x -. 0 is a strong function of v as detailed in Section 51:9. For a constant positive argument, K,(x) is an even function of its order v, as evidenced by equation 51:5:1. Moreover, for constant positive argument, K,(x) invariably increases as wj increases. Figure 51-1 shows maps of K,(x) for n = 0. 1, 2, 3. 4. 5 and 6. The curves for noninteger order Basset functions smoothly interpolate between these mapped curves: for example, Ka12(x) lies intermediate between K4(x) and K5(x).
51:3 DEFINITIONS Basset functions may be defined via those Tricomi functions [Chapter 481 in which the a parameter is a moiety of the c parameter. 51:3:1
K,(.,) _ \(2x)" exp(-x) U(12 + v; I + 2v; 2x)
v >- 0 499
THE BASSET FUNCTION K.(x)
51:3 e1,
,y0
40
O
U
Rr
4
00 y s $4' O
U
*
t
0 '. if
ti
O Nf
b
,V
V
r0 y'1.
0
ifL
0
4.,y.
i
rtj.'L
if
2.4
..:.. I .:... .......... .......: .... ............ .... :.... .... ............. .
2.2 2.0
0.2
or approaches infinity: 51:3:2
K,.(x) = Z
( 2 1,1 L {r(a - v) Ul a; 1 + v;
4a /
I
v
0
Equivalent to 51:3:1, but somewhat simpler, is the definition of a Basset function as a special case of Whittaker's W function [see 48:13:61. Among definite integrals that define the Basset function are 51:3:3
K,(x) _
exp(-x/)dt
v) -
2 (2x)' f (r' - 1)
F( , + v)
1
2
and 51:3:4
K,(x) _
c
r(3 + v)
cos(t)dt (
)" f. & +
I
v
2
Magnus, Oberhettinger and a large number of alternatives were assembled in the Bateman manuscript (see and Tricomi, Higher Transcendental Functions, Volume 2. pages 82 and 831. Because of relationship 51:5:1. the sign of v may be changed in definitions 51:3:3 and 51:3:4. Basset functions of noninteger order may be defined in terms of hyperbolic Bessel functions by
THE BASSET FUNCTION K,(x)
501
Tr[l_,(x) - l,(x)) 2 sin(vrt)
51:3:5
51:5
v*0,±1,-'2,...
but this expression must be written as a limit: 1.(x)1
IT
K(x)2limSl
51:3:6
sin(m)
Jr
when the order is an integer.
With c, and c, representing arbitrary constants, the differential equation
x. d/+xdf-(.Y+n')f=0
51:3:7
dx'
n=0,±1,±2....
dx
is solved by f = c,l (x) + c;K (.r). Other differential equations whose solutions involve the Ko or K, functions are listed as 49:3:10-49:3:13. Semiintegration with lower limit zero is a powerful method of generating Basset functions. Examples include
I _,v'rrexp\2x (1 d.r-"2 xeap d- uz
K.
51:3:8
K,
51:3:9
I
(
l
xvrt
'1
exp (
2
' /-
Kv< 2x> = x V to exp
51:3:10
1
r
2x
d- vx x=) d.r ''=
7
l _
I
1
eXp .r
51:4 SPECIAL CASES When its order is an odd multiple of , the Basset function reduces to a simple function involving an exponential. The simplest cases are rr
K,,_(.r) = K-, _(s) _
51:4:1
K3 .(x) = K_};.(x) _
51:4:2
exp(-x) 1
xL
l + xJ exp(-.r)
[see Section 26:13 and Figure 26-2] and others may be constructed via recurrence formula 51:5:2. The symbol k,(.r) and the name modified spherical Bessel function of the third kind are sometimes applied to the V a//1" function.
A Basset function of order 3 is related to an Airy function [Chapter 561: 51:4:3
K1/3(x) = K-113(x) _'r
X Ai(X)
X=
(3
while that of order ; is related to a parabolic cylinder function (Chapter 461:
51:5 INTRARELATIONSHIPS With respect to its order, the Basset function is even 51:5:1
K_,(x) = K,.(x)
2
THE BASSET FUNCTION K,(x)
51:6
502
so that v may be replaced by jv= in most of the formulas of this chapter. The Basset function obeys the recurrence relations 2v K,. ,(x) = K,_ ,(x) + - K,(x)
51:5:2
x and 1
K,.,r(x)1,(x) _ - - K,(x) l,.r(x)
51:5:3
X
The zero-order Basset function satisfies the argument-addition and -subtraction formulas
Ka(x±y)=lo(y)K0(y)+2(rIYI,(y)K,(x)
51:5:4
x>y>0
By sufficient applications of formula 51:5:2, any Basset function K"(x) of integer order may be expressed in and K,(x). The first two examples are
terms of
2
K,(x) = K((x) + - K,(x)
51:5:5
x
K3(x) =
51:5:6
4
x
/
Kd(x) + 1
\
\ x'/
I + 8, I K,(x)
and expressions for K,(x). K5(x) and Ke(x) are analogous to 49:5:13-49:5:15. but with uniformly positive signs.
51:6 EXPANSIONS The Basset function of noninteger order is expansible as the sum of two convergent series K,(x) =
51:6:1
--I
I(v) (xr 2
2
//+ --l (x'/4)1
_O j!(1 - v),
+
,
r(-v, Ix\
(x=/4)' =p j!(l + v),
I
2
2
v*0,'_1,±2....
that coalesce only if v is an odd multiple ofFor integer order the series are
r 51:6:2
K0(x)
/x\
y - In{ 2 I
-y + 1 + l
+
2
=-x 1
K1(x)
51:6:3
+l-
+
\/
I
3
- ln(x \2/
-y + I 2
2
x\\ 11 (x- 4)
ln(2
(x /4)
ln(x
(x2/4)r12
\2/
0!I!
I
x 11 (x' 4)-'
In(2)J (2 ).
+ ...
(3!)2
+I+2+6-1n\2.,J \1 HI
I (1 , + [-,y + I + 2
1J
- ---y + I + 2
1
4
In
x
(x-'14)312
2
l!2!
(x'/4)3'' 2!3!
+I
and generally
rr
(x/2)''" x Vb(l+j) dr(I+n+j) r + - x2) j!(n 2 2 \4 / + j)! L j! In these formulas r denotes the gamma function [Chapter 43], s the digamma function [Chapter 441, (1 - v), is
K"(x)=-(l
(n -j- 11.
1
51:6:4
2 \x/ ;=o
ln(J
a Pochhammer polynomial [Chapter 181 and y is Euler's constant (Section 1:71. Alternatively. Basset functions of integer order may be expanded as Neumann series in terms of hyperbolic Bessel functions: 51:6:5
K0(x) - {--
10(x) + 2 i
In J
;-r
L"_(L)
J
y = 0.5772156649
THE BASSET FUNCTION K,(x)
503
K,(x) _ - -y + I -
51:6:6
I
' (1 + 2j)
x
J=i J(1 + J)
10) + - lo(x) -
2l
51:8
It+2,(x)
-I(x)+-j x I(.r)+(-I)LLL 22 \x
and generally
51:6:7 K(x)I+(1+n)-In (x)] rr
(n + 2j)
1
k)
k! (n,
J(n
2
+
J)
The series 51:6:8
K,(x) -
- exp(-x) f I - 4
v' 2x
+
(1-y')(1v') - (11 -v')(i-y')(1-v') 48x'
8xz
-v),(1+v)i
0
l
j!(-2x)'
1
is generally asymptotic but it does terminate when of is an odd multiple of 1, and under these circumstances the expansion is exact. If vj is not an odd multiple of 1 and lies between the numbers n - ; and n + 1, where n is an integer, then the terms in series 51:6:8 corresponding to j = O. j = 1, j = 2..... j = n are uniformly positive (we are treating only positive x), whereas the j = (n + I)m term is negative; thereafter, the terms alternate in sign. Accordingly, it follows from the properties of alternating series [see Section 0:5] that if the series 51:6:8 is truncated, terms after j = J being ignored, then the partial sum is related to the true value of K,(x) by one of the inequalities 51:6:9
J ? n = lntl
(series of J terms) 5: K,(x) 5 (series of J + 1 terms)
+2
for large enough J.
51:7 PARTICULAR VALUES x - 0
K, x)
x * x
0
51:8 NUMERICAL VALUES Two algorithms are presented in this section. The first generates values of K0(x) and K1(x) for sufficiently small arguments, say x 5 3. From these, recurrence 51:5:2 shows how it is possible to compute the K,(x) function for any positive integer n. The second algorithm produces values of the Basset function for arguments exceeding 2.5 and for orders (integer or noninteger) less than about (24x/ln(x)J'". Neither algorithm is suitable for evaluating K,(x) for small x and noninteger v, but definition 51:3:5 provides easy access to these values via the algorithm of Section 50:8. With R, having the significance accorded to it in Section 49:8. the Neumann series 51:6:5 may be rewritten as the concatenation 51:8:1
K0(x) = lo(x) 4RoRi(2 + RR, 4 + R4R5 6 +
) 1 1- y - In (2)J
This is the formula used by the first algorithm to calculate K0(x).. the required value of 15(x) being determined via the similar concatenation 51:8:2
cosh(.r) = 10(x)[1 + 2RoR,(l + R:R}(l + R4R5(l +
)))1
which follows from series 49:5:7. In practice, both concatenations are terminated by setting R, = 0 for j greater
THE BASSET FUNCTION K,(x)
51:8
504
than an empirically chosen integer J. Recurrence 49:8:2 then serves to calculate all other R, values. The hyperbolic Bessel function 1,(x) is calculated by the algorithm as R010(x) and thence the Basset function K,(x) as R0{[I/xl,(x)] - Ko(x)}, an identity that follows from 51:5:3. Many problems in applied mathematics require values of the func-
tions 10(x) and 1,(x) as well as those of K0(x) and K,(x); the algorithm below delivers all four functions if the portions shown in green are included.
Input x >> >>>>
Storage needed: x, j, 1, K, r and R
Set j = 8
2 lnt(x)
SetI=K=r=R=0
(1) Replace
I by I + IrR
Input restrictions: x must exceed zero. If x exceeds about 3 (the precise limit
Replace K by (I 1j) + KrR
Setr= 1
/
A I
R+
2j)
depends on the computing device), accuracy may be impaired.
X
Replace j by j -
SetR= 1
Ir+
2j
Replace j by j -
Ifj*0goto(I) Replace I by [exp(x) + exp(-x)1/12 + 4IrR] Output I
10(x)< ««
Replace K by I14KrR - In(.890536209x)] Output K
K = K0(x) ««
Test values: 10(1) = 1.26606588
K0(l) = 0.421024438 1,(1) = 0.565 1 5 9 /04
Replace I by IR
Output I
K1(1) = 0.601907230
Replace K by R[(1/xl) - K]
K0(3) = 0.0347395044 K,(3) = 0.0401564311
h,..... h,,,. ha, Set J = 2 lnt(6 + ]Ivj"ln(x)/x]} (I) Replace t by r
//
v2
Set h, = r + h,_,
Setq=0 Set k=j
I
11
- I j- 2 1 ]/(2Jx)
\
//////
THE BASSET FUNCTION K.(x)
SOS
(2) Set p = h4_1 If p # hk go to (3) Set hk_, = l099 If p * 1099 go to (4) Set hk_, = q
51:10
Input restrictions:
x z 2.5 !vi < [24x/ln(x)J213
Go to (4) (3) Set he-i = q + [1/(h5 - p))
(4) Set q = p Replace k by k - 1
Ifk*0goto(2)
Test values:
Replace j by j + I
K,(5) = 0.0053089437/
If jsJgo to (1)
K,,,2(a) = 0.0305568546
Set p = h0V a/2x exp(-x)
K_,(10) = 1.86487735 x 10"5
Output P
K,0 (100) = 9.27452265 X 10-iO K16(5) = 186233.583
p - K,(x)
51:9 APPROXIMATIONS From expansions 51:6:1-51:6:4, the following two-term approximations, valid as x -' 0, may be derived:
I - y = 0.11593 - ln(x)
K,(x) -
51:9:1
v=0
small x
C/
r(v)x-"
K,(x) =
51:9:2
2' I
K,(x) =
51:9:3
X
+
+
r(-v)x'
x InrV = I 2
/2
K,(x) = r(wl)l
51:9:4
0-2 1
K,-,(x) f,.(x)]
v>0
J0
r 'K,,, (t)dt = z K,(x)
51:10:9 J
as well as for the following expressions involving exponential functions:
r J0
t` exp(ct) K,U)dt =
J
x'_' exp(±x ),(x) ± K,.,(x)] IK 2v + I
t-` cxp(t) K,(t)dt =
21'(v + 1)
v>-
2v + 1
x` exp(x IK,(x) * K,_,(x)] 2v- 1
1
2
v>-
2
The important King's integral is the v = 0 instance of 51:10:10 with upper signs selected.
For vi > I the definite integral fK,(t)dt between limits of r = 0 and t = x diverges. The convergent cases are the µ = 0 instances of the general formula 51:10:12
(µ+y+l\r(- v+I
f",
t" K jolt = 2"-' [ I\
/1
2
1\
namely
51:10:13 Jo
i K,(t)dt =
IT
sec(
\
-
f
/
IvI < I
51:11 COMPLEX ARGUMENT Definition 51:3:5 may be combined with equation 50:11:1 to produce an expression for the Basset function of complex argument when v is noninteger.
THE BASSET FUNCTION K,(x)
507
51:13
For purely imaginary argument, the Basset function is related to the functions of Chapters 53 and 54 by 51:11:1
K,(iy) =
I
2
+
2 Y,(Y) + sin (7-) 1.(Y)1 cosILT
2 Lsin(-2) Y,(Y) - cos( 2)J (Y)
When the real and imaginary components of the argument' of a Basset function are equal in magnitude, we ha
K,(x ± ix) = i'` [ker,(x y 2) t i kei,(x V G)]
51:11:2
where ker and kei are Kelvin functions [Chapter 55] and i'" is to be interpreted as cos(vw/2) ± i sin(vir/2). The Basset function is complex for negative real argument. The real and imaginary parts are given by 51:11:3
is
K,(-x) = cos(vir)K,(x) - Z [I,(x) + 1 _,(x)]
v * 0, ± 1, ± 2, .. .
51:12 GENERALIZATIONS The Basset function may be generalized to Whittaker's function [Section 48:131 51:12:1
K. \2/ =
x Wa,(x)
and thence to the Tricomi function [see Chapter 48].
51:13 COGNATE FUNCTIONS The Basset function is related to all the functions discussed in Chapters 49-57.
CHAPTER
52 THE BESSEL COEFFICIENTS Jo(x) AND JI(x)
Those Bessel functions [Chapter 53) in which the order is a nonncgative integer are known as "Bessel coefficients" and are the subject of this chapter. Emphasis is placed on the most important of the functions: those with n
= 0 and 1.
52:1 NOTATION Bessel coefficients are also known as Bessel functions of the first kind of nonnegative integer order.
52:2 BEHAVIOR All Bessel coefficients are oscillatory functions whose oscillations become increasingly damped as their arguments approach large values of either sign. As Figure 52-1 shows, all Bessel coefficients except J0(x) are zero at x = 0.
Moreover, as n increases, J,(x) retains near-zero values over an increasing range in the vicinity of x = 0; for example, J9(x) does not exceed 0.01 in magnitude in the range -5.4 5 x 5 5.4. Some regularities are apparent in Figure 52-1. Notice that each local maximum or minimum of J0 corresponds to a zero of J, but this rule does not extend to larger orders. On the other hand, for n z 1, each local maximum and J,,,, curves. Moreover, at each zero of J. or minimum of J. corresponds to a point of intersection of the 1,_, and 1,., have equal magnitudes but opposite signs. For x s 0, each Bessel coefficient encounters its first (and largest) maximum at an argument that, for small n, is close to nir/2. This rule fails for larger n, however, and for very large order, the Bessel coefficient attains its first maximum close to x = n. Subsequent, and ever-smaller, maxima are encountered with a spacing of approximately 2a. Local minima occur approximately midway between consecutive maxima. For x > 0 the first minimum is the largest and subsequent minima decrease progressively in size. A zero is encountered between each minimum and its adjacent maxima so that the spacing of the zeros is approximately IT.
52:3 DEFINITIONS Bessel coefficients are defined by a number of generating functions. Those of even order arise from
52:3:1
cos(x cos(t)) = Jo(x) - 2J,(x) cos(2t) + 2J,(x) cos(4t) -
= Jo(x) + 2
(- 1), J,t(x) cos(2jt) so
510
THE BESSEL COEFFICIENTS Jo(x) AND 11(x)
52:3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1.0 .
.
:Jo(x
........... ......................................................................... .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
FIG 52-1 : .......... Jt(x}:;J ....:....:....:....:....:....:...
:....:.. /.....:. \..:. !.:.. :...
:J.(x):
J ():
....
..
a B
Jz (x): - .Tw (z>' 0.4
and if the signs in this series are made uniformly positive, the terms are generated by cos(x sin(s)). Bessel coefficients of odd order arise from 52:3:2
sin[x cos(t)] = 2J,(x) cos(t) - 213(x) cos(3t) + 215(x) cos(5t) -
-
= 2 E (-I)'J,,,,(x) cos(t+2jt)
i-o or from the corresponding series for sin[x sin(r)], which differs from 52:3:2 in that sines replace cosines, and all terms are positive. The\ generating /function 52:3:3
expl xt=2r x I = 10(x) + I r -
J1(x) +
r= + r.
lz(x) +
t' - Ily
33(x) +
NJ; (z)
gives rise to Bcsscl coefficients of both parities.
The Bessel coefficient Jo may be represented as a definite integral in a number of ways, including 52:3:4
* cos[x cos(t))dt = 2
Jo(x) = I0
sin[x cosh(t)]dt 1o
THE BESSEL COEFFICIENTS Jo(x) AND J,(x)
S11
52:5
The following are among the integral definitions of the general Bessel coefficient 52:3:5
J"(x)
52:3:6
J"(x) = J
cos[x sin(t) - nt]dt
n Jo
2
x lin(t)] f(nr)dt j
'R
n = 0, 1. 2, .. .
n=0,2,4,...
f=cos f=sin
n1,3,5,...
and others may be obtained by specializing the formulas of Section 53:3.
The Bessel coefficients JO and 1, arise in a large number of physical contexts because they satisfy certain very simple second-order differential equations, as follows:
df
d2f
x- + - + xf = 0 dx' dx
52:3:7
52:3:8
xdzf+df+f=0 dx x dzf dx2
df - dx + xf = 0
f= c,4J,(x) + c2xY,(x)
'
'
-+f=0
x-dz
52:3:10
f=c,Jo(2V)+c2Yo(2Vx)
dx
52:3:9
f = c,J0(x) + c2Yo(x)
f=c, VxJo(2Vx)+c2VxYo(2Vx)
dX2
Here c, and c2 are arbitrary constants and Yo and Y, are Neumann functions [Chapter 541 of zero and unity orders. The Bessel coefficients JO and J, are generated by applying the operations of semiintegration or semidifferentiation, with lower limit zero, upon certain functions involving sinusoids. Examples include d-
sec(x)
52:3:11
Jo(x) =
() co s(2x)
dx'ir Jo(V x)
'
()
csc(x) d-'/' sin(2r) dx-"2
d"1
2 52:3:12
=
(sin(V x))
V. dx'2 d-"
52:3:13
1,(1f) _
l 1
52:3:14
Jo\
1
'rrX
dx uz
(sin(
- JxdxI d 12 (sin(l/\)/
Vrx
1\
x
52:4 SPECIAL CASES There are none.
52:5 INTRARELATIONSHIPS Bessel coefficients are even or odd
52:5:1
J"(-x)
(-1)' J"(x)
n = 0, 1, 2, .. .
according to the parity of n. A similar reflection formula 52:5:2
J -.(x) _ (-1)" J"(x)
exists for the order of Bessel functions of integer order.
n = 0, 1 , 2, .. .
THE BESSEL COEFFICIENTS Jo(x) AND J,(x)
52:5
512
Many formulas exist for infinite sums of Bessel coefficients. Some of these, such as 1
52:5:3
J,(x) - J3(x) + J5(x) - Jr(x) +
52:5:4
2 J0(x) - J,(x) + J,(x) - J6(x) +
= 2 sin(s)
1
1
= 2 cos(x)
and 1 1 1 J0(x) + J2(x) + l1(x) + J6(.r) + = J,(x) = 2 2 2 may be derived as special cases of formulas 52:3:1-52:3:3. Others, for example:
52:5:5
'+1) J2,- A)) = 2x
52:5:6 /=o
x 52:5:7
21,(x)
=4
4J2(x) + 161,(x) + 36J6(x) +
52:5:8
j2J.,(x) = 2
and, for n = 1, 2, 3. ...: 52:5:9
+ 1)!(n + 4)
(n- I) !nJ (x) + On + 2) J,,.,(x) + (n
2!
i-o
j!
x _-
1,.2,(x)
2
follow from the properties of Neumann series (see Section 53:14]. Yet others, such as 52:5:10
2 AX) + J-1(X) + 121(x) +
= 2 Jo(x) + i J,(x) = 2
and
52:5:11
(- I )` J,{x) J;.i(x)
J0(x) J,(x) - Jdx) J2(x) + J2(x) J3(x) - ... _
J,(2x)
;=o
may be derived from Neumann's addition formula
n=0,±1,±2,...
52:5:12
Included among still other types of expansions are 52:5:13
1,(x) + J2(2x) + J3(3x) +
x
_
J(ix) _ ,_,
'
2(1 - x)
and
52:5:14
121x)+1;(2x)+1;(3x)+...=iJ;(jx)=2[
1
- 1J
that are examples of Kapteyn series [see Erd6lyi, Magnus, Oberhcttinger and Tricomi, Higher Transcendental Functions, Volume 2, pages 66-68).
THE BESSEL COEFFICIENTS J6(x) AND J,(x)
513
52:6
By applying the recursion formula 2n
- J,(x) - J,-,(X)
52:5:15
x
for a sufficient number of times, any Bessel coefficient may be expressed in terms of J0 and J1. Examples are 52:5:16
2
- l,(x) - J0(x) x
J2(x)
/8 -
JAx)
52:5:17
48
Mx) _
52:5:18
J,(x) -
Zx
16(x)
1
+
x
- 768 + x
_ JO()
I
181
x
3
-
( x192
I) J,(x) - l
=si3840
(, x
X-)
- 72
JS(x) =
52:5:20
24
8
x -XI
I394 52:5:19
4
1
1 J Ji(x) - - Jdx) x
xz
-
J,(x)
`!1920 x
121 -J Jo(x)
x
-2 144 + 11 Jo(x) x
The ratio R. of two consecutive Bessel coefficients of common argument obeys the simple recursion formula J,+,(x)
52:5:21
2n
1
n= 1,2,3....
X -R--
J"(x)
which makes these ratios computationally useful [see Section 52:8]. Of course, all formulas in Section 53:5 can be applied to Bessel coefficients by setting v = n, a nonnegative integer. For example, the argument-multiplication formula yields
I /x - bzxlV b" F _ (
52:5:22
'-o J.
(x)
!
2
52:6 EXPANSIONS Any Bessel coefficient may be expanded as the power series 52:6:1
(.r) _
J(xll
I
2J
[n!
xz/4
(x'/4)'
(xz/4)z
3!(n + 3)! +
(n + 1)! + 2!(n + 2)!
(x^ ] - \2/
of which the first two instances are
.r- x6 + lo(x)=I-+ 4 64 2304 X2
52:6:2
B
x
147456
-...= (-xz/4);
and
x
52:6:3
JI(x)=-
2
x +
X5
384
16
-
xt
+...= x
(-x/4);
2
j!(j + 1)!
18432
Expansions as infinite products take the form 52:6:4
JO()
xz) ... _
I / \1
/
52:6:5
J,(x)=zll-
z
fIl
z Jo,
z
J'. :z
\I
z Joj
1 _ _L R=,
-Z J,a
Jos
z
=
r-,
where j,x denotes the k'° zero of the Bessel coefficient J,(x) [see Section 52:7].
JI.*
(-xz/4)' -o j '(n + J)!
514
THE BESSEL COEFFICIENTS Jo(x) AND 11(x)
52:7
By making use of formula 52:5:22, it is possible to expand a Bessel coefficient J,(x) in terms of the Bessel coefficients J J,,,,, J,-,, ... at any fixed nonzero argument. For example, choosing 2 for this fixed argument gives
(z\
J,(x)
52:6:6
` 1/ I
2J
-x
4 J,",(2)
1
-old
...
n = 0, 1, 2,
Such an expansion converges extremely rapidly. The Bessel coefficient ratio defined in equation 52:5:21 may be expanded in partial fractions 2x
,+ R, = , l+.1-x- j:.2
52:6:7
2x
2x
,+
1
ja-x
- x-
.-Ija-x'
or as the infinite continued fraction
R.
52:6:8
x'/4
x/2
_
x2/4
x2/4
l+n- 2+n- 3+n- 4+n-
J,(x)
The effect of curtailing this continued fraction at any point may be represented as another Bessel coefficient ratio. For example: R., _
52:6:9
x12
x'/4
xz/4
xR,.3
I +n- 2+n- 3+n-
2
For large arguments, the asymptotic expansions cos 52:6:10
J((x) -
nx
[
1
9
225
8x
128x"
3072x3
I - - - - +
1
+
+
si
1
8x
arx 1
9
-
I +
-
128x'
225 3072x3
X -' x 52:6:11
J()1
sin(x)
I +-+-3
15
315
8x
128x=
3072x3
cos(x)
1 -+-+ 3
15
315
8x
128x
3072x3
hold. See equations 53:6:6-53:6:8 for the general formulation.
52:7 PARTICULAR VALUES
1,(x)
1i(x), J,(x), J ,(X),
...
x= -x
x= 0
x= s
0 0
I
0
0 0
j ,, Those positive argument values that cause 1,(x) to acquire the value zero are denoted is called the km zero of the na Bessel coefficient. The value of the derivative d1,(x)/dx at the zero is a quantity whose value is needed in certain practical problems [see Section 52:141. Such a derivative is known as the associated value of the zero, and the usual notation is 1' .0.,.). although two alternative notations follow from the identity 52:7:1
J,(J,:A) = J,-1(jn:k) _ - 4*1(1.:A)
THE BESSEL COEFFICIENTS J0(x) AND 1,(x)
515
52:7
Table 52.7.1
i.
Vi,.)
ill
Jdjo,)
5.5201
-0.5191 +0.3403
3.8317 7.0156
-0.4028 +0.3001
8.6537
-0.2715
10.1735
-0.2497
11.7915
+0.2325
13.3237
14.9309
-0.2065 +0.1877
16.4706
18.0711
21.2116
-0.1733
22.7601
+0.2184 -0.1965 +0.1801 -0.1672
2.4048
19.6159
k-I k=2
k3 k=4 k5 k6 k=7
JiL
Jdja)
0.0000 3.8317 7.0156 10.1735 13.3237
+1.0000 -0.4028 +0.3001 -0.2497 +0.2184
16.4706 19.6159
jik
Ji(ju)
1.8412
+0.5819
5.3314
-0.3461
8.5363 11.7060 14.8636
+0.2733 -0.2333 +0.2070
-0.1965
18.0155
-0.1880
+0.1801
21.1644
+0.1735
which is a consequence of 52:10:1. Approximate values of the first seven zeros, and their associated values, for may be read from the n = 0 and 1. constitute the left-hand side of Table 52.7.1. Other approximate values of map in Section 53:7 (Figure 53-2). More precise values of j,.k and J,( j,:k) are available from the algorithm that follows.
The algorithm operates in two modes according to the relative values of n and k. If k >: n - 3, the algorithm and then uses Newton's method [Section 17:7] to generate the improved approximation
adopts x, = a(2k + n - 1)/2 as an approximate value of J,(x1)
52:7:2
x,
x,J,(x,)
_
x,
r' n - x,R. and this procedure is repeated until two successive approximants differ by less than l0-'. The method used to JR(xi) -
nJ,(xi) - x,J,,.1(x,)
compute R,,, the Bessel coefficient ratio, is described in Section 52:8. When k < n - 3, the algorithm first calculates n + 2n'"' [see 53:7] as a crude approximation to j,,, and refines this approximation by a single application of 52:7:2. A crude approximation to j.4, is then calculated by adding 4 to the j,,., approximant and this, in turn, is refined by a single application of Newton's method. The procedure of adding 4 and refining is repeated until a crude value of j,;k is reached. This is then improved by repeated applications of formula 52:7:2. The second half of the algorithm, which calculates -J,.,(j,;k), is essentially the same as the algorithm in Section 52:8.
Input n »> »»> Set J = 0
Input k »> »»> / n` Setx=a k+ n-1 [or =alk-I+n J
Storage needed: n, k, x, J, 1, R. L and D
Z
If .r=0goto(8)
11
If k > n - 4 go to (4)
Set x = n - 4 + 2n'13 [orn-4-0.8n"'[ (I) Replace x by x + 4 Set I = Int n2 + 2(x + 6)2 Replace J by J + 1 Go to (6)
121 If J * I go to (3) Replace.r by.r - (n114)
(3) IfJ 10-7 go to (5) (8) Output x
,,,
or }
Storage needed: n, J, D, R, x and I
SetJ=D=R= I
»»
Set 1 = 2 + 2 lnt [(kI + 7)-/2) + [n2/4) (1) Replace R by x/(2! - Rx)
If!>ngoto(I)
Replace J by JR
(2) Replace D by DR + 2 frac(1/2) Replace I by I - 1 If ! * 0 go to (2) Replace J by J/(2D - 1) Output J
Test values: Jo(8) = 0.171650807
J - J (x) < K
0 domain, whereas if the order takes large values of either sign there is a range of the argument. approximately 0 < x < v, during which J,(x) increases (if
v > 0) or decreases (if v < 0) in magnitude prior to breaking into oscillations. Once they are established, the oscillations steadily attenuate, as x increases, remaining centered about zero and eventually conforming to the asymptotic expression $21
53:2
THE BESSEL FUNCTION ],(x)
522
THE BESSEL FUNCTION J,(x)
523
J,(x) --
53:2:1
cosx -
2
\
arx
m 2
-4
53:3
x-+ m
If the argument x takes some constant value greater than zero, Figure 53-1 also illustrates how J,(x) varies with v. As v takes increasingly positive values, the oscillations of 1,(x) sooner or later cease and the Bessel function becomes a monotonically decreasing positive function that eventually obeys the asymptotic expression
rexl
I
1,(x) -
53:2:2
V-. =
x = constant > 0
2v/II
2av
On the other hand, as v acquires ever-more-negative values, 1,(x) remains oscillatory, and the oscillations become increasingly amplified and eventually satisfy 53:2:3
V/2 /-ex'"
J,(x)
avl\2v
v-- -x
sin(vir)
x = constant > 0
An advantage of Clifford's notation [equation 53:1:1 ] is that whereas J(x) is generally defined as a real function
only for x a 0, C,(x) is real for all values of its argument. In fact, for negative x. C,(x) becomes related to the hyperbolic Bessel function of Chapter 50:
C,(x) _ (-x)-"121.(2\)
53:2:4
x !5 0
53:3 DEFINITIONS Thus:
The function (x/2)" is a generating function for the Bessel functions J,(x), J,,,(x), (x/2)" r(1 + v)
53:3:1
k_o
k'
Bessel functions are defined by a number of definite integrals, including
(x/2),
1,(x)
53:3:2
J,(x) =
53:3:3
J
V; r(v+I) u 2(x/2)"
I
'Vrrr(v+0 0
(1
A
cos(x cos(t)) sin'"(t)dt
v>--
- t=)'-''' cos(xt)dt
2
and 2(2/x)" 53:3:4
r(I - V)
J.(x) -
f
sin(xt)
(t. - I)
I
di
-2Gv -1
0
and
t'-'7"(t)dr =
53:10:4 0
- x' '7._,(x)
2T(v)
Important definite integrals include J
53:10:5
T
v > -1
J,Ir)dt = I
0
r 53:10:6
exp(-bt) J.(t)dt =
(VI +b'-b)"
cos(v arcsin(b))/ 53:10:7
1
1-
O r b< 1
-sin(rv/2)
cos(bt)1.(t)dt =
b1(b+ b'-Ib>l Isin(v arcsin(b))/
53:10:8
v>-I
I + b'
J0
sin(bt) J"(t)dt =
1
b'
0 Bo:o = ; , Co,o = 0 and C,;o = J /(2h); for m = 1 . 2, 3, - - - n, A,,;o = A_ 1:0 - (1 /m), B",;o =
and C.,.:,, = hCo/m: and fork = 1. 2. 3..... j. A":x = A":k-, - (Ilk) - [11(n + k)], B,,
= -h2B":.-1/k(n + k) and C.., = k(n - k)C",,_,/h2. For a sufficiently large J. the terms 2t 2t,.,, 2r,.2, ... will alternate in sign and progressively diminish in magnitude. It then follows from the properties of convergent alternating series [Section 0:6] that the error in approximating irY,(x) by S, cannot exceed It, I in magnitude. The algorithm therefore
tests, for j = 1, 2, 3, ..., until jyI/js,I -< 5 x l0-e. Additionally, three other tests must be passed before J can be identified with certitude. First, j must be at least equal ton - 1, to ensure that all terms in the polynomial component of 54:8:1 have been included. Second, alternation of sign will not occur consistently until d.(j) + e(n + j) exceeds
2 ln(h) + 2: this is guaranteed if the sign of g is negative for even j. Third, it is necessary to confirm that the t terms are, indeed, alternating and convergent. Relationship 54:5:1 is invoked when the order is a negative integer. The asymptotic expansion 54:6:6 is employed whenever x exceeds both 8 and 5.6 + 0.71v1. or for all x when v is an odd multiple of ± 1. Provided (j + !t)2 > v= the terms in the series for P(v;x) and Q(v;x) alternate after the ja term, t;. Moreover, t, progressively diminishes in magnitude provided j < J = x + { + x= + -x+ v2. Incorporating these two provisos, the algorithm appends terms to the series for P(v:x) 2/lrx and Q(v;x) 2/rrx until
the j° term satisfies the inequality ft,l < 5 x 10-s {P(v;x) sin[x - vrr/2 - it/4] + Q(v;x) cos[x - yr/2 - w/ 4]}
2/ars.
THE NEUMANN FUNCTION Y,(x)
54:8
Storage needed: v, f, h, A, B, C, t, p, n, q,
s,jandJ
Set f = -109°
Set h = x/2
Iffracly+21
0 go to (16)
Ifh+0goto(1) Ifv>0goto(19) Replace f by f cos [180 Int(2 - v l
Go to (19) \ (1) If h 4 go to (2) If h > 0.35(8 + Jul) go to (17) (2) If frac(v) = 0 go to (11) Set B = cos(I80v)
////
Input restrictions: Order v may take any value but argument x may not be negative.
SetA=C=f=t=l If v > 0 go to (3) Set A = C = B Set B = 1 Replace v by -v
Use degree mode or change 180 to it and 90
to a/2.
(3) Set p = v
Set n=Int (2 + V) (4) Set q = h-° (5) Replace q by q/p Replace p by p + I
Ifp
bei,(x)
m{ Jl
l \
+
7/
fr
= Im{ I
l 111
/ \ /
+ i sinl 2 IJ I,1
\(!!)1
+
+ i sinII,I x + V2 \ I/
oos(2
\\\72
\ I55:3:4
I}
in terms of the Bessel function [Chapter 531 or hyperbolic Bessel function (Chapter 50] of argument whose real and imaginary parts are of equal magnitude. Similar formulas involving the Basset function [Chapter 51] define
THE KELVIN FUNCTIONS
548
55:4
the ker, and kei, functions
cos(2) - i sin(- )J K,1 s + 'X
55:3:5
ker,(x) = Re s I
55:3:6
kei,(x) = Imi I cos( Z) - i einl
V2 V1
2
IJ K,(
+
I}
V2= V1 )
}
The ker and kei Kelvin functions of zero order may be defined as the following definite integrals: 55:3:7
xZ r
a
Cil r I expl 8 J/o\ /1
ker(x)
kei(x) =
4
1 - 1J expl 4 ' \\in/Chapters
Jaw I Stl t
8
\
ln(1 + t°)1,(xt)dt
1
' + 4 dt = Z J f arctan(12)
Idt = -
J
38\\\
in terms of functions discussed
x 1 + t° dt = 4 J
:310(x[)
)dt o
/
r 55:3:8
-xrt
1,(xt)dt
and 52.
55:4 SPECIAL CASES Kelvin functions of orders equal to odd multiples of ± 1 /2 may be expressed as elementary transcendental functions. For example:
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
FIG 55-2
:
1.2
_8
THE KELVIN FUNCTIONS
55:5
55:4:1
ker112(x)
55:4:2
kei112(x)
\
exp
X
sin(72
\ J 81l
cos(
exp(
it 1
berln(x) =
55:4:3
exp (
2x
`V2
x
x- a +
cos
bei1/2(x) =
7(--) ( t I
exp
ax
x
"/
V2
x
sin
+
a 8
V2
)
1
+
- kei1,2(x)
+
- ker,,2(x)
8
taxx 55:4:4
546
1
The Kelvin functions ber, and bei, become somewhat simpler when v is a multiple of 3. Only for these orders is ber,(x) or bei,(x) expressible as a simple power series. When the order is an even multiple of 3 (i.e., a multiple + - + - + - ; when the order is an odd multiple of ;), the power series contains terms of alternating signs + + - - + + - - . . Equations 55:6:1-55:6:4 provide details. of 3, signs occur in the sequence
55:5 INTRARELATIONSHIPS The order-reflection formulas 2
55:5:1
ber_,(x) = cos(-) ber,(x) + sin(va) bei,(x) + - sin(vt) ker,(x)
55:5:2
bei_,r(x) = -sin(va) ber,(x) + cos(va) bei,(x) + - sin(va) kei,(x)
it
2
IT
55:5:3
ker_,(x) = cos(va) ker,(x) - sin(va) kei,(x)
55:5:4
kei_,(x) = sin(va) ker,(x) + cos(va) kei,(x)
all reduce to 55:5:5
n = 0, ± 1. ±2,
f = ber, bei, ker or kei
f_"(x) _ (-1)" f;,(x)
when the order is an integer. Negative arguments generally render the Kelvin functions complex. Exceptions are ber, and bei, for integer order, to which the argument-reflection formula
f,(-x)_(-1)"f(x)
55:5:6
f=berorbei
n=0,±1,±2,.
applies.
The recurrence relationships may be written -v V [ 55:5: 7
f
55 : 5 : 8
fe i ,,1(x)
f
fe i
f
I
fer
= b er
fe i
= be
k er
-0/2 _
x
(
fe i ,(x)
+ fer,(x)1 - fe i ,_1(x) I
i
or ke i
THE KELVIN FUNCTIONS
547
55:6
55:6 EXPANSIONS All four Kelvin functions are expansible in terms of two series for which we shall use the abbreviations 55:6:1
Vjr(x/4)'
(-x`/16)'
_(\2/l
Fe,(x)
1
j-, (1+2j)!f(2+2j+v)
2
1
v
((
v
(-x'/256)'
(r
v
3
v
(1) 2,(1);(2+2)'11+2)'
2 / (x/4)22'"
(-x4/16)'
Ge,(x)_(x)2*
55:6:2
(-x'/256)'
f\((2+2f(1+2) v ;_o
_o (2j)!f(I + 2j + v)
f`'+2)f(2+2) The second formulations above show that Fe, and Ge, are instances of K = 0, L = 4 hypergeometric functions [see Section 18:141. When v is a negative integer, formulas 55:6:1 and 55:6:2 are inconvenient and the identities Fe,-2,(x) = (-1)' Ge2,_1(x), Ge_,,,(x) _ Fe_ ,(x) = (-1)' may be used instead, where n = 1, 2, 3, .... The expansions
/3wr +L-
cost -
/
/ `
(3-Fe,(x) - sinm 1 \4 2 l x2 I' = cos___) \4 2)' _o j!r(1 +j+v) \4 3wn jir (X 4 2) 3 1 Ge,(x) sC(I+ j+v) 14) = sinI34) Fe,(x) + cost -o J 2
55:6:3
ber.(x) = I -
55:6:4
bei,(x) =
IV
j'.
\\\
are valid for all v, but a restriction to noninteger v is required to validate the expansions 55:6:5
ker,(x) =
/3va\
n csc(vtr) r
/va 4 I Fe,(x) -sin( 4) 4) Ge_,(x) - cost/vn\
/3vir
Lcosl 4 I Fe-.(x) + sin(
2
Ge,(x)
v#0. +1,±2.... 55:6:6
kei,(x) =
ac
2(m) I
sin - Fe_.(x) + cos - Ge_,(x) + sin - Fe,(x) -Ge,(.r) cos__) \34 4) 4 134 / J v*0.±1.--2....
For nonnegative integer order, ker,(x) and kei,(x) are given by the expansions
/2
55:6:7
ker,(x) = Inl
I ber,(x) + 4
X
+- (2)' = 2
55:6:8
1)!
j!(n + j)! n
I
cos
kei(x)=In )bei.(.r)-4ber(x)-2IXIn1 Irrr +2 \2 )^
(-+-j7r\I\fx2 (3na 4
(n
J!
4,0+j)+y(I+n+ J') -o
l(x'
J
k4(1 + j) + y(I + It + j)
=o
2
I /2\n 2 I x (nI!- cost/3nn4 + 2) \ 4 )'
j! (n+ j)!
3n7r
ll
2
n=0.1,2,...
4
sin(
n
3nar
4
x''
4)\4) J
a
x'
+J2)(-) n=0.1.2,--
THE KELVIN FUNCTIONS
55:6
548
where $ is the digamma function of Chapter 44. Asymptotic expansions of the Kelvin functions are provided by the formulas
exp(x/ V [) 55:6:9
(2 - v),Q
ber,(x)
tax bei,(x) -
55:6:10
(i - v), ( + v),
exp(x / V 1) 2Trx
55:6:11
55:6:12
c
ker r)-
exp (l
2x
v),+ v)j
-x V2
j,(- 2x )'
i-v
x-. x
\\
X_%
x +'7 -1n -
8J
4
2
va
+
2
x wn j n ) cos+-+-+2 4 8
x +x
V2
(1-v),(!+v),
x)
IT
( l
rx
sin
j!(2.)1
,moo
- exp(
kei,(x) - -
+ v)' cos
j!(2x)'
,=o
sit)
/x
m + 4 + -1 a
+
X _X
with suitably chosen upper limits to the summation index j. Although the series in 55:6:9 does, indeed, asymptotically represent ber,(x) for large x, it more closely represents ber,(x) + [sin(2va) ker,(x) + cos(2v,r) kei,(x)]/ I. Similarly, a better approximation to the series in 55:6:10 is bei,(x) - [cos(2var) ker,(x) - sin(2vir) kei,(x)]/a. When v = 0, the various expansions simplify to / 16Y
a
55:6:13
bei (x) =
55:6:14
55:6:15
55:6:16
55:6:17
ker(x) =
kei(x) =
ber
x2
4
[In() x
2304
x1o
4((2j + 1)!]2
14745600
1
-, `
exp(x/N/2) "
[(2j)!]2 cos (_L_ io (j!)2(32x)'sin
G
I(2J)!)2
2
,,2
+ 3Jn 4
I 3
I3
2
- !?r4 --8
cos(-x
o U!)'(32x)' sin
-+
1+
4;=u `
4
-x V 2x exp(72)
x'(-z'/16)' + ...
+
+
beix) - 7 ber(x) + sI
bei(.r)-
kc.(x) -
s°
[ln() - yJ ber(x) +'r bei(x) + x 4
2wx
55:6:18
_
[(2j)!]2
147456
64
\
- ?8 1
+
1 (-x`/16)'
+
2j1 I
1(2j)! I'
\ (-x'/16)'
2j + I J [(2j + 1)!]2 J = lnt(2x)
J = Int(2x)
As well, ber(x) and bei(x) are expansible in terms of the functions of Chapters 49 and 52: 55:6:19
ber(x) = 10(
)
o(
(_I))17J(
212(7) J,(--!z)
7) J2,(x)
+
x-. x x-+ x
THE KELVIN FUNCTIONS
549
55:6:20
bei(x)=21,\ x )J,I x 1-2131 x
55:7
x)+2Is( x)Js(x)-
111
)
)J2,,I(
The sum of squares of the her and bei functions is the simple series 55:6:21
ber=() + bei2(x) = I
z
xs
32
24576
(x ,
16) r
(j!)'I (j + :)
which is rapidly convergent for small arguments. Asymptotic expansions exist for ber'(x) + bei2(x), for kerz(x) + kei2(x) and for a number of similar functions; see Abramowitz and Stegun [Section 9. 101 for these.
55:7 PARTICULAR VALUES Kelvin functions display diverse behaviors as x -* 0, and this leads to very varied values for ber,(0), bei,(0), ker,(0)
and kei,(0) as functions of v. For the most part these particular values are +x, 0 or -, but the following finite nonzero values are also realized: 55:7:1
ber(0) = I
55:7:2
-ker,,(0) = kei(0)
55:7:3
2
a 4
Figure 55-3 shows what value represents the limit of the four Kelvin functions of order v as x -. 0. The color coding used in this diagram is that green represents zero, red represents +x, blue represents -x and black represents one of the values given in equations 55:7:1-55:7:3. Notice the very elaborate behavior of ber,(0) and bei,(0) for negative P. In this region these functions have a periodicity of 8 in their orders; that is: 55:7:4
f,(0) = G-00) = f,_16(0) =
f = her or bei
v < 0
As x approaches infinity, ber,(x) and bei,(x) oscillate with ever-increasing amplitude, but ker,(x) and kci,(x) converge towards zero:
ker,(x) = kei,(x) = 0
55:7:5
'
All four Kelvin functions have an infinite number of zeros. For a sufficiently large x, these are given by 3
55:7:6
ber,(r) = 0
r = V 2rr(k - 2 - 8
55:7:7
bei,(r) - 0
r - V to k -
55:7:8
ker,(r) = 0
r "- V
55:7:9
kei,(r) = 0
r = V 2x k - 2 -
v
+
k = large positive integer
18)
2
5
k - 2 - g) I
k - large positive integer
k =large positive integer k = large positive integer
-8
The first few zeros for v = 0 are included in Table 55.7.1 together with argument and function values of early extrema, as well as the function values at the x = 0 origin. The final line in Table 55.7.1 provides an approximation
formula for calculatintthe k'" zero of each Kelvin function; it uses the abbreviations a = N/2(8k - 3)sr. b = N/2-(8k + 1)a, c = V2(8k - 5)ir and d = V (8k Since the approximations are excellent, even for k as small as 4. the table permits all the zeros of the zero-order Kelvin functions to be estimated.
1
,4
_ci
obi
.y=5
.. y.4
550
THE KELVIN FUNCTIONS
551
55:8
Table 55.7.1 x
bei(x)
x
bei(x)
x
0
1.000000
0
0
0
2.848918
0
3.772674
6.038711
-8.864036
7.238829 10.51364
153.7818
5.026224 8.280989 9.455406
0 0 -2968.681 0
11.67396 14.96845 16.11356
a 8
+-+a' I
0
a
2.346147
-36.16540 0
12.74215
670.1602
13.89349 17.19343
-13305.52
0
+b- +b'8 b
I
0
11.63219 15.00269
C
8
x
0
3.914668 4.931811
'kei(x)
0
1.718543 2.665845 6.127279 7.172120 10.56294
0
ker(x)
-0.07102369
0
8.344225 9.404051 12.78256
0.001956681
0 -6.705969 x 10''
.7.i +--I
c
0.0003574233
0
13.85827
1.280427 x 10-'
17.22314
0
-+ ---
0
0 0
-0.7853982 0 0.01121607 0
d
I
V '32
8
d
d2
55:8 NUMERICAL VALUES The algorithm below calculates values of the zero order Kelvin function for all arguments x ? 0. Its precision is 24 bits (relative error less than 6 X 10-'). but some loss of significance may be encountered on computing devices that carry insufficient digits. The algorithm is based on expansions 55:6:13-55:6:18.
Input x >>
SetB=K=b=kj=r=0
-n)
Storage needed: B, K, b, k, j, r, x, s and r
1
Ifx 0 1
THE KELVIN FUNCTIONS
55:9
552
Replace K by K + r cos(s) Replace k by k + t sin(s) 70
x
Ifj2
axe
2 - 16
ker(x) = I n
55:9:7
55:9:10
(x2)
2
v * -1, -2, -3, ...
r(2 + v) 2) \x\` - r(v - 1) 2n(3vsr/4) (x)`-2 I
55:9:5
v + -1, -2, -3, ...
2)
cos(3va/4) (x
ker,(x) =
55:9:4
2
- y + 2]
v s ±2
v < -2 >2
55:9:11
55:10
THE KELVIN FUNCTIONS
553
kei,(z) =
\-
f(-v) sin(va/4) x
-f(v) sin(3vir/4) (X2)' 2
kei x)
55:9:12
+I =-_
x
kei(x) 4 +
55:9:13
[1,x
+
4IT
v=0
f(-v - 1) cos(vir/4) x
2) +
2
v * 0, ±1
v-_1
1
Z
LlnI-I -y+-I
4
r(- Y) sin(va/4) (xl
kei,(x) -
55:9:14
-2 < v < 2
(2)
2
(2)r2
v < -2
2
and these may be supplemented by using equation 55:5:5 for cases of negative integer order. Generally, equations 55:9:1-55:9:14 give the first two terms in expansions of the Kelvin functions in ascending powers of the argument x. For certain special values of v, however, some of these terms may vanish. The leading terms in equations 55:9:1. 55:9:3 and 55:9:5, for example, become zero when the order adopts one of the values 3, 2, IF, i, 6, .... For large arguments, the approximations large x
bei,(x) -
55:9:16
/x / x+va 2 - ag explV-2-)s nl -L
large x
exp
tax I
ker"(x) =
55:9:18
kei,(x)
+
Vt
%J
i
tax
55:9:17
coal
x
va
2 - a8
x
1
ber,(x) =
55:9:15
Zx
exp
(x
expl VVVV2x
\V-2
)
cos(x
+
sinl
+
`,r2
va 2
a
+
8
large x
)
+
large x
8
2
hold. From these, one may derive the order-independent relationships l=
[berr.(x) + bei,(x))[ker;(x) + kei(x)J =
55:9:19
large x
4x and 55:9:20
ber,(x) ker"(x) + bci,(x) kci,(x) = ber,(x) kei,(x) - bei"(x) ker"(x) =
x
7
large x
8
55:10 OPERATIONS OF THE CALCULUS The differentiation formulas
d 55:10:2
[fer.,(x) + fei,.,() - fer,,(x)
fer,(x)
55:10:1
l
fei"_,(x)] I
[fer = her or ker
1
- fei"(x) = dx Ve
fer,.,(x) - fei"_,(x) + fer,_,(x)J
fei = bei or kei
may be combined with recursions 55:5:7 and 55:5:8 to produce a number of alternative formulations. Among indefinite integrals are
-x' 55:10:3
ti-" fer,(t)dt = 0
;
[fcr,_i(x) - fei,_,(x))
THE KELVIN FUNCTIONS
55:11
554
while the following definite integrals establish links with the functions of Chapters 32 and 38: 55:10:4
a exp(-t/x) ber(2 V r)dt = x cos(x)
J0
exp(-t/x) bci(2V )dt = x sin(x)
55:10:5
(' 55:10:6
55:10:7
10
/
exp(-i/x) ker(2\ t)dt =
2z (l cos(x) Ci(x) + sin(x) r Si(x) - 2]r j {
L
111
exp(-t/x) kei(2Vt)dr = Z { sin(z) Ci(x) - cos(x) Si(x) -7 L
1}
2
Other indefinite integrals are given by Abramowitz and Stegun [Section 9.9] and other definite integrals by Gradshteyn and Ryzhik (Section 6.87).
55:11 COMPLEX ARGUMENT Kelvin functions are seldom encountered with complex arguments, and this Arias does not address this circumstance.
55:12 GENERALIZATIONS If the complex variable z = x + iy is represented by the polar equivalent z = r exp(i(l), one sees from definitions 55:3:3 and 55:3:4 that ber,(r) and bei,(r) are the real and imaginary parts of J,.(z) with 0 = 3a/4. Thus, one may generalize these Kelvin functions by allowing 0 to adopt an arbitrary angular value.
55:13 COGNATE FUNCTIONS Just as the her. and bei. functions are defined in terms of the Bessel J, function to satisfy 55:13:1
ber,(x) + i bei,(x) = J,
x+
so Kelvin functions of the third kind are defined by a similar relationship but with the Hankel function [Section 54:13] replacing the Bessel function. Thus:
One has the identities her,(x) _ (2/a) kei,(x) and hei,(x) = (-2/n) ker,(x).
CHAPTER
56 THE AIRY FUNCTIONS Ai(x) AND Bi(x)
The Airy functions Ai(x) and Bi(x) are related to Bessel functions of order 3 and -3, with resealed arguments. The two so-called auxiliary Airyfunctions are also important and are briefly discussed in this chapter.
56:1 NOTATION In formulas involving the Airy functions, the auxiliary argument 56:1:1
X = 3 (IxI)`/'
is frequently more convenient than the usual argument x. Note that dX = tix I dx. There appears to be no standard notation for the auxiliary Airy functions. This Atlas employs fai(x) and gai(x) for the functions that Abramowitz and Stegun [Section 10.41 denote by f(x) and g(x).
56:2 BEHAVIOR The functions Ai(x) and Bi(x) exist for all real arguments, but their behaviors depend crucially on the sign of x, as illustrated in Figure 56-I. For x z 0, both Airy functions are positive, but whereas Bi(x) increases rapidly as x --> x, Ai(x) steadily decays towards zero. For negative arguments, Ai(x) and Bi(x) are oscillatory functions with oscillations whose frequencies gradually increase and whose amplitudes gradually decrease as x -+ -x. The auxiliary Airy functions are mapped in Figure 56-2. For positive arguments, both increase exponentially as x -> x, while for negative arguments each function exhibits oscillations similar to those of the Airy functions themselves.
56:3 DEFINITIONS The Airy integral 56:3:1
Ai(x)-J o
`
3)dt
THE AIRY FUNCTIONS Ai(x) AND Bi(x)
56:3
iA
i0
,
.I
44
44
i', 44
44
FIG 56-1
.1
44
44
it
.1
44
IF
:
:
556
44~
:
............ :..............:......... :....:.... .......0.6 At (x)
......................
...:....:
.....0.4
Ell W
.....
...
...
..
..
:
.
.
p
:....:......-0.2
..:.
..
....:....:..
....
..
Bi (x)
:
:....:....:....:....:....:..
...:....:......-0.4
defines the Ai function for all arguments, while the similar definite integral sum 3
('
if s sin(xt + 3 Idr
Bi(x) _ ! J expl xt - 3 )d, +
56:3:2
serves the same purpose for Bi.
Using the 56:1:1 definition of an auxiliary argument, the two Airy functions may be defined in terms of hyperbolic Bessel functions [Chapter 501, or the Basset function [Chapter 51], for positive arguments. The Bessel functions [Chapter 531, or Neumann functions [Chapter 541, provide the corresponding definitions when the argument is negative:
56:3:3
Ai(x) _
[I-it3(X) - 11113M) =
3
x
[1-1p(X) + 1.,3(x) =
13 [I-1n(X) + 1113(X)1
K,n(X)
3
-x 3
56:3:4
1.
a
3
x> 0
[Y-113(X) - Y.1 3(X)1
X < 0
x>0
Bi(x) = P-1/3M - 1113(X )) = -
3
1Y_ '/)(X) + Yu3(X)1
x0
[J-,n(X) +
xr r( + v)
1 - 2v ' x
+
3!!(l - 2v)(3 - 2v)
-
x
+
(2j - 1)!! (2' - v), 2
(-x /2))
+
/
x- x
See Section 54:6 for an asymptotic expansion of Y,(x) that may be used in conjunction with the 57:6:7 series. The
important cases of orders -1. 0 and I are 45 1575 I2 _X,+s3 _X6+XB _...+ (2j + 1)!!(2j - 1)" +... 1
57:6:8
57:6:9
h_dx)-.(x)
(-x2))-'
2(1
h0(4-Y6(x)-rr
x
- -+;;1
.e
9
225
[(2j - I)"]2
xr
x(-x2)l
1
J
x-,x
r-,x
57:8
THE STRUVE FUNCTION
567 and
3 2 r h,(x)-Y,(x)-1+-n x' x' 1
57:6:10
45
(2j - 1)!!(2j - 3)!!
.r°
(-x2)
J
x
57:7 PARTICULAR VALUES x= -x h,(x) h,(x) h,(x)
n - 3. 5, 7.... n = 2, 4. 6,... I > v > - I: v
0
h.(x) h-.(x) h,(x)
-3
-1 > v > 22 .... 2
-3
0 0
2/v
0
undef
0
0 0
0 2/v
2/v 0 0 0
undef
x
0
undef
0
0
-
undef
-x
0
0
±x
0
undef
x
0
0
x
0
2
-5 -7
>v>-2,-2>v> -, - >v>-4.
h,(x) h,(x)
undef
-3 -5 -7
h.(x)
h.(x)
x=x
0
I v>-3.-3>v--,->v>-5,.. -5 2
2
2
h,(x)
Apart from its x = 0 value, the Struve function has no zeros for v > ;. For v ~= 1, however, there is an infinite set of zeros that, for large arguments, correspond to the roots of the equation 57:7:1
/
sinlr - 2 +
\
r'2
1=(I'(v)+J
h,(r)=0
larger
v0
n=1,2,3,...
permits extension of the argument to general negative values. Similarly, extension to arguments exceeding unity 573
THE INCOMPLETE BETA FUNCTION B(v; s;x)
58:3
574
is possible if the µ parameter is a positive integer: 58:2:2
x?0
B(v;m;x+1)=(m-1) -(-1),B(m;l-v-m;l+x (v),,,
m=1,2,3...
Here (v),.. is a Pochhammer polynomial [Chapter 181.
58:3 DEFINITIONS The incomplete beta function is defined by the indefinite integrals
0: x< 1
B(v;µ;x) = J s t"-'(1 - t)"-'dt
58:3:1
0
t' dt
B(v:µ;x) =
58:3:2
0sT=X
-l
-l<xsl
v >-1
may be used to define Lcgcndrc functions. Despite appearances to the contrary, the integrands in the final two definitions above are wholly real: all imaginary terms cancel after binomial expansion.
THE LEGENDRE FUNCTIONS P,(x) AND Q,(x)
59:3
O
0
4ICJ*
0 A*
4111
4
U
', ,yea 40 Oti Ob90'co O'0
4
4
4
4
4
4
'L
O
4ti' 4
4
584
0
co
y1 4
4
!....x.../...:/...1.1.. 11111 .1 A.v .. \.:\.
O
yti 4
4 .
'L
;. 4
2
V-5
by the indefinite integrals
The Legendre functions may also 59:3:7
V-2 rx P,(cosh(X)) = 'a J o 1
59:3:8
59:4
THE LEGENDRE FUNCTIONS P,(x) AND Q,(x)
585
Q,(cosh(X)) _ -
cosh((v +
r)
dt
s)
cosh(X) = x > 1
cosh([) - cosh(X)
exp(-(v + )t)
I
x > 1 cosh(:) - cosh(X)
and cos((+ 1)r)
0 I
x/ l+x\ Iyx 2-2E11 -lax
/
Q12(x) = Q_3;2(x) _
z >I
P_,,2(x)
2
Kl 59:4:7
xs
x+
_1
\ YY
\
/
2
2x + 2 EI
x+l
x>I
I
and others may be deduced with the aid of recursion formula 59:5:5.
59:5 INTRARELATIONSI{IPS The argument-reflection formulas
n = 0. 1. 2....
59:5:1
(-1), P,(x)
59:5:2
-(-1)" Q.(x)
n = 0, 1, 2. .. .
are valid only for integer degree, whereas the degree-reflection formulas
P-..-I(x) = P,(x)
59:5:3
a cot(vvr) P,(x)
59:5:4
have general validity for real arguments. The same recursion formula
f=PorQ
59:5:5
is obeyed by Legendre functions of both kinds. The equations 59:5:6
59:5:7
59:5:8
n Q,(±x) = Z [cot(vrr)P,(±x) - csc(va) P,(+x)]
-2
P,(±x) _ - [cot(vir) Q,(tx) + csc(vu) n
Qv(x) = P,(x)[Qo(x) - 4.(v + 1) - y] +
-1 <x< 1
v * 0, ± 1, t2, ...
-1<x 6. If v to + t, + t2 +
+ (v/2)]/r[l + (1 + v)/21} = {[I + (I + v)/2]/[l + I + (v/2)]}{r[/ + 2 + (v/2)]/r[l + (3 + v)/2]} has been applied L + 1 times with l = 0, 1, 2. .... L where 2 - (v/2) < L >
Storage needed: a, R, v. K. s, c, w, k, X,
u, 1,x,fandg
Set K = 4 011v - v-1
If 11 +vJ + Int(I +v)+Ogoto(1) Set a = 10" Replace v by -I - v (1) Set s = sin(90v) Set c = cos(90v)
Use degree mode or change 90 to n/2. Input restrictions: - I < x < 1
Setw=('),+ v)2 (2) If v > 6 go to (3) Replace v by v + 2 Replace R by R(v - 1)/v Go to (2) (3) Set X = 1/(4 + 4v)
Set g = I + 5X(I - 3X(.35 + 6.IX)) Replace R by R[I - X(1 - gX/2)]/V
To recalculate with unchanged v, simply input new x.
I
THE LEGENDRE FUNCTIONS P,(x) AND
59:9
Input x >>
590
Setg=u=2x Setf=t= I Setk - '
Set X = I + [10"/(1 - x2)] (4) Replace t by tx2(k2 - w)/[(k + 1)2 Replace k by k + I
11 4
Replace f by f + t Replace u by u--, (k2 - w)/[(k + 1)2 - 411
Replace k by k + I Replace g by g + u
If k < K go to (4) If (XtI > Ifi go to (4) Replace f by f + [x2t/(1 - x2)] Replace g by g + [xIu/(I l2)] p - P,-W
2 w2 vl the infinite series t, + t,_, + t,_. + + x2 + x4 + x° + ) and the sum is therefore approximately t,/(1 - x''). Hence, the fractional error in truncating at the t,_, term is approximately [t,/(l - x2)1/Ito + It + + t,_, + t,(I the infinite sum to + t, + t, + x)-']. The algorithm makes use of this principle in deciding when to truncate, thereby ensuring that lef, and leg, are computed to a precision of about 6 X 10-°. Close to the zeros of P,(x) and Q,,(x), this precision may not be carried over to the Legendre functions themselves. The algorithm is valid only for arguments in the range -1 < x < I and is extremely slow close to the boundaries of this range. -
59:9 APPROXIMATIONS Close to x = I, the Legendre function of the first kind is approximated by the linear function P,(x) =
59:9:1
[(1 - v)(2 + v) + v (I + v)x]
I I - x{ small
while that of the second kind obeys
Q,(x)-In
59:9:2
(1 +v)
II
II -xlsmall
The corresponding approximations close to x = -1 are sin(ir) Inr t 2+ xl I
59:9:3
59:9:4
P,.(x) = cos(vtr) +
+ y + 2d,(1 + v)
(I + x) small
y = 0.5772156649
1I
Qdx) -
coswa) r In(1 Z
x) + y + 2y(I +v)J -
17 si 2(vw)
(1 + x) small
Here 'h is the digamma function [ Chapter 44] and y is Euler's constant [see Section 1:4].
THE LEGENDRE FUNCTIONS P,(.r) AND Q,(x)
591
59:10
As x acquires very large values, we have
P,(x) =
59:9:5
V.- r(1+v)
v> -1
r(21 + v)(2x)' '
Q.(x)
-3 -5 -7
(-2v - 2)!!
IT
59:9:6
N/2
V'nr(i -v) r(-v)
x- z
v = 2 , 2 , 2 , ...
(4x)"
v)!
cot(var)(2x) "
-
'
1
>v*
Z,3 Z,5
27
..
59:10 OPERATIONS OF THE CALCULUS Differentiation or integration of a Legendre function gives an associated Legendre function [Section 59:13J: r
; U X) = S
59:10:1
-1
I
f
59:10:2
fj'(x)
-1 < x < 1
fl. ,'(x)
x>1
} f= P or Q
1
P,.(r)dr =
J x P,(t)dt =
59:10:3
x>
x= - 1 P;-"(x)
J x Q,(0(dl)
59:10.4
-1 <x< 1
1 - x= P;-"(x)
x>I
(.r)
Alternatively, the derivatives may be expressed as
vx+x v+ l f(x) 1 -x I -x'
d
-dxf(x)
59:10:5
v
l -x
f_I(W ) -
vx
I -x' f,(x)
a formulation that applies for either kind of Lcgcndre function and for any x exceeding - 1. Definite integrals of products of Legendre functions include i
59:10:6
T.
P,(t)P (t)dt =
P,Q)Q (t)dt=
59:10:7 _a
59:10:8
T.
Q,(t)
4ti(1 + v) - 44o(l + w) + 2w cot(ve) - 27r cot(a)a) I2(l + v + w)(w - v) csc(v7r) csc(w7r)
cos(wlr - vw) - it + sin(2vir)[ijr(l + v) - ,(1 + w)J 7r(1 + v + w)(w - v) IT
v + w * -1
v>0
2 + 2 cos(vrr) cos(wa)J[y(1 + v) - 4s(1 + w )1 - IT sin(m - wa)
all + v + w)((d - v)
w>0 v + w * -1
THE LEGENDRE FUNCTIONS P,(x) AND Q,(x)
59:11
592
when the integration limits are - I < t < 1. These formulas are indeterminate when w = v; in that event the three
integrals give [I - 2/1r2) sin2(vrr) ib'(I + v)]/(} + v), -sin(2vrr) qi'(1 + v)/[(I + 2v)tr] and {(w2/2) - (1 + cos2(v'rt)]y'(I + v)}/(l + 2v), respectively. Similarly, the integral between limits 1 < t < x: 59:10:9
Q.(t) Q.(t)dt =
J
4r(1 + w)
f
- ir(1 + v )
(I+v+w)(w-v)
has y'(1 + v)/(1 + 2v) as its special w = v case. but 59:10:10
F
Ji
P,(t)Q (t)dr =
I
w>V>0
(I+v+w)(w-v)
diverges if the degrees are equal. Since Pa(t) = 1. P,(t) = t, { P,(r) + } Po(t) = r', etc. [see Section 21:5], the above formulas may be adapted to give many definite integrals of the form ft'f,.(t)dt for f, = P, or Q,. Very many other definite integrals are listed by Gradshteyn and Ryzhik [Sections 7.1 and 7.2].
59:11 COMPLEX ARGUMENT With the real argument x replaced by x + iv. equation 59:12:1 serves to define the complex-valued function P,(x
+ iy) as F(-v,l + v;l;(l - x - iy)/2). The corresponding formula
1 - x - iy
it
59:11:1
Q,(x + iy) = 2 [cot(vrr) z i] F -v,1 + v:l;
2
-
/
2
csc(vrr) FI -v,1 + v;l;
I + x + iy1 2
provides a valid definition of the Legendre function of the second kind for all combinations of x and y except when y = 0 and > 1. The alternative signs in 59:11:1 apply accordingly as y > 0 (upper sign) or y < 0 (lower sign). For y = 0 and - I < x < 1, definition 59:11:1 yields a function that is multiple valued and complex. To avoid
this difficulty, it is usual to redefine Q,(x) as the average value of the two limits Q,(x + iv) and Q,(x - iy) as y -+ 0. This removes the i from 59:11:1, which then reduces to 59:5:6. This convention has been followed in all equations of this chapter (except 59:11: 1), so that our Q,(x) is always real for real argument.
59:12 GENERALIZATIONS Legendre functions may be generalized to the Gauss function of the next chapter. Included among the ways in which Legendre functions may be expressed as a single Gauss function are
I-x
59:12:1
59:12:3
V,
+ v;1; .2 _
/
I+v P,(x)=Flv 2, 2 ;1;1-x=)=xF(
v 1-v ,l+Z;l;l-x'
2
1
59:12:4
1,14) = [x +
(x -
x=
I J'' F
r' - 111/21.
+ x- I
v,
:1;
-2 x+ 1
`2
-1 <x < 3
)
-1
z'- 1
2V7- I x+
xr- 1
0<x'r
x>2+I
F(I+v,1+v,2+2s'; 2)'-" \ 1 ± x/
r(i + v)(2x
Q"(x) =
=
59:12:11
_
x'-l-x)
\
v I-v
x' - 1
2
1
v 1 P"(x) = x' F - -;1;1 - II = x-'-' FI -,1 + -:1:1 - - I 2' 2 -r\ 2 2 X-
59:12:8
59:12:9
'-1-
V77-111/2-
x- x'-1 59:12:6
x'- 1
2
x'
P"(x) _ [x -
59:12
.rl
/
x>3
V
r-I
x+
x>I
The restrictions on x attaching to the formulas above ensure that the Legendre functions are real and that the series expansion of the Gauss functions [see 60:6:11 converges. As well, there are many formulas that express the Legendre functions as the sum of two Gauss functions; some are 59:12:12
P"(x) =
r(-v - !)
r 2 1 Fl I + v.I + v:2 + 2v: - J
"" r(#+v)(2xt2)' /
V ir r(-,,)(2. +
r(-v)(2 x +
59:12:14
x>2+ I
``
/I+v I+v3- + v;
r(-v - D P"(x)
\
2 FI -v,-v;-2v;-I 1 ± x/
Vrrt r(1 + v) 59:12:13
t -r
1
2)
FI
1)'
2
2
r(; + v)(2 z' - 1)" ( -v
F 2'- 2-v
V;r(I+v)
I
- v:
'2
l-' 2
1
1
I
)
1-x'J
x > \/2
F -v l+ v :2,l x
P"(x) \\\
2
2Vnx
-
I2/
(l + v1
r
2'
2
2
FI
r (-v} \
2
v3 1-v ,1 +-;-;z' 2
22
-1 <x< 1
THE LEGENDRE FUNCTIONS P,(x) AND Q,(x)
59:13
P,(x) =
59:12:15
r(-v-1)
1`
F
/-v FI`2
C(- v 59:12:16
P,(s) =
(x -
r(-v) r(1+v)
P,(x) =
F
2a x- 1 (x-
r(+ v)
59:12:17
V'nr(l +v)
(1 11 \-,2:2
J
x
3
2V77- -1
Vis
+
2
1
x>1 x +
2
x=- 1
x' - 1
v) sinl
zr(l + z)
r(i+ZI/ \\
v\ 2
I 2
2 x-
3
2
2
Q,(x) =
r(1
2
F -,-v: - - v:
r(I + v)
r( l
1 Vc
+ v; - -
12
+
1
x>1
X.
Fr-1 + v; - + v;
fir(-v)(x+
-
I
r(-v-1.
r( + v)(x +
59:12:18
x
'2
2
222 F
2n x'-l
1
22
Iv I - v; -/1
(1 1 3
'
v3
+ -; - + v; ;
2
r(-v)(2x)' "
z + v)(2x)' +
/ l+v'I
594
2
2 v: J F( Zv, 1
2
3 cosva- xF 1-,- Iv + -;v22 -;xz
-1 < x < 1
2
/I
Many of these formulas become invalid, or require modification, if v is an integer. For x > I the substitution x = exp(tX/2) often yields simplifications. = cosh(X), x= I = sinh(X), s t V
59:13 COGNATE FUNCTIONS The associated Legendre functions P."(x) and Q,"'(s) represent a generalization of the Lcgcndre functions inasmuch as PY°'(x) = P.(x) and Q'.01(x) = Q.W. They are sometimes named "spherical harmonics" but we reserve that title for the extended functions discussed in Section 59:14. In linear combination, they satisfy the associated Legendre equation 59:13:1
+[v(1+v)-1 µ'x2Jf=0
f=c,P;"'(x)+c,Q;."'(x)
[see 59:14:4 for a trigonometric equivalent of this equation]. Replacement of x by I - 2x and then f by (x= x)"Rf leads to an example of the Gauss equation. 60:3:4, and accordingly associated Legendre functions are instances of the Gauss function of Chapter 60. Generally, these functions are complex even when their arguments are real. However, it is conventional to adopt redefinitions similar to that discussed in Section 59:11 to ensure that P;"'(x) and q."(x) are real for real x between - I and + 1. Here we discuss this range exclusively. Moreover, we shall emphasize cases in which v and µ are nonnegative integers, using n and m to represent the degree and order, respectively, in these cases. Calculation of values of the associated Legendre functions is possible via the definition
THE LEGENDRE FUNCTIONS P,(x) AND Q(x)
595
59:13
(l+V+µ\ 59:13:2
II
\
2
I
P;°'(x) =
V.-(1 -
vtr + µa\ I Ief,'(x)
cos r( I + vµ)
x'-)"r_
2
2
r(i+v+µ\
2
/I
\
Ileg;,"'(x)
sin
r(l+v-
2
J
2
V n2"-' 59:13:3
sin
(1 - x')"/'
(vir + µa\ 2
I
lef ,"'(r)
2
rll+v2IL
/
/
rrl+v-µl cos(m + µ J Ieg;1"(x)
+
1\
Fx2
2
J
-
where the auxiliary associated Legendre functions are defined by 59:13:4
59:13:5
Ief;"r(x) =
(-V - µ 1+v-µ 1
i yI+v-) µI (2x)2J
l-v-µ +v-;-µ ;x 3I_ - v - µv-I +µ(2x)"='
Ieg;"'(x) = 2xF
The algorithm of Section 59:8 may be modified to use these equations, although four distinct gamma function evaluations are required instead of the single gamma function ratio that sufficed with equations 59:6:1-59:6:4. Some interrelationships between associated Legendre functions are
()
59:13:6
Pt'-1W W x= 11."W W
59:13:7
(tan(") - tan(µa)] Q"0_,(x) _ [tan(var) + tan(An)1 Q."(.0 - TrPV`(x)
59 : 13 : 8
P
59 : 13 : 9
Q
59:13:10
59 : 13 : 11
r(1 + v - µ)
" (x) =
:_" (x)
P ,"(
r(1 + v + µ)
",
[cos(µa) P (x)
r(1 + v - µ)
-
= r(1 + v + µ) Icos(µ'n) Q :"'(x) +
2 sin(µa)
it
"
Q, (x)1
ir sin(µ-rr) 2
P;"' (x)1
-x) = cos(var + µn)P ` W - 2 sin(va + µa)
'' -
Q ( x)
;,
IT
_ -cos(av +
aµ) Q;"'(x)
constituting degree-, order- and argument-reflection formulas. Some examples of associated Legendre functions are
-
a sin(wv + nµ) 2
;
P ' (x)
59:14
59:13:12
59:13:13
THE LEGENDRE FUNCTIONS P.(x) AND Q.(.0 P'i'(x)
1
P;"(x) = -3x
Qi"(x) =
x
1 - x2
P':''(x) = 3(1 - x'-)
59:13:14
Q;"(x) =
Q," (x) =
1 - x'
2-3x' 5x - 3x'`
1 -x'
1 - x' artanh(x)
- 3x
1 - x' artanh(x)
+ 3(1 -xartanh(x)
n=0.1.2,...
P;°'(x)=(-I)"(2n- 1)!!(1 -x22)"'
59:13:15
-
5%
and others may be obtained via the differentiation
G -(X) = (- I)"(I - .r2)"/'
59:13:16
f,(x)
f = P or Q
or recursion
59:13:18
(1 + 2rt)x
(x) _
59:13:17
f,"
1 +n - m
- 2mx
(x) =
f' '(x) +
n + in
1 +n - m
f'; ',(x)
f=PorQ
f"'(x) - (I + n - m)(n + m) f'"-"(x)
71=-=T
formulas. Formulas 59:13:12-59:13:15 demonstrate the inappropriateness of the name "associated Legendre polynomials" sometimes given to these functions. Be aware that the (- 1)" factor in 59:13:16 is often omitted so that associated Legendre functions of odd order may be encountered with signs opposite to those employed in this Atlas. The associated Legendre functions of the first kind satisfies the orthogonality relationships
N+n
10 f P, '(t) P;,"' Will =
59:13:19
J
2
(n + m)!
2n + l (n - m)!
'
N=n
but no such relationship holds for the second kind.
59:14 RELATED TOPICS In Section 46:14 we discuss the solution of the Helmholtz equation in various orthogonal coordinate systems. In the spherical system one finds
-
2r dR I a'Y cot(o) ay csc2(o) d'Y - - + - - - kr- = h = - - - - - - R dr2 R dr Y ao' Y ao Y a8'
r2 d'R 59:14:1
where R is a function only of the radial coordinate r, Y is a function of the two angular coordinates, 0 and +, and h is a separation constant. With m2 denoting a second separation constant. the second equality in 59:14:1 may be further decomposed into 59:14:2
sin2(o) d26 A
do' +
sin(20) d6
29
do
+ h sin'(o) = m =
I d2I 11.$12
where 0 is a function only of o, and is a function only of 4). The second equality in 59:14:2 is satisfied by a sinusoidal function of the longitude (b 59:14:3
4) = C. cos(me) + S. sin(m$)
in = 0. 1. 2... .
where C. and S. are constants and m is constrained to be an integer by the geometric constraint that (P(-a) NW).
THE LEGENDRE FUNCTIONS P(x) AND Q,(x)
597
59:14
The first equality in 59:14:2 may be rewritten as d '8
de. + cot(O)
59:14:4
d8
+ [h - m' csc=(6)]8 = 0
d8
which, by the substitution x = cos(8), may be converted into the associated Legendre equation 59:13:1. Accordingly, the general solution is 0 = p, ,F,"(cos(0)) + q,,,,Q,"(cos(8))
59:14:5
v=
;+h-
where constant coefficients are denoted by p_ and The latter is usually constrained to be zero by physical considerations: the prohibition on Y, and hence 0, being infinite at 0 = 0. Because the latitude of the sphere is
(a/2) - 0, 0 = 0 represents its "north pole." At the "south pole," 0 = a, cos(8) = -1 and, since is infinite unless v is an integer, we are led to conclude that v = n = 0, ± 1, ±2, .... Because the P;"' and P;'_, functions are identical, we can ignore negative n values, leading to 59:14:6
n = 0, 1, 2....
0 = p,.,,P;"' (cos(8))
h = n(n + 1)
as the only physically significant solutions of 59:14:4. Moreover, since the associated Legendre function P.,` is zero if Imi > n, we can discount all values of m except 0, ± 1. t2.... ±n_ Where only certain values of separation constants are permitted, as n and m here, these values are named eigenvalues or quantum numbers. Because the eigenvalue in enters into the solutions for both and 0, it is often counterproductive to factor the Y function into its longitudinal and latitudinal components. The general solution of the second equality in 59:14:1 has now been shown to be
Y=±
59:14:7
[S"," sin(m4,) + C",,, cos(md,)J P;,"'(cos(0))
where S",,, and C",,, are redefined constants. The components of this solution are known as spherical harmonics or surface harmonics and, in light of orthogonality relationships 59:13:19, 32:10:25 and 32:10:26. they are usually defined in the normalized versions 59:14:8
2n +
Y";"(8:m)
I I
(n + m)! f(mm) P;,"'(cos(8))
21T
///f=sinorcos
n=0,1,2,...
However, normalization conventions sometimes differ from author to author. As their name suggests, spherical harmonics play a vital role in describing oscillatory behavior in systems of spherical symmetry: for example, in the quantum mechanics of atomic electrons. One distinguishes tonal surface harmonics when m = 0, sectoral surface harmonics when in = n and tesseral surface harmonics when m = I, 2,
Finally, let us return to the first equality in 59:14:1 with n(n + I) now substituted for the separation constant h. Replacement of the radial coordinate r by x/Vk (or by .r/' k for negative k) and the dependent variable R by 59:14:9
then leads to
d'f df x--+x--[(n+?)-'-x'Jf=O dxdx
n=0,1,2....
where the alternative upper/lower signs apply as k is positive/negative. This is the (hyperbolic Bessel)/(Bessel) equation 50:3:4/53:3:5. and the lower alternative is satisfied by 59:14:10
R =
f
A.
=
]
+
J.
1/2_(x) = a"j"(x) + b"y"(x)
where J, represents the Bessel function [Chapter 53J; A", B", a" and b, arc arbitrary constants; and j, and v" are spherical Bessel functions [see Sections 32:13, 53:4 and 54:4J.
CHAPTER
60 THE GAUSS FUNCTION F(a,b;c;x)
A surprisingly large number of simple intrarelationships make the quadrivariate Gauss function unusually flexible. It embraces many of the functions discussed in previous chapters and may be further generalized as explained in Section 18:14.
60:1 NOTATION The Gauss function is also known as the hypergeometric function or as the Gauss hypergeomerric function. The subscripted symbolism ;F1(a,b;c;x) is sometimes encountered; Section 60:13 contains an explanation of the "2" and "1" numerals. As usual, the x variable in the Gauss function is its argument. Because of their locations in expansion 60:6:1, the a and b variables are known as numeratorial parameters, while c is a denominatorial parameter. There is a second denominatorial parameter, equal to unity, whose presence is not explicitly displayed in the F(a,b;c;x) notation but is brought out in the first of the following alternative symbolisms: C(c) 60 1
f a- 1,b- 1
1
5a)1'(b) Lx
0, c
J = 2F11
a,b;x c
= F(a,b;c;-r)
60:2 BEHAVIOR Unless one of the four quantities a, b, c - a or c - b is a nonpositive integer (in which event, see 60:4:10, 60:4:11 or their analogs), the Gauss function is defined only for real values of its argument in the range -- < x < 1. The domain may be extended to embrace x = I provided that c > a + b. When c is a nonpositive integer the Gauss function adopts infinite values [unless a or b is also an integer such that a - c or b - c equals 0, 1, 2, .... so that 60:4:11 applies[- Nevertheless, significance can always be attributed
to the ratio F(a,b;c;x)f[(c), even when c = I - n = 0, -1, -2, ..., because of the limiting operation I F(a,b;c;x) (a)"(b)"x" n = 1, 2, 3, ... F(n + a,n + b;n + l;x) lim r(c) = (b 60:2:1 Because it is quadrivariate, the Gauss function displays such a wide variety of behaviors that it is impractical to depict graphically.
THE GAUSS FUNCTION F(a,b;c;x)
60:3
600
60:3 DEFINITIONS Expansion 60:6:1 provides the usual definition of the Gauss function for - I < x < 1. The transformation 60:5:3
permits extension to -x < x < ;. The Euler hypergeometric integral 60:3:1
F(a,b;c;x) =
r(c)
r(b) r(c - b)
?-'di (1 - t)1° `(1 - xt)°
f
c>b>0
x0
and
ax
F
v, 1 + v; I - µ;
p"(X)-(I+X)w
60.5.12
1-X
rU - µ)
1-X 2
X=I-2x>-I
The equality of the right-hand members of 60:5:11 and 60:5:12 constitutes a quadratic transformation, valid for X > 0. The Baseman manuscript [see reference above] includes a comprehensive listing of quadratic transformations and information on cubic transformations.
60:6
THE GAUSS FUNCTION F(a,b;c;x)
604
60:6 EXPANSIONS The Gauss series 60:6:1
abx
(I + a)(1 + b)x
c
2(1 + c)
F(a,b;c;x) = 1 + - I +
I+
(2 + a)(2 + b)x /
3(2 + c)
+(/+a-1)(J+b-I)xl+
(1 +
\
)'(b),Xj
.. ) /
I(J + c - 1)
-l<x
K(tIdt
1 61:10:9
_/
E(t)dl
61:10:10 o
E
V ' I - t-
1
a2
\Y'_I
4U:
_
K
V'1-t
I
V'j)
r
a
L'
2
8U2
2
( 6.14 and 6.15]. and more are listed by Gradshteyn and Ryzhik [Sections The operations of semidifferentiation and scmiintegration with respect to p-, when applied to complete elliptic integrals, produce elementary functions. Examples include 61 : 10 : 11
61 : 10 : 12
dir(dpz)i. E (p) =
(dp_)
,_
K (p)
('I-
)
p
= V-,r arcs i n(p)
THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p)
61:11
616
61:11 COMPLEX ARGUMENT The complete elliptic integrals of a complex modulus are themselves generally complex valued, but these functions have real values when their modulus is purely imaginary. The transformations 1
K(ip) =
61:11:1
N
P
'p V'YP p
K
and
E(ip) =V I + pr E(
61:11:2
p
1 +p' are valid for all real p. For example, K(i I = (l /\/2) K(l /\,2) and E(i) = V2 E(1 /V'2). The complete elliptic integrals acquire complex values when their real moduli exceed unity. The transformation formulas 61:11:3
p 61:11:4
p
P
F(P)Pfi\P,-Cp
P
permit the evaluation of the real and imaginary parts in this circumstance. They show, for example, that
K(\/)( I
61:11:5
61:11:6
E(VZ)=
+
i
1
V2 V2
+
[2E(
i )K(N/2) - =(l+i) V2-
1^ 1
1
\2
V8U
II =fl-i1-=
V2
AV2
Sri
2K(\ 2)
indicating that the real and imaginary parts are equal for a modular value of V2.
61:12 GENERALIZATIONS Allowing the upper limit to vary in the integral definitions 61:3:1 and 61:3:2 gives rise to the incomplete elliptic integrals of the next chapter. In the remainder of this section we discuss the complete elliptic integral of the third kind f1(v;p) defined by
-
dt 61:12:1
II(v;p) =
Jo
(I + vt2)
(l - t2)(1 - p2t2)
f tZ Jo
dB
[I + v sin2(9)]
l - p2 sin2(0)
It is a generalization of the first kind of complete elliptic integral inasmuch as [1(O;p) = K(p). Be aware of the variety of definitions and notation. Thus, the integral 61:12:1 is variously denoted 17,(v,k), f1(-iAaresin(p)), II(p,v, rr/
2). It(a/2,-v,p) or even (-1/v)fl(-1/v,p,w/2) by different authors. The characteristic v may take real values in the range -- < v < x, although the integral is infinite for v = -1. Interest concentrates on the range 0 5 p 5 1 of the modulus. Figure 61-3 shows the behavior of 11(v;p) in its most important domain. One may evaluate the complete elliptic integral of the third kind via the incomplete elliptic integrals of the
first and second kinds [Chapter 62]. The formulas permitting this are as follows: 61:12:2
I1(v;p) =
K+pv
+
(1 + vxP + v) { 2
+ [K(p) - E(p)]
K(p) E(q;+) 1
sin(41) =
1+v
v>0
a
oa Q
.,
a
,moo
.......................
5
TT(-0. 91p) :
Tr(-(L 85op)
1
.
rr}
n=0,±1,±2,
for reflection of the amplitude across any multiple of r/2. In the equation above. and that below. fi p,,-U12) is the complete elliptic integral K(p) or E(p). Both incomplete elliptic integrals obey the recurrence formula
f = For E
f(p;6 + nw) = 2nf P; 2) - f(P:d.)
62:5:3
n = ± 1, ±2, . .
As a result, the difference functions 62:2:1 are periodic. Let p_ po and p, be three moduli interrelated by 62 5 4 :
P- =
:
and dt- k and 62:5:5
1-VI -po 1+VI-Pa
2Vpo and
P, _
l+Po
be three amplitudes interrelated by 4,-, = 4ro + arctan{V I - p tan(4o)}
I
and
d+, _ - 14 + arcsin{p
Then the corresponding incomplete elliptic integrals of the first kind are interrelated by
F(p-i;4-i)
62:5:6
VI
[I
1 F(Po:d'o)
and
F(p,4,)
)
PoF(po:(iio)
and those of the second kind by
2
[E(po:dro) + \ P F(po:db)I -
E(P-A,-1)
(I -
1 po)
and
62:5:7
E(Po:4J 1 + po
1 - Po 2
F( Po: d+o I
PO sin(40)IPo cos(4) + (I + pot)
1 - pu sin2(db)1
I - Po + 2po cos(4MJ[Po cos(4,o) + VI - pp sin 2(4+o)1
Going from elliptic integrals of variables po, dte to those with variables p-,, 4b_, is known as descending Landen transformation, while the conversion from variables po. dro to P. dr, is called ascending Landen transformation. The adjectives 'descending" and "ascending" recognize the fact that, for 0 < po < 1. the parameter P-, is smaller than po, whereas p, is larger. Although it is not immediately apparent from 62:5:4, the numbers p_ : pr,: P, form
THE INCOMPLETE E t tv'FIC INTEGRALS F(p*) AND E(p;4)
627
62:7
a sequence, that is, the rule by which po is constructed from p_, is identical with the rule by which p, is formed from po. Similarly, though more obscurely, (b-1; din; (01 form another sequence, as do F(p-,;4o-,); F(Po;400); F(pl;4iJ
and E(p-1;4_i); E(po;0o); E(p,;dij). It is evident that each of these sequences may be extended indefinitely in both directions. In one popular method of computing numerical values of elliptic integrals, one or other of the Landen transformations is implemented repeatedly, starting with po = p. 4o = di, and generating a sequence of moduli p,) that satisfy the inequalities (either po; p-1; p-2; .. .; P-, or PO; Pi; p2; .
62:5:8
P < pi_, < ... < P-2 < P-1 < Pa < P1 < P2 < ... < P.-I < P.
The limits as n - x of p-, and p are 0 and 1, respectively; hence, if the transformation is carried out a sufficient number of times, it becomes possible to approximate and E(p..pbt.) by use of equations 62:4:1-62:4:3. In this way F(p;(b) and E(p;di), as well as K(p) and E(p) are calculable. The "common mean' technique used in Sections 61:8 and 62:8 is, in fact, an adaptation of the descending Landen transformation. Incomplete elliptic integrals with moduli exceeding unity may be related to ones in the standard 0 < p < I range by the transformations 62:5:9
= PFI p;aresin{
FI 1:(b P
\
(
62:5:10
(1 I E[ P:40) = p EI p;aresin
I.
sin(di)
- P ) -
p P
1
/I
1-p p
sin(tb)
p
F p;aresin
p s
Yet another transformation is 62:5:11
F(p;di) = K(p) - F(p;ii)
62:5:12
E(p;di) = E(p) - E(padi) +p sin((i) sin(i(i)
l - p '' sin'O.
where sin(g) = cos(4)/
62:6 EXPANSIONS For small values of the amplitude and modulus. there exist expansions of which 62:6:1 62:6:2
F(P 4b) = E(p; (b) =
2K(p)
it 2E(p)
it
4) -
[2KI p) -
r4K(P) _ 2
1
1J sin($) cos((b) - IL
3
IT
di +
I-
P2 1
6 Jsin3( ) cos(Qr)
-
31T
2E(p)
''
11
4E(p)
p2
I3 -
a
6
Jsin3(di) cos((b) +
are the leading terms. Similarly, when p is close to unity: 62:6:3
62:6:4
3K(q
F(p; 4)) =it ! mvgd(di) E(p: di) =
'K(
1 - cos((i)N,'l - p' sin'ldi) sinl(b)
)
IT
+
-1
sinl,.... be defined by the recurrence formula tan(21''6,_, - 2'4),) = G,tan(2'd)d
62:8:1
j = 0, I. 2,...
with db = 4), the amplitude of the sought elliptic integrals. Then. in addition to the results K(p) = a/(2G,) and
E(P)=K(2)2-p'-Y2'(A;-C)-
62:8:2
ro
J
-1
LLLLL1
which were given in the last chapter. we have 62:8:3
2K(p)4,,/,rr = 4)x./G
F(p;
and
62:8:4
E(p;dr1 =
E(p)F(p:d.) + K(p)
VA - G,2 sin(2'd,;)
I
(A, - G,) sin(2'-'d),.
These equations are the results of applying Landen's descending transformation an indefinite number of times. In practice, the angles are calculated via the formula
4),,4' -
62:8:5
1
are17'
((A, - C,) tan(2'd,;l
A + G tan'(2'dt, l j
and converge toward d): as rapidly as A, and G, converge toward their common mean G.
Input p > >
Sett=a= I
Storage needed: t. a. E, g. e.
Set E = 0
d) and T
Set e= 2(1+g2)
Use radian mode.
»»Setg=
Input 4 >> (1) Set T = tan(4) Replace dr by d, - {arctanl(a - g)T/(a + gT')1}/(2t) Replace e by e - t(a - g)2
Input restrictions: -1 < p < I
Replace t by 2t
4) must be in radians, not de-
Replace E by E + (a - g) sin(t4)
grees
Replace g by
Replace a by l(g2/a) + al/2 10-1 go to (I) Set It = 'R/(2g)
THE INCOMPLETE ELLIPTIC INTEGRALS F(p;4,) AND E(p;4,)
629
k-K(p)
>
Storage needed:
Jp»»
n. p. g, a. s, q. x. e, r and
Set g = p
(1) Seta= I
Input restrictions: 0 < p 5 0.99.
c may only be one of the 12
(2) Set s = ga
Replace a by (g + a)/2
codes listed below ..v * 0 if c = 21, 31. or 41. To reuse the
If a>
Use degree mode or change 90
Go to (1)
to r/2 and delete the (180/1T)
Replace n by exp(-rg/n)
factor.
Sets=0
Input code c Set a = 10 frac(c/5) - 5
Test values:
(4) Set v = gx(I80/7.)
P=
Set f = I If {al * I go to (5) Replace v by 90 - v
12
Set f =
(5) If a > 0 go tof(n/p{sin(v) 6)
Sc
ou ut 2.95528111
13
sd
1.07551806
+ n2[n4 sin(5v) - sin(3v)]}
14
sn
0.947240202
Go to (7) (6) Replace f by f {; + n4 cos(4v) - n[ne cos(6v) + cos(2v)]}
21
cs
23
cd
0.338377286 0.363930880
(7) If s * 0 go to (8)
Replace f by
24
en
Sets = I
31
ds
Set a = Int(c/5) - 5
32 34
do do
41
42
as nc
0.320524568 0.929784484 2.74777452 0.880729243 1.05569844 3.11988565
43
nd
1.13542273
Go to (4)
(8) Replace f by f/s Output f
f '. fg(p;x)
i and x
1.311139165 code, c -fQ
«
I and Y < I are quite distinct. For v > I, 4(v:u) is uniformly positive. When both the order and the parameter exceed unity, the Hurwitz function declines in value, asymptotically approaching zero, as either v or u increases. Conversely, l,(v;u) - x as v approaches unity from positive values or as u approaches zero from positive values. For the most part, the Hurwitz function assumes negative values for v < 1; however, there are "lobes" of positivity as mapped on the contour diagram. As a function of u. {(v;u) displays a small, finite number of zeros, mostly in the 0 < u < I range, for v < 1. 653
64:2
THE HURWTTZ FUNCTION {(v;w)
654
THE HURWITZ FUNCTION ;(v;u)
655
64:4
64:3 DEFINITIONS The Hurwitz function is defined by the integral transforms 64:3:1
I
Yv;u) =
r(v) Jo
t'-' exp(-ut)dt _ I 1 - exp(-t) ['(v)
In'-'(t)dt
tit - 1)
v>l
u>0
or by Hermite's integral
u-
64:3:2
u'-
(v;u)=-++2 v-1 2
sin(v arctan(t/u))dt (u' + t2)'/2[exp(2irt) - 11
o
u>0
Expansion 64:6:1 provides a definition of;(v;u) for v > 1, and Hurwitz's series 64:6:2, supplemented by recursion 64:5:1, can fill a similar role for v < 0. Notice that none of these definitions is valid in the important domain 0 < v < 1. Contour integration [see Erdelyi, Magnus, Oberhettinger and Tricomi, Higher Transcendental Functions, Volume 1, pages 25 and 261 can define the Hurwitz function in this region. More practically, the series 64:6:4 can be employed for all orders and parameters.
The difference i(u) - t,(u - t) of two digamma functions [Chapter 441 provides a generating function: 64:3:3
t[y(u) - 4.(u - t)] = 2 ;(n;u)t'
u>1
-2
for Hurwitz functions of integer orders 2, 3. 4, ...
64:4 SPECIAL CASES Here we discuss cases of 7(v;u) that correspond to special values of the order Y. See Section 64:7 for simplifications of the Hurwitz function that arise when the parameter u acquires particular values. When v is a positive integer greater than unity, the Hurwitz function is equivalent to a polygamma function [Section 44:121: 64:4:1
Un;u) _
(- I )" OR- u(u)
n = 2,3,4,...
(n - 1)!
When v is a nonpositive integer, the Hurwitz function can be expressed in terms of a Bernoulli polynomial [Chapter 19): 64:4:2
Y-n;u) =
n
n = 0, 1, 2, ...
+1
of which the first few instances are 1
I(0;u)=--u
64:4:3
64:4:4
64:4:5
>;(-l:u) =
(-2;u) =
-1 +6u-6u2 12
-u+3u2-2u3 6
Although the Hurwitz function is infinite for v = I, the quotient;(v;u)/r(1 - v) and the product [v - 1]i;(v;u) both remain well behaved in the vicinity of v = 1. This is evident from the limiting expressions 64:4:6
lim
VV*
-I r(1-v)
= _1
64:5
THE HURWITZ'FUNCTION Vv;u)
1 I_
64:4.7
v
656
Vu)
J
where 4' is the digamma function (Chapter 44).
64:5 INTRARELATIONSHIPS The recursion formula 64:5:1
1'(v;u + 1) = >;(v;u) -U.-
u Z. 0
may be iterated to 64:5:2
I,(v;u + J) + Z (j + u)-`
u a 0
J=O
and, provided v > I, becomes expansion 64:6:I when J = x. The duplication formula 64:5:3
Uv,2u) = 2 "LZ(v;u) + rl v;
+
u1
2
may be generalized to
{(v:mu)=m "m+u
64:5:4
m=2,3,4,...
and used to demonstrate , for example , th at
C('; 3
64:5:5
4
+ u) =
2"1'(v:2u) -
V;
I + ul
\\\
The following infinite series of Hurwitz functions of integer order 64:5:6
(2;u) + C(3u) + 4(4;u) + ... _
(n;u) =
1
u > I
U
64:5:7
i'(2;u) - 4(3;u) + Z(4;u)
- ... _
(- l)%(n;u) _ u
.=2
_ + - + Z(2;u)
64:5:8
2
3
,(4;u)
4
2
+
.-I
4(n:u)
n
u>0
=40)-In( u-1)
u>I
and
64 : 5 : 9
4(2;u)
2
1'(3;u)
3
+C(4;u) 4 _ ... = L, (-1)" ,=2
In(u) - tr(u)
n
u>0
may be combined in several ways: for example, to sum the series I{(k;u) where k is restricted to a single party
64:6 EXPANSIONS The important series 64:6:1
I
I
I
k`
(I + U),
(2 + u)`
po
THE HURWITZ FUNCTION Ov;u)
657
64:7
often serves as a definition of the Hurwitz function for orders exceeding unity. Hurwit_ formula:
/
av
sin (
4(v;u)=2r(1-v)
64:6:2
\
=i
2jau + 2/
1
VI
or ; are related to the lambda and beta functions of Chapter
3 by
Yv; i) =
64:7:8
a(v)1 LL
2
re(v) - (3(v)]
64:7:9
Adding equations 64:7:8, 64:7:6. 64:7:9 and 64:7:2 leads to Yv;',) + Y OZ) + {(v;,) + 4(v;l) = 4`I(v), which is
the J = 4 case of the general result
64:7:10v;j1J";(v)
J=1.2,3,...
As u increases, the following values are approached 0 64:7:11
{(v;x) =
-x
v>l v 1, the parameter is increased, if necessary, until it exceeds the order v. If v < 1. the parameter is invariably adjusted to he in the range from I through 2. Input V > Setf = 16"
Storage needed: v, f, u. k, r, Z, j, p, q and ho, h h2, ... hs9, hw
Input u >
If (v - 1)(u + wI - v) = 0 go to (10)
Setf=k=0
(I) Replace fbyf+u Replace u by u + I If v < I go to (6)
If u 0
1
659
THE HURWITZ FUNCTION
64:10
(3) Replace t by t(v + 2k)(1 - v - 2k)/(21ru)2 Replace k by k + I
Ifk:55 go to (7) Set Z = I + (I + 4-*)(4-' + 9-') + 25-* (4) Set hf = hk_ i + tZ
Setq=0 Setj = k
(5) Set p = h1_,
Ifh;*pgoto(8) Seth,-, = 1099
If p * l099 go to (9) Set h2_, = q Go to (9)
(6) IfuS2goto(2) Replace u by u - 1 Replace f by f - u Go to (6) (7) Replace Z by 10 r2Z/(504 - 301k + 74k2 - 6k3) Go to (4) (8) Set h;_, = q + I 1 /(h; - p)] (9) Set q = p Replace j by j - 1
Test values:
402;0) 1\
0;ZI=0 ?;(10;10) = 1.69268613 x 101°
Ifj+0goto(5)
If k * 60 go to (3) Replace f by f + [ I /(2u")] + [h°u'-'/(v - 1)] (10) Outputf
2= 0.00384424603
l
6:4)
f = C(v;u)
0.306886545
The algorithm returns 1090, instead of infinity, if v = I or if v a 0 = u. Otherwise. the algorithm generally has a precision of 24 bits although this may be degraded near the zeros of the Hurwitz function or at extreme values (say Ivj > 30) of the order.
64:9 APPROXIMATIONS A useful approximation to the J = 0 version of expansion 64:6:4 is provided by its first three terms: I
64:9:1
f,(v;u)
-u"
2u
I
[v - 1 + I + 6u]
8-bit precision
u>
(v2 - I)(v2 + 2v) I
3
Bear in mind, however, that the expansion is asymptotic and is not itself always accurate.
64:10 OPERATIONS OF THE CALCULUS Single and multiple differentiation with respect to the parameter obey the formulas a 64:10:1
au a. 64:10:2
au0
0v;u) _ -0V + 1;u)
Vv;u) _ (-1)"(v). {(v + n:u)
v0
THE HURWITZ FUNCTION ;(v;u)
64:11
660
Differentiation with respect to order gives 64:10:3
v>l
l(J (+ u),
d
1-0
and there is also the particular value 64:10:4
a W;u) = lnl r(u)
v = 0
I
for zero order.
Formulas for indefinite integration include 64:10:5
;(v r)dt =
r.
v-
v>2
1
IY;(v;r)dt=;(v- I;u)-;(v- 1)
64:10:6 (
J
64:10:7
v>
Input u >>
b»»
Storage needed:
v. u, j and f
r10u+2
Set j = 2 Int`
Input restrictions: u > 0; v must be large and positive.
2
Set f=-(u+j)" 1
(1) Replace jbyj-2 Replace f
by f - (u + j + 1)
Ifj*0goto(I)
+ (u + j)-" Test values: 1l(5;8) = 1.979 x
Output f
f = q(v;u)
:««
10-5
rl(8;0.2) = 390624.769
is based on definition 64:13:1 and uses the properties of alternating series [Section 0:61 to ensure an absolute accuracy of 6 x l0-8. Although the algorithm is valid for all positive v and u, it is tediously slow unless v is large. For orders that are not large and positive one may use identity 64:13:3 or 64:13:4 and the algorithm in Section 64:8.
64:14 RELATED TOPICS The differintegrals [Section 0:101 of periodic functions can be expressed as integrals involving the Hurwitz function, the integration range being a single period. The property that permits this simplification is illustrated by the transformation /-1 p /-I r JP (j + P) Idt r = u - jP 64:14:1 u" per(u)du f'.P-'p u"per(u)du = P. I'per(t) { J J l j°0 /-p
in which per(x) is a periodic function [Chapter 361 of period P. Differintegration with a lower limit of -x is especially facilitated by the Hurwitz transformation. Such diffcrintegrals are sometimes known as Weyl dijferintegrals. We report formulas for the two ranges, 0 < v < I and - I < v < 0, that embrace the important semidiffcrentiation and semiintegration cases 64:14:2
d° per(x)
[d(x + x)]" d" per(x)
=
=
I
P"1'(-v)
J
[per(x - Pt) - per(x)1 C(1 + v;t)dt
0