Special Functions of Mathematical Physics and Chemistry

Special Functions of Physics and Chemistry IAN N. M* C. QI O$c fRs •rrfj i)c OLIVIA ANO SOYC SPECIAL FUNCTIONS ...

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Special Functions of

Physics and Chemistry IAN N.

M*

C.

QI

O$c fRs

•rrfj i)c

OLIVIA ANO SOYC

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS AND CHEMISTRY

UNIVERSITY MATIIEMATICA I, TEXTS EI)ITORS

ALEXANDEIt C. AITNEN, D.Sc., F.R.S. I)ANJEL E. RUTI1ERFOItI), D.Sc., Dit. MAr11. DETERMINANTS ANI) MATRICES STATISTICAL MATHEMATICS • WAVES • ELECTRICITY l'no.j ECTIVE GEOM h:TIOY INTEGRATION PARTIAl. I)i FFEILENTIATION

•

• •

Prof. A. C. Aitken, i).Se., F.It.S. Prof. A. C. Aitkcn, I).Sc., F.it.S. Prof. C. A. Coulson, I).Sc., F.it.S. Prof. C. A. Coulson, l).Sc., F.lt.S. 'F. E. Faulkner, Pli.I). . . it. P. Gillespie, Pli.i). . . it. P. Gillespie, Ph.D. . . Prof. J. M. Ilysiop, l).Sc. .

.

INFINITE • INTEGRATION OF ORDINARY 1)IFFF:IIEN1IAI, EQUATIONS

E. L. Iiice, I).Sc.

TIlE ThEORY OF FINITE

INTROI)UCTION To

W. Le(lermann, Ph.D., I).Sc. ANALYTICAL GEOMETRY OF TIl REE

II. M'Crea, Ph.D., F.1LS.

Prof.

FUNCTIONS OF A \TARIAHI.E E. G. Plullips, M.A., M.Se. CLAssIcAl. MECHANICS . . D. E. Rutherford, D.Sc., Dr. Math. \'ECTOR METHODS . . D. B. Rutherford, D.Sc., Dr. Math. VOLUME AND INTEGRAl. . . Prof. \V. \V. Rogosinski, Plt.I). SI'ECIAI. FUNCTIONS OF MA°I'HEMATICAI. PIIYSICS ANI) TENsoR CALCUI.UH . 'L'IIEOIY OF EQUATIONS

.

.

I'rof. I. N. Sneddon, M.A., 1).Sc. Barry Spain, B.A., M.Sc., Ph.1). .

Prof. II. W. TLIrIlhull, F.lt.S.

Ill I'rcparalion TIIE0nY

Onni NARY DIFFERENTIAl. EQUAIIONS

J. C. ilurkill, Sc.D., F.R.S.

GER31AN.ENGI.IsII MAThEMATICAL VOCAUUI.ARY

ToroI.OoY

.

.

.

.

.

.

S. Mllcintyre, M.A., Ph.D. E. M. Patterson, Ph.D.

SPECIAL FUNCTIONS OF MATHEMATICAL

PHYSICS AND CHEMISTRY BY

IAN N. SNEDDON M.A.,

D.Sc.

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY COLLEGE OF NORTH STAFFORDSHIRE

OLIVER AND BOYD EDINBURGH AND LONDON NEW YORK: INTERSCIENCE PUBLISHERS, INC.

1956

PIRST EDITION

1950

PRINTED III HOLLAND BY XONINKUJRZ VERENIODE DRUKEERIJEN HOITEKEA W.V., GRONINGEN NOR OLIVER AND BOYD LTD. EDINBURGH

PREFACE

This hook is ifltefl(1C(l

for the student of applied

matheinat ics, physics, chemistry or engineering who wishes to USC t lie 'special' l'ii net ions associated with the iiamcs of Legendre, Bessel, lIerni(e and Lagtierre. It aims at ing in a compact form most of the propei't ics of' these functions which arise most frequent lv in applicat IOIIS, and at establishing these properties in the simplest 1)oSsiI)le way. For that. reason the methods it employs sholil(l be intelli— gible to anyone who has eoml)Icted a first course in calculus and has a slight acquaintance with t lie theory of (lifferentml equations. I Ise is ma(1e of the theory of functions of a corn—

plex variable only very sparingly, and most. of (lie book should be accessible to a reader who has no knowledge of an attenil)t is made to this theory. Throughout the show how t hese funet ions may he used in the discussion problems in classical physics and in quantum theory. A brief account is given iii an appen(liX of the main

ties of the Dirac delta 'function'. I should like to record my debt of gratitu(le to the late Sir John Lennard-.Jones, and to my colleagues Mi'. 13. Noble and I)r. J. G. Clunic fom' their generous help in reading the

first draft of the manuscript and making valtial )le sugges-

tions for its improvement. I am indebted to Miss ,Janet l3urchiiall for her assist aiice iii the pI'eJ)Lrat ion of the final

manuscript, to Mr. .J. S. Lowndcs for help in correcting proof shcet;s and to Miss Elizabeth Gildart for preparing the index. I should also like to thiatik I)r. D. E. Rutherford,

general editor of the series, for his advice and criticism throughout the preparation of' the 1)00k. My debt of gratitude to Professor 'I'. M. MacRobert is a much more general one. It was at his lectures that I first

vi

PRØACE

acquired a taste for the subject, and it will be obvious to

anyone who knows his published writings how much I have

been influenced by them. STAPPOEDSIIIUE.

20th August, 1955

CONTENTS i

INTRODUCTION PAGE

1. The Origin of Special Functions 2. Ordinary Points of a Linear Differential Equation 3. Regular Singular Points

4. The Point at Infinity 5. The Gamma Function and Related Functions Examples

1

4 6 9 10 14

CHAPTI'.R 11

HYPERGEOMETRIC FUNCTIONS

0. The Hypergeometric Series 7. An Integral Formula for the Ilypcrgcoinetric Series 8. The Hypergeometric Equation

18

20 23

9. Linear Relations between the Solutions of the Ilypergeonietric 28 Equation

10. Relations of Contiguity 11. The Confluent Ilypergeometric Function 12. Generalised Hypergeometric Series Examples

81

32 36 39

CHAPTER III

LEGENDRE FUNCTIONS 18. Legendre Polynomials 14. Recurrence Relations for the Legendre Polynomials 15. The Formulae of Murphy and Roderigues 16. Series of Legendre Polynomials 17. Legendrc's Differential Equation 18. Neumann's Formula for the Legendre Functions 19. Recurrence Relations for the Function

20. The Use of Legendre Functions in Potential Theory vii

46 52 53 57 60 65 69

70

viii 21.

CONTENTS

Legendre's Associated Functions

PAGE

78

22. Integral Expression for the Associated I.egendrc Function 79 28. Surface Spherical harmonics 80 24. Use of Examples

Legendre Functions in Wave Mechanics

88 85

ChAPTER IV

BESSEL FUNCTIONS

25. The Origin of Bessel Funet ions 26. Itceurrence Ilelations for the Bessel Coefficients 27. Series Expansions for the IIes4cl Coefficients 28. Integral Expressions for the Bessel Coefficients 29. The Addition Formula for the Bessel Coefficients 80. Bessel's Differential Equation 81. Spherical Bessel Functions 82. Integrals involving Bessel Functions 88. The Modified Bessel Functions 84. 'I'lie 11cr and Functions 85. Expansions in Series of Bessel Functions 86. The Use of Bessel Functions in Potential Theory 37. Asymptotic Expansions of Bessel Functions Examples

91

94 97 10() 101

102 108 110 113 117 119 121

124 127

CHAPTER V

TILE FUNCTIONS OF IIERMITE AND LAGUERRE 88. The Hermite Polynomials 89. llermite's Differential Equation 40. llermite Functions 41. The Occurrence of Ilermite Functions in Mechanics 42. The Polynomials 48. Laguerre's Differential Equation 44. 'I'he Associated Polyne)mials mine! Functions 45. The Wave Functions for the hydrogen Atom Examples

182 184 186 140 142 145 147 150 155

TIlE 1)IRAC I)ELTA FUNCTION

46. The Dirac Delta Function

159

CHAPTER I

INTRODUCTION

The Origin of Special Functions. The special functions of mathematical physics arise in the solution of partial differential equations governing the behaviour of certain physical quantities. Probably the most frequently occurring equation of this type in all physics is Laplace's equation 1.

V2tp=O

(1.1)

by a certain function descril)ing the physical situat ion under discussion. The mathematical problem satisfied

consists of finding those functions which satisfy equation (1.1 ) and also satisfy certain prescribe(l conditions on the surfaces bounding the region being considered. For exaniple, if denotes the electrostatic potential of a system, ip will

be constant over any conducting surface. The shape of these boundaries often makes it desirable to work in curvilinear coordinates q1, q2, q3 instead of in rectangular cartesian coordinates x, y, z. In this case we have relations (1.2) x = x(q1, q2, q3), y = y(q1, q2, q3), z = z(q1, q2, q3) expressing the cartesian coordinates in terms of the curvilinear coordinates. If equations (1.2) are such that

ax ax

ay ay

a:

——±——----t-——=o eq1 eq1 eq, aq1 j we say that the coordinates q1, q2, q3 are thogonal curvilinear coordinates. 1) The element of when i

1) 1). B. Rutherford, pp. 59—63.

Vector Methods, (Oliver & Boyd, 1989)

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

2

§I

length dl is then given by

+

d12 =

+

dqj

(1.3)

where

a and

v2, —

2

2

a

it can easily be shown that a

1

h1h2h3 1 aq1\

h1 8q1J +

a 1h3h1

a 1h1h2

h2 aq2J + aq3 \

/z3

aq3

One method of solving Laplace's equation consists of finding solutions of the type

= by

substituting from (1.5) into (1.1). We then find that a 1h3h1 aQ2\

a 1h2h3

i

Q1(q1)Q2(q2)Q3(q3)

Q1

h1 aq1)

'

h2 aq2I + Q3

Q3

a 1h1h2 '..

— —

If, further, it so happens that h2h3

etc.,

— /1(q1)

1(q2, q3)

then this last equation reduces to the form 1

d

d

I

+ F3(q1,

1

dr

dQ3i

=

0.

in certain circumstances, it is possible to find three functions g1(q1), g2(q2), g3(q3) with the property that Now,

F1(q2, q3)g1(q1) + P2(q3, q1)g2(q2) + P3(q1, q2)g3(q3) = 0.

When this is so, it follows immediately that the solution of Laplace's equation (1.1) reduces to the solution of three

INTRODUCTION

§I

3

seif-adjoint ordinary linear differential equations

(i=1, 2,8).

(1.6)

It is the study of differential equations of this kind which

leads to the special functions of mathematical physics. The adjective "special" is used in this connection because here we are not, as in analysis, concerned with the general

properties of functions, but only with the properties of functions which arise in the solution of special problems. To take a particular case, consider the cylindrical polar coordinates p, z) defined by the equations

z=z

x=ecosp,

1, h2 = h2 = 1. From equation (1.5) we see that, for these coordinates, Laplace's equation is of the form

for which h1 =

av'

1

+ ae2

e

+

1 a2v

+

— az2 —

1

If we now make the substitution

=

(1.8)

we find that equation (1.7) may be written in the form 11d2R R kd02 This

+

ldR\ e del

shows that if

I thp2

+

id2Z =

Z dz2

Z, B satisfy the equations

+ n2tli = —

+

0,

m2Z = 0,

(1.9a) (1.9b) (1.Oc)

4


rCS1)CetiVCIV,

§2

then the function (1.8) is a solution of Laplace's

equation (1.7). The study of these ordinary (lifferential e(luations will lead us to t lie special In nctions appropriate

to this coordinate system. For instance, equation (1 .9a) may be taken as the equation (1('fining the circular functions.

In this context sin (nq) iS (kliIIC(1 as that solution of (1 .9a) which has value 0 when = 0 and COS (iup) as that which has value I when = 0 and the properties of the fLinctions derive(l tliercfroni, ci. cx. 4 below. Similarly equal ion (1.91)) (lefines the eXl)onent nil hitict ions. In actual

pi'actice we do tiot proeec(l in this way ziierely because we have already elirollntere(l these functions in another context and from their fain iliar properties studied their relation to (quat ions (1 .9a ) and (1 .9h ). The situation with respect to equal ion (1 .9c) is different; We ('aflhlot its solution in terms of tue elenien tarv in netions of analysis,

as we were able to (10 with the other two equations. In this ease we define new fiinct ions in terms of the solLit ions of this equation and by invest igat ihig t lie series solutions of' the equal ions derive the propei't ics of I lie fLlnct ions so defined. Equation (1 .Oc) is called Bessel's equation and

solutions of it are callcd Bessel functions. Bessel functions are of great importance in theoretical physics; they are discussed in Chapter IV below. 2. Ordinary Points of a Linear Differential Equation. We sl ial I have occasion to (1 iseliss Or(linarv linear (Ii fI'eren— I ial e mat ions of the second order Lb varial)le coefficients SolLit 10115 (ahhhlot 1)e Ol)taihie(i in terms of I lie elenien—

tars' functions of iiiat hicmal ical analysis. In such cases one of the standard l)ro(c(ILlres is to derive a pair of linearly indcl)ehl(lenl sohi tions in the foriii of' infinite series and from these Series to eoml)Lhte tal)les of standard solutions. With the ai(i of' such tal)les the sohiitioii appropriate to given initial con(hit ions may then he read ilv found. 'I'he object of this tiote is to outline 1)riefly the procedure to be followed in these instances; for proofs of the theorems

INTRODUCTION

5

quoted the rea(lCr is referred to the standard textbooks.')

A function is called analytic at a point ii

it iS 1)OSSiblC

to expand it in a Taylor series Valid in SO1UC neighbourhood

of the point. This is equivalent to saying that the function derivatives of all orders at is single-valued and the point in question. In the equations we shall consider

the coefficients will be analytical functions of the independent variable except at certain isolated points.

An ordinary point x =

equation

a

of the Second order differential

=

+

y" +

(2.1) are analytical fuunctions 0

is one at which the coefficients It can be shown that at any ordinary point every solution of the equation is anal!/lic. Furthermore if the Taylor ea'pan—

sions of a(x) and fi(x) are valid in the range x — a < R of the solution is valul for the same the Taylor and fl(x) are polynomials range. As a consequence, if in x the series solution of (2.1 ) is valid for all values of x.

When, as is usually the case, a(x) and fl(x) are poiynomials of low degree, the solution is most easily found series of the form by assuming a y

=

—

(2.2)

for the solution and determining the coefficients c0, e1, c2,

by direct substitution of (2.2) into (2.1) and equating of x to zero. coefficients of successive The simplest C(lIIiktiOII this type is

y" ± y =

(2.3)

0.

Substituting a solution of the type (2.2) with a =

0

into this

1) See, for example, E. L. Inee, Ordinary Differential Equations, (Longmans, 1927), Chpt. VII; E. Goursat, A Course in

rlnalysis, Vol 11, Part II, (Gum, 1004), Clipt. III. See also J.

C.

Burkill, Theory of Ordinary Differential Equations (Oliver & Boyd;

to be published shortly).


6

equation

we find that, if the equation is to be satisfied, = 0.

— 1)CrXr_2 +

The

§3

series on the left is equivalent to (r + 1 )(r

+

that, equating coefficients of Xr, we see that the equation

so

is satisfied by a solution of type (2.2) provided that the coefficients arc connected by the relation

(r + l)(r + 2)Cr+o +

Cr

(2.4)

0.

determined by the prescribed and the others are determined by

The coefficients c0, C1 are

values of y, y' at x =

0,

equation (2.4). From this relation it follows that the solution is

(2.5)

An equation of the kind (2.4) which determines the subsequent coefficients in terms of the first two is called a

recurrence relation. 3.

Regular Singular Points. If either of the functions

ifix) is not analytic at the point x = a, we say that this point is a singular point of the differential equation. %SThen the functions are of such a nature that the differential equation may be written in the form (x — a)2y" + (x — a)p(x)y'

+

q(x)y

=0

(8.1)

where p(x) and q(x) are analytic at the point x = we

a

say that this point is a regular singular point of the

differential equation. If x = a is a regular singular point of the equation (8.1)

it can be shown that there exists at least one solution of the form

=

Cr(X — r-O

(8.2)

INTRODUCTION

43

7

is valid in some neighbourhood of x = a. More specifically, if the Taylor expansions for p(x), q(x) arc valid for x — a < R, the solution (8.2) is valid in the which

same range. Putting — a)t, q(x)

— a)t

(8.3)

and substituting the expansions (3.2) and (3.8) into (3.1) we see that for the equation (8.1) to be satisfied we must have + r)(p + r— 1)(x— a)Q+t + r)(x—a$'+"

+

— a)Q+t

—

=

0.

(8.4)

Equating to zero the coefficient of (x — a)Q we have the relation — 1)

so that if c0

+ p0c0

+ q0c0 = 0

0 we have the quadratic equation e2

+

(Po —

')e + q0 = 0

(8.5)

for the determination of This is known as the indicial equation. Similarly if we equate to zero the coefficient of (x — a)Q+? we obtain the relation

+

r

r — s)

=

+

0

which may be written in the form 1)

+ Po(Q+r)+ q0} 0.

(8.6)


8

If

Equation (3.5) gives the two possible values

we take one of these values,

§3

of

say, and substitute it

in the recurrence relation (3.6) we obtain the corand hence the

responding value of the coefficients solution

— arQi.

Y1(x)

In a similar way the root 02 of the indicial equation leads

to the solution Y2(x)

=

— a)'+Qa

distinct cases arisc according to the nature of the roots of the indicial equation. Three

Case (i)

— 02

neither zero nor an integer.

In these circumstances the solutions y1(x) and p2(x) are linearly independent and the general solution of equa-

tion (3.1) is of the form — a)t*Qa.

— a)"+Qi

Case

If

=

(ii)

=

(8.7)

02

the solutions y1(x) and y2(x) are identical

(except, possibly, for a multiplicative constant). The

general solution of the equation can be shown to be Yi(X) + y2(x) where y1(x) = (x —

Cr(X —

a)Qi

(r)

(3.8)

(x_a)r.

r-O Case

(iii)

02 =

01

— n where ii is a positive integer.

In this case all the coefficients in one of the solutions

INTRODUCTION

§4

fi'om

9

some point onwards arc either infinite or indeter-

minate. It can be shown that the appropriate solutions arc y1(x) = (x — a)Qi

Cr(X —

(39) log (x — a) + (x —

y2(x) =

r=O

where

is the coefficient of x" in the expansion of a))2

CXI)

[— i: up(u)duIJ.

It may happen that bU = 0 in which case y2(x) does not contain a logarithmic term. we wish The Point at Infinity. In many to find solutions of clilTercntial equations of the type (8.1) which are valid for large values of x. We seek solutions in 4.

the form

of infinite series with variable

If we make

the transformation 1

the 'point at infinity' is taken into the origin on the i-axis. With this change of variable equation (8.1) becomes (4.1) — If analytic in

then E =

0

0 and arc both 0(1) as is an ordinary point of equation

is an ordinary point of (4.1) and we say that x = equation (2.1). Returning to the original independent

variable we see that the condition for the point at infinity

to be an ordinary point of equation (2.1) are that

=

+ 0(x2), fl(x) = O(x4) as

(4.2)

10 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

The

§5

corresponding solutions arc of the form

= Similarly if, as x -* (4.8) arc constants we say that the point at infinity is a regular singular point of the equation (2.1). The cor• indicial equation is

where

0.

(1

If the root,s of this equation are

the solutions of (2.1)

valid for large values of x are of the form (4.4)

Yi(x)

The gamma function and related functions. In developing series solutions of differential equations and in other formal calculations it is often convenient to 5.

make use of properties of gamma and beta functions. The integral

=

(5.1)

converges if n>

0 and defines the gamma function. Similarly is in> 0, n> 0 the beta function is defined by the equation

B(,n,

It

ii) =

f

xm1(1 — x)"_'dx

(5.2)

is then easily shown that 1) (i)

1'(i)=l

(ii) 1'(n -f- 1) = nI'(n) 1) For proofs of these results the render is referred to II. P. Gillespie,

Integration, (Oliver and Boyd), 1951, pp. 90—95.

INTRODUCTION

11

(iii) 1'(n + 1) = n! if n is a positive integer, (iv)

B(m, n)

= 2 Jo1

(v)

B(m, n)

= r(m

(vi) (vii) (viii)

f(p)1'(i

— p)

sin2m_l 0 cos2"1 0 dO,

J'(rn)l'(n) —f- ii)'

=

cosec (pat),

0 < p < 1, — the

duplication

formula,

(ix) f(z+1)=lim

nI

(z+ 1 )(z + 2)... (z + n)

(z>0).

When n is a negative fraction 1'(n) is defined by means of

equation (ii); for example

—ï = (—ï)(—ï)3 3

$

=8

By means of the result (ix) we cami derive an interesting expression for Euler's constant, y, which is defined by the equation

y=

urn

(1 +

+ ... +

!

— log ii)

= 0.5772...

(5.8)

From (ix) we have urn (log

1

1

1

so that letting z -+ 0 we obtain the result —

1)]

(5.4)

and from (5.1) we find — IJo

etlogtdt.

(5.5)

12


§ 5

Integrating by parts we see that

Iogz+ so that

—y= iim(f°çu+iogz).

(5.6)

Closely related to the gamma function are the exponential-integral ei(x) defined by the equation ei(x)

f — du

(x> 0),

(5.7)

and the logarithmic-integral 11(x) defined by

li(x) =

I —,u

J0 log

(5.8)

—1

Fig. 1 Variation of Ci(z) and Si(x) with x.

which are themselves connected by the relation (5.0) ei(x) = — Other integrals of importance are the sine and cosine integrals Ci(x), Si(x), which are defined by the equations

INTRODUCTION

§5

Ci(x) =

—jS

U

dii,

.

Si(x)

=j

13

du

(5.10)

whose variation with x is shown in Fig. 1. In heat conduction problems solutions can often be expressed in terms of the error-function and

erf(x) =

(5.11)

whose variation with x is exhibited graphically in Fig. 2.*

—x Fig. 2

Variation of erf(x) with z.

Similarly in problems of wave motion the Fresnel in-

tegrais

* A. C. Altken, StatisticalMathematics, (Oliver & Boyd, Seventh Edition, 1052) p. 62 gives a short table of values of erf (x).

14


C(x)

S(x)

= s:cos (3iru2)du,

du

=

§5

(5.12)

The variation of these functions with x is shown in Fig. B.

occur.

2

—

:3

FIg. 8 Variation of the Fresnel integrals, C(x) and S(x), with x.

The importance of these functions lies in the fact that it is often possible to express solutions of physical problems in terms of them. The corresponding numerical values can

then be obtained from works such as E. Jahnke and F. Emde, 'Funklionenta/eln' (Teubner, Leipzig, 1983) in which they are tabulated. EXAMPLES 1.

Show that, in spherical polar coordinates r, 0, p defined by

x=

y=

Laplace's equation becomes

z—

rcos0,

INTRODUCTION

!

15

! (sin

+

=

bO/ brl sIn 0 bO \ sin2 0 and prove that it possesses solutions of the form

?.r \

0),

where ecu) satisfies the ordinary (Lifferentlal equation

2.

1—p

dp

d,u2

}Oo

Show that If

a=

a

cos ,j, y

cosh

=

a

slnh

sin

a

a

Laplace's equation assumes the form

+

+

=

—

Deduce that it has solutions of the form

where f(rj)

satisfies the equation

=

+ (0+ IOq con

0

In which 0 is a constant of separation and q

3.

Parabolic coordinates

z=

0.

—

a'y'/82.

are defined by

ij,

y=

5=

Show that in these coordinates Laplace's equation becomes

Prove

that if

is a solution of the equation dF I + In a— +— \ — dx' dx d2F

then

is

rnt\

—I F = 0

4x/

a solution of Laplace's equation.

4. Defining cos a, sin a to be the solutions of

+

=

dx' y which respectively are 1, 0 when a

0

0, prove


(i) cos(—x)=cosx, sln(—x)= —sinx;

(ii) cos(x + x') = cos x cos a,' — sin x sin x'; (iii) sin(x + a') = sin a cos a' + cos a sin a'; (iv) cos' a + sin' a 1;

(v)

=

dx

i(cosx) = —ama.

coax,

dx

The only singularities of the differential equation y" + p(x)y' + q(x)y = 0

5.

regular singularities at a = I of exponents a, a' and at a = — 1 of exponents fi, fi' the point at Infinity being an ordinary point. Prove are

that fi = — a, fi' = — a' and that the differential equation is

(x'—1)'y"+2(x—1)(x—a—a')y'+4aa'y=0 Show

that the solution is

= where 6.

Ix —

Ix —

fla'

c1 and c, arc constants.

Apply the method of solution In series to the equation dx'

showing

that, near a

series and v = 7.

+

C1

+

(a — a)

dx

—

y

=0

0, y = Au + Dv where (a is not an Integer).

u Is a Maclaurin

Find two solutions of the equation

(a' + in the form

dx'

dx

— k(k + 1)y = o

0

y=E n—O

Show

that, if k is a positive Integer, one of these solutions is the

polynomial I

1)! (2x)" (Ic — n + 1)! (2n)!

17

INTRODUCTION 8.

Prove that if 8

is

an Integer and a is fractional — a)

na — 9.

—a+s)

Show that (I)

(a-.1)(a),_$(a—1)R

(ii) (a),., = (—

(iii)

(t&_i__)!

(1 — a —

(._..

where

10. Prove that

as

ei(x)= and deduce that as

51(a) = a

—

as as — +— — 5.5! 8.8!

al

CIIAPTEIt II

HYPERGEOMETRIC FUNCTIONS 6.

The Hypergeometric Series. The series (6.1)

is of great importance in mathematics. Since it is an ob. vious generalisation of the geometric series

it is called the hypergeometric series. It is readily shown that, provided y is not zero or a negative integer the series is absolutely convergent if x < 1, divergent if x > 1, while if x = 1 the series converges ab. — 1, solutely if y > + *) It is convergent when x = provided that y> + fi — 1. If we introduce the notation (6.2) we

may write the series (6.1) in the form fi; y; x)

=

(6.8) ,-_o

the suffixes 2 and I denoting that there are two parameters

and one of the type y. We shall gencralise this concept at a later stage 12 below) but it is advisable

of the type

at this stage to denote the 'ordinary' hypergeometric function by the symbol 2F1 instead of simply F, if we are •) See J. M. Ilyslop, infinite Series, Fifth Edition, (Oliver & lloyd, 1954) p. 50. 18


§6

19

avoid confusion later. From the definition (0.3) it is obvious that to

2F1(fl, cc; y; a,) =

2F1(cc,

fi; y; x).

(6.4)

of the hypergcomctric series fol-

A significant

lows immediately from the definition (6.8). We have

P; y; x)

r

(r (cc

—

r+1 8 r+la,r

r-O

Now

+ I),. the right hand side of the last equation becomes (OC)r+i = cc(oc

so

ocj9

Vr—O

showing

(cc+1)r(fl+ ')ra,r

r!(y+1)r

that

T J1(cc, 8; v; =

2F1(cc+

8+1; y+ 1; x). (6.5)

It should also be observed that 2F1(cc, 8; y; 0) = 1

(6.6)

so that

[$-0F1(x,fl;y;

(6.7)

Several well-known elementary functions can be expressed as hypergeometric series; examples of them are given in cx. I below. It should be noted that, if we adopt a certain convention,

a hypergeometric series can stop and start again after a number of zero terms. For example, consider the hypergeometric series 2F1(—n; b; —n—rn; x) where both in and

_________


20

§

7

ii are 1)OSitiVC integers and b is neither zero nor a negative integer. Because of the occurrence of (—n)r in the numera-

tor in the expansion in powers of x it is obvious that the

+ )-th. term of the expansion will 1)c zero, and we are tempted to think that ever)' subsequent term is also zero. (n

If we note that, as a result of cx. 9(iii) of Chapter I, (—ii)

(i7i)r

(-4.-)! ('n—m—r)(n—f-in—r—l)

. .

. (n—r—f—l)

(6.8)

when the form on the left is not of type 0/0, and if, further,

we assume that it still has the value on the left when it is indeterminate, we see that we may write 2F1(— ii, b; — n — m; x)

r

r r \ 60 n+m—iJ n+1J r! that although the series stops at the n-th. term it starts

— r—O

so

n+?n)k

"

up again at the (n+m+1 ).th. term. For instance, 2

0F1(—2,1; 7.

1

2

1

An Integral Formula for the Hypergeometric

Series. In order to (lerive some further properties of the hypergcomctric series we shall first of all establish an expression for the series in the form of an integral. It is readily shown that

B(fJ+r, v—fl) = (Y)r

B(fl,

v—fl)

1

1(

B(fl, y—ThJo

—

tfi+r_1 dl

from which it follows that fi; v;

J(fi,

[3)

Interchanging the order in which the operations of sum-


§7

21

mation and integration are performed we see that 2F1(x,

j9;

y; x) (1—t)Y—P—'

= B(f3,

(xt)f} di.

iP' {

Using the fact that r—O

we have the integral formula /3; y; x) 1

(7.1)

.10 (1— = B(fl, valid if x < 1, y > /3> 0. The results hold if x is complex provided that we choose the branch of (1 —

in

such a way that

(1 —

—* 1

as I -÷ 0 and

The first application of (7.1) is the derivation of the value of the hypergeometric series with unit argument. Putting x = 1 in(7.1) we have 2F1(oc,

1)

jP1 dI

= B(fl, y—fl) s: (1— —

B(fI,

y — — /3> 0, /3> 0. If we express the beta function in terms of gamma functions we have Gauss' Theorem

(7.2)

/3; y;

Now if

= — n, a negative integer, we have — /3)

—

p

—

22


so

that equation (7.2) reduces to

§

7

'v,n

2F1(—n, j9, y, 1)— (y—19)n which is known, in elementary mathematics, as Vander-

monde's theorem.

Again, if we put x = from equation (7.1)

—1

and =

=

value is

1—

in this

jPl dt.

integral we see that its

Using this result and the relation

= r(1 + 4fl)/F(1 + /1) we have Kummer's

theorem 2

+ fi — v we have,

f'(l —

2F1(x, fi; fl—x+1; —1) =

If we write

1

1', —1) —

F

73

—

Further we can deduce from the formula (7.1) relations between hypergeometric series of argument x and those of argument x/(x — 1). Putting r = 1 — I in equation (7.1), and noting that x

= (1—

{i — x(l —

we see that 2F1(oc, fi; y; x)

= = B(fl, y_fl) whence 2F1(oc,

_r)fl1 T?41 { 1

—

—fi; y;

1

x

we have the relation v;

=

(cx, y—p; y; a,

(7.4)

j8


and,

23

by symmetry, the relation

2F1(cx, fi; y; x) = (1—x)P2F1

tV

fi; y;

(75)

)

Using the symmetry relation (6.4) and equation (7.4) with x replaced by x/(x —

1)

we see that

)

=

(1—

—

y; x),

fi, y —

so that x) = If we put x = in equation (7.4)

y—fl; y; x). (7.6)

2F1@, fi; y;

fi;

=

we

obtain the relation

v—fl; y; —1).

The series on the right hand side of this equation can be derived from equation (7.8) provided either that

or that We then obtain the formulae — r(4y) F(4y +4) 2

F

1

fi;

8.

.

Y'

4fl+ 4; 4)

=

.

7)

(7.8)

The Hypergeometric Equation. In certain pro-

blems it is possible to reduce the solution to that of solving

the second order linear differential equation (8.1)

in which

fi and y are constants. For instance, the Schrö-

24

THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 8

(linger equation for a Symmetrical-tOl) molecule, which is

of importance in the theory of molecular spectra,') can, by simple transformations, be reduced to this type. An equation of this type also arises in the study of the flow of compressible fluids. In addition certain other differential

equations (such as that occurring in cx. 1 of Chapter I) which arise in the solution of boundary value probletus in mathemaical physics can, by a simple change of variable, be transformed to an equation of type (8.1). Indeed it can

be shown that any ordinary linear differential equation of the scCon(l or(ler whose only singular points arc regular singular points, one of which may be the point at infinity,

can be transformed to the form (8.1). For that reason it is desirable to investigate time nature of the solutions of equation, which is called the hypergeometric equation. \\Te may write the hypergeomnetric equation in the form

so

that, in the notation of §

3,

we see that near x =

0

= y, q0 = the indicial equation is 02 + — ')o = 0 with roots = 0 and = I — Similarly, the equation can be put in the form (x—1 )2y"—(x—l) )(x—1)+ . an(l

+ zfl(.x— 1) {1— (x— 1) +

..

.

.) y =0,

with indicial equation

02+ of which the roots are

0,

=

=——

Finally in the notation of § 4 we have for large values of x 1)

See, for example, L. Pauling and E. 13. Wilson, Introduction to

Quantum Mechanics, with Applications to Chemistry, (McGraw-hill,

New York, 1085), pp. 275—280, and cx.

10

below.


§8

25

c41

x and

x

so the indicial equation appropriate to the point at

infinity is

—

= 0,

+ fl)i +

with roots Thus the regular singular points of the hypergeometric equation are:(i) x = 0 with exponents 0,1 — y. (ii) x = with exponents fi. (iii) x = 1 with eXpOnents 0, y — — j9. These facts are exhibited symbolically l)y denoting the most general solution of the hypergeometric equation by a scheme of the form oc

y=P

0

co

1

0

x.

0

(8.2)

13

symbol on the right is called the tion of the equation. We shall now consider the form of the solutions in the neighbourhood of the regular singular points. (a) x = 0: Corresponding to the root = 0 we have a solution of the form The

rO Substituting this series into equation (8.1) we obtain the relation (1 —x)

c,r(r—

+ {y — (oc + fJ+ I which

rvr -1 —

=0

is readily seen to be equivalent to )y] —

= 0,

26 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 8

that

so

(r+a)(r±fl) (r+

—

(8.8)

from which it follows that —

8.4)

It follows that the solution which reduces to unity when

x=O is

—1-'-

'

,,11

y = 2F1(x, fi; y; x).

i.e.

Similarly,

(8.5)

if 1 — y is not zero nor a positive nor negative y is

integer, the solution corresponding to the root = 1

where

_y)(r_y)atl' +

(1

_y)xT1'

+{v—

0,

which is equivalent to

1

—y)(r—y)+

1

implying that

(r+c—y+l)fr+fl—v+ 1) (r-+-1)(r+2—y) Comparing this relation with (8.8) and and taking c0 = 1


§8

we

27

see that this solution is

Combining equations (8.5) and (8.6) we see that the general solution valid in the neighbourhood of the origin is

y=A provided

If y =

x), (8.7)

i3—y+l;

fi; y;

that 1 — is not zero or a positive integer. the solutions (8.5) and (8.6) arc identical. If

1,

we write

j9; y; x),

Yi(X) =

and put

= we

y1(x) log x

find on substituting in (8.1), with y = 1, that

r(c+fl+ 1)Cr+

(r+

from which the coefficients c, may be determined. A similar procedure holds when 1 — y is a positive integer. (b)x = 1: If we let = 1 — x, equation (8.1) reduces to — cq9y

—

=0

which is identical with equation (8.1) with replaced by + fi — y + 1, and x by = 1 — x. Hence it follows from equation (8.7) that the required solution is

y=A

2F1(oc, fi;

+ (c)

x=

co:

1—x) y—fl;

Corresponding to the root

1—x). (8.8)

=

we

put


28

§9

which gives

(1— x)Ecr(r+ cc)(r+

1

i9+1)x}

= 0,

—

— 1

whence

it

follows

that

(r+ which

in turn is equivalent to Cr

Taking c0 1;

=1

!).

solution is

—

(OC)r(0Y+l)r

1)r we obtain the solution

From

—y

+ 1;

the symmetry we see that the other !), so that fi — y + 1; fi — + 1;

the required solution is (8.9)

fl—x+1;--).

9.

Linear Relations between the Solutions of the

Hypergeometric Equation. The series in the solution

(8.7) are convergent if x J < 1, i.e. in the interval (—1, 1) whereas those in the solution (8.8) are convergent in (0,2). There is therefore an interval, namely (0, 1), in which all four series converge, and since only two solutions of the

differential equation are linearly independent it follows

that there must be a linear relation valid if 0 < x < 1,

between solutions of type (8.7) and those of type (8.8).


§9

29

Let 2F1(oc, fi; y;

x) = A 2F1(x, j9;

1—x)

+

y—fl;

1—x),

then putting x = 0 we have 1= fi; x+fl—y+1; 1) + B 2F1(y—c', 1

1)

=

we assume that

(9.1)

Substituting for the series with unit argument from equa.

tion (7.2) we see that

and that

+

1—A —

so that —

whence

we find that

2F1(cx, fi;

y;

fi; v—fl;

I —x) 1—x),

(9.2)

provided that the condition (9.1) is satisfied and O<x y> + These

fi.

relations are typical of a larger number which

exist between the solutions of the hypergeometric equation (8.1). If we change the independent variable in this equation

to any one of

1—x,

1

x

1

i—x , -

x—i x

x ,

x—i -

the equation transforms to one of the same type (but, of course, with different parameters). The equation (8.1) therefore has twelve solutions of the types (8.5) and (8.6) — two for each independent variable — each convergent within the unit circle. Any one of these can be expressed in terms of two fundamental solutions. In addition twelve more solutions of the kinds (1_X)Y—2--fl2 F1(y—z v—fl; y; ar),

i—fl; 2—y; x) can be derived. The relations between these twenty-four solutions of the hypergcometric equation are of the types (9.2) and (9.8); for a full discussion of them the reader is


§ 10

31

to T. M. MacRobert, Functions of a Complex l'ariable, (Macmillan, 2nd edition) pp. 298-301.

referred

Relations of Contiguity. Certain simple relations exist between hypergeometric functions whose parameters 10.

differ by ± 1. For example if the parameters and fi remain fixed and y is varied we can prove that 2F1(cx,

y

+

fi;

x)

(y—x)(y—fl)x2F1(cx, j9;

iv)

— y(y—1 )(1—x)0F1(cc, fi; y—1; iv)

0.

(10.1)

The proof follows from the definition (0.3) for the coefficient of in the expansion of the function on the left of (10.1) is (

—1)

—

(9

+ (y—cL)(y—fl) _i_ I

—

v(v—')

i

and it is not difficult to show that this is zero.

In another kind fi

and

y are kept constant and

is

varied. One such is 0F1(c, fi;

+

fi;

iv)

iv) —

fi; y; x) = 0 (10.2)

the proof of which is similarly direct.

In the third type of relation y is kept constant and

and fi vary. One of the simplest among these relations is fi; y; iv)

=

fi; y; iv) — fl2F1(ot, fl-f-i; y; iv)

(10.8)

The proof of these relations is left to the reader; further examples are given below. (exs. 3, 4)

32

11.


The Confluent Hypergeometric Function. If we

replace x by x/fi in equation (8.1) we see that the hypergeometric function fi; y;

is a solution of the differential equation

so that letting

—* co

we see that the function x/fl)

lim0F1(x,

(11.1)

is a solution of the differential equation dy

d2y

(11.2)

1x2+ From the definition of (fl)r we see that 1

so that the function (11.1) is the series (c(\ r—O

Xr

rl

(11.3)

and this series we denote by the symbol y; x). This function is called a confluent hypergeometric tion, and the equation (11.2) is the confluent hyper—

geometric equation. Equations of the type (11.2) occur in mathematical

physics in the discussion of boundary value problems in potential theory, and in the theory of atomic collisions (see examples 13, 14 below). It is readily verified that the point = 0 is a regular point of the differential equation (11.2) and that, in the notation of § 8, Po = y and q0 = 0. The indicial equation is therefore


§ 11

with roots

=

0

Corresponding

the form

33

e(e + y — 1) = 0 and = 1 —

to the root

=

there is a solution of

0

Yi = substituting this solution in equation (11.2) and equating

to zero the coefficient of a? —

Putting c0 =

1

fiiid tllLLt

(x+r)cr

(y+r)(r+

1)

we see that Cr

(a),

1

r! and if y is neither zero nor a negative integer the solution is Y1(X)

= 1F1(x;

x).

(11.4)

Similarly, the root = 1 — y, leads, if 1 is neither zero nor a positive integer to a solution of the type y2(x) =

If

we write Y2fr) =

and

substitute in (11.2) we find that u(x) satisfies the

equation

d¼

du

which is the same as equation (11.2) with y replaced by 2— and replaced by — y + 1. We know from equation (11.4) that the solution of this equation which has value unity when x = 0 is u = 1F1(x — y + 1; 2 — x)

so that

2—y; x).

(11.5)

34


Thus if y is neither 0 nor an integer the general solution

of equation (11.2) is 2—v; x), (11.6)

y(x)=A1F1(ca; y;

where A and B are arbitrary constants. In the exceptional case = 1 we have (11.7) Yi(X) = 1F1(cx; 1; x) obtained simply by putting y = I in equation (11.4). For the second solution we write y1(x)logx

(11.8)

Substituting this expression in equation (11.2) and putting we find that the unknown coefficients

must be such

that 0.

—

Inserting the value of y(x) from equation (11.7) we see that these coefficients are determined by the recurrence relation

c1= 1

(r+I)2cr+i—rcr=

(11.9)

(1

The complete solution is therefore given by y = Ay1(x) +

By2(x) where A and B arc arbitrary constants and the functions y1(x), y2(x) are defined by equations (11.7), (11.8) and (11.9). The complete solution when is an integer may be found by a similar method. If in equation (11.2) we put (11.10) y(x) = arIVeFnIV(x) we find that the function W(.x) satisfies the differential equation

+{—

++

} W(x) =

0,

(11.11)

§ 11


35

we have written k for — and in for — The solutions of this equation are known as Whittaker's where

confluent hypergeometric functions. If 2m is neither 1 nor an integer the solutions of the

confluent hypergeometric equation corresponding to equation (11.11) are given by equation (11.6) with y = 1 + 2m and = — k + in. Thus the solutions of equation (11.11)

are the Whittaker functions Mkm(X) =

1 + 2rn; x), — Ic— in; 1 — 2rn; az).

Several of the properties of

(11.12a) (11.12b)

functions have analogues

for the 1F1 functions. Corresponding to equation (7.1) there is the integral formula

f1(i —

=

(11.18)

from which Kummer's relation 1F1(oc;

v; x) =

— x)

—

(11.14)

may be obtained by a simple change of variable. The analogue of equation (6.5) is x)}

= !.

+ 1; +

1; ar),

while corresponding to the contiguity relations of

have relations of the type

y+l; x)

oc1F1(oc+1;

y; x)=0,

y+l;

y+i; x)

v 1F1(a+1; v; x)=O, x) (x—y)1F1(x—i; y; x)=0,

(11.15) § 10

we

(11.16) (11.17)

—

y; (ca—y)x1F1(ot;

+

x)+y(x+y—1)1F1(a; y; x)

+ y(y— 1 )1F1(a; y— 1; x)=0.

(11.18) (11.19)


36

Cenerallsed Hypergeometrlc Series. There are two ways by which we may approach the problem of generalising the idea of a hypergeometric function. We may think of such a function as l)eing the solution of a 12.

linear differential equation which is an immediate generalisation of the equation (8.1) or we can define the function

by a series which is analogous to the series (6.1). At first sight it is (lifficult to see how the differential equation (8.1) can be generalised immediately, but if we introduce the operator 0

(1

=

(IX

and notice that (8.1) is equivalent to (12.1)

an obvious gcncralisation is {O(z9+ei—1)...

—

= 0,

x(O+c1)...

(12.2) are constants. Further. .., . .., more it is readily shown that this equation is satisfied by the series

where

. . . n—O

. .

(os,),,

n!'

(19 8)

which is, itself, a gencralisation of the series (6.1). Such

a series is called a generalised hypergeometric series and is denoted by the symbol

.

.,

It is left as an exercise to the reader to show that, if no two of the numbers 1, differ by an integer U2' . . ., (or zero) that the other v linearly independent solutions of equation (12.2) are

(i=1, 2, ...,n). As it stands (12.8) is a generalisation of the series (6.1)

§ 12


37

but it is not sufficiently wide to cover a simple series of the ty1)e (11.8). To cover such cases we generahise, not the differential equation, the series defining the function. The generalisation of (6.1) which includes (12.3) is the series . . .

ii!'

. . .

which

(1" 4)

we denote by the symbol Ui'

.

.

x),

.

or, if we wish particularly to throw into relief the difference

between the numerator and the denominator parameters, by the symbol Fq

x].

. .

'flic suffix p in front of the F denotes that there are p numerator parameters . . ., ocr; similarly the suffix q indicates the number of denonui nator parameters. Gencralised hmypergeometric series do not usually arise in mathematical physics because we have to solve equations of

the type (12.2). Their muse is more indirect. Such series occur normally only in the evaluation of integrals involving

special functions. In certain cases these series reduce to series of the type

F

.

., cs,;

1

which have iuuiit argument. For this reason it is desirable

to have information about sums of t his tv1)e. An account of the theory of such sums is given in \V. N. Bailey, Generalised II/fl)ergconzetrzc Series, (Cani I )ri(lge University

Press, 1935). Here we shall consider only one such calculation because it illustrates the use of time theorems of Gauss and Kummner proved above (equations (7.2) and (7.3) respectively). Other results of this kind are given in examples 18 and 20 below.


By expanding the 3F2 series involved we see that

s—

r(cx)r(p)

F r

y;

fi'

1

1+cX—V;

+ n)1'(fl + n)l'(y + n)

+cx—y+n) — — n-on

Now by Gauss' theorem' (7.2) the expression inside the

curly bracket is equal to 2F1(fl+n, y+n;

1)

which may be written f(P+n+m)F(v+n+m)1'(1+cc+2n) 1'(fl+ whence we find that — —

Interchanging

the order of summation and putting

p = + m we see that —

+ n)

p)

1'(fl +

+ — — y) ,,..on!(p — n)!r(1±x+n+p)'

Now by example O(iii) of Chapter I 1

(p—n)! so that the inner sum is equal to r(cx)

F r

p1

—P; —11

which by Kummer's theorem (7.8) is equal to

p!r(1

+


39

Therefore

1'(cc)f'(iI + p)1'(y + i')l'(1 + ,_op!(1+c'—flv)f(l r

S—E

y;

1

This 2F1 series with unit argument can be summed by Gauss' formula (7.2) and the expression for S found. It the follows that 3

1

fi,

2

—

1'(i

— a

result which is known as Dixon's

theorem.

EXAMPLES 1.

Show that (i) ,F1(a, fi; fi; z) = (1 —

(ii)

÷

z)

=

— z)

•; zt) =

(iii)

1

z) 1;

z z

(vii) ,P1(4, 1; (viii)

— z') =

2F1(4, 4; 1; k') =

tan-1:

2

(ix) 1F1(— 4, 4; 1; k') = — E(k).

+ (1 +

{(1 —

— (1

+

40


By transforming the equation y"-I- n'y = 0 to hypergeometric z form by the substitution = sin's, Irove that, if — (I) cos(nz) = ,F1(4n, — 4n; 4; sin's) — 4n, 4 + 4n; -; sin's) n sins (Ii) sln(nz) 2.

and that,

a

if 0

(iii) cos

—4n; 4; cos'z)

(nz)

— 4n, 4+jn; •; cos'z);

+ (iv)

sin

(na)

3.

—in; 4; cos's)

sin

—

ncos(4n.7T)cos(z),F1(4—4n,

4+jn;

eos'z);

Prove the relations: (i) (a—fl)(I —x),F1(c, fi; y; x) (y—/3),F1(a, fl—I; y; a,) — (y—cz),F(cx—1, /1; y; x); (Ii)

(y—fl—I),F1(s,fl;y;x) 1; y;x) + a(1 —x),F1(a+1, fl+1; y; Z y; a =(a—fl—I )(I —x),P1(x,fl+I ;y; x)+

(lii)

fi; y; x) (y—a),F1(x—I,fl; y; .r)—fl(I

fl+I; y; x);

(iv)

= (cs-I-i —y),Fi(cs, fi; y; x); (v)

(1—x),F1(cs,fl;y;x)—,F1(cz—1,fl—I;y;x) —

(vi)

(1—fl)x

x);

(I —a') ,F1(cs, fi; y; a')

Prove

(I)

y+i;

,P:(cs—I,P1;y;x),F:(cs,fl—I;y;x);

= ,F1(x.fl—1;y;x)+ 4.

x,F1(cz, fi;

,F1(cs,fl;y+I;x) —

(vii)

cs+fl—y—1

cs

)a,

,F1(rz, fi;

y+1;.r);

that

,F1(cs,fl+1;y+I;z) —,F1(cs,fl;y;z) -I-

;F1(oc +1, fl+ 1; y+2; a),


(ii)

41

1P1(x, fi; y; z)

,F1(cz+I,fl—1;y;z) +

;F1(cz+1;P;y+1;z).

Deduce a simple expression for the hypergcometric series ,P1(a, fi;

a—I; z). 5.

If n

a positive integer, prove that f'(y) a+n; y; J'(y+n) = is

_.X)cx—V+n},

and deduce that

I

I—p

a+n;

—i-— —

d,ii"

@2_1)"+4.

6.

If n Is a positive integer, and > 1, prove that (_1)Rxn+l (n+I n+2 f 1 1; — = n! 2 ' 2

7.

Establish the following formulae:

I\

x) x)x,F1(y; ö; x) x fl+e; -12F1(a+1; fi; y; kx))=

2F1(x;

(I)

= (Ii)

Da(x'2F1(ct,

(III)

JJ(A,

where 8.

/1; y;

fi; A; xt)dt,

0, y — A> 0.

Prove that if fi>

,F1(a, fi; 2/1;

x)=f

0,

(I 2:P-1B(p,

j

.

(sin

r ('+

cos q,}'9

L +{1

cos

—

q,)-aJ dp,

where = z/(2 — z). Deduce that

5F1(a, /1; 2/1; z) = 9.

Prove that CI"

J

cosmOcos"OdO

=

C').

THE SPECIAL FUNCTiONS OF PHYSICS AND CHEMISTRY

42

and evaluate I

Jo m Is a positive Integer.

where

10. Schrodlngcr's equation for the rotation of a symmetrical-top molecule Is

bI

1

b!p\

?J'tp

1

A\b'tp

I

8n'AW

2cosO

h' where A, C, TV, h are constants. Show that It possesses solutions of the form — — )5P,(cx, fi; z), where n ,n, z — 4(1 — cosO), y = n m + 1, and a, fi are the

roots of the equation

A

z'—(2n+l)z+ën2+n—

8n'AW ht

—0.

Prove that: 3F3(a; a; x) = e'.

11.

(I)

(ii)

1F1(a+ l;a;x)= (i

(III)

1F1(4; t; — r')

(iv)

1F1(a4-1; y; x) — 1F1(a; y; x) = !_1P1(rz+1; y.fI; as).

(v)

1F1(—4; 4; —x')

(vi) x"1F1(n; n+1; 12.

crf(x).

—

— —

Prove that the equation

possesses

1W

solutions of the type I 4;

where

m and C are constants.

____ HYPERGEOMETRIC FUNCTIONS

43

Show that the Schrodingcr equation

13.

172v'

possesses

+ (k

—

a solution of the form eta.

a

p 1/

—.

1;

ikr_-ikz).

14. The Schrodinger equation governing the radial wave functions for positive energy states In a Coulomb field Is n(n 1 d I [8alrn

—(r--)-f —( rdr' dri

TI

T1

—

J

that It possesses a solution

Show

L — r"e9'1P1(irz where k' — 8a'mW/h, a

+n+

1; 2n + 2; — 2ikr)

Show that the equation

15.

idy I — + m' — x xcix dx' + — I. d'y possesses

a solution

y = x rI'1P1(4n +

— ifi; 2inix)

and hence that a solution of equation (1.Oc) Is R = 01"e49P(In ÷ j; n + 1; 16.

Show that (1)

JO

x''(l

. ., a,,

,P,

J

dx

(ii)

f'xs.1(1_x)P'S—1,F5 Ia1,

1; u

raa,

B(1, iii)

l+m;J

1—x—1

.

—i-—. I dx

Lp,,. — B(I, in)

1-x,

(Iii)

(1

rx1,...,a,, in; .

i + mJ

Iaa, . . ., a,; —i-- I J — B(j,

rai,. . .. a,, rn; ii Ifla,...,P.,2m; J

_ 44


17.

Prove that (')

dx

::

flxi,. . ., a,, .

(ii)

b/a

P.;

±b'x'] 5 ., a,, Ijs, ,,1F, Ia1,.. P.; . .

±4b11a

±b'x']

(Hi)

rai,. . .,a,, 4/i; ±b'/p ., P.; 18.

By equating coefficients of z in the relation

(1 — 1P1(a, fi; y; x) prove Saalschutz's theorem

ía,

— a, y — fi; y; x)

/3, —n; 1 i+a+fl—y--—nJ

Hence prove that /3; x

if

4+ ja—fl; H-a--fl;

]

—

(1_x)a]

0 is odd.

0,

f

57

=

—I

iif

(1 —

—1

—

1

which, on account of the duplication formula, is equivalent to I

(15.10)

16. Series of Legendre Polynomials. In certain problems of potential theory it is desirable to be able to

express a given function in the form of a series of Legendre

polynomials. We can readily show that this is possible in the ease in which the given function is a simple polynomial. For example, from the equations (13.4a) we have 1

= P0(4u),

/2

=

/22=

+ +

= =

+ +

so that any cubic c04u3 + c,jz2 + c2p + c3 can be written as the series

+

P0cu) +

+

P1(ii)+

P0(1u)

It is obvious that we could proceed in this way for a polynomial of any given degree n though if ii were large the arithmetic involved might become cumbersome. Since

58


is a polynomial of degree n in 1u, it emerges as a result of an extension of the above argument, that any can be expressed as a series polynomial of degree n in of the type ( r—()

1

1)

(10.1)

'I'lic problem which now arises is that of expressing any function /ca), define(l in the interval — 1 series of Legendre functions of the form

1 as a

(10.2)

if it is assumed that the infinite series (10.2) converges we may uniformly in the range (— 1, 1) to the sum and integrate multiply the terms of the series by term by term with rcsl)eet to over the range (— 1, 1) to obtain the relation I

I

f /(p)P,,(/L)(l/1 =

Cr f Pr(fl)Pn(/t)d/t —1

r—O

—1

by equation (15.8) the sum on the right hand side is equal to and

Cr

'2n+ 1

6r.

n=

which shows that the series (ii —1— r—O

converges

(10.8)

f

—1

uniformly to the sum

in the range (—

1,

1).

The series (16.3) is called the Legendre series of the We shall not discuss here the conditions function which must be satisfied by the function /(p) if this series

to be uniformly convergent; for such a discussion the reader is referred to Chapter VII of E. W. llobson, The Theory 0/ Spherical and Ellipsoidal harmonics (Cambridge University Press, 1981).

LEGENDRE FUNCTIONS

§ 16

59

The possibility of expanding a function in the form of a series of tYPe (16.2) is a consequence of the relation (15.0). In the theory of special functions we frequently encounter sequences of functions p2(x), . . ., p,,(x), . . . which

have the property

=

0,

(iii

?Z)

then say that the functions (r 1, an orthogonal sequence for the interval If, in addition, the functions are such that \Ve

form

(16.4)

2, . . .), (a, b).

b

1,

(16.5)

for all values of it, we say that the functions of the sequence

are normalised, and form an orthonormal set. Given a set of orthogonal functions it is obviously a simple mutter to construct a normahse(l sequence. For example we see from

equation (15.8) that the sequence of functions (it = 0, 1, 2, . . .) is orthogonal but not normahised. By multiplying each function by + we find that the functions

(ii

+

(16.6)

form a sequence of normalised orthogonal functions in the interval (— 1, 1).

There is another property of importance which such a sequence of functions may possess. If there is no integrable

function W(x), different from zero, such that

=

0.

(16.7)

for all values of ii we say that the sequence is a plete orthogonal sequence. It may be shown in the case of the functions (16.6) 1) by considering the Fourier coefficients of in (— that if (16.7) 1)

E. W. Hobson, Op. cit. p. 40.

60


holds this function is a null-function and hence that is a null-function in (— 1, 1). In otherwords it can be shown that the functions (16.6) form a complete sequence of normalised orthogonal functions.

Legendre's Differential Equation. If we write —1)" v= then it is readily shown that 17.

(1 —

1u2)

d/L

+ 2/Lnv = 0

and if we differentiate this equation n +

1

times using

Lcibnitz's theorem we find that dn+2v (1—4a2)

dn'+lv —

+ n(n+

d"v

=0

which when written in the form

n(n+

(1

{

shows that d"(p2 — I equation (1

is —

a solution of the differential + ;z(n

+ l)y = 0

(17.1)

so that we conclude from Rodrigucs' formula (15.3) that is one solution of the equation when n is an integer (17.1). This equation, which we shall now consider in a little

more detail, is called Legendre's differential equation. We saw in example 1 of Chapter I how such an equation arises in the solution of Laplace's equation when solutions of the type R(r)O(cos 0), (i.e. in = 0) are sought. It is obvious by inspection that the point = 0 is an ordinary point of the equation (17.1). Writing the equation in the form

(,a_

—

________ LEGENDRE FUNCTIONS

§ 17

61

observing that in the notation of equation (3.1) with

and

a=1 p(1u)

=

=

1

is a regular singular point with indicial

equation e2 = 0.

Furthermore in the notation of equation (4.2)

= so

that as ii

=

n(n+1) —

co

n(n+1)

2

2

that the point = co is a regular singular point with indicial equation — (n + 1 )} + n) = 0. We thus sec that the equation is defined by the scheme —1 1 co showing

y=P

0 o

n+1 0 —n

,u

(17.2)

.

0

If, however, we put x= Jc(1 —'u) in equation (17.1) we find that it reduces to the form x(1

i)y=O,

(17.8)

which is equation (8.1) with = n + 1, = — n and y = 1, so that the scheme (17.2) is equivalent to the scheme

62

THE SPECIAL

OF PHYSICS AND CHEMISTRY § 17

o

00

1

y=P 0 n+1 0 o

(17.4)

.

0

It should be noticed that the values along the top row are

those assumed by — not by 1u. We consider first the solution corresponding to the singular point at infinity. We write Yi(/t) =

which on substitution into (17.1) leads to the recurrence

relation

=1

On taking

we obtain the solution

n(n—1)

,,2..!...n(n—1)(n—2)(n—3)

2. (2n—1)'t' —

1

1

1

-I-

•

+

which may be written in the form

+—+n; i—n;

(17.5)

Also, if' we write for the second solution

= we

find that

(L) Y2/

/L

(n+1)(n+2) 2.(2n-f-8) ) (n+2) (n+8)(n+4) 4U

+

2.4.(2n+3)(2n+5)

+ (17.6)

LEGENDRE FUNCTIONS

§ 17

63

provided n is any number other than a negative integer or half a negative integer. These solutions are valid for all values of n for which

the 2F1 series have a meaning, not only for integral values of n. If n is an integer the series y1(p) terminates — in other words, is a polynomial of degree n in If we multiply this polynomial by (2ii)! 2"(n! )2 we

obtain the Legendre polynomial of degree n (equation

(18.4b) above).

On the other hand the series for y2(u) does not terminate when ii > — i so there is no point in restricting n to be

an integer. This series solution when multiplied by a factor

± 1)

+ gives

the function

= (17.7)

The function

=

1

I (17.8)

is a solution of the Legcndre equation (17.4) even when n is not an integer and it reduces to the Legendre polynomial

when n is an integer. We shall continue to denote it by but when n is not an integer shall refer to it as the

Legendre function of the first kind of degree n. The function

defined by e(luation (16.7) will be referred

to as the Legendre function of the second kind of degree n; even when a is an integer it is not a polynomial.


With these definitions we may write the solution of Legendre's equation (17.1) as

y= (I iz

I>

1).

+

(17.9)

In some problems we know that the solution

should be a polynomial in 1u; in that case we must take the

solution to the form y = The variation of with from equation (17.7) when ii>

1.

may be computed easily Tables calculated in this

way are contained in the volume quoted at the end of The following Fig. 7, which was prepared from these for a few values of ii. §

18.

tables, shows the variation of Q,,(p) with

0.

0.2

3

1

4

— Fig. 7

Variation of

with

4u.

Since Legendre's equation has a regular singular point 1 we may on the basis of equation (3.8) take the

at =

LEGENDRE FUNCTIONS

§ 18

second solution

65

of Legendre's equation to be proportional to log

— 1)

+

—

now be obtained by substituting this expression into the differential equation (17.1) and equating to zero coefficients of successive powers of The coefficients Cr can — 1).

Instead of proceeding in this way we shall derive

this second solution by means of a method due to F. E. Neuniann.

18. Neumann's Formula for the Legendre Functions. Let us now consider the integral ç' where

> 1, and n is a positive integer. Expanding

the denominator by the binomial theorem, we have, for the value of the integral, the series 1

1

J —1

s—0

From equation (15.9), it follows that the integrals in this n, or if 8 = n + 2r where r is a positive series are zero if s integer, so that the above series is equivalent to 1

1 I

n+1+2rJ —1

which,

by the same formula, is equal to 1 4Ufl+1

; 2"(2r)!1'(n + r +

Now from the duplication formula

1 +r) 2"(2r)I so

—

that the series reduces to

r!


66 1

which is equal to 1)

+ 1;

oF1

Noticing that and

=

+

1'(jn+

1)/2"l'(n+ comparing the result with equation (17.7) we see Hence we have shown

that this series is merely

that if

>

1 1

=

(18.1)

/1 —

a result

which is known as Neumann's formula. Equation (18.1) holds not only for real values of greater than 1, but for all values of p which are not real. For this reason equation (18.1) may be regarded as defining the second solution, Q,,(p), of Legendre's equation. Certain related formulae, due to MacRobert, follow readily from this result. If p > 1 and in is a positive integer then I

J

—

f

=

f1

and the integral on the right is equivalent to the finite sum tn—I

pnslrJ

1

(18.2) —1

r—O

n it follows from equation (15.9) that each term of this series vanishes so that we have If rn

= provided

in, n arc integers and in

(18.8) n.

On the other hand if in = n + 1, the series (18.2)

reduces to the single term

LEGENDRE FUNCTIONS

§ 18

I so

67

= (+

that 1

= f_j

+

1)!'

(18.4)

is a polynomial of degree Now if in is an integer in in p so that it follows from (18.8) by finite summation that, if Ui 'U,

=

(18.5)

Similarly from equation

(18.4) and the definition of

we have the formula

=

)

±

(18.6)

1 and in by n and subtract from equation (18.6) we obtain the relation

If we replace n in equation (18.5) by ii -4— —

=

(18.7)

Other formulae of a similar nature are contained in cx. 24 below.

We shall now make use of NeUmann's formula to derive the form of the second solution of Legendre's equation in the neighbourhood of the points p = ± 1. From (18.1)

we have the result that if p > 1,

=

log

—

(18.8)

denotes the integral

where

—

ç'

ii—E Now when ii is an integer,

n in /t so that degree n — degree ii —

1 1

—

in p. Hence in p.

is

a polynomial of degree is a polyirninial of

—

is a polynomial of

68


If we substitute from (18.8) into Legendre's equation (17.1) we find that W,,_1 satisfies the differential equation (18.9) + n(n+1)Wn_i = is a polynomial of degree n — 1 in /4 we may write (cf. § 16 above) (1

— 24uW,_1

Now since

=

n—i r—O

Using the fact that

— 4r —

1) 1 according as ii is odd or even (which follows from equation (14.6)), and the result —

2/2Pr(/L) + n(n+

= (ii — r)(n + r + 0, (s=0, ..., p)

we find, from equation (18.9), that

and that

—

2ii—4s—1

(2s+ 1)(n—s) Substituting these values in (18.8) we obtain the formula

=

sO (2s+lXn—s) (18.10)

Another expression for fact that both and equation (17.1) so that

may be derived from the are solutions of Legendre's

_1j2)P

Qn(i')

0

which is equivalent to [(1

—

=0,

showing that (1

—

= C,

(18d1)

LEGENDRE FUNCTIONS

§ 19

69

C is a constant. Now from equations (17.7) and (17.8), we can readily show that for large values of

where

— — 1,

co

d JQn(/L)l

P(p)J —

and

2-f-i

the left-hand side of (18.11) tends to showing that C = — 1. Writing (18.11) in the form

so that as Au

noting from (17.7) that

1

— —>

0 as 4u

co

we have (18.12)

= Recurrence Relations for the Function

19.

Recurrence relations for the Legendre function of the second kind can be derived from Neumann's formula (18.1) and the corresponding recurrence relations for the Legendre

P,(u). From the recurrence relation (14.2) and Neumann's formula we have (n +

+ nQfl..4(Au) = (n

+

Now

=

— (n

j'

+

If we write the second term on the right as f

we see from (15.6) that it vanishes if a

0. Hence we

have

= 0 (19.1) — + for three consecutive showing that the functions values of n satisfy a relation of the same form as that for (equation (14.2) above). the functions From Neumann's formula (18.1) we have

=—

i11 Cu—E)2

70


if we integrate by parts on the right hand side we find that and

(—

1

= * Jf

1 p

1

(1

—

1f

=

—i /i — by virtue of equation (14.6). The integral on the right is

by Neumann's formula so that, fiiially, (19.2) = (2,i+ The Use of Legendre Functions in Potential —

20.

Theory. In potential theory we have frequently to

= 0 which determine solutions of Laplace's equation satisfy certain prescribed boundary conditions. If we

have a problem in which the natural boundaries are

spheres with centre at the origin of coordinates it is natural to employ polar coordinates r, 0, q. In cases in which there is symmetry about the polar axis, will not depend on so we may write = 0). It then follows from example

I of Chapter I that

r"')

(A,, r" + v, will be a solution of Laplace's equation tluLt V,, is a solution of Legcndrc's equation (17.1). Taking v,, to be

0) we sec that we may write

0) +

y(r, 0) =

0)

(C,,r" +

0)

(20.1)

n—O

the quantities constants. where

= 0,

1, 2, . . .) arc all

Now it is obvious from equation (18.8) that Q,(cos 0) is infinite when 0 = 00, and we know on physical grounds

LEGENDRE FUNCTIONS

§ 20

71

that in the case of spherical boundaries tp remains finite along the axis 0— 0. IlcnccweniusttakeC,, = Oforall values of ii and obtain the 1)otefltial function (A,,r"

tp =

+

(20.2)

which is valid as long as r is neither zero nor infinite, i.e.

r b where a an(I b are finite and non-zero. If the region under discussion is the interior of a sphere, i.e. if if a 0

take

r a, then to avoid ip becoming infinite we must to be zero to give

(20.3)

0).

tp

On the other hand if the region being consi(Iercd lies

outside this sphere, we must take (20.4) 8, 4) in potential theory are given in Coulson's Electricity (Oliver & Boyd, 80; a further example is given below. 1948) The Legendre functions of the second kind, Q,,(cos 0),

Examples of the use of the solutions (20.2,

which are absent from problems involving spherical boundaries, enter into the expressions for potential functions appropriate to the space between two coaxial cones.

If 0 < < 0 < /3

we

must take a solution of the

form (20.1 ). Suppose, for example, that tp = and = on 0 = /1 then we iiiust have

/3)

+

on 0 =

=0

+

and

0

/3)

= a,,, B,,

D,,

= 0,

the latter results following from the fact that if /3, /3) — /3) does not vanish. and C,, and inserting the Solving these equations for solutions in equation (20.1) we find that in the space between the two cones

72


—

f Q,,(cos cx)P,,(cos 0)—

0)

fi) —

fi)

n—O

(20.5) To illustrate the use of the solution (20.1) and of some of the properties of Legendre functions we shall now consider the problem in which an insulated conducting sphere of radius a is placed with its centre at the origin of coordinates in an electric field whose potential is known to be

(20.6)

0)

and we wish to determine the force on the sphere. The conditions to be satisfied by the potential functions are (i) that tp is a solution of Laplace's equation; (ii) that tp has the form (20.0) for large values of r; (iii) = 0 on r = a.

The conditions (i) and (ii) are satisfied if we take

+

B

and (iii) is satisfied if we write

0)

= — cc,,

We

therefore have (rn —

=

P,,(cos 0).

The surface density of charge on the conductor is

(2n+1)a"'cc,,P,,(cos 0) =— and since the force per unit area on the conductor is 22ra2 the resultant force on the sphere is in the 0 = 0 direction, and is of magnitude — 47r ar

F

= =

2:a2 sin 0 cos 0 dO Ia2

cc,,,cc,, am+nsIm,, n—i rn—i

(20.7)

LEGENDRE FUNCTIONS

§ 21

where

73

I,,,,, denotes the integral

(2n + 1 )(2rn +

1) fcos 0 sin 0 Pm(COS

0)dO.

the variable of integration to /z = cos 0 and using the recurrcnce relation (14.2) we find that Changing

=

(n+ i)f

(2m+

Pm(/i)Pn+i(ii)d/i

+n

P,2_1(1z)

}

by the orthogonality property (15.8) this reduces to the form and

'mn = 2(fl + 1)ôm.n÷i + 21U3rn.n_i•

(20.8)

Substituting from (20.8) into (20.7) we find that the total force on the sphere is

+

F' = 21.

1

Legendre's Associated Functions. We saw in

example 1 of Chapter I that the solution of Laplace's equation in spherical polar coordinates reduces to the solution of the ordinary differential equation, (1 —

—

+ { n(n+1)—

@=o

(21.1)

which reduces to Legendre's equation when in = 0. This equation is known as Legendre's Associated Equation. To solve this equation we may write

0=

— 1)—imy.

(21.2)

Substituting this expression in the differential equation we find that y satisfies the equation (1

— 2(1

+ (n+ m)(n—m+ 1)y= 0

and differentiating this equation in times with respect to by Lcibnitz's theorem we find that


(21.8)

{

showing that if dtmy/d1um is a solution of Legendre's equation

(17.1) the function 0, defined by equation (21.2), is a solution of Legendre's associated equation (21.1). Similarly if we put 0 = (/42 — 1)Imy in equation (21.1)

we find that (1—4u2)

djj

— 2(1

+ (n—m)(n+m+1 )y= 0. (21.4)

If now we differentiate equation (17.1) in times with respect to we obtain the equation

=0

2(1+m)_f_ + (ii—

{

showing

that if

is a solution of Lcgendrc's equation

then

(21.5)

is a solution of Lcgcndre's associated equation (21.1). the two solutions of Legendre's equation to be and it follows from (21.5) that the functions

= Qmcu)

—

= (/42.

(21.6)

are solutions of Legendre's associated equation. As a con-

sequence of equation (21.3) we see that so also are the functions

Pm(/4)_(p2

—

1)+'

J

11J

. . .

J

(21.7) 1

and

=

—

1

)_jns

f

(21.8)

LEGENDRE FUNCTIONS

§ 21

75

It is an immediate generalisation of (6.5) that dm

(oc)m(fl)m

=

(v)m

+

fi +

we see that

so that using Murphy's form (15.2) for —

Hence

(—2)min!

2

in; y + in; x)

'2/

F (in_n, n+m+l; m+1' 1

as defined in (21.6) may be written in the

form

r(ta+nm+1) 2mml 1'(n—rn+l)

n+m+ 1; rn+ 1

X 2F1

—1

(21.9)

Other expressions for the first of the two functions (21.6) can be easily derived. If we make use of the result (7.4)

we see that F(n+rn+1)

X 2"l

—ii; 111+1;

/L+1 (21.10)

and similarly, if we make use of relation (7.6) we may derive the expression

1'(n+m+l)

Im

X 2F1

rnLr(n—m+1)

—ii;

1

(21.11)

From Rodrigues' formula (15.3) we derive the simple expression 1

=

2

—

The simple differentiation

(21.12)

76 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY §

— (—1)"

I'(m+n+1+2r) 1'(n+1+2r)

21

—m—n—1--2r

may (because of the duplication formula) be written in the form 1'("+ 1

dm f d,e"

—'

)r

1

/

r ,fl+n+i'

)r

/

(

'

'*')

showing that, by term by term differentiation of the soluti-

on (17.7) of Legendre's equation, we obtain the solution Qm(4u)

= (—1 )"

x (21.18)

2F1

of Legendre's associated equation. Solutions of the type (21.7) and (21.8) can be derived in a similar fashion. Using the result ,.e 1 rI "' I I 1___ 2 2 / ' ' / in equation (21.7), with Murphy's expression (15.2) for we derive the expression I

I

2F1 (— n, n + 1; in

—

+

1;

1

(21.14)

for P!1

,.e

j 11 J and

. . . J

1

this yields the solution ml

n+1;

(21.15)

The solution provided by Rodrigues' formula (15.8) can obviously be written as

LEGENDRE FUNCTIONS

§ 21

=

(

—

1

/

77

(u2—

In a similar way the solution ) 1'(it — in + 1) \—

1

)nl.

(21.16)

1 )+"

—

!)

(21.17)

is (Icrived from equation (21.8) and the result 1'

1 '(gui

+ 1) J' J =

. .

.

(Ii,z

f

—L

1) (1,i _L- 1) r

use(I in equation (17.7). defined The four functions by equations (21.9), 21.13), (21.14) and (21.17) respectively are therefore solut ions of Legendre's associated equation.

They are known as Legendre's Associated functions. Although the expressions above have been found by assuming in and ii to he integers it is readily shown that the solutions quoted are valid even when in and ii are not integers. Since Legendre's associated equation is of the SCC0n(1 degree it follows that only two of these four

functions are linearly in(lcpcndeut, and that the other two may he expressed simply in terms of them. It follows imrne(liately from equations (21.11) and (21.15) that if in, ii arc integers (/1)

— 111+ 1)

=

I)m (fl).

Furthermore. if we apply the result (7.6) to the hypergeometric series on the right hand side of equation (21.17)

we find that

=

— in+ 1

(—1 )Tfl 2

—1

•

n+;


78

which, on comparison with equation (21.18), reveals the

relation

I'(n—nz+l)

—m

(21.19)

It is now a simple matter to prove that when in and ii are integers, and in is fixed, the polynomials form an orthogonal sequence for the interval (— 1, 1). Making use of the results (21.18), (21.12) and (21.16) we find that 1'

1

J —1

—

r(ii+,ii+ 1)

—

r(n—in+1)

1

1

ii'! f_1

d1in'+m

after integrating by parts n — in times tile CXprCSSiOfl

and

on the right reduces to

'

1'(n+m+l) (— iy'-" J'(n—in+ 1)

n In1! .1_i

d/it)+th'

(u2—1)"d4u

This integral is evaluated by the method used at the end

and we find that r(iz+in+1) (_1)m 2 ô (21.20) J'(n—m+l) 2n+1 'b" In many physical problems = cos 0 so that —1 1. It is then not always convenient to have a factor of the form — 1 )4tfl We use instead Ferrer's function of §

15

I

=

(21.21)

(1

With this notation we may write (21.20) in the form

J'(n+rn+l) 5'

2

= .l'(n—m+1) 2,i+1 The other formulae are ammended similarly. 1

('21.22)

79

LEGENDRE FUNCTIONS

§ 22

Integral Expression for the Associated Legendre Cauchy's theorem in the form 1) Function. If we 22.

= where

is

an analytic function of the complex variable

in a certain domain R which includes the point =

where the integral is taken along a closed contour C which includes = and lies wholly within the domain U, then by differentiating both sides of the equation r times with respect to we obtain the result and

—

(2° 1

I'

d/2"

Substituting in + ii for r and

— I )" for in this equation we find that, as a result of equation (21.1), — (m+n)! (2° 2 Pm( " / ) — 2" . ii! i 0 we may take the contour C to be the circle If

=

1)1.

I

Integrating round this contour we obtain from equation (22.2) the equation

+

— 1) cos ((p — tp)}"

(m(p) thp

nl cos Pn (1u) in)! sin from which follows immediately the Fourier expansion

=

(n +

1)cos((p—lp)}"

=

id

+ 2rn—i (n—f- iii)!

m(tp—p). (22.8)

1) E. G. Phillips, Functions 0/a Complete Variable, (Oliver & Boyd, 1940,) p. 98.


80

Changing

n to — (n + 1) we obtain the expansion

+ v'(i-i'2 — 1) cos

=

(n

+ 2 E (—1 )m

(22.4)

rn-i

Applying Parseval's theorem for Fourier series to the series (22.8) and (22.4) we find that the series 'I (n — m).

+ '2 E (..1

cos imp

(n—f-rn)!

converges to the sum

r

— 1) cos

{/L' +

J

integral may be evaluated by means of Cauchy's theorem.') Its value is found to be

This

+

= cos 0,

Writing

COS & =

COS

—1

— 1)1 eos p)}

= cos 0' and 0 cos 0' + sin 0 sin 0' cos

we obtain the result

0) =

0') —

t1

+ 2 rn—i(_1)m (n+rn)!

0') cos (imp) (22.5)

which is often of value in the solution of problems in wave

mechanics.

23.

Surface Spherical Harmonics. From the two

sets of orthogonal functions

0), cos ('imp) we can

form a third set of functions Xnm(0, L)

=

0) eos mq {

(ni ii)

For details see E. W. Hobson, Op. cit. pp. 865—71.

(28.1)

81

LEGENDRE FUNCTIONS

§ 23

which is an orthogonal set of functions on the unit sphere,

i.e. the functions of the set satisfy the normalization relation f 51fl 0 dO j' Xnm

(28.2)

= â,,n'

,,,' dip

0

o

In a similar way we can construct a set (n—rn)! 1'n ,m(0t ip) =

(—.j;--)

A

0) sin nup

{

n)

(28.8)

sinOdOf Yn.rnYn#.rn'thp=ônn'âmm'

(28.4)

(iii

which satisfies the relations n

$o

2n

0 2n

n

(28.5)

f sinodOf.10

all integral values of n, n', in and in' with 7fl

for in'

ii'.

Because

71,

of these orthogonality relationships we can

establish an expansion theorem which is a straightforward generalization of the Legendre series (16.8). It is readily shown that for a large class of functions /, the function 1(0, p) can be represented by the series

c,,P,(cos 0) +

{XnrnXn,m(0, (p) + Ynrn Ynm(0, p)} n—i rn—i

n—0

(28.6)

arc given by the ex-

where the coefficients pressions

=

('

/(0,

dip

0) sin 0 dO,

(28.7)

.10

0 0

0

Zn

n

Y11m = $ 0

Sifl OdO

Yn.m(01 473)1(0, (p)dip.

(28.9)

82


For any given function /(O, (p) the series (28.6) can therefore in principle be computed by a series of simple integrations.

which are known as The functions Xfl.m and surface spherical harmonics can be constructed easily from the known expressions for the associated functions

We find, for instance, that X1,1(0,

=—

X21(0, tp )=

sin 0 COS 4

sin 0 cos 0 cos

— (i—)

X2,2(0, q) = X3,1(0, (p) =

0 cos 21 —

=

sin 0 (5 cos2 0 — 1) cos

0 cos 0 cos

X3,2(0, q,) = X313(0,

1

—

0 cos

are the corresponding expressions for the tamed by replacing cos (mp) by sin (mr) in the expressions and

for the Xnm. The functions Xn,m and Yn,m

have

the important

property that they arc solutions of the partial differential equation (sin 0

+

+ 71(11+

so that the function (Ar" + BT"')Xnm(O, p) + (Cr" ÷

i)X= 0

(28.10)

v')'

where A, B, C and D arc constants, is a solution of Laplace's

equation. It follows immediately from equations (28.7)— (28.9) taken with the expansion (28.6) that the function

83

LEGENDRE FUNCTIONS

§ 26

tp(r, 0, p)

(L)

=

+ n—i rn—i

0) + X,,

(I

(p)

+

Ynni

Ynrn(0,

Laplaces equation in the region 0 r a, is finite at r = 0, and takes the value /(0, on the sphere satisfies

r = a. For example, suppose we wish to find the solution of

Laplace's equation which takes on the value x2 on the surface of the sphcic r = a. Here we have

=

f(0, q,) =

=

a2

—

a2 3cos2O—1 ( P2(cos

—

+

0) +

cos X2,2(0, q,)a2.

Thus the required solution is 0,

=

a2 —

r2P2(cos 0)

+

Substituting the values of P2 and X22 and transforming

back to cartesian coordinates we see that the required solution is 24.

ip=

Use of Associated Legendre Functions in Wave

Mechanics. To illustrate the use of associated Legendre functions in wave mechanics, we shall consider one of the simplest problems in that subject — that of solving Schrodingcr's equation

+

8212m

(II' —

for the rotator with free axis, that

=0 is

(24.1)

for a particle

IV moving on the surface of a sphere. In equation represents the total energy of the system, V the potential energy. In the case under consideration V is a constant, V0


84

will be a function of 0, If the radius of the sphere is denoted by a then equation (24.1) is of the form say,

and the wave function

only.

cot 0

1

TV— V0)

1

h2

0

a2

— —

(24.2)

If we consider solutions of the form

= then

eu +

—l's)

+

/,—

—

sin 0

e

o.

Substituting 87thna2(lI'—

= n(n + 1)

(24.8)

find that this equation reduces to Legendre's associated equation (21.1) and hence has solution we

0=

0). 0) + However for the same reason as in the case of potential theory 20 above) we must take B = 0. The solutions of equation (24.2) will therefore be made up of com-

binations of solutions of the form (24.4) 0), p) = where ii,,,,, is a constant. The physical conditions imposed on thc wave fLinction are that it should be single—valued and continuous.

Obviously then the 'physical' solutions will have in an must equal v,,, ,,(0, (7)). integer, since v,,, +

should converge Further in order that the series for it is necessary that it should have for the values = only a finite number of terms. This is possible only if ii is a positive integer. If therefore the solution (24.4) is to be

valid for 0 =

0

and 0 =

we must have a a positive

LEGENDRE FUNCTIONS

85

The physical conditions on the wave function are therefore not satisfied by systems with an arbitrary value for the energy W but only by systems for which integer.

II' =

V0

h2

+

n(n

+ 1),

(24.5)

where n is a positive integer. In other words, the energy of such a mechanical system does not vary continuously, but is capable of assuming values taken from the discrete set (24.5). EXAMPLES

Show that, if n

1.

odd P,,(O) =

is

P,,(O)

=

0,

n

ii is even,

(—i)I"(I)I,, (In)! =

Prove that

2.

and that, If

(i+

nil

Li—p 3. If

—

p

+

—

(1)

(1 — hC)—I(i —

(ii)

P,,(p)

Deduce

1)

show that

—n; I—n; c-')

that

(ill) ,F1(I, —n; I—n; I) 4.

'1 —

f(n + 1)

If n is a positive integer prove that — 2ph + h')-k dp

and

=

hence, making use of Rodrigues' formula, deduce that

f(1 1

—

— 2ph + h')-"-4

2"1(n!)' (2n + I)!


Prove that

5.

P'R(x)

p

=

(2n — 4r —

r—O

n is odd or even. — 1) or p= Deduce that for all x in the closed interval (—1, 1) and for nil positive integers ii, the values of the functions P"(x) , I, 1r9 can never exceed unity. where

I

6. Prove that t'l

1

= 2n + 1 {PN_l(u) — deduce, from example 1, that if n Is an odd integer J

and

1.1

J0P,1(,L)dp

=

What is the value of the integral when ii

Is

even?

Using equation (14.2) and the results of the last example show

7.

that if n is even (1

(n—2)!

and

that the integral has the value zero if ii is odd.

8.

If

= fi

= 2. Hence evaluate prove that (n + 1 + 9. If m and n arc positive integers, prove that I

J—1

10.

(1+

I

If n is even and m> — Jo1'

Deduce

)d —

2m+n+l{(m + n)!)2

— m!(rn+2n-f-1)!

1,

prove that

d1'

that

f0(1

=

1

87

LEGENDRE FUNCTIONS 11.

Show that

I/4+1\"

I

k—n; —ii, 1;

and hence that S

=

(1

r—0

Deduce that

P, (cosh u)

1

— I )" show, by using Rolle's theorem that 12. If f(js) = must have at least one zero between — 1 and I. Proceeding in this way deduce that f(n)(p) has n zeros between —1 and I. hence show that when ii is even the zeros of P,(1i) occur in pairs, equal in magnitude but opposite in sign, and that when n is odd, = 0 is a zero and the others occur in equal and Opposite pairs.

13.

If y(,i)

is

any solution of the linear differential equation dy

d2y

+

y are continuous functions of ,i whose (lerivat'ves of all orders are continuous, prove that yip) cannot have any repeated which satisfy the equation zeros except possibly for values of = 0. Deduce that all the zeros of P,(u) are distinct.

14. Prove that the Legendre polynomial P,(z) has the smallest distance in the mean from zero of all polynomials of degree n with leading coefficient 2"( 1),,/n! 15.

If R denotes the operator

d( (1

dx

prove

—Xe)

d dx

that

f

P,(x)R{f(x))dx = — n(n-f 1)

provided that f(x) and

Prove that if ii

f(x)

are

finite at x = ±

1.

1

flog (1 — x)P,,(x)d.r

—

1)


Prove that

16.

—x)2n+l

i—i

r'

(n)

2V2

PR(x)dx

r'

•

If R'=l —2hx+h', prove t.hatlf IhI rn;

= 1

=

Prove that, if n is a positive integer and nP (cosO)sinOdO

fl

=+

> —

1),


90

Deduce

that, if

I

I > 1, —

0

171.1

By evaluating this integral show that

n+1; n+4; c—').

= fln+f) 26.

Prove

that, if in Is a positive integer, "

)

2mm!(1_2ph+h*)M+*

27. Show that, if Ii' I> 1, n > —I then r(n+m+1) m f1

Jl

2R+1 I'(n+l) + 11>2 then

Deduce that, if V'7il'(fl+m+I) (/4)

=

4)

/

n+m+ 1; 2n+2;

(jz+

2

Find a simple expression for 28. Prove the following recurrence relations for Ferrer's Associated Legendre Functions:. (I)

(ii) (iii)

(2n+1)(1

— —

—

—

+ (n+m)T'_1(p)

0;

=

29. Derive the expressions for T'(O, q,) and X'(O, q,) for in— 1,2,8,4. sin 0 cos' 0 cos Express the functions sin' 0 sIn' sin' 0 cos' In terms of surface spherical hamonics. 30. FInd the function which satisfies Laplace's equation in the

interior of the sphere x' + y' + a' = a', remains finite at the origin on the surface of the sphere. and takes the value ocx' + +

CIIAI'TER IV BESSEL FUNCTIONS

The Origin of Bessel Functions. Bessel functions were first introduced by Bessel, in 1824, in the discussion 25.

of a problem in (lynamical astronomy, which may be

described as follows. If P is a planet moving in an ellipse whose focus S in the Still and whose centre and major axis

are C and A'A respectively (ef. Fig. 8), then the angle

:1

C

S

A

Fig. 8

ASP is called the true anomal!, of the planet. It is found that, in astronomical calculations, the true anomaly is not a very convenient angle with which to deal. Instead we use the mean anomaly, which is defined to be times

the ratio of the area of the elliptic sector ASP to the

area of the ellipse. Another angle of significance is the eccentric anomaly, it, of the planet defined to he the angle ACQ where Q is the point in which the ordinate through P meets the auxiliary circle of the ellipse. It is readily shown1) by a simple geometrical argument 1)

Cf: D. E. Rutherford, Classical Mechanics, (Oliver & lloyd, 1051) *42. 91

92


that, if e is the eccentricity of the ellipse, the relation

between the mean anomaly and the eccentric anomaly is (25.1) c'= u—e.sinu. The problem set by Bessel was that of expressing the difference between the mean and eccentric anomalies, as a series of sines of multiples of the mean anomaly, i.e. that of determining the coefficients cr(r = 1, 2, 8,.. .)

u—

such

that

u—

(25.2)

sin (rC).

To obtain the values of the coefficients cr we multiply both sides of equation (25.2) by sin (se) and integrate We then obtain with respect to from 0 to —

r..150

Now

s:siii (re) sin and

=

an integration by parts shows that sin

=

!

ii) cos

From (25.2),

— u

.11 (dli

eos

is zero when = 0 and when =

that the square bracket vanishes; the integral can be written in the form so

it" j cos

—

du

and hence, using equation (25.1) we obtain the result C1

=

$cos (s(u — e sin u)}du.

(25.8)

o

The integral on the right-hand side of equation (25.8) is

93

BESSEL FUNCTIONS

§ 25

function of s and of the eccentricity e of the planet's orbit. If we write a

= — j o cos(x sin 0 — nO)dO

(25.4)

it follows from equations (25.8) and (25.2) that U —

=

2

Jr(CT)

sin (re) . r

(25.5)

so defined is called Bessel's coeffi-

The function

cient or order n. We shall now show that of I" in the expansion of exp

is equal to the coefficient — r')); in other words

by means of the expansion

we may define

1

exp {

(25.6)

('— T) }

this we need only show that the .J,,(x) of (25.6) can be expressed in the form (25.4). We first of all observe To

that (—

1

=

(—

since both expansions are equal to cxp Equating coefficients of t" we have (— 1

=

— 1/1)).

(25.7)

In the expansion (25.6) we may write I = e'0 to obtain the relation exp (ix sin 0)

C"°

Making use of the result (25.7) we see that the series on

the right can be put into the form J0(x) +

cos (2m0) +

sin (2m+1 )0

so that by equating real and imaginary parts we obtain the expansions

94 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 26 cos (x sin 0)

= J0(x)

sin (x sin 0)

+

=

cos (2rn0),

(25.8)

Sill (2rn + 1)0.

(25.9)

If we now multiply (25.8) by cos nO, (25.9) by sin nO, integrate with respect to 0 from 0 to 'r, and use the formulae

j'cos (inO)cos (nO)dO = J'sin (mO)sin (nO)dO =

obtain the formulae

we

I

t7'

J cos (x sin 0) cos ('nO)dO,

(n even), (25.10)

o

J o

sin(xsinO)sin(nO)dO,

(ii odd).

(25.11)

Because of the periodic properties of the trigonometric functions we know that the integral on the right of equation (25.10) is zero if ii is odd, while that on the right of equation (25.11) is zero if n is even. Thus for all integral values of n, we have

= !r{eos(x sin O)cos (nO) + sin (x sin O)sin (nO))dO which is identical with the expression (25.4). In particular J0(x) = ! fcos (x sin O)dO

(25.12)

Jn what follows we shall assume that the Bessel functions

of the first kind are defined by equation (25.6) or, which is equivalent, by equation (25.4).

Recurrence Relations for the Bessel coefficients. If we differentiate the generating equation (25.6) with respect to x obtain the relation

26.

-. (t — 2,.)

{

— 2,-)

} =J,(x)i"

BESSEL FUNCTIONS

§ 26

95

which is equivalent to

= 0.

—

Equating to zero the coefficient of t" we obtain the relation

=

(20.1)

On the other hand, if we differentiate (25.0) with respect

to t the resulting equation is

Fr (1 +

exp {

(1

—

and this is equivalent to the relation

+ t"2)J,,(x) ...EiaJ

= 0.

Equating the coefficient of 1n1 to zero we obtain the

recurrence relation

+

J7,(x) =

(20.2)

Adding equations (20.1) and (20.2) we find that

= and

(26.8)

—

subtracting equation (20.1) from (20.2) we obtain

= n= special case

0

in this last equation we have the important

= — J1(x), and putting n =

1

(26.5)

in equation (26.8) we find that

J(x) = J0(cv) —

J1(x).

(26.6)

Differentiating both sides of (26.5) with respect to x and

96


making use of the result (26.6) we have 1

Jo (a,) = —J0(a,) + —J1(x) which, as a consequence of equation (26.6), may be written

+ J0(x) = 0

+

(26.7)

showing that y = J0(x) is a solution of the differential equation

268

—0

We can show similarly that the Bessel function satisfies the differential equation

+

!

+ (i

y

=0 (n, integral).

(26.9)

For, from equation (26.4) we find, as a result of differen-

tiating both sides with respect to x, that

+

= nJ2(x) —

—

Now, from equation (26.4), n11,(x)

= !J(tv)

and putting n + 1 in place of n in equation (26.3) we see

that zvJ,÷1(x) + (ii

+1

= xJ(x)

a solution of equation (26.9) that provided, of course, that n is an integer. As we pointed out in § 1, equation (26.9) is known as

BESSEL FUNCTIONS

§ 27

97

Bessel's equation. WThat we have shown is that if the n which occurs in Bessel's equation is an integer, one solution of the equation is It is because of this fact that

Bessel coefficients arc of such importance in mathematical

physics, for as we saw in § 1, the equation (26.9) arises naturally in boundary value Problems in mathematical physics.

27. Series Expansion for the Bessel Coefficient. We

shal now find the power series expansion for the Bessel coefficient If we write exp { and

(t

—

-i-) } = exp

xl) exp (_

make use of the power series for the exponential

function we obtain the expansion (xl)t

1

(—x)

2't's!

=

x

E (—

r+a1r—1

2

r—O

—. rls!

(27.1)

By our definition (25.6), the Bessel coefficient is the coefficient of in this expansion. If n is zero or a positive

integer, we find that

(—1)'

(X\12+2$

s—O s!(it + s)l

272

and when n is a negative integer we can deduce the series for from equation (25.7).

Writing equation (27.2) in the form

=

2!

1),

we see that

= 2"n! 0F1(n + 1; —

(27.8)


The

variation of the Bessel coefficients J0(x), J1(x),

J2(x) for 0

x 20 is shown graphically in Fig.

9.

These are the Bessel coefficients which occur most fre-

Fig. 0 Variation of J0(x), J1(x) and

with x.

quently in physical problems and their behaviour, is similar

to that of the general coefficient

Simple relations for the Bessel coefficients may be derived

easily from the series expansion (27.2). For example, since this equation is equivalent to

=E

(—1

1

fl+2'

s!(n+s)!

(27.4)

it follows, as a result of differentiating both sides of this equation with respect to x, and making use of the fact

that (2n +

2s)/(n

+ a)! = 2/(n

1+

a)!,

that

BESSEL FUNCTIONS

§ 27

d

(.—. 1

99 1

n1+2'

—1+ s)! which, by comparison with (27.4), shows that

=

(27.5)

If we write this result in the form

Id

-;

=

we see that if in is a positive integer less than ii, then

ldtm (—

=

(27.6)

Similarly we can establish that

{x"J,,(x)} =

(27.7)

—

or, which is the same thing,

(!#) a

=—

result which may be generaliscd to the form

Idtm fr"J,,(x)}

= (— I)mX"mJn+m(X).

In particular we have the relation 1

x"J,,(x) = (— 1)" (—

J0(x),

(27.8)

may be which shows how the Bessel coefficients derived from J0(x). Another interesting property of the Bessel coefficients also follows from the power series expansion (27.3). This concerns their behaviour for small values of the argument Since

lim0F1(n+ 1,

1


it follows from equation (27.8) that

=

lim

1

(27.9)

In other words, for small values of x, the Bessel coefficient behaves like x"/2"nl. 28. Integral Expressions for the Bessel Coefficients.

We have already derived one integral expression for the Bessel coefficient of order n (equation (25.4) above). In this section we shall consider other simple integral expressions for these coefficients. We shall consider the integral

in

I = f(1 — If we develop exp (ix!) in ascending which n> —

powers of ix! we see that the value of this integral is 1

a—O

(1 — 12)n—1 1' dl.

sI

If 8 is an 0(1(1 integer then the corresponding integral occurring in this series is zero, and if .c is an even integer,

2r, say, then the integral has the value 1'(n + 1(1 — u)"4u'4 dii =

+

1'(n+r+l)

so that (—

(2r)!

— rPr'

i

(_ r!r(n+r+ i)22r

since, by the duplication formula for the gamma function, It follows immediately from = 22tr!1'(r + that the series expansion (27.2) for

= 1'(f'(+

—

(28.1)

BESSEL FUNCTIONS

§ 29

101

it is easily shown that this is equivalent to the formula

and

f (1

=

—

cos (xt) dl.

(28.2)

particular cos

1

J0(x) =

(rl)

(28.8)

15d1

The result (28.2) may be expressed in a slightly different form by means of a simple change of variable. If we put I = cos 0 we obtain the integral expression cos(x cos 0) sin2" 0 dO, (28.4)

while if we make the substitution I = sin 0 we get the formula

=

(x sin 0) cos2" 0 dO. (28.5)

+U

The particular forms appropriate to ii = 0 are J0(x) = 29.

0

o

cos 0)dO

$o cos

cos(x sin O)dO. (28.6)

=

o

The Addition Formula for the Bessel Coeffi-

cients. In certain physical problems we have to reduce a Bessel coefficient of type + y) to a form more amenable to computation. We shall now derive an addition

formula which is of great use in these circumstances. From the (lefinition (25.6) we have the expansion exp {

+ y) (t —

} Writing the left-hand side as a product CXI) {

and

that

(t

—

}

. CXI)

{

(i

inserting the appropriate series from

—

+ y) I".

} (25.6) we find


-4- y)t"

Equating coefficients of I" we obtain the addition formula (29.1)

+ Y)

To put this in a form which involves only Bessel coefficients of positive order we write the right hand side in the form n

—1

+r—n+1Jr(3)Jn_r(Y)

Jrfr1')Jn-r(Y) +

r—co

and

r—O

note that because of the relation (25.7) the first term

can be written as —1

(— 1

=

(

1

Similarly the third term is equal to

so that finally we have

+ ii) + E(—

+

(29.2)

30. Bessel's Differential Equation. We showed preis a viously 26 above) that, if n is an integer, solution of Bessel's equation (26.9). We shall now examine

the solutions of that equation when the parameter n is not necessarily an integer. To emphasise that this parameter is, in general, non-integral, we shall replace it by the symbol v, so that we now consider the solutions of the second order linear differential equation (80.1)

BESSEL FUNCTIONS

§ 30

103

Writing the equation in the form (80.2)

we see that the point x =

0

is a regular singular point and

that in the notation of § m = I and q0 = — v2. The indicial equation (cf. (3.5) above) is therefore =0 — and this has roots = ± v. First of all we shall suppose that v is neither zero nor an integer. Then the first solution is of the form (30.8)

Substituting this series in equation (80.2) we see that the coefficients must be such that

r)(v+ r —1)+(v+r) — Hence

we must have c1 {(v +

and

0.

1)2 — v2}

=0

in general

(r= 2,8,...).

ct{(r+v)2_v2}=

(80.4)

must therefore take c1 to be zero and hence, in order that (30.4) may be satisfied for all r 2, we must take C2r+1 = 0 We

and (.—

—

(2r+ 2v)(2r+ 2v—2) ... (2v-f- 2)2r(2r—2) ... 2

an expression which may be put in the form c0

=

r!(v + 1 )r

T)

Taking c0 = 1/2'v! we see that the basic solution of type (80.3) may be taken as


(_

y=

+

(80.5)

1)r

Comparing this series with the series (27.2) we see that it

is of precisely the same form as that equation, the only difference being that n is replaced here by v. If we take the series (27.2) to (lefine the Bessel function of the first kind of order n, even wimeim is not an integer, then we

may write the solution (80.5) in the form

y= Similarly, if we substitute a series of type Y

to correspond to the second root of the indicial equation,

we find that it must be of the type

=

(80 6)

r!(— 3+ 1)r and with the extension of the definition (27.2) to nonintegral values of v we may write this solution in the form r-O

y= Thus when v is not an integer we may write the general solution of equation (30.1) in the form (80.7) y = AJ, (x) + where J, (x) is defined by the equation

J,(x)

= f(v+

+ 1; —

kx2).

(80.8)

It should be observed that the results of § 27, 28, with the exception of (27.8) are true when n is not an integer, since they were derived directly from the definition (27.2), which is equivalent to (30.8). Time transition is effected merely by replacing n! by P(;' + 1). When 1' is zero or an integer we know from equation (25.7) that the solutions and are not linearly

BESSEL FUNCTIONS

§ 30

105

independent. We must therefore use the formulae (8.8) to

calculate the second solution. We shall consider first the case in which v = 0. If we let

in order to satisfy the recurrence relation (30.4) we must have then

—

r_or!(e+ l)r and

putting e =

0

wc obtain the first solution

= J0(x). Using the result i a

1

(e+ we

(80.10)

')rla-ie+s

see that r

Putting e =

0

r_IT!(O+1)r and substituting the value (30.10) for w0

we find that the second solution (aw/ae )Q—O Y0(x) = J0(x)

— r—1

(r!)

(30.11)

where

q(r)

=

—.

(80.12)

s—I S

is called Neumann's second kind of zero order. Obviously if we add to Y0(x) a function which is a constant

The function Y0(x)

so

Bessel function of the

obtained

multiple of J0(x) the resulting function is also a solution of the differential equation (80.13)


In particular the function Y0(x) =

— (log 2 — y)J0(x))

where y denotes Euler's constant, will be a second solution

of the equation. Substituting from equation (80.11) for Y0(x) in this equation we obtain the expression Y0(x)

= —2

+ y)J0(x) —

p(r) (80.14) /

where (p(r) is defined by equation (80.12). The function Y0(x), so defined is known as Weber's Bessel

function of the second kind of zero order.

Thus the complete solution of the equation (80.18) is

y = AJ0(x) + BY0(x)

(80.15)

where A, B are arbitrary constants and J0(x), Y0(x) are given by equations (30.8) and (80.14) respectively. It can be shown by an exactly similar process that when

v is an integer the complete solution of the equation (80.1) is

y = AJ,,(x) +

(80.16)

where A, B are arbitrary constants, J, (x) is defined by equation (80.8) and (x) is given by 2

(x) = — {y + log (ix))

2 v—2r

1

(x) — —

I

,

/

(—)

(80.17)

r!(v+r)! The function Y, (x) so defined') reduces to the Y0(x) of equation (80.14) as v —. 0 and is known as Weber's Bessel —

function of the second kind of order v.

The variation of Y0(x) and Y1(x) for a range of values of x is shown graphically in Fig. 10. ') This function is denoted as N,(x) by Courant and hubert.

BESSEL FUNCTIONS

30 1

i

i

Fig. 10

I

I

I

I

107 I

I

I

-

Variation of Y0(x) and Y1(x) with x.

The functions J, (x) and

(x) are independent solutions

of the equation (80.1), but in certain circumstances it is advantageous to define, in terms of them, two new independent solutions. If we write iY,(x)

= J, (a,) — iY, (x)

(80.18) (80.19)

then it is obvious that we can take the general solution of Bessel's differential equation (80.1) to be

y= A1

A2 arc arbitrary constants. The functions defined by equations (80.18) and (80.19)

are called Hankel's Bessel functions of the third kind

of order v. The Hankel functions and JJ12)(a,) bear the same relation to the Bessel functions J, (x), Y, (x) as the functions exp (± ivx) bear to cos vx and sin vx, and are used

in analysis for similar reasons. It should also be noted JJ(2)(a,) satisfy that the Bessel functions Y, (x)'


the same (liffercfltial equal ions and recurrence relations

as the function 31.

SpherIcal Bessel Functions. A iroblcm which

arises in mathematical physics is that of the solution of the wave cqLlatiofl in spherical polar coordinates a

1

1

F

r ar

r2 sin 0

0

a solution of this equation of the form (81.2)

p) is the surface spherical harmonic defined by equation (28.3) an(l is a function of r alone which satisfies the equation wherc 1'm

+

dr2

2

n(n + 1)

r dr

r2

c2

—0

81

3)

putting

W= r41€

(81.4)

we see that equation (31.8) becomes 1 dR d2R w2 (n + dr2

+

r dr

+ 1c2

r2

R— 0

whose general solution is readily seen to l)C

+

1? =

(81.5)

Hence the function

= Y?)L

(81.6)

a solution of the equation (31.1). The functions which occur in the solution (81.6) are called spherical Bessel functions. We shall is

now show that they are related simply to the circular functions. First of all we consider the Bessel function J1(x). If we let n = in equation (80.6) and make use of' the duplication formula for the ganinia function we obtain the result

BESSEL FUNCTIONS

§ 31

109

Icc —1 rt2r+J (2r+ 1)! which shows that A

J1(x)

Again, if we put n =

(31.7)

x.

==

in equation (30.6) we obtain

—

the relation

(81.8)

The other functions

where m is half an odd integer

nitty be worked out in a similar fashion. It is left as an exercise to the reader to show that cos

Siti X

= =

sin X+/m(X)

where the functions I,,,'

g,,,

are given in Table 1.

'l'ablc 11?

COS

I

tIn

I ()

11

T

105

45

I

42015

045 —

105

945

10

105

±1


functions which arise in the way described have been tabulated in Tables of Spherical Bessel Functions 2 vols. (Columbia Univ. Press, 1947) prepared by the Mathematical Tables Project of the National Bureau of These

Standards. 32.

Integrals involving Bessel Functions. In this

section we shall derive the values of some integrals involving Bessel functions which arise in practical applications. In the first instance we shall consider definite integrals.

From equation (27.5) we have the relation

= If n> 0,

0 so that the lower limit is x zero and we obtain the integral

=

(n > 0),

(32.1)

which, by a simple change of variable, gives the result a

=

f Jo

a"

(n> 0).

(82.2)

A particular case of this result which is of frequent use in

mathematical physics is obtained by putting n = 1 in equation (32.2). In this way we obtain the integral a

a =— J1(afl.

I

(82.8)

Further results may be obtained from (32.2) by familiar devices such as integration by parts. For example, making use of equation (27.5) we may write

f

a o

=

ala

f r2 —

dr,

o

an(1, integrating by parts, we see that the right hand side of this equation becomes

BESSEL FUNCTIONS

§ 32

111

a3 I

which

reduces, by virtue of (82.2), to —

Now by the recurrence relation (26.2) we have the expression

so

—

=

J2(Ea)

that finally we have the result

=

+

{

—

}

(82.4)

Combining this result with equation (82.8) we obtain the integral a I

Jo

r(a2

The

= 4a

—

—

"a2 :__

(82.5)

most commonly occurring infinite integrals are

most easily evaluated by means of substituting the formula

(27.8) in parts (ii) and (iii) of example 17 of Chapter II. From part (ii) of that example we see that

2'l'(v+ 1) foFi(v +

1;

—

=

v+1; (82.6)

If we make use of equation (80.8) on the left-hand side of this equation and of equation (7.4) on the right-hand side, •

we see that this result is equivalent to the formula F(iz+v+1)a' IJ,(ax)xPtr.Pzdx= Jo 2'l'(v-f- 1

where p> 0,

+ + v>

4v— 0.

v+1; a2±p2)' (82.7)


The hypcrgeornetric series occurring on the right-hand of

this equation assumes a particularly simple form if either = v or = i' + 1, and we obtain the formulae I

j

(82.8)

—

= 2'+'!'(j, +

J, (ax)x'+1

Two

a'

J,(ax)xre_Pxdx

(32.9)

(a2 +

special cases of the formula (82.8) which occur

frequently are

jo J

1

J0(ax)e_Pzdx=

= o

(82.10)

v'(a2+p2)' a

(82.11)

(a2 + p2)3!2

Integrating both sides of equation (82.11) with respect to

we find that

p from p to

i,Ji(ax)e_Pzdx =

p

—

a aV(a2+p2)' A special case of (82.0) which is often needed is

5xJ0(ax)tr1'xdx = o

(82.12)

(82.18)

(a2+p2)312

If we let p tend to zero on both sides of equation (32.7)

we find that we can sum the hypergeometric series by Gauss's theorem (7.2) proVi(Icd that v + 1 . We then have the result (ax)xPdx =

+

+

(82.14)

Similarly from part (iii) of example 17 of Chapter II we have the equation j

'F

P± 1;

—

BESSEL FUNCTIONS

§ 33

which, J0

113

because of (27.8), is equivalent to

J, (ax)e_P'n'xP_ldx a2

v+1;

—

(82.15)

parts (i) and (ii) of Ex. 11 of Chapter II we have

From

the special cases I Jo

dx

-

=

(2p2)'+''

(82.16)

and

=

Jo

(Zr

—

4p2/

(32.17)

of which the most frequently used are

dx = _L

(82.18)

2p2

and

i°x3J0(ax) eP'z'dx =

Jo

33.

(i —

a

(82.19)

The Modified Bessel Functions. By an argument

similar to that employed in § 1 we can readily show that Laplace's equation in cylindrical coordinates lä2ip

=0

possesses

solutions of the form

= where

R(e) satisfies the ordinary differential equation d2R

ldR

0

(88.1)

Writing x in place of me we see that this equation is


equivalent to the equation

(88.2)

If we proceed in exactly the same way as in §

80 we

can show that if v is neither zero nor an integer the solution

of this equation is (88.3) R = Al, (x) + BI_,(x) where A and B are arbitrary constants and the function I, (x) is defined by the equation r!(v+1)r = 2'r(v+l)

= 2'p(V+l)

0F1(v+1; (83.4)

Comparing equation (88.4) with equation (80.8) we see that

4(x) = i'J,(ix)

(83.5)

a result which might have been conjectured from the differential equation itself.

is a multiple of If v is an integer, n say, then so that the solution (88.8) in effect contains only one arbitrary constant. By a process, similar to that outlined in § 80 we can show that in these circumstances the general solution of equation (84.2) is

R=

+

is defined by the equation

where the function

K8(x) =

{log (ix) + y)

+

+

(33.6)

1

r—1

The functions

r!(n+ r)!

n—i r—O

'

/

'

' (38.7)

defined by equations (84.4)

BESSEL FUNCTIONS

§ 33

115

(88.7) respectively are known as modified Bessel functions of the first and second kinds. and

The result (83.5) is very useful for deducing properties of the modified Bessel function from those of the Bessel function For instance, when n is an integer it follows from equation (25.7) that

= and

(83.8)

from equations (26.1) to (26.5) respectively that

= 2,i -

= = = =

+

(33.9)

—

(88.10)

—

(88.11) (83.12) (88.18)

+

Fig. 11 VariatIon of e—'10(x), eZ11(x) and

with. x


Similarly equations (27.5) and (27.7) imply the relations

{x"I,,(x)} =

(88.14)

{a.r"I,,(x)) +

(83.15)

All of these relations can, of course, be derived directly from the definition (83.4) of I, (x) and it is suggested as an exercise to the reader to derive them in this way. It should also be observed that satisfies the same recurrence relations as 5

4

3

2

1

0

1

2

3

4

5

6

FIg. 12 VariatIon of #K0(z), eK1(x) and eaK,(x) with x.

The variation of !0(x), (a,) and 12(x) with x is hown graphically in Fig. 11 and that of K1(x) and K2(x) is shown in Fig. 12.

BESSEL FUNCTIONS

§ 34

34.

117

The Ber and Bei Functions. If we wish to find

solutions of the form

= of the (liffilsion equation —

°r have

0°0

——

to solve the or(linarv differential equation &R idft jo)J?

d02+

0 (10

changing the independent variable to x = e we see that this latter equation is equivalent to the equation On

(84.1)

Formally we may take the independent solutions of this equation to be 10(i4x) and K0(i&x). Kelvin introduced two new functions ber(x) and bei(x) which are respec. tively the real and imaginary parts of i.e. (84.2) ber(x) + i bei(x) = 10(i&x). From the definition (83.4) of 10(x) we see that her(x)

and that hci(x)

=

,

(_l)'(fx2)23+'

=

(2s + I ) 12

(84.8)

(84.4)

The variation of the functions ber(x) and bci(x) with x is shown diagrammatically in Fig. 18.

In a similar way the functions ker(x) and kei(x) are

defined to be respectively the real and imaginary parts of the complex function K0(ilx), i.e. kcr(x) ± i kci(x) = (84.5)

118 THE SPECIAL FUNCTIONS OF

AND CHEMISTRY

34

Fig. 13 Variation of ber(x) and bei(x) with x.

From the definition (83.7) of K0(x) we can readily show

that

ker (iv) = — (log (lx) + y} ber (iv)

+

+

(iv)

(2r)!2

and that kei (iv) = — {log (jx) + y}bei (iv) —

+E

r_o

(2r.-f-l)12

q(2r),

(84.0)

ber(x) (84.7)

Fig. 14 shows the variation of the functions ker(x) and kei(x) over a range of values of the independent variable iv. The four functions ber, bci, ker and kei are used more often in electrical engineering than they are in physics or chemistry. For a full account of their properties and those of their generalization to higher order and of their cation to engineering problems the reader is referred to

119

BESSEL FUNCTIONS

§ 35

N. W. McLachlan's

Bessel

Functions br Engineers,

(Oxford University Press, 1034).

Fig. 14 VariatIon of ker(x) and kel(x) with x.

35. ExpansIons In Series of Bessel Functions. We

know from § {

{ x2

80

that

+x

+ (22x2 — m2)

+x

+ (1u2x2 — fl2)}

}

= 0,

(85.1)

= 0,

(85.2)

so that multiplying equation (85.1) by (85.2) by integrating with respect to x from 0 to a and subtracting we find that (22

/22)f

a

a

dx X

= if n> —1, m> —1.

—

(85.8)


Putting In

= ii

in this result

we find that if 2

22_/L2 IizJ,,(2a

(85.4) The corresponding expression for 2

= p is obtained by putting p = 2 + e, where e is small, using Taylor's theorem and then letting e tend to zero. We find that (85.5)

22a211

Suppose now that 2 an(l p are positive roots

dental equation

of the transcen-

+ k

arc constants. It follows than that

=

(85.7)

wherc CA

=

{k222a2

+ h2 —

k2n2).

(35.8)

that we can expand an arbitrary If we now function /(x) in the form (85.0) /(x) = E where the sum is taken over the positive roots of the equation (35.6) then we can determine the coefficients

as follows: Multiply both sides of equation (85.9) by x from 0 to a then

= from which it follows that 1 a,=_J CA,

(85.10) 0

BESSEL FUNCTIONS

§ 36

121

Because of its similarity to a Fourier series a series of the

type (85.9) is called a Fourier-Bessel series. In particular if the sum is taken over the roots of the equation (85.11)

then the coefficients of the sum (85.9) are given by a a5

=

{J (2;a)}2

—

2

(35.12)

.))

Similarly if the sum is taken over the positive roots of the equation

=0

(35.18)

we find that the coefficients a5 are given by the formula a5

(85.14)

=

In this section no attempt has been made to discuss

the very (liffleult of the convergence of FourierBessel series. For a very full discussion of this topic the

reader is referred to Chapter XVIII of G. N. Watson's

A Treatise on the Theory of Bessel Functions, 2nd. edit.,

(Cambridge University Press, 1944).

36. The Use of Bessel Functions in Potential Theory. As an example of the use of Bessel functions in potential

theory we shall consider the problem of determining a 0, z 0 function v'(o' z) for the half-space a satisfying the differential equation 1

e

and

the boundary conditions: (i)

(ii)

tp=/(e),

on

z=0;

—

(q


(iii)

remains finite as —* 0. We saw in § 1 that a function of the form = R(e)Z(z) is a solution of equation (86.1) provided that (iv)

d2Z

that

and

d2R

where

2

(86.2)

ldR

+ — 71;; +

2

=0

(86.8)

is a constant of separation. To satisfy the boundary

condition (ii) we must take solutions of equation (86.2) of the form

Z= to satisfy the condition (iv) we must take as the

and

solutions of equation (86.8) functions of the form

B= since

the second solutions

in the region of the axis

=

would become infinite 0.

The differential equation (86.1) and the boundary conditions (ii) and (iv) are satisfied by any sums of the form (86.4) tp(r, z) = and are constants. But if we are to satisfy the boundary condition (iii) we must take the sum over the positive roots of the equation 1)

where the

(86.5) = 0. + The solution is determined therefore if we can find constants

such that condition (i) is satisfied, i.e. such that

= 1)

(86.6)

For properties of the roots of this equation see example 10 below.

BESSEL FUNCTIONS

§ 36

123

From equations (35.10) and (85.8) we see that we must take

=

feRe')Jo(21e')de'

+

(86.7)

Hence the required solution is tp(r, z)

4

(36.8)

where the sum is taken over the positive roots of the equation (86.5). If, instead of the boundary condition (iii), we had the condition v = 0 on = a then it is easily seen that the solution would have been z) =

Z

(86.0)

the sum is taken over the positive roots of the transcendental equation J0(Aa) = 0. (86.10) For example suppose that satisfies the conditions where

tp=O on as z = 0 a, then the solution to the Problem is given by equation (86.9) with equation (82.5) we have

=

—

By

—

since is a root of equation (86.10). Thus the required solution is tp(r,

z) =

(86.11)

a

124 THE SPECIAL. FUNCTIONS OF PHYSICS AND CHEMISTRY § 37

Tables of the first forty zeros of the function with the corresponding values of arc available') so that it is convenient to express results of the kind (86.11) in terms of theni. It is readily seen that in this case tp(r,

where = z/a

z) =

=

and

8a-i1 0

i

(86.12)

ce

e/a.

37. Asymptotic Expansions of Bessel Functions. In certain physical problems it is desirable to know the value

of a Bessel function for large values of its argument. In this sectiOn WC shall (ICI'iVC the asymptotic expansion of the Bessel function of the first kind and merely indicate the results for the othci' Bessel occurring in matliematical I)liYSiCS.

We take equation ) as our definition of the function Applying the theory of functions of a complex variable it is readily sliots'ii that this definition is equivalent to eixldt + fL,0

= 1'(UJ'(n+4) { where

e"tdi } (37.1)

L1 is the straight line

=

in the tipper

—1

half of complex 1-plane and L2 is the corresponding pitrt of the straight line = 4— 1. By changing the variable from 1 to a = ix(1 — 1) in the first integral and to

u=

— ix(1 —

1)

in the second we see that

=

+ j(x)}

(87.2)

where

= 1)

on

A.

(1

+

Gray, C. B. Mathews and T. M. MacRobert, A Treatise Functions and Their ilpplications to Physics, 2nd. edit.,

(Macmillan, 1931).

BESSEL FUNCTIONS

§ 37

125

denotes its complex conjugate. Expanding by the binomial theorem and integrating (1 + term by term we find that and

=

0F0

± ii,

—

ii;

.

(87.8)

If we adopt ilankel's convention of writing

',

—

/

+

—

71)r

iii equation (87.8) and substitute the result in equation (37.2) we find fitially that for large values of x the asymptotic expansion of the Bessel function is

(—1Y(n, 2r)

eos (a,

—

{

1)

1

— sin (x —

}

r..O

(87.4)

The corresponding expansion for the Bessel function of

the second kind is found to be

2r)

(—

+ eos (x —

—

+

}.

+ (87.5)

Substituting these asymptotic expressions in equations

(80.18) and (30.19) we find that as x

(87.6)

where j,,(x)

is

given l)y equation (37.3).

In certain problems only a very crude approximation to the behaviour of the Bessel function is desired. In these

circumstances the following formulae arc usually suf-


ficient:-

(87.7)

1/!

j/iiet4Ta7U+hi,

(87.8)

Similar formulae exist for the modified Bessel functions.

Proceeding in the same way as in the establishment of equation (28.1) we can show that

=

— 12)fl—Idt

+

which becomes {e

+ezlJo

du

by a simple change of variable. By a method similar to that employed above to obtain the asymptotic expansion we can then show that if — < arg x of (n, r) r) 1 e

(2x)r

+

r-O (2xY

the factor exp{—x+ (n+ <arg The corresponding is replaced by exp {— — (n + formula for the modified Bessel of the second kind is and that if —

(2xY

as

BESSEL FUNCTIONS

127

EXAMPLES

Making use of Example 2 of Chapter II and of the expansion (25.8) expand cos (x sin 0) as a power series in sin 0 In two ways. Hence by equating powers of 0 show that ifs Is a positive Integer 'n+s 1'! 1.

Z'

(n — a)!

n—a

Derive the corresponding result for

' J, (x).

and show that the two

results may be combined Into the single formula

(r+ fl — 1)!

2.

If u

show

and

denote the

eccentric

(r + 2fl)Jr+,R(x)

and mean anomalies

of a planet

that

E— J,1_,(me) cos (me) m 1

cos

(nu) = n

sin

(nu) = n E —Jm_n(me) sin (m*).

1

m

Making use of the expression for P,, of Chapter III show that

3.

E — P,,(cos 0) 4.

in Example 19

el'cOSOJ0(r sin 0).

Show that

=

(i)

+ 8Ja÷i(Z) —

—

+ 3J(z) + J5(z)

(ii)

5.

(cos 0) given

0.

Prove that

2

+

(—1)'J0(rz)

(._i)N

du .10

coslu

and deduce that I 2

—

+ E (_I)rJ0frx) =

0.

v'(x' .-.u')

______— 128 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

6. Show that the curve with freedom equations

y—cost/(1 —cost) may be represented in the Interval 0 <S ashowthat cos(x sin 0)dO

— a2)}

lf—iIandm+n+1>O,provethat (x'—I

Qm(x)

denotes the associated Legendre function of the second

where

kind. 18.

Prove that + Y)

E m—O

+

rn—i

+

130 THE SPECIAL FUNCTIONS OF PHYSCIS AND CHEMISTRY 19.

Show that

20.

Prove that

=

E and

deduce that J1(re'0) = Z

(1)

m

(ii) 4(x) = — Jn+m(X). ml 21. Using the expansion of the last question prove that is the coefficient of r in the expansion of — i(a sin a + b sin fi) exp {

} rn——co n—co

By putting R = a cos a + b cos fi, 0 a sin a + b sin fi, prove Neumann's addition theorem a—be-'0 }n co J,,(R) z JR+m(a)J,..s(b)r"° — ( a — be where R' a' + b' 2abcosO. 22.

Prove that J,(ax)J0(bx)

where

where

R'

a' + b' —

2ab

-!

cos 0 and deduce that

4

K(k)—I J0 /(t—k'sin'ip)' Prove, in a similar way, that

=

4ab

(a+b)' -f-c' ((1 — 4k')K(k)

—

E(k)}

BESSEL FUNCTIONS

where

k and K(k) are as defined above and

= Jo I

E(k) 23.

131

—

Show that JM(x)Jn(x) —

sin 0)

dO

and hence that

= I j"J21(2x sin 0) dO Deduce that (2n+2s)!

(ix)" 24. Prove

(

that

=

exp

a

+b)1

(_

define the Bessel..Integral function of order n by the

25.

equation

Ji,,(x) Prove

that if n is even Ji,,(x) —

—

I —du

Jo

!

U

(nO) ci (x sin 0) dO,

where ci(x) denotes the cosine integral, and derive the corresponding

expression when n is odd. Show (i)

ii

(ii) (iii) (iv)

(n

(v) (vi)

=E = J,,(x)/x; — 1 )Ji,,...1(x) — (n

+I

=

2J1,(x) -4-. 2Ji4(x) — ...; si(x) = 2.J11(x) — 2Ji,(x) + 2J1,(x) — . .

ci(x) = Ji,(x)

—

Ji,(x) = + log(4x) —

a!'

1; 2, 2, 2; —x'/4).

ChAPTER V

THE FUNCTIONS OF HERMITE AND LAGUERRE

The Hermite Polynomials. The Hermite polyis defined for integral values of n and all nomial real values of x by the identity 38.

H,,(x)

=

If

(88.1)

we write

=

/ (x, t) =

then it follows from Taylor's theorem that

=

=

Now it is obvious from the form of the function exp {— (x — g)2} that ,In

I

Late

and

I

C

— (— — dx"

so we have the form (e_z*)

= (—

for the calculation of the polynomial It follows from this formula that the first eight Hermitc polynomials are:H0(x) = 1, J11(x) = 2x,

H2(x)= 4x2— 2 H3(x) =8x3 — 12x. 182


§ 38

II4(x) = 115(x) 116(x) 117(x)

= = =

133

16x4 — 48x2 + 12, 82x5 — 160x3 + 120x, 64x6 — 480x4 720x2 — 120, 128x7 — 1844x5 8300x3 — 1680x.

±

+

In general we have

—

1)

+ or,

(2x)"2

— 1)(n—

— 4)

+

in the notation of Section 12,

=

(2x)"

—

(—

—

(88.8)

Recurrence formulae for the Hermite polynomials follow

directly from the defining relation (88.1). If we tiate both sides of that equation with respect to x we obtain the relation 2tC2xt_t*

=

/

nl from which it follows directly that

=

(88.4)

On the other hand if we differentiate both sides of the identity (88.1) with respect to £ we obtain the relation 2(x —

t)c2tx_12

=

gn.-1

(n— 1)!

which can be written in the form 2x

Ii,,(x)

— 2

£8—i

to yield the identity 2x118(x) =

+

by the identification of coefficients of 1".

(88.5)


from equations (88.4) and (88.5)

Eliminating

we obtain the relation

=

(88.0)

—

Differentiating both sides of this identity we find that

=

+

= 2(n + 1 = 0. +

and, by equation (88.4), —

In other words y = ferential equation

—

so that (88.7)

is a solution of the linear dif-

y" — 2xy'

+ 2ny =

0.

(88.8)

39. Hermite's Differential Equation. We saw in the

is a solution of the differential equation (88.8). Replacing the integer n in that equation

last section that

by the parameter v we obtain Herinite's differential equation dzy

dy

2vy= 0.

(89.1)

If we assume a solution of this equation in the form y ==E substitute in the equation (39.1) we obtain the recurrence relation and

—

392 )

Equating to on equating to zero the coefficient of zero the coefficient of xQ2 we obtain the indicial equation — 1)

=

Corresponding to the root = relation

0

0.

(89.8)

we have the recurrence

§ 39

THE FUNCTIONS OF HERMITE AND LAGUERRE a,4.2

2(r

135

v)

= (r + l)(r + 2) a,

(89.4)

which gives the solution

ai(i

6+

..)

(89.5)

a1 is a constant. Similarly, corresponding to the root = 1 of the indicial equation we have the recurrence relation 2(r+ 1—v)2)a? a,4.2 (80.6) — (r + 8)(r+ from which is derived the solution (12(V—. 1) where

()

)

(89.7)

a2 is a constant. The general solution of Hermite's differential equation is therefore where

11 = Yi(X)

+ y2(x).

(89.8)

For general values of the parameter v the two series for y1(x) and y2(x) are infinite. From equations (89.4) and (80.6) it follows that for both series a,4.2

115

(89.9)

T

If we write exp (x2) =

b0

+

+

. .

. + b,x' +

+

then

as r-+cii

(89.10)

Suppose that aN/bN is equal to a constant y, which may be

small or large, then it follows form equations (89.9) and


(89.10) that, for large enough values of N,

In other wor(ls the higher terms of the series for y1(x),

y2(x) differ from those of exp (x2) only by multiplicative

constants Yi' )'2'

that for large values of x

so

y1(x)

yie'2, Y2(X)

for such values the lower ternis are unimportant. We shall see later (Section 41 below), that in quantum mechanics we require solutions of 1-lermite's differential equation which do not become infinite more rapidly than as x J It follows from the above consiclerations that such solutions are possible only if either or i,2(x) reduce to Siml)le polynomials and it is obvious from equations (39.5) and (39.7) that this occurs only if v is a positive integer. For example if v is an even integer n we get the solution since

y(x) where c is

=

(39.11)

a constant, by taking a2 = 0, a1

= (—

Similarly, if v is an odd integer ii we get the solution (89.11) by taking a1 = 0 and a2

= (—

c.

Hermite's differential equation therefore possesses solutions

which do not become infinite more rapidly than as x —÷ if and only if i' is a positive integer 'ii. When this is so the required solution of Ilermite's equation is given by equation (39.11).

Hermite Functions. A differential equation closely related to 1-lermite's equation is

40.

dx2

+—

=0

(40.1)

40


If We transform the (lependent variable from ip to y =

137

(40.2)

and put 2 = I 2z' then it is readily shown that y satisfies Hermite's equat (89.1). 'l'he general solution of equation (40.1) is therefore given equations (40.2) and (39.8) with y1(x), given by equations (39.5) and (39.7) respectively. The argument at the end of the last section shows that the equation (40.1 ) solutions which tend to zero as x —* if and only if the parameter 2 is of the form 1 -4— 2n where n is

a positive integer. When 2 is of this

form the rcquire(l solution of (40.1) is a constant multiple of the function (lefined l)y the equation (40.8)

where II,,(x) is t lie Ilerinite polvnoniial of (legrec ii. 'I'hc function 1J',1(x)

is

called a Hermite function of order n.

The recurrence relations for follow imnied iately from those for II,4(x). For instance equation (38.4) is equivalent to the relation

= and eqLlation (:38.5) is

±

unaltered in form so that

= Eliminating

have the relation

(40.4)

+

(40.5)

from equations (40.4) and (40.5) we

(40.0)

—

From the point, of' view of mat heniatieal

the most

important properties of' I termite functions concern integrals involving pi'oducts of two of t hem. In establishing most of these properties the starting point is the observation that the function satisfies the relation f- (2ii + I —

=

0

(4().7)

138 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 40 as

is obvious merely by substituting 2n +

1

for

in

equation (40.1). Writing down the corresponding relation for !Pm,

=0 (40.8) multiplying it by and subtracting it from equation (40.7) multiplied by Wm we obtain, as a result of in. + (2m + 1

tegrating over (—

cx)), the relation

cxi,

2(m — n) f WmWn

=

Now an integration by parts shows that the right hand side of this equation has the value —

—

and, if we remember that, for all positive integers n, —* 0 as x -÷ x), we see that this has the value zero. Hence if we let

= we see that Im,n =

0,

(40.9)

fl.

7fl

In particular

=0 that from equation (40.5) we have 'n—l,n+l

so

=

(40.10)

Now if in equation (40.8) we substitute for equation (88.2) we have

=

(—

(ex')

from (40.11)

§ 40


that the left-hand side of equation (40.10) is equal to

so

dn—1

dx"' and an integration by parts shows that this is equal to n

i.e. to I,,

,,. Hence

+ 'n+l• n—I

from equation (40.10) we have 1

—01

Repeating this operation n times and noting that

=

10,0 = we

find that tn,n =

(40.12)

Combining equations (40.9) and (40.12) we have finally (40.18) 'm,n = The evaluation of more complicated integrals can be

effected by combining this result with the recurrence formulae we have already established for tile ilermite functions. For instance, it follows from equation (40.5) that

=

+

showing that

=

0

if rn

n±1

(40.14)

and that

= 2"(n +

(40.15)

Similarly, making use of equation (40.4) and equations


(40.18—15)

we can show that 0

00

if

f

1

if 'in = ii ± 1.

41. The Occurrence of Hermite Functions in Wave

Mechanics. The Hermite functions which we have dISCUSSCd in the last section occur iii the wave mechanical

treatment of the harmonic oscillator 1). Although this is a very simple mechanical system the analysis of its l)roperties

is of great importance because of its application to the quantum theory of radiation. 'rue Schrodinger equation corresponding to a harmonic oscillator of point mass in with vibrational frequency v is dx2

= 0,

(II' —

+

(41.1)

where W is the total energy of the oscillator and h is Planck's constant. The l)r011C111 is to (letcrnline the wave which have the property that functions (i) (ii)

as

f_j tp

=

1.

If we let nii'

then the equation (41.1) becomes °W d2

(41.2)

and the conditions (i) and (ii) become 1) N. F. Mott and 1. N. Sneddon, Wave Mechanics and its .4p. plications. (Oxford, 1018), p. 50.

141

THE FUNCTIONS OF HERFILTE AND LAGUERRE

§ 41

(i')

as

=

(ii')

The argument given at the beginning of Section 40 shows that equation (4L2) possesses solutions which satisfy the con(Iition (ii') if and only if the pitranicter (21V//n') which occurs in the equation takes one of the values 1 + 2n where n is a positive integer. In other words by the probal)ihty

solutions of this type, which arc

interpretation of the wave function ip to correspond to stationary states of the system can exist if and only if

= lir(n +

W

(41.8)

where n is a positive integer. When this is the case the form

of the wave function ip is known from Section 40 to be (41.4)

tp =

where C is a constant. Applying condition (ii') and equa-

tion (40.12) we see that — jtrmv\k —\

1

)

Thus the wave function corresponding to an admissablc is energy (n +

In

=

x

(41.6)

theory the matrix elements

(ii x p)

E

=

I

defined by the equation (ii

I

x

=

arc of considerable importance in the case of the harmonic

oscillator. In terms of the variable (n

Ii I

I p) = i—,——

we have

142 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY § 42 so

that substituting from equation (41.6) we have

(n I x

)1}.

It follows then from equations (40.14) and (40.15) that

(n

+ 1)/i

(41.8)

}, nh

')={82

(ii

42.

(41.9)

}.

The Laguerre Polynomials. The Laguerre poly-

nomials L,1(x) are defined for n a positive integer and x a positive real number by the equation

= (1—

(—

In.

(42.!)

Expanding the exponential function we see that xl

(— 1

v

—

r!s! The coefficient of I" in this expansion is r!(n —

'o

Using the relations

+ )n—r we

—

n!

—

r!'

—

(— n)r

(n — r)! —

see that this sum can be written in the form

n (_ '.

r—O

(rI)2

Frxr

r1r-f-s

143


§ 42

Identifying the coefficient of with L,1(x)/n! and adopting

the notation of Section ii we see that

= nl 1F1(— ii;

1;

(42.2)

x)

is a polynomial of degree

It should be observed that

n in x, and that the coefficient of xt' is (—

1

)".

can be obtained by finding a new representation for the confluent A useful formula for the polynomial

hypergeometric function on the right hand side of equation (42.2). By Leibnitz's theorem for the n-th derivative of a

product of two functions we have 1)

(Dn_rxn)(Dre—x)

(—

rl D denotes the operator d/dx. Using the relations = n!xn/r! exDr(e_x) = I r—O

where

we

see that

=

(42.8)

It follows immediately from equation (42.2) that = Cr (xne_z). dx

(42.4)

The first five Laguerre polynomials can be calculated easily from this equation; we find that L0(x) = I, L1(x) =

1

—

= 2 — 4x + x2, L3(x) = 6 — 18x + 9x2 — L4(x) = 24 — 96x + 72x2 — 16x3 ± x4. L0(x)

Equations (42.4) can be used to show that the functions

=

e+rL,1(x)

(42.5)

form an orthonormal system. From (42.4) we have as a


result of in integrations by parts

= fm = (_1)m7n!f

d's—" —rn

and this is zero if n> in. Since Lm(X) is a polynomial of degree in in x it follows that

= 0 if Since the term of degree n in

that, then in =

in

ii.

(42.6)

is (— 1)t'x" it follows

ii, 1

= =

f

r

0

= (n!)2 Combining this result with equation (42.6) we find that

=

(42.7)

showing that the

form an orthonormal set. Recurrence formulae for the Laguerre polynomials may be derived directly from the definition (42.1). Differen-

tiating both sides of this equation with respect to I we obtain the identity x

exp

—

which

xl

— — n-O (n—i)!

ii!

may be written in the form L / fl! / +(i—l)2E u..O (n—i)!

L n'

L it!

/

=0.

§ 43


145

in the expansion on the left we obtain the recurrence relation Equating to zero the coefficient of

+

=

+

— 2n —

0.

(42.8)

Similarly if we (lifferentiate both sides of (42.1 ) with respect to x we obtain the identity + (1 —

I

'-'1" =

1)

0

which yields the recurrence relation 0.

± nL71.1(x)

—

(42.9)

Differentiating equation (42.8) twice with respect to x and replacing n by ii + 1 we find that (a') = 0. (a' )+ (ii + 1 (x— 2iz—8 (x)+ (42.10)

Now from (42.9) (a') = (ii + 1 and hence

=

(n

— L,3(x)},

+ 1)

—

and its in terms of derivatives can be readily obtained. Substituting these iii equation (42.10) we and values of find that A similar expression for

+

+ (1 —

=

0.

(42.11)

43. Laguerre's Differential Equation. Equation (42.8)

shows that y =

differential equation d2y

xr+ in

is

a solution of Laguerre's dy

(43.1)

the case in which v is a positive integer n. If we put


= 1, & = — i' in equation (11.2) we see that it takes the form (43.1) so that it follows from equation (11.4) that one solution of equation (48.1) is Y1(X) =

v;

1; x),

(48.2)

Similarly we see from equation (11.8) that the second solution is Y2(X) = Yi(X) . log x

+

(43.3)

where the coefficients arc defined by equations of the type (11.9). The general solution of Laguerre's (11ff erential

equation may therefore be written in the form y = A111(x)

+

By2(x)

A and B are constants and y1(x), by equations (48.2) and (48.3) respectively. where

are defined

If we are interested only in solutions which remain finite at x = 0 it is obvious from equation (48.8) that we must take the constant B to be zero. Further if a,. is the coefficient of

in the series expansion for y1(x) it is easily

shown that ar+l/a,. r1. As the result of a discussion similar to that advanced in Section 89, it follows that if is not an integer y1(a') cx as x If, therefore, we are looking for solutions which increase less rapidly than this we take 3' to a positive integer, in which case y1(.r) reduces to a 'l'he equation (48.1 ) a solution which increases

less rapidly than c as x 3'

occurring it

if and only if the parameter

is a 1)ositivc integer, n say. If it is also

required that the solution shall remain finite at x = solution is of the form

=

0,

the

(43.4) where A is a constant and L,,(x) is the Laguerre polynomial y

of degree ii.

§ 44

147


The Associated Laguerre Polynomials and Functions. If we dilferentiate Laguerre's dilferential equation 44.

in times with respect to a' we find that it becomes f-i1,

(1"?,

which shows that Lm(.r) (ICIIIIC(l by

=

L,(x),

(ii

(44.1)

in)

is a solution of the differential equation (121/

a'

4— (in ± 1 — a')

(11/

(LX

+ (,,

— in )y

=

defined liv equal ion (44.1 )

The l)01Yn0n11a1

(44.2)

0. is

called

the associated Laguerre polynomial. It follows from

equation (42.2) that it may he represented as a series by the equation

=

n+ in; ?n+ 1; a') (ii in). (44.8)

Similarly equation (42.4) leads to (lie formula (1"

=

.

{

(44.4)

}

The simplest associated Luguirre polvnoniials are: —1,

=2 = — 4 + 2r, = 18—6z', = —18+ 18x—3r2, = —t)6-r 144x—48x2--- 4.r3, = 24. = — 96 ± 24x,

—6. 12x2,

The definition (44.1 ) for the assoeiate(l Laguerre polynomial is the one usually taken in applied mathematics.


In pure mathematics the function 1F1(— n;

=

m 4—

1; x)

(44.5)

which is a solution of the differential equation

("+ I —x)+ny=O is

often taken as the definition of the associated Lngucrrc

so that care must be taken in reading the literature to ensure that the particular convention being

polynomial 1)

followed is understood. It is readily shown from equation (42.1) which defines the generating function for Laguerre polynomials that the associated Laguerre polynomials may be defined l)y the equation = (1 —

(— 1 )mjm exp (_-

t".

(44.6)

This identity can then be used to derive recurrence relations for the associated Laguerre polynomials similar to those of equations (42.8) and (42.9) (ef. Examples 6(u), (iii) below). The Laguerre functions J?Hj(X) are defined by the equation

=

1 + 1) (44.7) If in equation (44.2) we replace in by 21 + 1, n by n + 1 (n

is a and y by we see that the function solution of the ordinary linear differential equation x (LX

1.4

(44.8) X2

X

J

In most physical l)rol)lenls in which this equation arises

it is known that 1 is an integer. By reasoning similar to 1) See,

for instance, E. T. Copson, Functions of a Cauiple,r J'ariablc,

(Oxford University Press, 1935)

269.

§ 44


149

that outlined in Section 43 it can readily be shown that the a solution which is finite at equation (44.8) if, and only if, the parax = 0 and tends to zero as x meter n which occurs in it assumes integral values. When this does occur the solution is A where the function is given by equation (44.7) and it is an arbitrary constant. We shall now evaluate the integral

= which arises in wave mechanics. From equation (44.6) we

have the identity 12i+11 -

'n+Z

=

jn+l

n—l+1 n'—i+L (n41)!(n' -4— 1)! I expj— I.

xi

xt-

i—i

(1 —

—

from which it follows that ,i'—l+I (n—4—1)!(n —f-i)!

=

f

(tr)21+'

(l_j)2L+2(l_r)21+2JoX

ri xr 1 I -eXl)1—x—j------j—j--—--Jdx

This last integration is elementary and gives for the expression on the right — t)(1 —

(21 +

r)

(1 — and

by means of the binomial theorem we may expand

this function in the form 1— r — IT)

-i-- r TI

'


Now the coefficient of (1r)'*l in this expansion is

2n{(n+ 1)1) (n—l——- 1)!

this is equal to

and

f

+ 1)1)2. Hence

=

/

(44.9)

The Wave Functions for the Hydrogen Atom.

45.

We shall conclude this chapter by discussing the motion

of a single electron of mass m and charge — Coulomb field of force with potential

in the

e

Ze2 V(r)=——,

due to a nucleus of charge Ze. In a first approximation we

may treat the mass of the nucleus as infinite and this

case the wave function of the system is governed by the Schrodingcr equation 1) (45.1)

and the conditions: (i) (ii)

0,

0, q,) for all r, 0, p;

must be bounded in the range 0 all r, p;

tp—+Oasr—.cx; tp remains finite as r—*O; (v) ip is normalized to unity, i.e. $1

for

0

(lii) (iv)

12 dr =

1

where

the integral is taken throughout the whole of space.

W is the total energy of the system. 1)

N.

F. Mott and I. N. Sneddon, bc. cit. p.

54.


§ 45

The

equation (45.1) may be solved by setting

= where,

151

(45.2)

by the metho(l of separation of variables we have

=

4- +

(45.8)

0,

(45.4)

(iv

+

+ zt2)

—

1(1

+ 1) }

R—O (45 5)

To satisfy condition (1) we must choose as a solution of = such that (45.3) a function for +

all (p. Thus u occurring in equation (45.8) must be an

integer and a convenient solution will be

=

(45.6)

A is an arbitrary constant. Equation (43.5) is the well—known equation of which the solution is the associated Legendre polynomial 0). If 1 is integral and 1 u , then 0) is the only solution which is bounded in the range 0 0 and is therefore the only one leading to a wave function which satisfies condition (ii) above; if 1 is not an integer no bounded solution exists. To solve equation (43.5) we write

= and

=

.)' V

—

Zc2

;' = —--—— h-ct

(4o.7)

change the independent variable from r to x where We than find that (45.8)

where, in order that conditions (iii) and (iv) should be satisfied, II must be such that U 0, as x and as

152 THE SPECIAL FUNCTIONS OF PHYSiCS AND CHEMISTRY § 4S —> 0. From the arguments of Section 44 it follows that this is possible only if 1 is a positive integer and only if v is an integer, ii say, which is greater than 1 4— 1.

x

this is so we may write the solution of equation (45.8)

the form (44.7) so that the solution of (45.5) is proporNow by the second of the equations tional to (45.7) we have in

for the possible values of the total energy TV. The wave functions corresponding to the value of energy given by

the integer n are

(45.10)

0,

where C,,1,, is a constant determined by the condition (v).

In polar coordinates dr = r2

that this

sin 0 dr(l0 thp so

condition gives

1= so

f:sin 0 {P' (cos 0)J2d0

that it follows from equations (21.20) and (44.9) that

C

— (Q I.

Now

8ru(l+u)!{(n+1)!}3

(4511) J

from equations (45.7) and (45.9) we find that

= 4n2Ze2m/(h2n) so that if we introduce the Bohr radius a by means of the equation It2

(45.12)

we find that = Z/an. Introducing this result into (45.11) into equation and substituting the value obtained for (45.10) we find that


§ 45

0,

153

=

{ (2Zs)3

an'

(1—l-u)!{(n+l)l}38nn (45.18)

n — 1 are the 1 corresponding to the energy (45.9) of the hydrogen atom. If we write = — 2'I2Z2e'ln/h2 then correspondmg to the energy 1V0 we have the wave function

with u

=

0,

(e = Zr/a),

\ a'

and to the energy level

we have the three wave

functions 1

=

0,

0, ç,)

1

=

Z3 (2 Z3

COS

(—)' I

v211(r, 0, p)

—

0,

sin 0

=

From the last of these functions we can construct two functions I

——

QsinO

Sm

cos

Similarly corresponding to the energy W0/9 six wave functions Z'3 = I (—)* 0,

0, tp) tp311(r,

a

'8e +

Z3

(—)' (6 — 0)0e1Q cos 0, (i

=

0, p) =

(27 —

we

1

Z'3

— /r (—P a (6 —

sin 0

have the

154 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY §45 cos2 0 — 1),

= 81i/6n(a) 0,

Sm 0 cos 0 e±ic',

=

V'324(r, 0, p)

sin2 0

=

In wave mechanics v(r, 0, p) I2dr represents the probability that the electron whose wave-function is 0, I

is to be found in a small volume dr centred at the point To make the total whose polar coordinates are (r, 0, probability unity we must have I2

$ where

(1x=

1,

the integral is taken throughout the whole space.

Since, in polar coordinates, dv =

r2

sin 0 drd 0 dp it follows

that the probability that the electron is at a distance between r — from the electron is q(r)dr and r + where 2n

'p(r)=r2 I dtp Jo Jo I

for an electron in the state defined by quantum numbers n, 1, u (equation (45.18) above) we have Hence

(n—I—i)!

= 'rue

2n{(n +

2-

mean value of any function / which depends on r

alone is given by

1= that is, by the formula

=

2n{(n

For examples of the use of this formula see example 17 below.


155

E X A M P L ES

1.

Prove that if us < n d"

2.

{tI,,(x))

2mn!

=

— in)!

If K(x, y, I) = n—O

are the orthonormal set of Ilerinite functions defined

where the

by the relation

24 prove that y, 1)d.r Assuming

that K(x, y, 1) is of the form A exp(fLr + D are functions of I prove that. C I f.iayt — (x+y3)(1+1')

K(x, y, 1) = 3.

-=

CXJ)

2(1 —

I)'

Show that 4x'!

I"

I

I

n!

—4(s)

—

LI 1-I

(If (x))'— n—O

4. The Sclirödinger equation for the thrce.dimensional oscillator in cylindrical coordinates a) is 8z'ns

p's, + —--

4- w'z2))vp = 0.

Show that the solutions v' of this equation such that. -÷ 0 as and a u and tp is finite at. the origin arc of the form Cexp {iu — — where C is a constant, 1, it, u arc positive integers, and f, is a polynomial of degree 21 in which fi = satisfies the equation d'f I 1\df I — uI + 21 0. +

_ 156 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY

Show also

that the corresponding values of W are given by the

equation

(2! + Iu + 1)1w + (is + 3)hw.

W

Prove that

5.

,,

n!

—

Prove that

6.

d

(I)

M

(x) (x

(iii)

7.

If

—

in

—

2n —

nL'_1(x)

+

+

+

0.

is prove that (_1)?*

(n—rn)!

J0

8.

is !

Prove that

if

is

1+1

RR, (z)dx

9.

10.

0

= (n—I—i)!

(..I)is+i

Show that

Prove that the function

is a solution of the differential equation 1 d'y a—y+i dy f 2fl+a+I 2x

—

y(2x—y) 4x'

0


ii.

157

The potential energy for the nuclear motion of a diatomic

molecule is closely approximated by the Morse function V(r) = — Show that the sJ)herically symmetrical solutions of the Schrodingcr equation with this potential arc

=

(0

—

b= C,, is a normalization constant, and the corresponding values of the energy are where

("+4) + b

r

II

12. Prove that. the normaliz.ation constant C,, in the previous example has the value

2ba N,,

where

N,, = {V(2b

— "n

—

r

— 1)

r-.O

13.

Show

that in paral)olic coor(linates

q.

defined by the

equations

x=

cos q',

y

sin

z

—

SchrUdingcr's equation for a hydrogenlike atom of nuclear charge Ze takes the form

+

+

1/1 +

+

in

+ ij)

+

=

0

Show that. this equation has solutions of the forzit

'l"i') = where

a is the Bohr radius, n

Determine

k-f- i-f-a f—i and JI'= —2n2mc'z2/(n'/i2).

the constant C so that ip

is

normalized to unity.

158 THE SPECIAL FUNCTIONS OF PHYSICS AND CHEMISTRY 14. In the theory of the rotation and vibration spectrum of a diiitomic molecule there arises the problem of solving the Sehrodingcr equation with potential energy

B

Ze2

Show that the energy levels are given by II' = — 2ntnsZe'/(h'a') where

a = n 4 -1- iJ{b + (1 +4)2) with n, integers and b and that the corresponding wave functions arc 1

(s—) aa

where a is the Bohr radius, malisation factor.

+ (1

Show that the constant the value 15.

5

O)eiuq

(i—-) aa

occurring

+ 4)2)

is

a nor-

in the last example has

(21+])n!(l—u)!

2

taal'(n-s-rz-l1) I I 16.

and

I

If 1(1; ii; a)

I

Jo

x'42{R,,5(x))dx

prove that 1(1; n; a) ± ((n + 1)1)2 is the coefficient of

expansion of

in the

(°l+s+r r

Deduce that. {(n

—I-

1

1(1;

i—i)!

(On2 — 21(1 + I))

Show that in a hydrogen-like atom of nuclear charge Ze the average distance of the electron front the nucleus, in the state 17.

described by quantum numbers 1, ii, is i 1 1(1+1) ls'a r

Find the average value of hr and show that the total energy of a hydrogen atom is just one-half of the average potential energy.

APPENDiX

THE DIRAC DELTA FUNCTION

The Dirac Delta Function. In mathematical physics we often encounter functions which have non-zero values in very short intervals. For example, an impulsive force is envisaged as acting for only a very short interval of time. The Dirac delta function, which is used extensively in quantum mechanics and classical applied mathematics, 46.

may be thought of as a gcncralisation of this concept. If we consider the function

IxIa, then

(46.1)

it is readily shown that

=

1.

(40.2)

Also, if /(x) is any function which is integrable in the interval (— a, a) then, by using the mean value theorem of the integral calculus, we see that

f(x)ôa(x)dx =

f /(x)dx = /(Oa), I 0 1.

(40.8)

We now define

6(x) =

urn Oa(X).

(46.4)

Letting a tend to zero in equations (46.1) and (46.2) we see that 6(x) satisfies the relations (40.5) 6(x) = 0, if x 0, 150


fo(x)dx = 1

§

46

(46.6)

The "function" 6(x), (lefincd by equations (46.5) and (46.6), is known as the Dirac (lelta function. It is unlike the functions we normally encounter in mathematics; the latter are clef mccl to have a definite value (or values) at each point of a certain (lomain. For this reason 1)irac has called the delta function an ''improper function'' and

has emphasised that it may be used in mathematical

analysis only no inconsistency can possibly arise from its use. The (lelta function could l)c (liSpdnSc(l with entirely by using a limiting l)I'occdlure involving ordinary functions of the kind 60(x), but the "function" 6(x) and its "derivatives" play such a useful role in the formulation and solution of boundary value Prot)lcms in classical mathematical physics as well as in quantum mechanics that it is important to derive the formal properties of the Dime delta function. It should he emphasised, however, that these properties are purely formal. First of all it should be observed that the precise variation of 6 (x) in the neighbourhood of the origin is not important

provided that its oscillations, if it has any, are not too violent. For instance, the function 6(x) =

Inn

sin(27rnx)

satisfies the equations (46.5) and (46.6) and has the same formal properties as the function clef med by equation (46.4).

If we let a tend to zero in equation (46.3) we obtain the relation 1(0),

(46.7)

which a simple change of variable transforms to —

a)dx = /(a).

(46.8)

161

THE DIRAC DELTA FUNCTIONS

§ 46

In other words the operation of multiplying /(x) by ô(x—a)

integrating over all x is merely equivalent to substituting a for x in the original function. Symbolically we may write and

— a)

= /(a)ô(x — a)

(46.9)

if we remember that this equation has a meaning only in the sense that its two sides give equivalent results when used as factors in an integrand. As a special case we have (46.10) xô(x) = 0. In a similar way we can prove the relations 6(— x) = ô(ax) =

ô(x),

ô(a2 — x2) =

a> 0,

(46.12)

{ô(x — a) + ô(x + a)), a> 0. (46.18)

Let us now consider the interpretation we must put upon the "derivatives" of â(x). If we assume that exists and that both it and ô(x) can be regarded as ordinary functions in the rule for integrating by parts we see that =—

f/(x)a'(x)dx = Repeating

this process we find that

=

(—

(46.14)

The statement is often made that the Dirac delta function

is the derivative of the Hcaviside unit function 11(x) defined by the equations H

_Ji, ifx>0; if

and it is easy to see on geometrical grounds that there are


for conjecturing such a relationship. To make it precise we note that if, in the definition 1) of the Stieltjes integral reasons

we take F(x) to be the Heavisidc function H(x), we find immediately that, for any integrable function 1(x),

= /(0).

(46.15)

Comparing equation (46.15) with equation (46.7) we sce the relation between 11(x) and 6(x). It may be seen from these equations that 6(x) is not a function but a Stieltjcs measure, and that the use of the Dirac delta function could be avoided entirely by a systematic use of Stieltjes integration. 1)

The simplest definition of a Stieltjes integral f f(x)dF(x) is as

the limit of approximative sums x, are points of subdivision of(a, b)and

— F(x,_1)], where the in the interval xe).

INDEX formula for coefficients 101, 181).

Ad(ljt ion

I lessel

C. 18.

Analytic function 5. Anomaly 81 functions 147; POlynomials 147. Asyinptot ic expansions of Bessel functions 124. Associ:mted

I.agLICrre

N. 87. Jlailev, Bei funet ions 117.

Ber functions 117 Bessel coefficients 93; functions 4; integral function 1:11. Bessel's differential equations 4,

)irac delta function 1 59. 1)ixon's theorem 89.

1

l)uplication formula 11. Elliptic integrals 39, 1:10, 131. Etude, M. 14. Error—function 13. F:milcr's constant 1 1.

Exponential-integral 12, 17.

Ferrer's function 78, 90. lourier-ltcssel series 121. Fresnel integrals 13. (;muuma function 10.

(;auss' theorem 21.

Beta function 10. Bohr radius 152.

(;encntlised luvpergeometric series 80. Gillespie, H. P. 10.

Cauchy's theorem 79.

Gray, A.

Complete sequtence 50. Cot pressilile flow 24.

I lankel's I tessel functions 107. I larmonie oscillator 144), 155. I lcaviside's unit funct ion 101 I lernuite functions 186; polynomials 1:12. I lermite's (liffercntial equation

97, 102.

j.; 5.

Circular functions 4, 15. Conduct iOU of heat cu1tiatiot* 42.

nc equaConfluent tion 82; hvpcrgeonietric fune— tiomi 32.

Contiguity relations 81 , 85. Copson, E. '1'. 148. Cosine-integral 12, 17. Coulomb field 48, 150. Coulson, C. A. 71. Curvilinear co-ordinates 1, 15.

124.

184.

Ilobsotu, B. \V. 55, 59, 80. hydrogen atom

150, 157, 158.

equation functions 18; series 18. Ilvslop, .L M. 18. 11 ypergeomctric

Cylindrical to-ordinates 8, 113, 121, 155.

Imuce, B.

28;

5.

lndieial e(luatiofl 7.

l)elta function 159.

Integral formmumlac 20, 85, 70, 100,

1)iatomic molecule 157,

110.

163

164


Heeurrence relation 6; relations

Jahnke, B. 14.

for Legendre functions 52, (19;

lCuniner's relation 85.

relations for Bessel functions 94.

Lagtierre's (lifferential equat ion 146.

Laguerre polynonnals 142. Laplace's e(luation 1, 14, 15, 46, 70, 83, 90, 121. Legcndre functions 63; poly— nonüals 46; series 58. Legendre's associated e((IULtIOn 78; ncSOCiatC(1 functions 77, 83;

(lifferential equation 60. Linear different jail equation 4, 16. Logarithmic integral 12. MlIclLOl)ert '1'.

M. 81, 66, 124.

Mathews, G. 11. 124. Matrix elements 141. 119. N. MO(lified Bessel functions 118. Molceiiliir sI)ectru 24, 157.

Morse function 157. Mott, N. F. 140. 130.

itegular singular point 6. lticmann P—function 25.

lloderigues' formula 55. Ilotator 88. itutherford, 1). B. 1, Dl.

Saalsehfltz's theorem 41. Schrddinger equation 24, 42, 48, 88, 140, 150, 155, 157. Series of Legendre functions 57; of Bessel functions 11 9. Sine-integral 12, 17. Singular point 6. Sneddon, I. N. 140, 150. Special functions 3. Spherical Bessel functions 108; co-ordinates 14, 70. Sticltjes integral 162. Surface spherical harmonies 80. Symnetrical-top molecule 22, 42.

Murphy's formula 54.

Tables of functions 13, 14, 51,

Neumann, F. E. 65.

Taylor series 5.

110, 124.

Neumann's addition theorem 130

Bessel function 105; formula (16.

Vandermnonde's theorem 22.

Nornmlised fund ions 59.

Watson, C. N. 121. Walson'S theorem 45.

Ordinary point 5.

Wave mechanics 42, 43, 88, 140,

59. Orthogonal Orthonorinal set 59.

Parabolic co-or(linates 15, 157. Pauling, L., 24. P-function 25. Phillips, B. G. 79. Point at infinity 0. Potential theory .16, 57, 70, 83, 121.

149.

Bessel functions 10(1.

theorem 45. uittakcr's functions 85. Wilson, B. B. 24. Zeros of Legendre functions 87: of Bessel functions 129.

Special Functions of Mathematical Physics and Chemistry

Special Functions of Mathematical Physics and Chemistry

Special Functions of Mathematical Physics and Chemistry (LMT)

Special functions: an introduction to classical functions of mathematical physics

Special matrices of mathematical physics

Special Functions of Mathematical Physics: A UNIFIED INTRODUCTION with applications

Generalized functions in mathematical physics