SPECIAL
FUNCTIONS
FOR
SCIENTISTS AND ENGINEERS
W. W. BELL
lecturer in Theoretical Phyncs
Department of Natural Philosophy
University of Aberdeen
D. VAN NOSTRAND COMPANY LTD
PRINCETON, NEW JERSEY TORONTO MELBOURNE

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Copyright (?) 1968 D. Van Nostrand Company, Ltd
Published simultaneously in Canada by
D. Van Nostrand Company (Canada), Ltd
No reproduction in any form of this book, in whole or
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reviews), may be made without written authorisation
from the publisher.
Library of Congress Catalog Card No. 68-26453
Printed in Great Britain by
Butler & Tanner Ltd, Fromc and London

PREFACE
This book is primarily intended for use in undergraduate courses by
physics and engineering students in universities and technical colleges. It
should, however, also prove useful to those working in other sciences,
since it does not contain any specific examples drawn from the fields of
physics, engineering, chemistry, etc. We believe it to be preferable to give
here the principal results concerning special functions likely to be encoun-
encountered in applications, and to leave the applications themselves to the par-
particular context in which they arise. The book is designed for those who,
although they have to use mathematics, are not mathematicians themselves,
and may not even have any great mathematical aptitude or ability. Hence
an attempt has been made, especially in the earlier chapters, to sacrifice
brevity for completeness of argument. The level of mathematical know-
knowledge required of the reader is no more than that of an elementary calculus
course; in particular, no use is made of complex variable theory. Equally,
no attempt is made to attain a high level of mathematical rigour; for
example, we use results like 1/0 = со, l/oo = 0, which, to be rigorous,
should really be written in terms of limits.
In Chapter 1 we discuss the solution of second-order differential equa-
equations in terms of power series, a topic of crucial importance for the follow-
following chapters. This is because very often in applications we have a partial
differential equation for some quantity of interest, and this equation may,
by the method of separation of variables, be split into several ordinary
differential equations whose solutions we thus require. If, as is often the
case, these equations do not have solutions in terms of elementary func-
functions, we have to investigate these solutions, and this is best done by the
method of series solutions. These solutions lead to the definitions of new
functions, and it is the study of the properties of such 'special' functions
which is the main subject-matter of this book.
Chapter 2 is devoted to the gamma and beta functions, two functions
tlcfmcd by integrals and closely related to one another; these functions
arc not only used in later chapters, but are also encountered in many other
contexts.
Chapter 3 is concerned with a study of the Legend re polynomials.

VI PREFACE
They are introduced as solutions of Legendre's equation, which vrry often
arises when a problem possesses spherical symmetry. Such problems can
occur, for example, in quantum mechanics, electromagnetic theory,
hydrodynamics and heat conduction.
Chapter 4 is devoted to Bessel functions. These tuiu tкш» occur in a
very wide range of applications, such as loudspeaker design, optical dif-
diffraction, the vibration of circular membranes and plates, the scattering (»f
sound by circular cylinders, and in general in many problems in both
classical and quantum physics involving circular or cylindrical boundaries
of some kind. Kelvin's functions, introduced in Section 4.°, are of most
interest to the electrical engineer, while the spherical Hcssel functions of
Section 4.10 are of prime importance in quantum mechanical scattering
theory.
The Hermite polynomials, discussed in Chapter 5, have their main
application in the quantum-mechanical harmonic oscillator, but other
applications do occur. Likewise, the Laguerre polynomials of Chapter 6
are most useful in the quantum mechanical study of the hydrogen atom,
but they also find applications in, for example, transmission line theory
and seismological investigations.
Chapter 7 treats the Chebyshev polynomials which not only arc of
interest for their use in polynomial approximations to arbitrary functions,
but also occur in electrical circuit theory.
Chapter 8 is devoted to two generalisations of the Legendre polynomials
—the Gegenbauer and Jacobi polynomials. These occur less frequently
than the functions mentioned above, but they do have application in various
branches of physics and engineering, e.g., transformation of spherical har-
harmonics under co-ordinate rotations.
In Chapter 9 we discuss the hypergeometric function. This is the most
general of all the special functions considered. Indeed, all the other special
functions considered, and many elementary functions, are just special
cases of the hypergeometric function.
Chapter 10 contains a brief summary of various other functions which
occur in applications. These are often defined by integrals which cannot
be evaluated in terms of known functions, and very often all the useful
information concerning these integrals is contained in a table of values of
the integral.
Appendix 1 is concerned with the definition of Euler's constant, encoun-
encountered in the chapter on Bessel functions, and Appendix 2 treats explicitly
the convergence of the solutions to Legendre's equation found in Chapter 3.
The remaining appendices summarise certain properties of the special
functions which have been proved throughout the book.

PREFACE Vll
The problems at the end of each chapter should be considered as an
integral part of the text: every reader is urged to attempt most, if not all,
of the questions. Hints and solutions are provided for all but the easiest
of the problems, but the reader should not seek help from the hints before
he has made an effort to solve the problem for himself.
The proofs of certain theorems are indicated as being able to be omitted
at a first reading. This does not mean to imply that the proofs of such
theorems are difficult, but rather that they are fairly lengthy and that
inclusion of such a proof on a first reading would tend to destroy the con-
continuity of presentation of the material.
W. W. B.
Aberdeen, 1967

LIST OF SYMBOLS
Symbol Function Section in which
symbol first appears
B(x,y) Beta function 2.1
bern#, bein# Kelvin's functions 4.10
C(x) Fresnel integral 10.3
Cjfcc) Gegenbauer polynomial or ultraspherical 8.1
polynomial
Ci(tf) Cosine integral 10.2
Dn(x) Debye functions 10.5
E(k, ф) Elliptic integral of second kind 10.6
E(k) Complete elliptic integral of second kind 10.6
Ei(#), Ej(x) Exponential integrals 10.2
En(x) Generalised error function 10.3
erf x Error function 10.3
F(k, ф) Elliptic integral of first kind 10.6
Hypergeometric function 9.1
Hn(x) Hermite polynomial 5.1
Hnw(x), Я„B)(л) Hankel functions or Bessel functions of the 4.5
third kind
А„<1)(я;), An<2>(x) Spherical Hankel functions 4.11
In(x) Modified Bessel function 4.7
Jn(x) Bessel function of the first kind 4.1
jn(x) Spherical Bessel function 4.11
K(k) Complete elliptic integral of first kind 10.6
Kn(x) Modified Bessel function 4.7
kern #, kein x Kelvin's functions 4.10
Ln{x) Laguerre polynomial 6.1
L*(x) Associated Laguerre polynomial 6.7
li (x) Logarithmic integral 10.2
Л/(а, р, x) Confluent hypergeometric function 9.1
^k,m{x) Whittaker's confluent hypergeometric 9.3
function
yVn(#) Neumann function or Bessel function of the 4.1
second kind

LIST OF SYMBOLS
Symbol Function
pfln/ \
n (*xt p\/ \
Qi(x)
S(x)
si(tf), Si(*)
ЗД
ад
ад
Ц*)
7
Я(А, ф, а)
Щк, а)
Legendre polynomial
Associated Legendre function
Jacobi polynomial
Legendre function of the second kind
Fresnel integral
Sine integrals
Chebyshev polynomial of the first kind
Chebyshev polynomial of the second kind
Spherical harmonic
Bessel function of the second kind
Spherical Bessel function
Gamma function
, a) Incomplete gamma functions
Euler's constant
Riemann's zeta function
Elliptic integral of third kind
Complete elliptic integral of third kind
Weber-Hermite function
Section ii
i which
symbol first appears
3.1
3.8
8.2
3.10
10.3
10.2
7.1
7.
3.
4.
4.
2.1
10.
4.
1
1
10.4
10.6
10.6
5.'
l

CONTENTS
Preface v
List of Symbols ix
Chapter 1 Series Solution of Differential Equations
1.1 Method of Frobenius 1
1.2 Examples 8
Problems 21
Chapter 2 Gamma and Beta Functions
2.1 Definitions 23
2.2 Properties of the beta and gamma functions 24
2.3 Definition of the gamma function for negative values of the
argument 30
2.4 Examples 37
Problems 40
Chapter 3 Legendre Polynomials and Functions
3.1 Legendre's equation and its solution 42
3.2 Generating function for the Legendre polynomials 46
3.3 Further expressions for the Legendre polynomials 48
3.4 Explicit expressions for and special values of the Legendre
polynomials 50
3.5 Orthogonality properties of the Legendre polynomials 52
3.6 Legendre series 55
3.7 Relations between the Legendre polynomials and their deriva-
derivatives; recurrence relations 58
3.8 Associated Legendre functions 62
3.9 Properties of the associated Legendre functions 65
3.10 Legendre functions of the second kind 70
3.11 Spherical harmonics 78
3.12 Graphs of the Legendre functions 82
3.13 Examples 85
Problems 90

Xll CONTENTS
Chapter 4 Bessel Functions
4.1 Bessel's equation and its solutions; Bessel functions of tlic first
and second kind 92
4.2 Generating function for the Bessel functions 99
4.3 Integral representations for Bessel functions 101
4.4 Recurrence relations 104
4.5 Hankel functions 107
4.6 Equations reducible to Bessel's equation 108
4.7 Modified Bessel functions 110
4.8 Recurrence relations for the modified Bessel functions 113
4.9 Integral representations for the modified Bessel functions 116
4.10 Kelvin's functions 120
4.11 Spherical Bessel functions 121
4.12 Behaviour of the Bessel functions for large and small values of
the argument 127
4.13 Graphs of the Bessel functions 133
4.14 Orthonormality of the Bessel functions; Bessel series 137
4.15 Integrals involving Bessel functions 141
4.16 Examples 148
Problems 154
Chapter 5 Hermite Polynomials
5.1 Hermite's equation and its solution 156
5.2 Generating function 157
5.3 Other expressions for the Hermite polynomials 158
5.4 Explicit expressions for, and special values of, the Hermite
polynomials 160
5.5 Orthogonality properties of the Hermite polynomials 161
5.6 Relations between Hermite polynomials and their derivatives;
recurrence relations 162
5.7 Weber-Hermite functions 163
5.8 Examples 164
Problems 166
Chapter 6 Laguerre Polynomials
6.1 Laguerre's equation and its solution 168
6.2 Generating function 169
6.3 Alternative expression for the Laguerre polynomials 170
6.4 Explicit expressions for, and special values of, the Laguerre
polynomials 171

CONTENTS Xlll
6.5
6.6
6.7
6.8
6.9
6.10
7.1
7.2
7.3
7.4
7.5
8.1
8.2
8.3
9.1
9.2
9.3
9.4
10.1
10.2
10.3
10.4
10.5
10.6
10.7
Orthogonality properties of the Laguerre polynomials
Relations between Laguerre polynomials and their derivatives;
recurrence relations
Associated Laguerre polynomials
Properties of the associated Laguerre polynomials
Notation
Examples
Problems
Chapter 7 Chebyshev Polynomials
Definition of Chebyshev polynomials; Chebyshev's equation
Generating function
Orthogonality properties
Recurrence relations
Examples
Problems
Chapter 8 Gegenbauer and Jacobi Polynomials
Gegenbauer polynomials
Jacobi polynomials
Examples
Problems
Chapter 9 Hypergeometric Functions
Definition of hypergeometric functions
Properties of the hypergeometric function
Properties of the confluent hypergeometric function
Examples
Problems
Chapter 10 Other Special Functions
Incomplete gamma functions
Exponential integral and related functions
The error function and related functions
Riemann's zeta function
Debye functions
Elliptic integrals
Examples
Problems
172
173
176
177
182
182
185
187
190
192
193
194
196
197
198
200
201
203
207
210
212
216
218
218
221
223
224
224
225
228

contents
Appendices
1 Convergence of Legendre series 230
2 Euler's constant 231
3 Differential equations 233
4 Orthogonality relations 234
5 Generating functions 236
Hints and Solutions to Problems 237
Bibliography 243
Index 245

1
SERIES SOLUTION
OF DIFFERENTIAL EQUATIONS
1.1 METHOD OF FROBENIUS
Many special functions arise in the consideration of the solutions of
equations of the form
We shall restrict ourselves to equations of the type
where q(x) and r(x) may be expanded as power series in x,
т»=0
convergent for some range of x including the point x = 0.
The basis of Frobenius' method is to try for a solution of equation A.2)
of the form
z(x,s) = x'(a0 -|- axx + a#? + . . . + anxn + ...)
n-0
with «„ .-< 0. (We may always take aa ¦/- 0, since otherwise we should just

2 SKHIUS SOLUTION ОЬ' I) I Г К I! Kl- NTI Л I. KQIJATIONS CH. 1
have another scries of the type A.5) with a different value of s, the first
coefficient of which we could now call <¦/„.)
From equation A.5) we have
^ an(s I n)x'
and . .- -
d2
я О
so that if we require л to satisfy equation A.2) we must have
and this reduces to
00 00
2 an{s 4- n)(s + n - l)x>™ + q{x) ^ «»(* + я) ж'1П
0
+ r(x) > anx>+n = 0
n-0
which, on using equations A.3) and A.4) for q(x) and r(x) and cancelling a
common factor of xs, becomes
GO 00 00
> У anqm{s + n)x +
n = 0 n=-0
CO 00
22
For the infinite series on the left-hand side of equation A.6) to be zero
for all values of л; in some range, we must have the coefficient of each power
of x equal to zero. This requirement gives rise to a set of equations as
follows.
Requiring that the coefficient of x° be zero, and noting that tf° arises
only from the choice m = 0, n = 0, gives
aos(s - 1) + a0qos + aoro = 0. A.7)
Requiring that the coefficient of ж1 be zero, and noting that x1 arises in
equation A.6) by choosing in the first term n == 1, and in the second and
third terms n -f- m = 1 (i.e., m = 0, я = 1 or»i = 1, n =0) gives
«,(j I 1)* + {aiqo(s + 1) + «oft*} + {axr0 + a^} = 0. A.8)
We may now write down a general equation from the requirement that
tin* coefficient of x* should be zero. Remembering that in equation A.6) x*
arises in the first term by choosing n = i and in the second and third terms

§1.1 METHOD OF FROBENIUS 3
by choosing n + m = i (i.e., n = i, m — 0 or n = i — 1, m = 1 or
n = г — 2, m = 2, etc., up to я = 0, m = t) we obtain
aj(j + «X* + * -1)
+{a*>(* + i) + ai-i4i(s + i — 1) + at_2q2(s + i — 2) + ... + «<#**}
+ {агг0 + at^rx + . . . + aurt) = 0 (i > 1). A.9)
Thus
*,/(* + 0
+ {ai-iid? + г - 1) + e<_2?2(j + г - 2) + • • • + во?<4
+{««-г»'1 + «*-Л + • • • + «о^Л = 0 (i > 1) A.10)
where we have collected all the terms in at together and denoted their
coefficient by
f(s + i) ^ (s + i)(s + i - 1) + ^(* -f- 0 + r0. A.11)
We now see that equation A.7) reduces to
«o{*2 + (!7o - IX* + 'o)} = 0
which, with the given assumption that a0 ^ 0, becomes
s* + (q0 - l)(s + r0) = 0. A.12)
This is called the indicial equation; it is quadratic in s and hence will yield
two roots which we shall denote by sx and s2. In many cases of interest these
roots will be real; this we shall henceforth assume, together with the fact
that s2 > sv
Noting that equation A.12) is just/(s) = 0, we have immediately
f(s) =(s- Sl)(s - *2). A.13)
We might now expect that these two values of s would lead to the two
independent solutions of the original differential equation. We shall see
that this is true apart from certain exceptional cases, viz. when
(a) the two roots of the indicial equation are equal; or
(b) the two roots of the indicial equation differ by an integer.
The set of equations A.10) can now be used to determine the coefficients
ax, a2, . . . in terms of a0.
Equation A.10) with г = 1 gives
a,f(s + 1) + «(#1* + <Vi = 0
so that we have
a, =
aJiAs)
-ЛТП) <U4>
where hy(s) (</,.? | r,) is a polynomial of first degree in s.

4 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
Equation A.10) with г = 2 gives
aafis + 2) + {fli51(s + 1) + auq2s} + {fliri + аогг] — 0)
which, when we use equation A.14) for alf becomes
yielding
"flo{?i(J + l)^i(J) + ?2J/(S + 1) + f-Ji-^s) + ''г/С^ + 1)}
a, = -
(US)
where h2(s) = -{g,(* + 1L*) + яЖ* + 1) + гЛ(*) + r2f(s + 1)} is a
polynomial in $.
In general it is not difficult to see that we obtain
aa + lf(s + i) <U6>
where A,-(j) is a polynomial in 5.
If in equation A.16) we now substitute s = s1 we shall obtain expressions
for щ in terms of a0, leading to a solution with a0 as an arbitrary multipli-
multiplicative constant; similarly we may substitute * = s2, so that we have the
two solutions required. This will work, however, only if the denominator
of equation A.16) is never zero, i.e., provided
/(
and f(s2 +i) 5^0 for i equal to any positive integer. (Ы7)
Now, equation A.13) tells us that
f(s) = (s - sj)(s - s2)
so that
f(s + i) = (s + i- Sl)(s + г - s2)
and hence
f(h + l) = K'i ~ S2 + 0
and f(st + i) = (*, - Ji + i)b A.18)
so that equation A.17) will be satisfied if, and only if, s% — ^ is not a
positive integer, i.e., provided the roots of the indicial equation do not
differ by an integer. (We recall that we have labelled the roots so that
*2 > h-)
If the roots do differ by an integer, the procedure given above will work
for the larger root: if s2 — st = r (a positive integer), then from equations
A.18) we have

§ 1.1 METHOD OF FROBENIUS 5
/(*!+*) = *¦(-'+*) 0-19)
f(st+i) = i(r+i), A.20)
and we see that f(sl + i) = 0 for г = r, while /(j2 + ») 7^ 0 for any
positive integral i. Hence the procedure will work with s = s2 but not with
s = х^ How, in fact, do we find the second solution? If we use the relation-
relationship A.16) before fixing the value of s we have the series
/(» + If /(• + !)/(• + 2)
We know by the method of construction of au a2, . . . that if all the terms
on the right-hand side are well defined, then this must satisfy
(nince equations A.10) are satisfied, the coefficient of each power xs+i with
I > 1 must vanish, and the coefficient of Xs is, by equations A.7) and
A.11) just aof(s)). This equation may be rewritten in the form
*2 dS + xq{x)ix+ r{x)}z(x's) = a°f{s)x*- A-22)
Unfortunately, all the terms on the right-hand side of equation A.21)
•re not well defined for s = s^, there are zeros in the denominators for
I > r. However, if we multiply z(x, s) by f(s + r) we shall just cancel the
factor in the denominators of the later terms which vanishes when * = sv
And since f(s + r) = (s + r — st)(s + r — s2) = (s + sz — 2*!)(* — Sj)
we might just as well multiply by (s — sx). Also, since this factor is inde-
independent of x, equation A.22) now takes the form
+ ^ +
= ao(s - ^J(* - h)x\ A-23)
using equation A.13).
Setting s = st, we have
which shows that z(x, st) is a solution, a fact we already know.
Similarly, setting s st gives [(s — s,)z(x, s)]t ,8i as a solution. In fact

6 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
it turns out that this series is not independent of z(x, s2) but is just a multi-
multiple of it. This comes about as follows. The factor j — sk cancels zeros in the
denominator for terms with i > r, but will provide zero in the numerator
for all terms with i < r, thus the first power in the series is just x'1+r = xs',
which is just the first term in z(x, s2). And since we use the same rules for
calculating any coefficient in terms of preceding ones in the two cases, we
must just obtain the same series in both cases, apart from a constant
multiplicative factor.
If we differentiate both sides of equation A.23) with respect to s, we
obtain
\x^ + x{x)±+r(x)\\.
I d*2 }dx ') L
s {{S - h)z(x'
= a0 [(, - s,r js{(s - sjx*} + 2(s - Sl)(s - s2)x^ A.24)
and we see that the right-hand side of this equation is zero when $ = su so
that we must have [(d/d$){($ — Sj)z{x, s)}]SssSi as a solution of the differential
equation. It may be proved that this solution is in fact independent of the
first solution (i.e., it is not merely a constant multiple of the first solution),
but we shall not do so here.
Thus we can take as independent solutions
[(*-*i)*(*, *)].-., A.25)
and [&'-***,')}]._.. A-26)
Another type of situation may arise when the roots of the indicial equa-
equation differ by an integer. As well as f(s + i) = 0 for s = st and / = r, it
may happen that hr(sx) = 0. In this case ar is indeterminate (since it has a
zero in both numerator and denominator) so that we may in fact use it
as another arbitrary constant, thus obtaining the two independent solutions
from the one series. We shall see in detail how this happens when we come
to consider particular examples.
The one remaining case is when the indicial equation has equal roots,
say s = sv Here/(j) = (s — s^2, and if we make use of equation A.22) we
obtain
{ ^ LЬ*'s) = <s ~ *i)v- A -27)
Setting s = Si makes the right-hand side zero, so that z{x, st) is a solu-
solution. But if we differentiate both sides with respect to s, we obtain

§ 1.1 METHOD OF FROBENIUS
s - sjx' + (j - Siy jxs>. A.28)
Again the right-hand side is zero when j = sx, so that we have for a
solution [(d/ds){z(x, s)}]s=sh. This, in fact, may be shown to be independent
from z(x, Sj), so that in this case we have the two independent solutions
z(x, sx)
Г ,
and
Let us now summarise briefly the methods of obtaining independent
solutions which have been described above.
Form the series z(x, s) by the use of the relations A.10). Then there are
four distinct cases.
A) If the roots sx and s2 of the indicial equation are distinct and do not
differ by an integer, then the two independent solutions are given by
Z{X, Sj)
and z(x, s2).
B) If the roots st and s2 (sz > h) differ by an integer and one of the
coefficients in the series for z(x, s) is infinite when * = su the two indepen-
independent solutions are given by
Г,
and
C) If the roots s1 and s2 (sz > si) differ by an integer and one of the
coefficients in the series for z(x, s), say ar, is indeterminate when s = slt
the two independent solutions are obtained from z(x, Sj), keeping я0 and я,
as arbitrary constants.
D) If the roots of the indicial equation are equal, say, s = su then the
two independent solutions are given by
z(x, sx)
Г А
and
The question of convergence of the series obtained must, of course, be
considered. It may be proved (although we shall not do so here) that the
rungc of values of x for which the solution is convergent is at least as large

8 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
as the range of values of x for which both q(x) and r(x) may be expanded
as convergent power series.
In certain cases q(x) or r(x) may not be expanded as power series in x for
any range of x (e.g., \/x or exp (l/x)). It may perhaps then be possible to
make a change of variable, such as x' = x — a, or x' — 1 /x, so that q(x)
and r(x) may be expanded in powers of x' and a solution found in the form
A.5) with x' replacing x.
Finally, let us remark that when carrying out the actual solution of a
differential equation of type A.1), it is not usually profitable to transform
it to an equation of the type A.2); all that we require for the above results
to hold is that this transformation should be possible.
Let us now illustrate the above methods by means of some examples.
1.2 EXAMPLES
Example 1
d2y dy
We must first verify that this equation is of the form
4 2
where q(x) and r(x) may be expanded as power series. Multiplied by x/2,
the given equation becomes
d2y 1 dy 1
+x +xy=°
* a» +x-o-a +ix
ax2 2 ax 2
so that q(x) = \, r(x) = \x and the required condition is obviously
satisfied.
Since q{x) and r(x) are already in power series form, valid for all values of
x, it follows from the remarks made at the end of the previous section that
any series solutions obtained will be convergent for all values of x.
We now revert to the original equation.
Set z = x ^ 2
n—0
so that — = _2, a^s + n)xs+n-~L
Я-0
and ^ - J a'is + n)(s I" n -

§ 1.2 EXAMPLES
Then we have
d*z dz
+ +
00 00
= V 2an(s -|- n)(j + и - 1)х"+"-г + V an{s
0
n)(S
A-30)
where in the last summation we have replaced the label я by the label
я — 1 in order that the general power should look the same in all terms.
In order that z should satisfy the differential equation, we must have
the coefficient of each power of x in A.30) equal to zero. The power with
я = 0 appears only in the first two terms; thereafter the powers with
я > 1 appear in all three terms. Hence we have the equations:
2ao(s + 0)(s + 0 - 1) + ao(s + 0) = 0 A.31)
and
2an(s + n)(s + n - 1) + вя(* + n) + an_x = 0 (n > 1). A.32)
These equations simplify to
aosBs - 1) = 0 A.33)
and
an(s + n){2{s + n) - 1} + в„_г = 0. A.34)
Equation A.33) gives the indicial equation
sBs - 1) = 0
with roots j = 0 and * = \. Since these are distinct and do not differ by an
integer, we know that we may proceed according to prescription A) on
page 7.
Equation A.34) gives the recurrence relation for the coefficients in
s(x,s):
A35)
and hence a, ----- - , rv^-^
(s + \)Bs

10 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
a* = ~ (s + 2)Bs + 3) = GT3xr+2)p7Ti)B* + 3)'
a2 a0
a* =~^ + 3X2^ + 5)= "G+ 1)(* + 2)(s -\ 3)(ТГ+"Т)"B7 Ь 3)B* + 5)
and in general
/_1V, ^o
N I? -4- I и с —I— /i к Ц-. vt if /c —U I w / с 4— \ l i / с —I- /« I i
Thus we have the series z[x, s) given by
z(x, s) = a^-i 1 — -rrr Г7 + 7
I (s-|-l)Bs + l) (
xn
~'~(~ ' (^+1)^+2) . . . (s+w)Bs + l)Bs-f3). . .
A.36)
Since the roots of the indicial equation are s = 0 and s = \, the two
independent solutions are given by z(x, 0) and z(x, 4) which, by equation
A.36), are
{X X2 X3
1 - U + Г2Л73~ 1.2.3.1.3.5 +•••
J /1\ \ ) Л О ^^ О С
+ ^3S 2и + 1 9, f 9 +--- . A.38)
2"* 2 J
We may rewrite these series more compactly if we note the following
results:
1.2.3. ...n = n!, A.39)
1.2.3.4 ...Bw - 1).2и
1.3.5...(и- )_ 2.4... Bя -2).2п
= 2.1.2.2.2.3...2(и - 1).2я
= (о-?];. 0-40)

§1.2 EXAMPLES 11
_ 1 2.3.4.5.6.7...Bw + 1)
= 2n 2.4.6 In
1 Bh + 1)! = B_«_fJ_)J
2" 2". n\ 2гпп\ ' K >
and 2.4.6. . .In = 2"n!. A.42)
Using these results we see that equation A.37) reduces to
while equation A.38) becomes
Thus the general solution is given by the general linear combination of
s{x, 0) and z(x, \), viz.
. 1V, B«)"
where Л and В are arbitrary constants.
Example 2
We first verify that this equation is of the form
where #(д:) and r{x) may be expanded as power series in x.
The given equation is obviously equivalent to
„d2y u> x
ux- d.v 1 яг

12 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
Hence q(x) = 1 and r(x) = —x/(l — x). q(x) is already in power series
form, convergent for all values of x, but r(x) is not. However, it may be
expanded as a power series in д; by the binomial theorem, and this series
will be convergent for those values of x such that | x | < 1.
It follows from the remarks at the end of the previous section that any
series solutions which we obtain will be convergent for at least the same
values of x, viz. | д; | < 1.
We now revert to the original equation.
CO
Set z = ^ a»xS+n
ti=0
, oo
so that ~ = V aJs + я)*84"-1
dx *->
n-0
,2 00
and -j^- = У an(s + n)(s + n - 1)х^п~г.
dx ??o
Hence,
,d2# .. ^dz
= T an(s+n)(s-\-n-\)x'+n-1 — ^ an(s+n)(s+n~l)x'+n
n=0 n=0
an(s+n)xs+n~1 - 2an(^+«2
n=0 n=0 n=0
2
For s: to be a solution, we require the coefficient of each power of x to be
zero, so that we have
а„*(* - 1) + я0* = О A.45)
and
«„(* + n){s + n - 1) - «„_!(* + n - l)(s + n - 2) + д„(* + я)
-an^(s + n - 1) - ап_г = 0 (n > 1). A.46)
These equations simplify to
<W2 = 0 A.47)

§ 1.2 EXAMPLES 13
and an(s + n? - an^{{s + n - IJ + 1}. A.48)
Equation A.47) gives the indicial equation
*2=0 A.49)
while equation A.48) gives the recurrence relation
{(s + n- 1Г + 1}
*-*-!- {t + n)i • О'50)
Equation A.49) has coincident roots s = 0; so we know (by prescription
D) on page 7) that the two independent solutions are given by z(x, 0)
and [(d/ds)z(x, s)],=0.
Equation A.50) gives
{(s + 1)*
and in general
{s* + 1} {(* -!- I)8 + 1} {(s + 2J + 1} . . . {(s + n - IJ + 1}
"n a° " (s+l)%s + 2)*...(s+ny
so that
? (,+1)^+2)*... (И-*J
A.51)
The first solution is then
a(*,0) = «0[ 2 1.2V3
V 1.2-5.10.17...{»-
= OeyiC»), eay. A.52)
For the second solution we require (d/ds)#(a:, s). Now, from equation
A.51) we have
tl 1 \S -\- 1 / \\S \- 1 j , - , . . . lv_ i -- -/i - у I n
' - — — — — —. ' - I Л .
/1С iCl— ll**(C-i-/i 1С l^H1 I
Ui) ^ Id ill IД i X* I • t ¦ 10 \Ilfj _|

14 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
For the first term we note that
ds ds ds
= (In x) es ln * = (In x)xs A.54)
and hence —ж' = ln ж. A.55)
Lds Js=o
The second term we evaluate by the technique of logarithmic differen-
differentiation.
If we denote the term in square brackets by/n(.s) we have
so that ~fn(s) =fn(s) ~-lnfn(s). A.56)
But
ln/n(*) = ln {s* + 1} + ln {(s + \y + 1} + .. . + ln {(s + n- IJ 4- 1}
- ln (s + IJ - ln (s 4- 2J. . . - ln (s 4- яJ
ln {(s + m - \J 4- 1} - V ln (s 4- mJ.
m=l m—1
Hence -r- ln fJs) = / — — /
ds f-r. u + от — IJ 4- 1 ¦f-' * 4- от
_2yf * + т~1 L
^-i \(s + от — lJ Ь 1 s -\- m
so that
(s+l)K..{s + ny ^ [(s + m ~\y
and
1*.2*7.п* ^{(m- \y + 1 m.
f Wo recall that x - cln ', where we are using the notation ln x 1orl. .y.

§ 1.2 EXAMPLES 15
Denoting this expression by cn, we have
_2.1.2.5.10.17...{(it-l)' + l} у m-2
" t~,\\l Z^. milm 142 I 1V У1'-*')
Thus, setting s = 0 in equation A.53) gives the second solution
2
n=l
— аоуг(х) In x + a0
= йо^О*). say»
where
(*) = Ух{х)
Then the general solution is given by
у = Ayx{x) + By2(x)
where A and В are arbitrary constants.
Example 3
., + C — x)y = 0.
ax
We see immediately that this is of the form A.2) with q(x) = — 3,
r(x) =3—x, which are already in power series form, so that they have a
power series expansion valid for all values of x.
It follows from the remarks at the end of the previous section that any
scries solutions obtained will be convergent for all values of x.
00
Writing z = у апх"+н leads in the same manner as in the previous
>l=0
examples to the following equations, which have to be satisfied if z is to be a
solution of the given equation:
ao(s - 1)(* - 3) = 0 A.59)
«„(* I n 1)(v I n 3) «,...! О (и ¦¦ 1). A.60)

16 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
Equation A.59) gives the indicial equation
(* - 1)(* - 3) = 0 A.61)
with roots s = 1, s = 3. These differ by an integer, so that we know that
we are dealing with an exceptional case.
Equation A.60) gives the recurrence relation
so that we have
a
1 ~ (
-2)'
a,
2 (s + \)(s~- 1) s(s -\-'l)(s -2)(s
and in general
" s(s + 1)(* + 2). . . (* + я - l)(s - 2)(* - 1)j . . . (* + я - 3)'
A.63)
We see explicitly from equation A.63) that when s = 1 all the an with
и > 2 are infinite, so that we have to apply method B) described above on
page 7.
We have
xn
j&^+l) • ¦ • (*+я-1Х*-2)(*-1) . . . (Н-я-3).
so that
ys — l)s(#, s) = a0
s(s - 2)
xn
(s + 1) . . . {s + n - \)(s - 2)s{s + 1). . . (s + n - 3)J'
A.64)
The first solution is given by
CO
= aoX 2 Г2Т7Г|К-1)Л.2...(я-
U n\(n-2)\
-= aay^x), say.
The second solution is given by [(d/ds){(s - \)z{x,

§ 1.2 EXAMPLES 17
From equation A.64) we have
2Xn "I
n=2 s(s + 1) . . . (s + n - \)(s - 2)s(s + 1) ...(, + „_ 3)/
i d
1 +
— 2)
1
dSS(J + l)...(I+«- 1)(j - 2)s(s + 1)...(, + я _ 3)J
A.65)
To evaluate the last derivatives we use logarithmic differentiation:
denoting
1_
*(* + 1) ...(*+ n - 1)(j - 2K* + 1) ...(* + и - 3)
by /„(^), we have again
and since
ln/n(j) = - In j - In (j 4- 1) - In (j + 2). . . - In (j + я - 1)
- Ь (j - 2) - In j - In (j + 1) ¦ . . - In (j + я - 3)
we have
1
', + 1'---', + я_3',-2
1 1
1 s + n -2 s +n-1
so that
¦-«')]..,
+ UU...-Ц
2 3 и — 2
2V-1 .i+JL+I\

18 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
and ftflU = r^_^_!5L___
1
n\(n -2)!
so that
11
- . г ~т~ ~
w — 1 n
fd 1 1 Г v? 1
(«-1I
(s -2)J»=i
= cn, say
It is also easy to show that
(*-
2).
and, as before, we have
—xs — (In x)xs.
Thus, finally, we have from equation A.65)
fd 1 Г
T{(s - \)z{x, s)} = ЯоAп x)yx(x) + aox< 1 - x
Ids J.-i I n=2
= aoy2,(x), say,
and the general solution is then
у = Ауг(х) + Вуъ{х)
where A and В are arbitrary constants.
Example 4
d*2 ' v 'dx J ~
Again this is of the form A.2) withg(x) = —2 + хгът\&г(х) = —2, so
that the series solutions to be obtained will be valid for all values of x.
Writing z = Xs У^ апх" and requiring г to be a solution of the given
n=0
equation leads, as before, to the system of equations:
ati(s - 2)(s - 1) = 0 A.66)
e,(*-l)j = 0 A.67)
an{s + n - 2){s + n 1L- «e..8(* I я 2) --= 0 (и Г= 2). A.68)

§ 1.2 EXAMPLES 19
Equation A.66) gives the indicial equation
(*-2)(,-I)=0
with roots s = 2,s — \. These differ by an integer, so we are dealing with
an exceptional case. When * = 1, equation A.67) is satisfied irrespective
of the value of a^; i.e., ax is indeterminate, so that we are dealing with case
C) of page 7. We then know that the two independent solutions are
obtained from the one value of s, namely s = 1.
Equation A.68) gives the recurrence relation
«•--(ТТГГц (">2> (L69)
(the factor s + n — 2 may be cancelled since it is non-zero for n > 2 and
$ = 1 or s = 2), which with s = 1 becomes
°n='"^ (И>2)- (L70)
a0
1 hus a., = --„-, я4 = =
2' 4~ 4 ~2.4'
У пь~ 5 ~3.5'
and in general
an ч ' 2.4.6... 2я
and
v ; 3.5.7... Bи + 1)'
so that we have
2 2^
That is, we have the general solution given by
у = Aylx)
where
_i '2.4.6... a.
iiiul .'J and /i are arbitrary constants.

20 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
In the examples given so far, it has always been relatively easy to write
down the general term of any series appearing in the solutions. This is by
no means always so; it may be impossible for us to find in any simple
manner an expression for the general coefficient from the recurrence
relation. This is so, in general, when the recurrence relation contains three
or more terms instead of the two they have had up till now. All that we can
do in such a situation is to give the first few terms of the power series
solution, and hope that this will be of some use. We illustrate by the
following example.
Example 5
(ж2 -
Rewriting this in the form
„2&У , „ 3*2 АУ
we see that it is of the form A.2) with q(x) = (Ъх2)/(хг — 1) and
r(x) = xs/(x2 — 1). Both q(x) and r(x) may be expanded as power series in
x (by the binomial theorem) convergent for | x \ < 1. Hence any series
solutions we obtain will be convergent for at least this range of values of x.
We now revert to the original equation, and trying for a solution of the
form
z =
leads, in the same manner as in previous examples, to the set of equations
aos(s - 1) = 0 A.71)
Ol(s + 1> = 0 A.72)
a<fi(s + 2) - a2(s + 2)(s + 1) = 0 A.73)
««-3 + ««-2 (s + n-2)- an(s + n){s + n - 1) = 0 (и > 3). A.74)
Equation A.71) gives the indicial equation
*(* - 1) = 0
with roots s = 0 and * = 1. Equation A.72) then shows that ax is inde-
indeterminate if s = 0, so that we are dealing with case C) above—the two
independent solutions may both be obtained from the root s = 0, taking
a0 and ax as arbitrary constants.

PROBLEMS 21
With s = 0, equation A.73) gives
a2 = 0
and equation A.74) becomes
а„-3 + n{n - 2)й„_2 - n{n - \)an = 0,
giving
ап-з + n{n - 2)an_2
It is impossible for us to use the recurrence relation A.75) to obtain a
general expression for an; but we may proceed to calculate as many terms
as we please.
T. a0 + Ъаг
1 hus a3 =
8a2
12
and so on as far as we please.
We have calculated the series solution up to terms in xb:
{ l № }
PROBLEMS
A) Obtain solutions of the following differential equations in ascending
powers of x, stating for what values of x the series are convergent:
dx

22 SERIES SOLUTION OF DIFFERENTIAL EQUATIONS CH. 1
(v) A-^ + 2*1+^=0;
In J
(vi) x* d^+(*a - 3x)?+D ~2ф=0;
B) Determine whether or not it is possible to obtain solutions of the
following equations in terms of ascending powers of x:
dv dv
(iii) x*-^+x<>-f
' dx2 dx
C) Obtain solutions of the following equations in terms of ascending
powers of l/x:
i) 2*4* - l)g + *C* + 1)^ - 2y = 0;
) 2*3 ^ + B* + *2)~ + 2j = 0.

GAMMA AND BETA FUNCTIONS
2.1 DEFINITIONS
We define the gamma and beta functions respectively by:
T(x) = pe-'^d* B.1)
J о
B(x,y) = f F-i(l - i)*-i&t. B.2)
J о
The first definition is valid only for x > 0, and the second only for
x > 0 and у > 0, because it is for just these values of x and у that the
above integrals are convergent. We shall not prove this statement, but shall
at least make it plausible.
Consider the integral in definition B.1). It is known that at infinity the
behaviour of an exponential dominates the behaviour of any power, so
that e~' f~x —> 0 as t —> oo for any value of x, and hence no trouble is
expected from the upper limit of the integral. Near the lower limit of the
integral we have e~* — 1; hence if this approximation is good between
t = 0 and t = c, we may write equation B.1) in the form
T(x) ^ Г f-i d* + f
Jo J
f00
e
J с
f -'**-1 dt
J e
1
+
so that if the first term is to remain finite at the lower limit we must have
x > 0.
A similar argument applies to the beta function, the behaviour at t = 0
leading to the restriction on x and at t 1 to the restriction on y.

24 GAMMA AND BETA FUNCTIONS CH. 2
2.2 PROPERTIES OF THE BETA AND GAMMA FUNCTIONS
Theorem 2.1
ГA) = 1.
Proof
We have, from the definition B.1),
г<1) - J."
j:
- м:
= e-Jdf
Theorem 2.2
Г(ж + 1) = жГ(«). (ж > 0)
Proof
Г(х + 1) = f°e-'/"d*
J о
(by the definition B.1))
= Г(-е-')*""Г - ["(-e-')*^^
L Jo J о
(on integrating by parts)
= 0 + x fV
(where the first term vanishes at
the upper limit since the exponential dominates the power, and at the
lower limit since x > 0)
= хГ(х)
(using definition B.1) again).
Theorem 2.3
If x is a non-negative integer, then Г(х + 1) = x\
Proof
Г(х + 1) = хГ(х)
(by theorem 2.2)

§2.2 PROPERTIES OF THE BETA AND GAMMA FUNCTIONS 25
= x(x - 1)Г(ж - 1)
(using theorem 2.2 again)
= x(x - IX* - 2)Г(ж - 2)
(by further use of theorem 2.2)
= x{x - IX* ~ 2)(* - 3) . . . 3.2.1ГA)
(by repeated use of theorem 2.2,
and remembering that x is integral, so that continued subtraction of unity
will eventually lead to 1)
Theorem 2.4
= x\
(by theorem 2.1).
Г(ж) =2 Г е~*'12х-1 dt.
J о
Proof
In definition B.1) substitute a2 for t: t — u2, so that At = 2м du; when
t = 0, м = 0 and when f = oo, и = оо, so that we have
(ж) = fa>e-"!()a!-1.
Jo
= 2 ["е^'и^-^
J о
= 2 ["e-*1**1-1^
J о
(since in a definite integral we may choose the variable of integration to be
anything we please).
Theorem 2.5
f * i
cos2*-1 0 sin2"-1 в dd =
f cos 0 sin в dd = .
Jo 2V(x+y)
Proof
We prove this result by considering in two different ways the double
integral
/ = Г Г exp ( t2 - M2)f2a!-1M21'-1 At Аи
(where R is the first quadrant of the fM-plane, shown unshaded in Fig.
2.1).

26
GAMMA AND BETA FUNCTIONS
и
сн. 2
Fig. 2.1 The to-plane
Firstly, we have
/= Г™ [C°exp(-^ — m2)^-^2»-1 d«
= f" e-(I*I"-:d*.fee-|lV
Jo Jo
»-1da
(using theorem 2.4)
= 1Г(*)ГЫ. B.3)
Next we change variables to plane polar co-ordinates r and 0 in the
to-plane so that t = r cos 0, и = r sin 0 and the element of area dt Аи is
replaced by the element of area r dr АО. Then we shall have
/ = [ [ exp (-r2 cos2 0 - r2 sin2 0)(r cos ву-\г sin 0J*-V dr d0
= f° Г l e-'V24 cos2*-1 в г25' sin2"-1 в rdrAO
J r=0J в=0
= [°°e-rV2(a!+'')-1dr Г cos2*-1 0 sin2*-10 d0
Jo Jo
= $Г(* + у) Г cos231-1 0 sin2"-1 0 d0 B.4)
J о
(using theorem 2.4 again).
Equating the two expressions B.3) and B.4) for / leads immediately to
the required result, which will be true for x > 0 and у > 0, since it is for
this range of values that the gamma functions involved are defined.

§2.2 PROPERTIES OF THE BETA AND GAMMA FUNCTIONS 27
Theorem 2.6
Щ) = y/it.
Proof
Put x = у = \ in theorem 2.5, and we obtain
Jo 2ГA)
= НЩ)}2.
(using the fact that ГA) = 1).
Performing the integration on the left-hand side gives
\ = unw
and hence Г( J) = ± yVr.
But from the definition B.1) we see that Г(х) must be positive (since the
integrand is positive), so that we can discard the negative square root and
obtain Г(|) = л/тс as required.
Corollary
P
J о
Proof
Setting x = \ in theorem 2.4 gives Щ) =2 e"' At, so that the
J о
required result follows immediately on the use of theorem 2.6.
Theorem 2.7
Proof
Substitute t = cos2 0 in the definition B.2) of B(x,y): t = cos2 0 so
that d? = — 2 cos 0 sin в dd; also, when t = 0, cos в = 0 so that 0 = тг/2
and when t = 1, cos 0 = 1 so that 0=0.
Hence we have:
Щх,у)= \ (cos2 0)*-1 (sin2 ву-1. -2 cos6 sin в dd
J л/2
-2
J о
я/2
я/2
cos2*-1 0 sin2"-1 0 dO

28
GAMMA AND BETA FUNCTIONS
= 2
Ц«)гоо
2Цх+у)
(by theorem 2.5)
Theorem 2.8
Proof
B(x,y)=B(y,x).
This result follows immediately from theorem 2.7.
Theorem 2.9
Proof
(i) We have
x -f- у
T(x
(by theorem 2.7)
хЦх)Г(у)
(х+у)Г{х+у)
(by theorem 2.2)
x +уГ(х+у)
(by theorem 2.7).
(ii) Exactly similar to (i).

§2.2 PROPERTIES OF THE BETA AND GAMMA FUNCTIONS
Theorem 2.10 (Legendre Duplication Formula)
Ц2х) =2^Цх)Г{х + J).
Proof
We have, by theorem 2.7,
= [ t*-\\ - t)*-1 dt
J о
(by definition B.2))
(on making the substitution t = \{\ + s))
[ A - и)х-Ци-1/2 Аи
o
(on making the substitution и = s2)
= 2~2X+1 \ A - иу-Ч-1'2 du
J о
(by theorem 2.7).
Hence, using theorem B.6),
___
2л) Г(ж
and thus
,

30 gamma and beta functions ch. 2
Corollary
If л; is a positive integer
Proof
In the theorem we may use theorem 2.3 to rewrite ГBж) as Bx — 1)!
and Г(х) as (ж — 1)! Hence we have
which, on multiplying both sides by 2x, becomes
22X
2*B*-l)! = —-*(*-1)!Г(*-1-?);
Л/71
this equation may be expressed more simply in the form
22X
B*)!=—*!Г(* + ?)
and thus
2.3 DEFINITION OF THE GAMMA FUNCTION FOR NEGATIVE
VALUES OF THE ARGUMENT
From theorem 2.2 we have
Г(х) = -Г(х + 1). B.5)
ОС
Apart from x = 0, where the denominator vanishes, the right-hand side
of this equation is well defined for those values of x such that the argument
of the gamma function is positive (since this was the condition for definition
B.1) to hold), that is, for x + 1 > 0, i.e. for x > — 1. We also note that we
may say that Г@) is infinite, for as x —-> 0 we have T{x + 1) —> ГA) = 1,
and hence Y{x) = (\/x)Y{x + 1) —*¦ со. So far the left-hand side is only
defined for x > 0, but we can now use the right-hand side to extend this
definition to x > — 1. The argument we are using is as follows:
(i) Equation B.5) was proved true for x > 0;
(ii) Right-hand side is well defined for x > — 1, left-hand side for
x > 0;
(iii) Use right-hand side to define left-hand side for x > 1.

§2.3
DEFINITION OF THE GAMMA FUNCTION
31
This means that we now have Г(х) defined for x > — 1, so that the right-
hand side of equation B.5) is well defined for x + 1 > — 1, i.e. x > —2;
we may thus use it to define the left-hand side for x > —2; and this
process may be repeated to define Г(х) for all negative values of x.
Theorem 2.11
T(m) = oo, if m is zero or a negative integer.
Proof
We have already remarked above that Г@) = oo. From equation B.5)
we have
Г(-1) =Д-Г(О) = oo
and Г(-2) = —Г(-1) = oo,
etc.
It is now possible to draw a graph of Г(х). Plotting of points combined
with information from the various theorems leads to a graph as shown in
Fig. 2.2.
1
1
1
-5
1
1
1
1
r\
4 -
3 "
A
II
\]
\j
I
2 -
1
Г
4
3
2
1
A
U)
г /
1 /
i i i i
12 3 4 1
— 1
--2
— 3
—4
—5
Fic;. 2.2 The gamma function

32 GAMMA AND BETA FUNCTIONS CH. 2
Theorem 2.12
Г(*)ГA - *) = -r^—.
sin ял;
PROOFf
We shall prove this result in three stages:
(i) We shall prove the infinite product expansion for sin в, viz.
sin б = I j{l - @7n27c2)};
(ii) We shall show that this implies that
sin б ? 0-nn
n= —oo
(iii) We shall use result (ii) to prove the required result.
(i) We know that
sin 0=2 sin — cos -
n . 0 . /л 0\
= 2 sin - sin (^ + 2) B-6)
and hence, by applying this result to both factors on the right-hand side,
we obtain
sin в = 2B sin - sin (- + -j)B sin (- + -j am (^ + ^ +
n, . 0 . л + в . 2л + в . Ъл + в
= 2s sin - sin —ф- sin —-— sin 2a ¦ B.7)
It is left to the reader to verify that applying the result B.6) to each of
the factors on the right-hand side of equation B.7) gives us
„ . в . л + в . 2л + в . Ъл + в . Ал + в
sin в = 27 sin - sin —p- sin —^— sin —^— sm -~^—
. 5л + в . вл + в . 7л + 0
If this process is carried out n times altogether, the general result will be
f The proof of this theorem may be omitted on a first reading; alternatively,
parts (i) and (ii), which are standard trigonometric identities, may be assumed and
used to prove the result in part (iii).

§2.3 DEFINITION OF THE GAMMA FUNCTION 33
„ , • в . л + 0 . 2л + в . (р - 1 )л + в
sin 0 = 2"-1 sin - sm •—— sin — ... sinv-^- B.9)
P P P P
with p = 2".
The form of this equation is plausible as the correct generalization of
equations B.6), B.7) and B.8); if desired it may be proved by the method
of induction.
The last factor in equation B.9) is
. (p-l)n + 0 . J {n -в)]
sm \L 1 = sin i JT — ¦
p I p
¦ (л~в\
= sin 1.
V p )
Similarly, the second last factor is
B* - 0I
. {р) + . I
sm — = sm < л —
p I
=sin
P
In - 0\
and in general the rth factor before the end is
(m -
sin
P
We now regroup the factors appearing in equation B.9) by taking
together the second and the last, the third and the second last, and so on.
Then equation B.9) becomes
in в = 2р~г sm -< sin sin M sm sin г ¦ ¦ ¦
pl P p J I p p )
. B.10)
rin |sin
p p J p
The factor inside each curly bracket is of the form
sin {A + B) sin (A - B) = i{cos 25 - cos 2A}
= sin2 A - sin2 В
and hence equation B.10) becomes
sin0 = 23-1 sin--Ism8- - sin2-W sin2 — - sin2-!-. . .
pip p) I p p)
{s
{sin2 (J^i^-sin-0) cos°. B.11)
I p pf P K '
Dividing both sides by sin @/p), letting 0 —> 0 and remembering that
lim (sin x/x) — 1 so that

34 GAMMA AND BETA FUNCTIONS CH. 2
.. sinO ,. sinO 0/p
lim = lim . ——.p
o^osin@/p) в^о в sin @/p) y
we obtain p = 2»-1 sin2 - sin2 — . . . sin2 '-^—^—^. B.12)
P P P
If we now divide equation B.11) by equation B.12), we obtain
sinO . 0Г sin2 @ /p)\ Г sin* @/p) "I
p pi sin2 (я/р)/ I sin2 Bjz/p)J " ' '
h~l ГГ I cos~. B.13)
sin
p J
We now let p —> oo. We take into account the three results
(a) lim p sin - = lim ^l^L^.fl
p^co P p-,00 0/p
= 0,
?->co sin2 (гл/р) p^oo I 0/p J \pj Isin (r?r/p)J (гл;/р)а
(с) lim cos - = 1
and see that therefore equation B.13) gives
sin 0 =
Я = 1 '
(ii) It is trivial to verify the result that
1 d , t
Hence
1 d , sin @/2)
sin 0 dO cos @/2)
d 2 sin2 @/2)
dO 2 sin @/2) cos @/2)
d 2_sin»_@/2)
d() " "~s"in 0

5 2.3 DEFINITION OF THE GAMMA FUNCTION 35
= Afln 2 + In sin2 @/2) - In sin в}
do
= ^{2 In sin @ /2) -In sin в}
_ d
~~d0
j ^ CO CO
= — jln 0 + 2 ^ Ь Bял: - в) + 2 J Ь Bтт + б)
?г=1 й=1
со со .
(ил; -е) - 2ln (nn+вч
п=1
1СО1 00 ^ 00- О),
i зТ ~ i зТ — V -1 v l
О
j2V ' у l
1 CO i
_J у __L
+ 2ия ^-i 0 + яя.
1 °° / 1 \П "
1 "^ (~)
{~\y
- ял;  б + ия
п1
n —
(iii) Let
Then
6 0-nn
¦ — CO
us first assume
Г(ж)ГA -
that
ж) =
0
= 1
--1
1
<
3(л
Г1
lo
X
', 1
¦, 1
< 1.
- ж)Г(ж + 1
(by theorem
-X)
{ — t) Lt
~x)
2.7)
(by definition B.2)).

36 GAMMA AND BETA FUNCTIONS CH. 2
We now make the substitution t = 1/A + u); then
d* = {-1/A + wJ} du and и = A - t)/t.
Also when t = 0, и — oo and when / = 1, м = 0.
f° 1 / м W 1 \
Then Г(ж)ГA - x) = j му-ЛГТй/ \ ГГТ«72 /
_ froJi"-!-dM
—- I Lit*
Jo 1+M
Г1 м-ж , f00 м~ж
= z—-— du + du.
J 0 1 + U J 1 1 + И
In the second integral we make the substitution и — 1/v. Then
du — ( — 1/w2) dw; also when и = 1, w = 1 and when м = oo, » = 0.
Thus Г и-* , Г° vx I 1
Hence, from equation B.14) we have
f1 v^
= (м-^ + м^-1) У ())n
Jo ,tr0
= > (-1)" {м"-ж J-«"+*-
" Г 1
= > (-1)" — „n-^M
^ v ; U - x + 1
n0
undu
dw
1
o
yn. — x + 1 я + л
(remembering that 0 < .v < 1)
x -¦ n
n-— — oo

§ 2.4 EXAMPLES 37
_ X" (-1)пл
п-
7t
(by part (ii) of the proof).
We now remove the restriction 0 < x < 1. Suppose that x = у + N
where N is an integer and that 0 < у < 1.
Then
Г(л;)ГA - x) = T(N + у)ГA - у - N)
= (N + y - \)(N + у - 2) . . . з>Г(;у)
1 1 1
\—у — N2—у — N у ^ У'
(by repeated use of the result T(x + 1) = xT(x))
sin ny
(since we know this result to be true for 0 < у < 1)
n
sin (Ntt 4- лгу)
sin nx
as required.
2.4 EXAMPLES
Example 1
Express each of the following integrals in terms of gamma or beta functions
and simplify where possible.
(i) V(tanO)dO. (ii) r7?f—3.-
Jo Jo 8v(l— x3)
f=° fl /1 I r\l/2
(iii) t-*>*{l - e-<) At. (iv) (i-t^ d*.
Jo J -l \1 — ay
(i) We use theorem 2.5 where, in order to get \/(ta.ne)=sms/20 cos~1/20
as integrand, we take 2x — 1 = --.V and 2y -- 1 = +-|, i.e. x = ? and
>• - 3-

38 GAMMA AND BETA FUNCTIONS CH. 2
(by theorem 2.1)
1 n
2 sin (я/4)
(by theorem 2.12)
1 л
21Д/2
We change the variable to t = ж3. When ж = 0, t = 0 and when x — 1,
= 1. Also df = Ъхг dx, i.e., d»; = |г~2/3 df.
Hence
Г1 dx Г1
j-jr- ~=\
J о 3y A — x3) Jo
1 n
(by definition B.2))
(by theorems 2.7 and 2.8)
3 sin(n;/3)
(by theorem 2.12)
3 (V3)/2
3(V3)"
(iii) Г t~3/2(l -e-l)dt.
Jo
We cannot relate this immediately to the gamma function; we must first
integrate by parts:
V^l -e-')df = Г 2*~lrt(l --с-')Г Г 2</2.e-'d<
L JJ
')df = Г 2*~lrt(l --с-')~Г Г
L J» J о

§ 2.4 EXAMPLES 39
= 0+2 f"V»'«e-«d* t
Jo
(by definition B.1))
(by theorem 2.6).
In order to change this into a form resembling the beta function it is
necessary that the range of integration be changed to 0 to 1. This is
accomplished by the change of variable t — \[\ + x). Then x = 2t — 1,
dx = 2d* and we have
1/2
= 2 I"
2B{h i)
(by definition B.2))
ГB)
(by theorem 2.7)
1!
(by theorems 2.2 and 2.3)
л (by theorem 2.6).
t The behaviour of <~1/2A — e~"') at t = 0 requires some explanation. We
recall l'Hopital's rule which states that if i(a) = g(a) = 0, then
W f (a)
provided f'(a) and g'(«) are not both zero (or infinite). Hence we have
A-е-) [c-1, -o 1 n

40 GAMMA AND BETA FUNCTIONS CH. 1
Example 2
Show that B(n, и + 1) =-*¦)"'* and hence deduce that
Z 1 (Zn)
ГП_1
Jo \sin30 sin2
ГЖ„» =
2e/ ly/n
We have 5(«, . + 1) - ™^
(by theorem 2.7)
Г(я).яГ(»)
2иГBи)
(by theorem 2.2)
2ГBи)'
Setting n — \ gives
which, on using definition B.2) and theorem 2.6 becomes
f t~m(l - t)Vi At =
J о
Now set f = sin в in the integral on the left-hand side. When t = 0,
0 = 0 and when t = 1, в = тг/2; also d/ = cos 0 d0.
Hence we have
Г1 t-3/4(i - 01Д d< = f * " (sin 0)-3M(l - sin 0)l/i cos 0 d0
Jo Jo
["'* /1 -
(—^
Jo \ si
()
sin30
Гл/2 /1 1 Vм
= (-r—z - -¦—~n) cos 0 d0,
Jo Vsm30 sin20/
so that we have proved the required result.
PROBLEMS
A) Prove that
(i) JV-^dx-^^ + l)
(n > -l.e^-O);

PROBLEMS 41
(ii) I xm e~*n dx = - Г {(т + \)/n}
Jo я
(m > — 1,я >0).
/•ею
(iii) exp Bаж — x2) dx = | V^ exP (я2)-
J я
B) Prove that Г tan" 0 dO = ^Г{A -|- н)/2} Г{A - н)/2} if | и | < 1.
J о
C, Prove that ?" sin- , d» - ff со,- ,»-% Щ?Щ
D) Express each of the following integrals in terms of the gamma or beta
functions and simplify when possible:
Г1 /1 \1/4 f1 / IV-1
«Jot-1) **; <fi)J0H ^ (fl>°)=
r*
(iii) (b - лг)™-1^ - a)"-1 dx, (b > a, m > 0, n > 0);
J я
(iv) I *m(l - Xя)" dx, (m > -l,p > -1, и > 0);
J о
E) Show that the area enclosed by the curve x4 + у* = 1 is {ТA)}2/2л/л.
F) Evaluate Г(-|) and Г(-?).
G) Show that
(i) Г(*)Г(-*) = -,
(ii) Щ +
x sin nx
л

3
LEGENDRE POLYNOMIALS
AND FUNCTIONS
3.1 LEGENDRE'S EQUATION AND ITS SOLUTIONS
Legendre's differential equation is
Нё-1-*-0- (зл)
We shall write k in the form 1A + 1) for reasons which will become clear
as we proceed.
This equation,
A - **)% ~ 2xfx + /(/ + \)y = 0, C.2)
is of the form of equation A.2) with q(x) = — 2хг/(\ — л;2) and
r(x) = {1A + 1)#2}/A — хг). The binomial theorem may be used to ex-
expand these as power series in x for values of x such that x2 < 1, i.e., such
that —1 < x < 1. Thus the methods of Chapter 1 will be applicable to
the solution of this equation, any power series obtained being valid for
at least — 1 < x < 1.
00
Writing z(x, s) = xs Ъ апхп and requiring z to be a solution of equation
C.2) leads, as in Chapter 1, to the system of equations
*„»(*-l)=0, C.3)
в1(* + 1)*=0 C.4)

§3.1 legendre's equation and its solutions 43
and
an+i(s + n+ 2)(s + n + 1) - an{(s + n)(s + n + 1) - 1A + 1)}
= 0 (и > О). C.5)
Equation C.3) gives the indicial equation s(s — 1) = 0 with roots s = 0
and s = 1 (which differ by an integer, so that we know we are dealing with
one of the exceptional cases).
Equation C.4), with * = 0, is satisfied irrespective of the value of ax\
this means that at is indeterminate and hence the one root of the indicial
equation (s = 0) leads to two independent solutions with the two arbitrary
constants a0 and av
Equation C.5) with s = 0 becomes
tin + 1) - /(/ + 1)
«•¦• = ^-оГмйгтг) - <3-6>
which, since
n(n + 1) - /(/ + 1) = и2 + и - /2 - /
= (и2 - /2) +(n~l)
= (n- l)(n + /) + (н - /)
= (H - Z)(H + / + 1),
may be written in the form
Thus we have
/(/ + 1
= а
_ ()( )
a4 - -e, 1T4— - at - -a
3 —
3) _ (/ - 1)(/ - 3)(/ + 2)(/ + 4)
_
~ °~ ГгТз^ ' ~ai 2X475
and in general
«2П =(-1)"Яо
/(/-2X/-4) . . . (/~2И+2)(/+1)(/+3). ¦ ¦ G+2H-1)
__ C.8)
and
B« -1-1)!

44 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
Therefore
/(/-2) . . . (/-:
2
n=l
Bh + 1)!
= aoyi(x) + а!уг(х), say,
so that Ух(х) a.nd y^(x) are the linearly independent solutions.
We know already that yt(x) and у%(х) will be convergent for — 1 < x < 1.
For many applications, solutions to Legendre's equation are required
which are finite for — 1 < ж < 1. The theorem quoted at the end of
Section 1.1 guarantees convergence only for —1 < x < 1, and says nothing
at all about convergence for x = ±1. In fact standard methods of
convergence theory may be used to show that the series above are
divergent for x = ±1 (see Appendix 1). How then is it possible to
obtain solutions which are finite for x = ±1? The only way is to
make the infinite series reduce to finite series, and this will happen for
any positive integral value of /. For, from equation C.8) we see that if
I — 2n (an even positive integer) we have a2n 5^ 0 but a2n+2 = 0, and hence
all subsequent even coefficients must also equal zero. Similarly equation
C.9) shows that if / = 2и + 1 (an odd positive integer) then ain+1 ?±0
but a2n+3 and all subsequent odd coefficients are zero. Thus we see that if /
is an even integer уг(х) reduces to a polynomial, and hence is finite for all
finite x, while if / is an odd integer the same occurs for y2(x).f In both
cases the highest power of x appearing is xl, so that since equation C.7)
applies to the series for both y^x) and у2(х), we may obtain a single series
valid for both even and odd / if we write it in descending powers of #. The
f When we consider integral values of /, we need consider only positive values
of I, since the constant appearing in the equation was /(/ + 1) and if / were a
negative integer we could write m = — (Z + 1) and use the easily verified fact
thatm(m + 1) = 1A + 1).
This arrival at integral values of / is, of course, the reason for writing the
original constant k in the form 1A + 1).

§3.1 legendre's equation and its solutions 45
first term is щх1 and subsequent terms are given by equation C.7) written
in the form
«»--«„+2(/_и)(/ + и + 1). (зло)
so that
(I - 2)(/ - 3)
4B/ - 3)
= a,
and, in general,
¦'2.4.B/ — l)B/-3)
a ( lVe
«i-* - A) «i 2A7. . 2rBl - 1)B/ - 3) ... B/ - 2VTT) (J'H)
Thus for / even, y^x) reduces to the following expression, while for
/ odd, y2(x) reduces to the same expression:
y(x) =
tin
i-2 , ii, , fao for / even
+ в| V* + • • • + {a[x for z odd
r = 0
, , r, Л fi/ for / even
(where [I/] = ||(/ _1)for/odd> (ЗЛ2)
i.e., [^/] is the largest integer less than or equal to |7)
y iy /(/-1) . ¦ . G-2Г + 1)
r=0 ' 2.4 ... 2rB/-l)B/-3) . . . B/-2r+l) v '
from equation C.11).
We may rewrite this series more compactly if we note the following
results:
(Т^Щ (ЗЛ4)
2.4.6 . . . 2r -. B.1X2.2X2.3) . . . B.r)
-2r. 1.2.3 . . . r
Ъг\ C.15)

46 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
and
B1 - l)Bl - 3) . . . B1 - 2r + 1)
= 2/B/-l)B/-2X2/-3) . . . Bl-2r+\) Bl-2r)\
~ 2/B/-2)B/-4) . . . B/-2r+2) 'Bl-2r)\
Bl)\
24A - 1) ...(/- г
= 2'B/-2г)!Л" (ЗЛ6)
Using equations C.14), C.15) and C.16) in equation C.13) gives us
"" '-xy^n 1 2rBl-2r)\l\
r=0 V
(Z!JB/-2r)!
^ ; r!(/-2r)!(/-r)!BZ)!
This is a solution for any value of аг. If we choose al = B/)!/{2'(/!J}
we obtain the solution which we denote by Pi(x) and call the Legendre
polynomial of order /:
Bl-2r)l ,_2r ,ri7v
Thus this is the solution of Legendre's equation which is finite for
— 1 < x < 1; it is the only such solution, apart from an arbitrary multi-
multiplicative constant.
3.2 GENERATING FUNCTION FOR THE LEGENDRE
POLYNOMIALS
Theorem 3.1
This means that when A — 2tx + t2)~in is expanded in powers of t, the co-
coefficient of tl is Рг(х); A — 2tx + t2)-in is called the generating function of
the Legendre polynomials.
Proof
Expand A — 2tx + *2)-1/2 by the binomial theorem:
A - Itx -I- <2)/a = {1 - t{2x - <)}~1/2

§ 3.2 GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS 47
= i + (-Ш-42» -1)} + (~*Х-*> {-кг* -1)}'
Now expand Bx — t)r by the binomial theorem (remembering that r is
integral):
where ГС„ is the binomial coefficient —r.—'-—— •
p PK'-PV-
Hence we now have
A -2tx +t*)-u> = 2^^^гСр(-1)Ч'+*Bху-». C.18)
We wish to find the coefficient of ?¦, so we must take r + p = I, and
hence for a fixed value of r we must take p = I — r. But p only takes on
values such that 0 < p < r, so we must only consider those values of r
satisfying 0 < I — r < r, i.e. \l < r < /. Hence if / is even, r can take on
values between \l and /, while if / is odd, r can take on values between
\{l +1) and /. For any of these values of r the coefficient of tl in equation
C.18), obtained by taking/» = / — r, is
and the total coefficient of tl is obtained by summing over all appropriate
values of r so that we obtain
coefficient of tl = j ъШ
, f//2 (/ even) * v

48 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
If we now change the variable of summation from r to k = I — r, we
obtain
coefficient of t?
* Bl~2k)\ (_k ik
Ck{ l
2«-»{(/ - /г)!}2 (/ - 2/г)!ЛГ ;
(where [1/2] is as defined above in equation C.12))
B/-2ft)l ^
~ ' 2\l - k)\(l - 2K)\kf
(by the definition C.17)).
Hence the required result is proved.
The restriction on the values of x comes from the condition for
convergence of the binomial expansion of {1 — tBx — t)}~1/2, viz.
| t{2x — t) | < 1. When | x | < 1 this condition may be shown to be
equivalent to | t \ < 1.
3.3 FURTHER EXPRESSIONS FOR THE LEGENDRE POLYNOMIALS
Theorem 3.2 (Rodrigues* Formula)
Proof
We may expand (x* — II by the binomial theorem:
i
(X* - 1)' =
Hence,
2?Л T/Xi - V = 2T! ^2;Сг(-1)^-. C.19)
l)ut the /th derivative of a power of x less than / is zero, so that
^a? -'r ^ 0 if 21 - 2r < I, i.e., if r > 1/2.

§3.3 FURTHER EXPRESSIONS FOR LEGENDRE POLYNOMIALS 49
i m (j-o/2
Thus we may replace 2, by 2. ^ ' *s even an(-^ by ^^ if / is odd, i.e.,
r=0 r=0 r=0
[г/21
by 2. where [1/2] is as defined above.
r=0
Also,
?jx> =p{p~ \){p - 2) ...
so that d' r B/-2r)l
dxr (I — 2r)\
and hence, substituting into equation C.19), we have
Li(x _ п f (ir
2'Л сЬг( ^ ~ 2'/! Z rl(/ - r)P ] (I - 2r)!
2r!(Zr)!(/2r)!
= Pt(x) (by the definition C.17)).
Theorem 3.3 (Laplace's Integral Representation)
Pt(x) = - f" {x + V(x2 - 1) cos О}1 dO.
л j о
Proof
It may be shown by elementary methods (for example by means of the
standard substitution t = tan @/2)) that
f" d0— = —* C.20)
J о 1 + Я cos в V(l ~ Я2)
If we now write Я = —\и-\/{хъ — 1)}/A — их), expand both sides of
equation C.20) in powers of и and equate the coefficients of corresponding
powers of м, we shall obtain the desired result:
1 = 1
Г-|- Я cos 0 Л uv'(x2 - 1)
1 тл ^ cos 0
A - их)
= A - ux)[l - u{x + V(x2 - 1) cos в}]-1
ГО
= A - их) 2 «"{* + V(x2 - 1) cos 0}'

50 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
oo
(since, by the binomial theorem A — a) = y^ an).
»=0
1
A - их)
A - их)
V{A - иху - и*(х* - 1)}
A - их)
Hence, substituting into equation C.20) gives
Г У ul{x + V№ - 1) cos в}1 dd = -jpr~ r~^
i.e., 7. »' I (ж + VW — i) cos i
(by theorem 3.1).
Equating coefficients of ul gives
rn
_p/v\ . 1 (v _L_ ,//v2 1\ rn<s fl\ rlfl
ГСХМЛ1 ¦— I 1Л H^ д/ IЛ II CUo V t 111/
J 0
which is the required result.
3.4 EXPLICIT EXPRESSIONS FOR AND SPECIAL VALUES OF THE
LEGENDRE POLYNOMIALS
From the definition C.17) we may write down explicitly the Legendre
polynomial of any given order. We give here the first few polynomials:
Рг{х) = x,
PI \ 1/C3 T\
Theorem 3.4
(i) P;(l) = 1.
A1)^-1) =(-1)'.

§ 3.4 VALUES OF THE LEGENDRE POLYNOMIALS 51
(vi) Ри+1@) = 0.
(By P{{\) we mean [{dPl(x)}/dx}^1).
Proof
(i) Set x = 1 in theorem 3.1 above and we obtain
1 = j ^(i),
, . i
that is =
But 1/A ~ i) = 2.^ (by the binomial theorem or by considering the
right-hand side as the sum to infinity of a geometric progression), so that
CO CO
we have /^ = /^ ^Piity- F°r tms t0 be true for all values of t in some
(=0 1 = 0
range (in this case \t\ < 1) we must have the coefficients of corresponding
powers of t equal, i.e., JP^l) = 1.
(ii) Exactly similar to (i) but setting x = — 1 in theorem 3.1.
(iii) Pi(x) satisfies Legendre's equation C.2), so that we have
& d% = ° C<22)
and setting x = 1 in this equation gives
which reduces on using part (i) above, to P[(\) = \l(l + 1).
(iv) Exactly similar to (iii), but setting x = —1 in equation C.22) and
using part (ii) above.
(v, vi) Set x = 0 in theorem 3.1 above and we obtain
Expanding the left-hand side by the binomial theorem gives us
= 1 н-(-

52 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
.21
, 1.3.5 . . .{21- \)i
_iV 1.2.3.4.5 ... B/ - 2)B/ - 1J/
J 24\. 2.4.6... B/ -2J/
*—< 22(<7П2 '
Thus we have
and equating coefficients of corresponding powers of t on both sides gives
= o.
3.5 ORTHOGONALITY PROPERTIES OF THE LEGENDRE POLYNO-
POLYNOMIALS
Theorem 3.5
rl Г 0 ifl^m
J -i {2l + ltfl-m-
{This result may be written more concisely if we introduce the Kronecker delta,
defined by
\ifl = m.
Then the statement of the theorem is
2 .

§3.5 properties of legendre polynomials 53
Proof
Pt(x) and Pm(x) satisfy Legendre's equation C.2), so that we have
dP dP
and A - x*)-^ - 2x-^ + m(m + \)Pm = 0, f
which may be rewritten in the form
° C-23)
and ~{A - x*)^} + m(m + \)Pm = 0. C.24)
Multiplying equation C.23) by Pm(x), equation C.24) by Pi(x), sub-
subtracting equation C.24) from equation C.23) and integrating with respect
to x from —1 to +1 gives
But since
+ {/(/ + 1) - m(m + 1)} f' РгРт dx = 0.
- x^ + P ЦA - x*&
dx dxL dx
A x^l A x^ + P ЦA x&
d#J dx dx dxL dx
we shall have
^(i *)^l dx + (/« + / - m^ - m) f РгРт dx = 0.
ax dx J J -l
On integrating the first term we obtain
(/ - «)(/ -|- ж + 1) f PtPm dx = 0;
J -l
X Throughout this proof we denote Pi(x) by Pi and Pm(x) by Pm. The argument
ж is to be understood in every case.

54 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
but now the first term vanishes at both upper and lower limits, because of
the A — x2) factor, so that we have
(I — m)(l + m + 1) PbPm &x — 0, and when Mmwe can cancel
the factor outside the integral to obtain
Г РгРт dx = 0.
J -i
f1
It remains to prove that {Рг(х)}2 dx = 2/B1 + 1). For this we use
J -l
the generating function of theorem 3.1:
1 °°
г» + *¦> = 2 *ЭД
2
s°that Ь{1
1=0
2
l,m=0
We now integrate both sides with respect to x between —1 and -\-l:
But the left-hand side is a standard integral of the form
(a + to)} d* = A/b) In (a + *«).
while the right-hand side contains only terms for which / = m, by the first
part of this proof.
Hence
ln(l+*
J 1 1=0
which gives
У t* С {Pt(x)Y dx = -1 In A + t» - 20 + i In A + f2 + 20
(~o J -1 zr zr
- - 2\ln(l - 02 b^ In A-1 02

§ 3.6 LEGENDRE SERIES 55
(using the series expansion of In A -f-t))
t«
Equating coefficients of corresponding powers of t on both sides gives
2
' 21 + 1
as required.
3.6 LEGENDRE SERIES
Theorem 3.6
If i(x) is a polynomial of degree n, then
^ Г1
i(x) = ? crPr(x) with cr = (r + i) f (x)Pr(x) dx.
Also, if t(x) is even (or odd), only those cr with even (or odd) suffixes are
non-zero.
Proof
We have i(x) = anxn + an_xxn~x + • • • + axx + a0, say, and we may
write Pn(x) = knxn + kn_2xn-2 4-... (since by equation C.17) Pn(x) is a
polynomial of degree n, containing only odd or even powers of ж according
to whether n is odd or even), so that if we take i(x) — (an/kn)Pn(x) we
obtain either zero (in which case the first part of the theorem is proved)
or else a polynomial of degree n — 1. This means that
f (*) = cnPn(x) 4- &.-!(*)
where cn = an/kn and gn-X(x) is a polynomial of degree n — 1.
The same argument may be applied to gn-i(#) to give

56 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
and hence f(x) = cnPn(x) + сп_лРп_1(х) + gn_2(*).
The same argument may now be applied to gn_2(*), etc., to yield
eventually
f (*) = cnPn(x) + «W-A-iC») + • • • + CjPjUx) -i- c0P0{x)
= 2
If we now multiply both sides of this equation by Ps(x) and integrate
from — 1 to + 1, we obtain
J _ f(x)Ps{x) dx = 2, cr J ^Pr(x)Ps(x) dx.
But, by theorem 3.6, the integral on the right-hand side is non-zero only
for the value of r which equab s, in which case it is 2/Bs + 1).
f1 2
Therefore f (x)Ps(x) dx = cs r—,->
J -1 /5 + 1
giving cr = (r + ?) f (x)Pr(x), as required.
Now suppose that f (x) is even. Then, since Pr(x) is even when r is even,
and odd when r is odd, so the integrand i(x)Pr(x) is also even when r is
even and odd when r is odd. But an odd function integrated over the range
—1 to +1 will give zero, since positive and negative values cancel one
another. Hence c, is zero when r is odd. Similarly when i(x) is odd, cr is
zero when r is even.
Corollary
If f (x) is a polynomial of degree less than /, then
f1
f(a;)P,(^) dx = 0.
J -l
Proof
Suppose f (x) is of degree n; then by the theorem we have
r=0
so that
f (x)Pt(x) dx = У ст\ Pr(x)Pi{x) dx
= 0
by theorem 3.5, since r < n < I, so that r is never equal to /.

§3.6
LEGENDRE SERIES
57
The results of the above theorem may be extended to functions which
are not polynomials. We shall not prove this extension, but will merely
quote the following result (the proof is not difficult, but is fairly lengthy).
Theorem 3.7
Suppose i(x) satisfies the following conditions in the interval —1 < x < 1:
(i) f(x) is continuous apart from a finite number of finite discontinuities
{we then say that f(x) is piecewise continuous); and
(ii) f (x) has a finite number of maxima and minima.
CO
Then the series /^ с,Рт{х)^ where
r=0
converges to i(x) if x is not a point of discontinuity of f (x) and to
if x is a point of discontinuity, f Also, at the end points of the range, x — ±1,
the series converges to f(l—) and f(—1+) respectively. This series is called
the Legendre series for i(x).
t By f(xo+) we mean Lim f(x0 + «) where e —>¦ 0 through positive values and
«->O
by f(x0—) we mean Lim f(x0 — e) where again e—> 0 through positive values.
See, for example, Fig. 3.1.
У
fU<r>"
у-fix)
f(jro+)
Fig. 3.1 Left- and right-hand limits at a point of
discontinuity

58 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
3.7 RELATIONS BETWEEN THE LEGENDRE POLYNOMIALS AND
THEIR DERIVATIVES; RECURRENCE RELATIONS
Theorem 3.8
(i) Р\{х) =
«•=¦0
(ii) xPix) = l
(iii) (I + 1)PI+1(*) - B1 + 1)*P,(«) + /P;_^) = 0.
(iv) p;+1(«) - p;_^) = B/
(
(vi) PJ(*)
(vii) (ж2 -
*=0
Proof
(i) We know that Р;(ж) is a polynomial of degree /, containing only even
powers of x if Z is even, and only odd powers of x if / is odd. Hence P[{x)
is a polynomial of degree / — 1 containing either odd or even powers of л:
according to whether / is even or odd. Hence by theorem 3.6 we have
Р'г(х) = c,-i^-i(«) + Ci-iPt-J*)
г
= (s + I) f P%x)PAx
with cs = (s + I) f P%x)PAx) df
(on integrating by parts)
P«(-1)P.(-1) - 0}
(where the integral vanishes by the corollary
to theorem 3.6, since Ps(x) is a polynomial of
degree s — 1, and ^ — 1 is always less than /)
(by theorem 3.4 (i) and (ii)).

§3.7 LEGENDRE POLYNOMIALS AND THEIR DERIVATIVES 59
But s takes on the values / — 1, / — 3 so that s + I takes on the
values 21 — 1, 2/ — 3, . . ., which are always odd, irrespective of the
values of I or s. Hence ( — l)s+! = — 1 and we have
c. = {s+i){l-(-l)}
= Bs + 1).
Hence
ci-zr-i = 2A - 2r - 1) + 1
= 21 - Ar - 1
ЗЛ(«) ^'»even
and so
П*) -
we have
= B1 - \)I
+ B/
' + B/ -
-4r -:
r=0
(ii) хРг(х) is a polynomial of degree / + 1, odd if / is even and even if /
is odd. Hence by theorem 3.6 we have
т>( \ d ( \ , т> < \ ^ , JVi(*) (/even)
where
^r = (»¦ + i) J t хРг(х)Рг(х) Ах
But this integral is zero by the corollary to theorem 3.6 if r + 1 < I
(since then xPr(x) is a polynomial of degree less than /), i.e., if r < / — 1.
Hence we have
xPl(x) = cniPl+1(x) + c^P^x). C.25)
To determine cl+1 and сг„х we set x = 1 in equation C.25) and in the
same equation differentiated with respect to x, viz.
Pt(x) + xPl(x) = cMP'l+1(x) + c^P'^x). C.26)
Setting x = 1 in equations C.25) and C.26) and using theorem 3.4 (i)
and (iii) gives
1 = cl+l + <:,_, C.27)
and 1 |- U(l -|- 1) = fJH ,{1A -f- 1)(/ + 2)} -I f| ,{-i(/ 1)/}. C.28)

60 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
Solving equations C.27) and C.28) for cl+1 and cl_1 gives
1+1 2/ + 1* l~x 2/ + 1
so that on insertion of these values in equation C.25) we obtain
xPl{x) = 1±1р1+1(я) + -J-pu*).
(iii) Multiply both sides of equation (ii) above by B/ + 1) to obtain
B1 + l)xPt(x) = (I + 1)P,+1(*) + HW*).
On rearrangement, this equation becomes
which is the required result.
(iv) We may use result (i) for both P/ +1(x) and P't _ 1(л>):
p' /r\ _ o] i прм i_ /о/ э\р cr\ I со/ _
P' (v\ — C2/ — З^P Cr"» 4- f?/ —
Subtracting these two equations gives
(v) If we differentiate result (iii) with respect to x we obtain
(/ + 1)Рг'+хМ - B1 + l){Pt(x) + xPl(x)} + №,'_!(*) = 0
and if we now use (iv) to substitute for P\ +1(ж) we shall have
(/+i){p;-i(*)+B/+i)p,(«)> - B/+i){p,(*)+*p/(*)} + /p;_x(x) = o.
Collecting terms in Р,'_г, Рг and Рг' gives
B/ + 1)P;_!W + B1 + 1)(/ + 1 - 1)P,(«) - B/ + 1)«P,'(*) = 0,
which reduces to P/_i(«) + /Рг(ж) - жРДж) = 0,
and on rearrangement this equation becomes
*ВД-Р;_г(*)=Р,(*).
(vi) If we multiply (iv) by x we obtain
*P;+1(x) - xPi^ix) = B/ + \)хРг(х)
and substituting this expression for B1 + l)xPj(x) in (iii) gives
(/ + l)Pl+l{x) + lPt^(x) = *p;+1(») - xPU(x). C.29)
If we now rewrite (v) with / replaced by / + 1, we have
and substituting this value of (/ + l)Pm(;c) into equation C.29) gives
*p;,,(*) p;(x) ч /p, ,(*) = .тр;м(ж) ^р; ,w.

§3.7 LEGENDRE POLYNOMIALS AND THEIR DERIVATIVES 61
When rearranged, this equation gives
which is the required result.
(vii) If we multiply (v) by x we obtain
ж2Р;(х) - **>,'_!(*) = lxl\(x)
and on subtracting (vi) from this we obtain
*»/>,'(*) - P[{x) = lxPt(x) - IPU
which may be rewritten in the form
(viii) If we replace / by / + 1 in results (v) and (vi) we obtain
and P'l+1(x) - xP[{x) = (/
Eliminating P'i+i{x) between these (by multiplying the second equation
by x and subtracting from the first) gives
which reduces to
(ix) Using (iii) we have
- Pk{x){{2k + \)yPk{y) - *iW
multiplying both sides of this equation by k + 1, we obtain
(ft + 1){Рк+г(х)Рь(у) - ВДП+ibO}
= BЙ+1Х*-3')Р»(*)Р*(У)+Л{Л(*)^-1(У)--Р*-1(*)Р^У)}. C.30)
If (^ -b l){Pt+1(«)Pi(y) - Р*(*)Р*+1Ы} is denoted by fto equation
C.30) may be written in the form
f, r : BA I I)(* -)')Р^)Л(г) I f, t. C.31)

62 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
(Strictly, this equation is true for k > 1, since it is for this range of values
of k that all quantities involved are well defined; but we may also make it
true for k = 0 provided we define i_x = 0, for then
= x — у
(by equations C.21))
and x — у is also equal to
Bk + 1)(* - у)Рк(х)Рк(у) + fk-i with k = 0.)
If we now sum the set of equations C.31) from k = 0 to k = l, we
obtain
i i i
f > OJ, _i_ IVv. лЛР (v\P (лЛ 4- > f
^_! \ ' ^^
*=0 *=0 *=0
I I
B* +1)(* -^)вд%) + 2 f^-i
* = 0 *=1
= 2 B* + ххл - y)p*(x)p*(y) + 2ffc-
*o
Hence
i i-1 i
fк- У ik = > Bk + l)(x
so that
i
ifc=0
and if we remember the definition of f, we obtain
i
(x-y) !+1 l l l+1 По
3.8 ASSOCIATED LEGENDRE FUNCTIONS
Theorem 3.9
If z is a solution of Legendre's equation
then A — х2)тП (dm^/dxm) is a solution of the equation
d2v dv ( тг
0—ri^-^ + lt' + D-rrs
(known as the associated Legendre equation).

§3.8 associated legendre functions 63
Proof
Since z is a solution of Legendre's equation, we must have
n , x2)— 2x + 1A + \)s = 0 C 32)
Now let us differentiate equation C.32) m times with respect to x:
dm (,. ^d2z] „dmfd;srl ,., _dm^
A - x2)-r— > — 2-—{ x— \ + 1A + IV— = 0
v }dx2\ dxm\ dx) K 'dx™
which, when we use Leibniz's theorem for the »2th derivative of a product,f
becomes
,Am+2z d ,„ ,. dm+lz m(m - 1) d2 „ o4 dmz
A - x2)- Ь m—(l - ж2).-; Ь -Ц; -,— A — * )•-;—
v Мжт+2 ^ dxK ' dxm+1 2 dx2K ' dxm
С dm+1^ d dmz) ... ,,dmz
- 2<x- + m~-x. — > +1A+ 1)^— = 0
I dxT+1 dx dxrj K 'dx™
(since higher derivatives of 1 — x2 and x vanish).
Collecting terms in dm+2z/dxm+2, dm+1z/dxm+1 and dmz/dxm, we obtain
A2
m{m-\)~2m}— = 0,
which, on denoting dmz/dxm by zly becomes
A ~ *2)? ~ 2{m + 1)Ж^ + {/(/ + 1)~ m{m + 1)}Zl = °- A33)
If we now write
z2 = A _ x2)ml2zx = A - x2)m/2 ~~
(XX
equation C.33) becomes
A - х2)^{Ф ~ ж2)"™72} - 2(ж + l)«^{*i(l - *2)-m/2}
+ {/(/ + 1) - «(от + 1)}*,A - х2)-™'2 = 0. C.34)
But
+ mz%x(\ - л;2
d.v'"
2, raC^d^-

64 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
SO that
%(A - ж2)-'™'2'-1 4- «^ж
Hence equation C.34) becomes
Н- 1) - т{т + 1)}г2A - х*)~«ч* = О
which, on cancelling a common factor of A — x2)~m/2 and collecting like
terms, becomes
m(m+2) „ 2(т+1)тхг „, ,,
The coefficient of d^2/d* is just — 2x, while the coefficient of z2 is
... , t, , (m2 + 2m - 2m2 - 2m)x2
x. X
Thus equation C.35) reduces to

§3.9 PROPERTIES OF ASSOCIATED LEGENDRE FUNCTIONS 65
so that z2 satisfies the associated Legendre equation which, by the defini-
definition of z2, proves the theorem.
Corollary
The associated Legendre functions P™(x) denned by
РГ(х) = A-х*Г/а'^яР1{х) C-36)
satisfy the associated Legendre equation.
Proof
This result follows immediately from the theorem, since Pt(x) satisfies
Legendre's equation.
Using Rodrigues' formula (theorem 3.2), it is possible to rewrite
definition C.36) in the form
1 dl+m
pm/Y\ __Л _ v.2\m/2 . (V2 __ iy
rlKx) — 2Ц\У X > d^+m^ ' '
The right-hand side of this expression is well defined for negative values
of m such that / + m > 0, i.e., m > — /, whereas the original definition
C.36) of P™(x) was only valid for m > 0. Thus we may use this new form
to define P™(x) for values of m such that m > — /.
It is easy to verify that if we consider m positive, the function Pfm{x)
defined in such a way is a solution of Legendre's associated equation as
well as P™(x). In fact, it is not an independent solution; it may be proved
that
I^ C-37)
(see problem 3 at the end of this chapter).
3.9 PROPERTIES OF THE ASSOCIATED LEGENDRE FUNCTIONS
Theorem 3.10
(i)tf(*)=P,(«).
(ii) PJ"(ar) = 0 if m > I.
Proof
(i) This result is immediately obvious from definition C.36).
(ii) Since Pt(x) is a polynomial of degree /, it will reduce to zero when
dm
differentiated more than / times. Thus - P,(x) = 0 for m > I, and the
dxm
required result then follows from definition C.36).

66 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
Theorem 3.11 (Orthogonality Relation)
I Pi\XiPu\X\ U.X — On'
Proof
We first prove that if / 9^ Г
J P?{x)P$ (x) dx = 0.
This proof follows exactly the same lines as that of the first part of theorem
3.5, so we shall not repeat it here.
All that now remains to be proved is that
Г.,
Assume first that m > 0; then from definition C.36) we have
1
1 If dm 11+1
1 A ') V ' dxm IX ;JJ-i
fl f(Jm-l I d Г dm 1
J-i Idx"»-1 lK ;Jdxlv ; d*m lK ')
(on integrating by parts)
(the first term vanishing at upper and lower limits because of the A — л;2)
factor).
Now, from equation C.33) with m replaced by m — 1 we have that
Am+l
which, when multiplied by A — ж2)™'1, becomes
Am+l
A Г ?М
= 0

§3.9 PROPERTIES OF ASSOCIATED LEGENDRE FUNCTIONS 67
and this equation may be rewritten in the form
s{(i"хГ <?ад}= -(/+m){i ~m+i)( ?^
Substituting this result in equation C.38) gives
P {P?(x)}*dx
J —1
= J -1
Applying this result again gives
— 1 J — 1
and repeating the process m times in all we obtain
. . . /. J1
2/
(using theorem 3.5)
m)! 2
(/-«)! 2/+ 1
which is the required result.
Suppose now that m < 0, say иг = —n with и > 0.
Then j {РГ(х)}2 dx = f {Рг "(ж)}2 dx
(by equation C.37))

68 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
I 2
l + n)l) G-я)!2/ + 1
(by the result just proved, since n > 0)
-n)\ 2
which is the required result.
Theorem 3.12 (Recurrence relations)
(i) РГ\х) - -^-—-^P-Xx) + {/(/+1)-«(«-1)}Р--*(ж) = 0.
(ii) B/ + 1)*РГ(Ж) = (/ + m)Pr_iW + (Z - m
(iv)
Proof
(i) This is the fundamental relationship linking three associated Legendre
functions with the same / values and consecutive m values.
Let us denote (dm/djcm)P!(x) by P\m)(x) so that definition C.36) may be
written in the form
Р|»(ж) = A - х*)тПР1т\х). C.39)
Now, in equation C.33) we know that we may take z = Рг(х) and hence
Zl = P[m)(x), so that we obtain
+ {/(/ + 1) - m(m + l)}Pim\x) = 0.
Using the definition of Plm\x), this equation becomes
A - *2)P<m+2)(*) - 2(m + \)xPt+1\x)
+ {/(/! !)-«(» | !)}/*»>(*) = 0

§3.9 PROPERTIES OF ASSOCIATED LEGENDRE FUNCTIONS 69
which, on multiplying throughout by A — х2)тП, gives
A - *2)(m/2) + lp(m+2)^ _ 2(ttl + 1)ЖA - X2)
+ {/(/ + 1) - m(m +
Hence, using equation C.39), we have
х) - 2(m + \
which, when m is replaced by m — 1, becomes
^ 4*) = 0;
this is the required result.
(ii) This is the fundamental relationship between associated Legendre
functions with equal m values but consecutive / values.
By theorem 3.8 (iii) we have
(/ + 1)Рг^(*) - B1 + \)xPl{x) + WU*) = 0
which, when differentiated m times (making use of Leibniz's theorem for
the second term), gives
(/ + \)P\™\(x) - B1 + \){xP\m\x) + mP^-^x)}
+ lPt\(x) = 0. C.40)
Similarly by theorem 3.8 (iv) we have
which when differentiated m — 1 times gives
P\1\(x) - Pt\(x) = B1 + l)P$m-l\x). C.41)
Using equation C.41) to substitute for P[m~l\x) in equation C.40) gives
(I + l)^fi(«) ~ B1 + \)xP\m\x) - m{P\X\(x) - P?\[x)} + &П(х) = 0.
Multiplying this equation throughout by A — x2)ml2 and using equation
C.39) gives
(l+l)P7+i(*) ~ B1+1)хРГ(х) ~ mPT+l(x) + mir_i(*) + lPT-i(*) = 0-
Collecting like terms gives
(/ + 1 - т)РГ+1(х) - B1 + Y)xPT(x) +(l + т)РГ^(х) = 0
which, when rearranged, is the required result.
(iii) Multiply equation C.41) throughout by A — x2)m/i and we obtain
A x»)»«^7>(x) A x*)M'*P}m\(x) B1 | 1)A • ж»)»'»/»}-""(a-)

70 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
which, on using equation C.39), becomes
P?+i(*) ~ РГ-М = B/ + 1) V(l - х*)РГ\х). C.42)
Replacing m by m + 1 gives
PT+V(x) ~ pt_\\x) = B/ + i)v(i - *s)i?(*)
which, when divided by 2/ + 1, is just the required result,
(iv) We use (ii) to replace xP?(x) in (i) by
х) + (/ - « 4 )}
so that we obtain
+ {/(/ + 1) - m(m - 1)}РГ-\х) = О.
If we now use equation C.42) for Pfl~1(x), we obtain
+ {1A + 1) - m(m - 1)}
= 0.
By straightforward algebraic manipulation this reduces to
which, when m is replaced by m — 1, is just the required result.
3.10 LEGENDRE FUNCTIONS OF THE SECOND KIND
In the first section of this chapter we obtained two independent series
solutions yx(x) and y2(x) of Legendre's equation. We obtained solutions
finite for —1 < x < 1 (indeed, finite for all finite values of x) by taking /
integral; then for / even Ji(#) reduced to a polynomial, while for /
odd y2(x) reduced to a polynomial. In both these cases the other series
remains infinite; it may be shown to be convergent for | x | < 1 and diver-
divergent for | x | > 1. In some physical situations we wish two independent
solutions valid for the region | x \ > 1; one of these is, of course given by
Pi(x), while a second solution is given by the following theorem (note that
it is still infinite for x = ±1).

§ 3.10 LEGENDRE FUNCTIONS OF THE SECOND KIND 71
Theorem 3.13
A second independent solution of Legendre's equation is given by
where
17 l 'П is odd
I 2
if I is even.
2
Qi(x) is called the Legendre function of the second kind.
PROOFf
In Legendre's equation, set у = zP\{x) so that z is a new dependent
variable. We have
dj _ , ,d^ dPj
dx dx dx
and hence the equation becomes
Collecting terms in z, ds/dx and d2s:/dx2, we have
A2P dP
which, on using the fact that Рг satisfies Legendre's equation, becomes
f The proof of this theorem may be omitted on a first reading. The proof ends
on page 77.

72 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
2x
Hence dVb~ + ~PlxY ~ Г- x<
and this is equivalent to
which when integrated gives
In -p -f In {Pi{x)Y + In A - x2) = constant.
Therefore -^{Pi(x)}%\ - x2) = constant = A, say,
dz A
so that — =
v — A
This means that we have a solution of Legendre's equation given by
We must now show that this is of the form stated in the theorem. We
first dispose of the case / = 0:
d*
= Г dx
Jl -ж2
2Vl - x + 1 + х
-ln(l -*)+
. 1 +x
ln
If now / y± 0, we remember that Pt(x) is a polynomial of degree /, so that
it may be written in the form
Pi(x) = 4X - «х)(ж - ag) .. . (ж - аг).
Thus
1 1

§3.10 LEGENDRE FUNCTIONS OF THE SECOND KIND 73
1 _ ж ^ 1 + ж ^ г4Д(ж - «,) + (* - a,)»/
C.44)
on splitting into partial fractions.
We may determine a0, b0 and cr fairly simply. Multiplying both sides of
equation C.44) by A — ж2){Рг(ж)}2 gives
1 = ao(l + *){/»,(*)}« + Ы1 -
Setting ж = 1 in this equation and remembering that РгA) = 1 gives
a0 = |, while setting x = 1 and remembering that P,( —1) = ( — 1)'
gives &0 = J.
We now show that
To prove this we note that
~{(x -
= 0 when x = аг, provided that
i(x) is finite at x = аг-.
The only terms on the right-hand side of equation C.44) which are not
finite at x = oq are c,-/(# — a,-) and dt/{{x — агJ}.
Hence we have
*
Thus, if we write Рг(ж) = (ж — a4)L(«) we have
d l
* A X2){/.(^)

74 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
2x 2L'(x)
У2хЬ{х)-2{\-х*)ЬЩ
L A -**){?(*)}• J,=«,
^-}. C.45)
But we know by setting Pt(x) = (x — cf.t)L{x) in Legendre's equation
that
which, on performing the differentiations, becomes
A - x*){(x - y.t)U{x) + 2L'(*)} - 2x{(x ~ ^)L'(x) + L(x)}
+ /(/ + 1)(ж - х,)Цх) = О,
and setting x = a,- in this equation gives
A - <x?JL'(a() - 2а,Ца,) = О
so that by substituting back into equation C.45) we obtain ct = 0.
Thus from equation C.44) we have
? 2A - *) + 2A + *) + ^ (x -
where the dr are constants whose values will not concern us.
Thus
у <¦
so that from equation C.43) we have
But (# — <xr) is, for all ar, a factor of Р,(ж), so that Pt(x)/(x — ar) is a
polynomial in x of degree / — 1. Thus 2_, dr {-Pj(*)}/(* — ar) is a poly-

§ 3.10 LEGENDRE FUNCTIONS OF THE SECOND KIND 75
normal of degree / — 1; let us denote it by Wl_1{x). Then we have
Ь j-±J - Wl.1{x). C.46)
1 X
To determine Р^_1(ж) we remember that Qt(x) is a solution of Legendre's
equation so that
which, on use of equation C.46), gives
V2\ P (v\ In
i-i = 0. C.47)
But
so that
ln
Hence equation C.47) becomes
which, on remembering that Pt{x) satisfies Legendre's equation, reduces
to
d Г dW, A dP,
- U\ Xй) -—1\ -4 /(/ I l)^_, =2 , '. C.48)
ил; l ил; J d.v

76 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
Now, by theorem 3.8 (i) we have
-И = B/ - IIP, .Л*) + B1 -
ах
-\-B1 — 4г — ЛР, „ (х\ 4-
Htf-Di
= 2J.21 ~4r - lJP,.*.^*) C.49)
r=0
so that if we assume for Wl_1(x) (which we know to be a polynomial of
degree / — 1) an expression of the form
and substitute equations C.49) and C.50) into equation C.48), we shall
obtain
r=0 N ¦" г=0
= 22B/-4г-1)Р|_8г_1(«). C.51)
But by Legendre's equation we have
{A — ж2)Р,'_ _ (#)) -f- (/ — 2r — l)(l — 2r)Pj_ _i(*) = 0
dx
so that equation C.51) becomes
Hfl-Dl
r=0
The coefficient of each polynomial must be the same on both sides,
so that we obtain
{-(/ - 2r - 1)(/ - 2r) + /(/ -1- 1)K = 2B/ - 4r - 1). C.52)

§ 3.10 LEGENDRE FUNCTIONS OF THE SECOND KIND 77
But -(/ - 2r - 1)(/ - 2r) + /(/ + 1)
= -(I - 2rY + {I - 2r) + /(/ 4- i)
= -I2 + 4r/ - 4r2 + / -2r -t- /2 + /
= 4r(/ - r) + 2(/ - r)
= 2(/-r)Br + l).
Hence equation C.52) reduces to
2(/ - r)Br + IK = 2B/ - 4r - 1)
which gives
and now, by combining equations C.53), C.50) and C.46), we obtain
immediately the result of the theorem.
That the solution of Legendre's equation Qt(x) which we have obtained
1 + x
is independent of Pt(x) is readily seen: because of the factor In , Qi(x)
X X
is infinite at both x = ± 1, whereas we know that Pi(x) is finite for these
values of x.
We may use this theorem to write down explicitly the first few Legendre
functions of the second kind:
&(*) = * In ii?
i. X
/~\ / \ 1 /^ 1 1 \ 1 ~T~ о i
f I I Л* I — i/kuil Л VI In ля2 _!
}cz2>\ ) — 4V O&) -ill ~1 — ~л ~р ~,
A X ? о
We now state without proof several theorems concerning Legendre
functions of the second kind.
Theorem 3.14
i=0
z/ .v > 1 and \ у | < 1.

78 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
Theorem 3.15 (Neumann's Formula)
Theorem 3.16
The results contained in theorem 3.8 (ii — ix) remain true when Pi(x) is
replaced by Qt(x).
Theorem 3.17
The associated Legendre functions of the second kind defined by
satisfy Legendre's associated equation.
3.11 SPHERICAL HARMONICS
In many branches of physics and engineering there is interest in the
equation
Ь (ID s^eW + Kl + 1)<F = O- C'54)
solutions of which we shall call spherical harmonics. (This equation usually
arises in the solution of a differential equation such as Laplace's or Schro-
dinger's in terms of spherical polar co-ordinates г, в, <f>, so that the range of
the variables involved is 0< 0< я, 0< <? < 2эт and we often require a
solution which is finite and continuous for these values—the continuity
implying that the value of W at <f> = 2л is the same as at <f> = 0.)
One method of finding a solution of equation C.54) is the so-called
method of separation of variables—we look for a solution of the form
WF, ф) = &F)Ф(ф). Inserting this expression into the given equation gives
or, dividing throughout by &F)Ф(ф) and multiplying by sin2 в,
sine d
which, when rearranged, becomes
sinO d / . „d<9\ , 14 . I d20
{O) H/ + i)'o

§ 3.11 SPHERICAL HARMONICS 79
Now the left-hand side of this equation is a function only of the variable
в, while the right-hand side is a function only of the variable ф. Since these
two variables are independent, it follows that left-hand side and right-hand
side must separately be a constant which we shall denote by m2.
Thus we have
and ф^ ()
Equation C.56) is just
d20
w = -»4> C.57)
while equation C.55) simplifies to
sin в d
Equation C.57) has the general solution
ф = Аётф + Ве-1тф
where, if the solution is to be continuous, we require ФBл) = Ф@), so that
m must be an integer (which we may take conventionally to be positive).
In equation C.58) we make the change of variable cos в — x. Then we
have — sin 0 dO = dx
.. 1 d d
and hence ^— — = — - --
sin v da dx
d d d
and sin 0— = sin2 в. r = — A — x2)-r-.
dd dx dx
Accordingly, equation C.58) becomes
which we recognise as Legendre's associated equation; it will have a
solution finite at в = 0 and n{x — +1 and 1) only if / is integral. In that
case the finite solution is given by 0 = РГ(Х) = iT(C0S 0).
Thus the general solution which is finite at both 0=0 and ж and is
continuous must be
W{0, ф) = (Ле'м* + J3e-im*)PT(cos 0)
which, because of equation C.37), we may write in the form
XF --¦ /lle""*P',"(cos 0) |- AiC~""*P-f(cos 0)

80 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
where A^ = A
If now we denote
y?{Q, ф) = eim*Pf(cos в) C.59)
we may write the general solution in the form
V = А1УТ{6, ф) + А,уГ(в, ф).
Of course, this is a solution of the original equation C.54) for any value
of от, and since C.54) is homogeneous we have the solution
ATWO, ф) + АРуГ{в, ф)}.
For many purposes it is more useful to consider a multiple of yf (which
we shall denote by Yf1) as the basic solution; a multiple chosen so that the
solutions are orthogonal and normalised in the sense that
Г &ф Г d0 sin 0{?Т(в, ф)}* Yjf@, ф) = dnSmm, C.60)
Jo Jo
(where the * denotes complex conjugation).
We may readily prove that this is accomplished by taking
For then we have
Аф ? дв sin в{?ГF, ф)}*??'(в, ф)
2п\1
4(/ + т)\ (/' + т'I
Г еКт--т)ф йф Г рт (cos 0) p^'(cos Q) s\
Jo Jo
(using the fact that P™(x) is real),
+ \){1-т)\{1'-т'У\
')! I m'
2^VI
I \Л1± /' IA 1 ил
- 1

§3.11 SPHERICAL HARMONICS 81
(since the first integral vanishes unless m' = m, in which case it is equal to
2тг; and in the second integral we have made the substitution x = cos в)
Щх)Р?(х) dx
f
J
//B/ + 1)BZ' + 1)(/ - «)I (/' - «I\. 2(l + m)l
VI 4(/ + m)! (/' + m)! / m'm B/ + 1)(/ - m)! "'
== "(Am''
4(/ + m)! (/' + m)! / m'm B/ + 1)(/ - m)!
(by theorem 3.11)
The factor (—l)m in definition C.61) of Yf(d, ф), which we shatf take as
the basic spherical harmonic, was not necessary for the orthonormality
property; however, its introduction is conventional (although the reader
is warned that in the topic of spherical harmonics different authors may
employ different conventions).
Theorem 3.18
Proof
me,
(by equation C.61))
2 A+т)\
(by equation C.37))
(by equation C.61)).

82
LEGENDRE POLYNOMIALS AND FUNCTIONS
CH. 3
We may use the definitions of Yf(в, </>) and Pf(cos в) to obtain the
following explicit expressions for the first few spherical harmonics:
3.12 GRAPHS OF THE LEGENDRE FUNCTIONS
In this section we give graphs of some of the functions encountered in
this chapter.
0-5 -
-0-5
Fig. 3.2 P,(x), 0 < x < 1, / = 1, 2, 3, 4

§3.12
GRAPHS OF THE LEGENDRE FUNCTIONS
83
Fig. 3.3 Pi(x), x> 1, I = 1,2, 3, 4 (Note that the scale
on the vertical axis is logarithmic.)
-0-5 -
1'h;. 3.4 P, (ins 0), 0 • ¦ 0 ¦ ' ,-r/2, / 1, 2, 3, 4

LEGENDRE POLYNOMIALS AND FUNCTIONS
P,'M
20 -
1-5 -
0'5 -
-0-5 -
-10 -
-1-5
Fig. 3.5 Pi\x), 0 < ж < 1, / = 1, 2, 3
-i-o
Fig. 3.6 Q,(x), 0 ¦¦. x < 1, I 0, 1, 2, 3, 4

§3.13
EXAMPLES
85
\a'-
Fig. 3.7 Qi(x), x> 1, I = 0, 1, 2, 3, 4 (Note that the scale
on the vertical axis is logarithmic.)
3.13 EXAMPLES
Example 1
Show that
Deduce the value of xiPl+1{x)Pl_i{x) 6.x.
J о
^
We use theorem 3.8(ii) to dispose of the x2 appearing in the integrand,
and we may then use the orthonormality property of theorem 3.5.
Thus,
^ хЧ\ , Л*)?,. ,{x) dx = J' {xl\ , l(x)}{xPl л(х)} dx

86 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
(the other terms vanishing by theorem 3.5, since they are of the form
(" Pl{x)Pm(x) dx with / ^ m)
J -l
1A + 1)
B1 + 3)B/ - 1) 2/ + 1
(by theorem 3.5)
21A + 1)
D/2 - 1)B/ +'3)"
We know that P((#) is a polynomial of degree /, so that the above inte-
integrand is a polynomial of degree 2 + (I + 1) + (/ — 1) = 2A + 1), i.e., of
even degree. Thus the integrand is even, so that
Г хФ^МР^х) dx =2 [lx*Pl+1(x)Pl.1(x)dx
J -1 Jo
and hence
x'PWPU*) dx =
Example 2
Г1
Evaluate Pi(x) dx when I is odd.
Jo
By theorem 3.8 (iv) we have
Pt(x) dx = -^

§3.13 EXAMPLES 87
(-lfnw (/+_1)!
2/ + 1 Г 2;+1 [{(/ + 1)/2}!]2
(_l)(i-i)/g (/-1)!
(by theorem 3.4(i) and (v), remembering that / + 1 and / — 1
are both even)
(/~1)' Г, _
- l)/2}!]2 L
27 + 1 21'1 [{(I - l)/2}!]2 L 23{(/ + 1)/2}2J
_1_(Z1)(^/2 (
2/ -|- 1 2l~l "[{(/
(_l)(^D/2(/_ 1)!
Example 3
QO
Ifi(x) = | # |/or —1 < ж < 1, expand i(x) in the form V c,Pr(#).
r 0
By theorem 3.7 we know that such an expansion is possible and that
cr is given by
cT = {r+l)\ \x\ Pr(x) dx.
J -l
Now, | ж | is an even function and Pr(x) is even if r is even and odd if r
is odd. Hence if r is odd | x \ Pr{x) is odd, so that | x | Pr{x) dx = 0
and hence cr = 0. On the other hand, if r is even, | x | Pr(x) is even, and
hence
•l
f | x | Рг(ж) d* = 2 f |*| Pr(x) dx
J -i Jo
= 2 Г*РГ(*)(
J о
I dx
J о
so that now
ct Br | 1) f.TP,.(.T)dx C.63)
J

88 LEGENDRE POLYNOMIALS AND FUNCTIONS CH. 3
To evaluate this integral we use theorem 3.8(ii):
= [ {(r + \)Pr+l(x) +rPr_l(x)}Ax.
Jo
Now, r is even, so that both r + 1 and r — 1 are odd, and we may use
the result of example 2 to obtain
c =
+ l)l(Jr)! ^ 2'~Hir)l(Jr - 1)!
-2)! /(r + \)r{r -
_
(-lf'\r -2)! /(r + l)(r-l)
_ J—7TP2 '
- 1 - r« - 2r"
!(|r - 1)! I 2(FTl)
+ l)(r -2)!
C.64)
Thus we have
CO
w-Z
п=0
Example 4
Ifx>\, show that P?x) < Pt+1(x).
We prove this by the method of induction: assume first that
Pi-i(x) < Pt{x) and use this to prove that Рг(х) < Рг+1(ж). Then since
the result is trivially true for I = 0 (P0(x) = 1 and Рх(ж) = x) it will be
true for all /.
We may assume throughout this proof that P,(x) > 0. For, from Rod-
riguee' formula (theorem 3.2), it is readily seen that if x > 1 (which it is
in this example) then Рг(х) > 0 for all values of /.
Now, by theorem 3.8(iii) we have
(/ + 1)^1 - B/ + 1)* I lP? - 0.

§ 3.13 EXAMPLES 89
Thus
p, / + 1 / + 1 p{
B/ + 1) _ J_
(since x > 1 and we have
assumed that Pj-^/P, < 1)
= 1.
Since P, is positive for all /, this implies that P,+1 > Pt and hence the
method of induction guarantees the result to be true for all values of /.
Example 5
Prove that /{0,(*)P,-,(*) - ??,-,(*)P,(*)}
}
and deduce that /{(?,(«)P,-i(a:) - Qt-iMPjO*)} = -1-
From theorem 3.8(iii) with / replaced by / — 1 we have
lPt(x) - B1 - 1)*P,_X(*) + (/ - 1)Р,_2(ж) = 0 C.65)
and by theorem 3.16 we have also
lQt(x) - B1 - \)xQU*) -t- U - l)Qi-i(*) = 0. C.66)
Multiplying equation C.65) by 0{_i(*), equation C.66) by Pi~i(x) and
subtracting gives
which is equivalent to
as required.
If we now define
Щ
this result may be written in the form
F(l)
and hence we have
F(l)-
Thus F(l)-~FA)

90 LF.GENDRE POLYNOMIALS AND FUNCTIONS CH.
en that ISOIvW (v\ (~> (\ЛР{\Л\ О (v\T>(v\ О (\ЛР(л»\
But we know that Po(x) = 1>
Рх{х) = x,
Qlx) = \\n\±^,
q tx\ .-- xx jn __JL x_ _ i
so that we obtain
which is the required result.
PROBLEMS
A) Show that I xPJxWiJx) fa
J -l
A 2l
dx = 4/^" Г
B) Show that A - x*)Pi(x)P'm(x) dx =
C) If D =. -—, use Leibniz's theorem to prove that
ax
m(x2 - 1)' = (-1)«[/-
@ < m < /)
and hence deduce that P^x) = (-l)m [,—-:-.PT(x).
(I + wz)!
D)Iff(*) = ( x ,пч
^ y w \—J (-1 < x < 0)
GO
expand f(x) in the form y^ crPr(x).
r= о
E) Show ,ha,

PROBLEMS 91
Г1
F) If и„ = j x-WJWPn^x) дх, show that nun + (я - l)Mn_! = 2,
and hence evaluate un.
G) Show that J Br + l)PJx) = P;+1(*) + P'n(x).
(8) Show that A - x) J Bj- + *)ВД = (n + 1){ВД - P»+i(*)}.
(9) Assuming the result of theorem 3.14, prove theorem 3.15.
A0) Show that Р,(*Ш*) - P[{x)Ql{x) = r-I-.
A1) Show that 17@,0 =
A2) If л; > 1, prove that
and deduce that
(i) by makmg the substitution t = ?
we obtain
l{-X) = J о {x + V(*2-

4
BESSEL FUNCTIONS
4.1 BESSEL'S EQUATION AND ITS SOLUTIONS; BESSEL FUNCTIONS
OF THE FIRST AND SECOND KIND
Bessel's equation of order n is
(where, since it is only пг that enters the equation, we may always take n
to be non-negative).
Since this is of the form of equation A.2) with q{x) — 1 and r(x) =
x2 — я2, we may apply the methods of Chapter 1 and be assured that
any series solution obtained will be convergent for all values of x.
Setting z(x, s) — ^^ a,xs+r and requiring z to be a solution of equation
f=0
D.1) leads, by the same method as in previous cases, to the system of
equations
{s(s - 1) + s - и2}а0 = 0, D.2)
{(s + l)s + (, + 1) - n*}ai = 0 D.3)
and {(* + r)(s + r - 1) + (s + r) - n*}ar + ar_a == 0
(r > 2). D.4)
Equation D.2) gives the indicial equation s2 — n2 = 0 with roots
s = ± n. We shall obtain two solutions from these which will be inde-
independent, apart, possibly, from the case when the two roots differ by an
integer, i.e., when 2n is integral.

§4.1 bessel's equation and its solutions 93
Equation D.3) is
{(s + IJ - n*}ai = 0
and, since s2 = и2, we cannot also have (s -f IJ = n2f, so that
(s + IJ - n2 ^ 0 and hence ax = 0.
Equation D.4) is just
{{s + rJ - и2}аг + ar_2 = 0
which gives
Ur== {sTrf-n*-
Taking s = n, this becomes
Or-2
ar = — 7 - - -
(я + rJ - я2
(я + r — n)(n + r + я)
Thus we have
п* = ~ 2Bя +) " ~ 22.Т(я -i- 1)'
4Bя + 4) 22.2(я + 2) 2*.2!(и + 1)(я + 2)'
^4 = а*.
6Bя+6) 22.3(я + 3)
2в.З!(я + 1)(я+2)(и-ГЗ)
and in general
а =/ Л\т а»
2r V ; 22гг!(я -1-1 )(я +2)... (и ] -г/
f The only situation in which we can have s2 —- (s + IJ = n2 is n = i, s = — J.
For this case it is true that s "-- — J makes at indeterminate and leads to two inde-
independent solutions, but the results of the text still hold good, since zero is a
possible choice for aL and we obtain the second independent solution from s = J.

94 BESSEL FUNCTIONS CH. 4
We simplify the appearance of this expression somewhat by noting that
(n + 1)(и + 2). . . (n + r)
= (n + r)(n + r - 1) . . . (я + 2)(n + l)~~{j
^ Г(я + r + 1)
Г(я + 1)
on using repeatedly the fact that xT(x) = T(x + 1) (theorem 2.2).
Thus we now have
a -( IV _?¦?<> +])_
Also, if we use the fact that аг — 0 together with equation D.5), we
obtain
ai = аз = «5 = • • • = «гг+г = • • • = 0.
Hence, substituting these values for the ar into the series for z(x, s), we
obtain as a solution of Bessel's equation
Z ^w^+r + if
This is a solution for any value of a0; let us choose a0 = 1 /{2"Г(я + 1)}
and we obtain the solution which we shall denote Ьу/„(х) and shall call the
Bessel function of the first kind of order n:
2r+n
¦ D-6)
From the remarks at the beginning of this section it is clear that the
infinite series in equation D.6) will be convergent for all values of x.
So far we have dealt with the root of the indicial equation s = n. The
other root s = —n will also give a solution of Bessel's equation, and the form
of this solution is obtained just by replacing n in all the equations above by
—n, so that we obtain the solution to Bessel's equation
2r~n
. D.7)
Suppose now that n is non-integral. Then, since r is always integral, the
factor Г(— n + r + 1) in equation D.7) cannot have its argument equal
to a negative integer or zero, and hence must always be finite and non-
nonzero. Thus equation D.7) shows that/_„(#) contains negative powers of л;
(they arise for those values of r such that 2r < я), whereas equation D.6)

§4.1 bessel's equation and its solutions 95
shows that Jn{x) does not contain any negative powers. Hence, at x = 0,
Jn(x) is finite while/_n(x) is infinite, so that one cannot be a constant multiple
of the other, and we have shown that JJx) and J -n{x) are independent
solutions of Bessel's equation for n non-integral (which is a stronger con-
condition than 2w non-integral, obtained from the general theory). The
explicit relationship between/„(x) and/^n(x) for integral я is shown in the
following theorem.
Theorem 4.1
When n is an integer (positive or negative),
/-(*) = (-1Ш*)-
Proof
First consider n > 0.
Then
from equation D.7).
But Г(- и + r + 1) is infinite (and hence 1/{Г(— n + r + 1)} is zero)
for those values of r which make the argument a negative integer or zero,
i.e., for r = 0, 1, 2, ... (я — 1) (remembering that this is possible because
n is integral).
Hence the sum over r in the above expression for/_n(#) can equally well
be taken from n to infinity, and then
2r~n
2 1 (xy2(m+n)-n
v.. , n)\F(m + l)\2j
(where we have changed the variable of summation to m = r — n)
i 2m+n
-0
But
-<-'>¦! (-ir^rasrnj©
2m+n
(by equation D.6))

96 BESSEL FUNCTIONS CH. 4
so all that remains in order to complete the proof is to show that
(m + и)!Г(я! + 1) = т\Цп + т + 1)
for n and m integral.
But
(m + пIГ(т + 1) = (m + n)(m + n — 1). . . (m + l)m\F(m + 1)
= т!Г(т +и + 1)
(on using repeatedly the result that Г(д; + 1) = хГ(х)), and thus the
result is proved.
Now consider n < 0; in this case we may write n — — p with p > 0.
Then what we require to prove is that
Л(*) = (-!)-'./-,(*)
or (
which, of course, since /> is positive, is just the result we have proved
above.
Let us summarise what we have proved so far: we have shown that if n
is not an integer then JJx) and J-n{x) (denned by equations D.6) and
D.7), respectively) are independent solutions of Bessel's equation (so that
the general solution is given by AJn(x) + BJ_n{x)) while if n is an integer
they are still solutions of Bessel's equation but are related by
/-»(*) = (-!)"/«(*)•
Theorem 4.2
The two independent solutions of Bessel's equation may be taken to be
and YJLx) = t^JZ
for all values of n.
Proof
We consider separately the cases n non-integral and n integral.
(i) n non-integral.
Here sin nn Ф 0, so that Yn(x) is just a linear combination of Jn(x) and
7 -n(x). But from the above discussion we know that in this case Jn{x) and
J-n(x) are independent solutions, so that/„(я) and a linear combination of
Jn(x) and/__n(#) must also be independent solutions. Hence/n(x) and Yn(x)
must be independent solutions.

§4.1 bessel's equation and its solutions 97
(ii) n integral.
In this case sin пл = 0 and cos пл = ( — 1)™, so that by theorem 4.1 we
have
costmjn(x) -/_„(*) = (-1Ш*) - (-1Ш*) = 0.
Hence Yn{x) has the form 0/0 and so is undefined. However, we may give it
a meaning by defining it as
Yn(x) = lim Yv(x)
sin
[(д/dv) sin m]v=n
(by L'Hopital's rule)f
(x) + cosm(d/dv)Jv(x) -
[л cos wr],,=re
D.9)
We must now prove two things; firstly that Yn(x) as defined by equation
D.9) is in fact a solution of Bessel's equation and, secondly, that it is a
solution independent from Jn(.x)- To accomplish the first of these we note
that/,,(x) obeys Bessel's equation of order v:
which, when differentiated with respect to v, gives
x*.f 9Ji +X±f + (*¦ - ,# - */, = 0. D.10)
dx2 dv Ax dv dv ч '
Also, of course, J-,{x) satisfies Bessel's equation of order v. so that we
have in exactly the same way
f L'Hopital's rule states that if f(a) = g(a) = 0, then
where f*r'(«) and g^r'(a) arc the lowest-order derivatives of f(x) and к(х), respec-
respectively, whieh are not both zero at x — a.

98 BESSEL FUNCTIONS CH. 4
Multiplying equation D.11) by (— I)' and subtracting from equation
D.10) gives
_ (-1)'^} - 2v{Jv - (-1Г/-Л = 0.
Setting v = n in this last equation and using equation D.9) gives
™ Уп(«) + *^Y»(*) + (^2 - я»)ВД
(*)} = 0.
But now и is an integer, so that we may use the result of theorem 4.1 that
-n{x) = (-l)n/«(«) to obtain
which just states that Yn(x) satisfies Bessel's equation of order n.
That Yn(x) is independent from Jn(x) may be seen from the result of the
next theorem, which implies that at x = 0 Yn(x) is infinite, while we know
that/n(x) is finite.
Theorem 4.3 (Explicit expression for Yn (x) for n integral)
n+2s
n+2s
у is Euler's constant (see Appendix 2).
Proof
We shall omit the proof of this theorem, merely noting that the method
to follow is to use the series expansions D.6) and D.7) in equation D.9).
Euler's constant appears because of properties of the derivative of the
gamma function.

§4.2 GENERATING FUNCTION FOR THE BESSEL FUNCTIONS 99
Theorem 4.4
When n is integral, Y_n(x) = (—1)" Yn{x).
Proof
From equation D.9) we have
We shall call/n(x) and Y«(#) Bessel functions of order n of the first and
second kinds respectively. (Yn(x) is sometimes called the Neumann
function of order n and denoted by Nn(x).) As we have seen, Jn{x) and
Yn(x) (with n considered positive) provide us with two independent
solutions of Bessel's equation, Jn{x) always being finite at x = 0 and
Yn{x) always infinite at a: = 0.
4.2 GENERATING FUNCTION FOR THE BESSEL FUNCTIONS
Theorem 4.5
fl= — GO
Proof
We expand exp i -x\ t ) > in powers of t and show that the coeffi-
\L \ t/ )
cient of tn is Jn(x):
_ v №Y у ( Wty
r 0 « (I

100 BESSEL FUNCTIONS CH. 4
Д <4Л2)
We now pick out the coefficient of tn, where first we consider n > 0.
For a fixed value of r, to obtain the power of t as tn we must have s = r ~ n.
Thus for this particular value of r the coefficient of tn is
A\ 2r—n
)
2
We get the total coefficient of tn by summing over all allowed values of
r. Since s = r — n and we require s > 0, we must have r > и. Hence the
total coefficient of tn is
(where we have set ^ = r — n)
("/2)"+"
-f ( D
(remembering that both p and ?г
are integral, so that we may use
the result of theorem 2.3 that
n + \)=(p+n)\)
(by equation D.6)).
IfnowM < 0, we still have the coefficient off" for a fixed value of r given by
'_1У-п(Ч Х
' ' \2/ r\(r-n)\
but now the requirement that s > 0 with s = r ~ n is satisfied for a//
values of r. Thus the coefficient of tn is just
^ /1\2r-n V2r~n
2 (-¦>'-© л(^т-
(by equation D.7))
= /»(*)
(by theorem 4.1).

§4.3 INTEGRAL REPRESENTATIONS FOR BESSEL FUNCTIONS 101
4.3 INTEGRAL REPRESENTATIONS FOR BESSEL FUNCTIONS
Theorem 4.6
1 f"
Jn(x) — - cos (пф л1 sin ф) дф
л J о
(n integral).
Proof
Since/_n(x) = (— 1 )nJn{x) for n integral, the result of theorem 4.5 may
be written in the form
If we now write t = e1* so that
f = e1* - е^!* = 2i sin <
this equation becomes
But when и is even
ei«* + (-l)n е-'1Пф = e1"^ + e"il!* = 2 cos пф,
while when n is odd we have
Thus we have
ft odd
=/oW + 22 cos щм*)+i 22 sin B^
Equating real and imaginary parts of this equation gives
со
cos (* sin ф) =J0(x) + 2 2 cos 2Ц]^{х) D.13)
sin (л; sin ф) = 2 2 sin B^ " ^)<t>Jik-i{x). D.14)
If we multiply both sides of equation D.13) by cos пф (п \- 0), both sides

102 BESSEL FUNCTIONS CH. 4
of equation D.14) by sin пф (n > 1), integrate from 0 to л and use the
identities
Г
(О (т ^ п)
cos тф cos пф &ф = < л/2 (от = я 5^ 0)
[я: (ш = я = 0)
ancj I „:_ ...j „:„ „j jj j " \m j?n)
sin тф sin я</> d^
Jo
л/l (m = n^ 0),
we obtain the results
X/n(«) (я even)
г
J
cos пф cos (ж sin ф)(\ф =
o 10 (и odd)
, Гл . , • / • ,ч j, f° («even)
and sin пф sin (x sm ф) аф ==<
Jo U/nW («odd).
Adding these last two equations gives
Сл
{cos пф cos (x sin ф) -(- sin я^ sin (x sin ^>)} d^> = njn{x)
J о
for all positive integral n.
rn
Hence cos {пф — x sin ф) <1ф = njjx)
J о
which is the required result for positive n.
If n is negative, we may set n — —m where m is positive, so that the
required result is
гл
cos (— тф — x sin ф) Аф = nj__m{x)
J о
(where m is positive).
But cos {—тф — x sin ф) йф
J о
f°
= cos {—т{тс — 0) ~~ x sin (n — 0)}. — dO
J л
(where we have changed the variable by setting
в = ж - ф)
Гл
= cos {—тл + тв — х sin 6} dG
J о
= {cos (mO — х sin 0) cos mn
J о
Н- sin (mO — х sin 0) sin тл} dO

§4.3 INTEGRAL REPRESENTATIONS FOR BESSEL FUNCTIONS 103
= (_l)m Г cos (mO ~ x sin 0) dO
J о
(since we know the result to be true for positive m)
Theorem 4.7
(lv\n ft
T fv\ — ^2 ' П — Г2У~*еы dt (n ~> — l\
Jn\X) — ,, yp~( Т1Л J
Proof
Consider the integral / defined by
/=f_i(l-*.)-*e«d,
^Л df
Now, if r is odd, the integrand in A — f 2)"-fyr d^ is an odd function
of t, so that the integral is zero; while if r is even (say, equal to 2s), the
integrand is even, so that we have
j* A - t*)n-kr dt = [ A -t*)n-k*»dt
= 2 (* A - t2)n-h*s dt
J о
f1
= A — и)"-*и'-5 du
J о
(where we have made the change
of variable и = t2, du = It dt)
= B(n + i,s+ I)
(by the definition of the beta
function; we must have n > — \
to ensure the convergence of the
integral (see Section 2.1))

104 BESSEL FUNCTIONS CH. 4
Г(п
Thus I =
Г(и + s + 1)
(by theorem 2.7).
(using the corollary to theorem
2.10)
(by equation D.6)).
1 /v\"
Thus TJx) = -—, t_.
= ^ f*V f1 A - ^V'-'e^d/.
4.4 RECURRENCE RELATIONS
Theorem 4.8
(ii)
(iv)/;(*) = ?/.(*)-Л+х(*)-
(v) Л*) = i{/»-i(«)-/.«(*)}¦
x(x) = ~-Jn{x).

§4.4 recurrence relations 105
Proof
(i) From equation D.6) we have
so that
dco л л
w _____ V /IV 1 1_
CO л
= > (-1)г-,„; rrriz—Br
*-> K ' r\Vn A-r + П22г+« v
rI(,+r)r?,+rJ
(using the result of theorem 2.2 that Г(л: + 1) = хГ(х))
(by equation D.6)).
(Note that in this theorem no restriction is placed on n: it may be integral
or non-integral, positive or negative.)
(ii) We have
" .. 1 1
1)
2r
r=0 ' " v- '
ly 1 1 0rv2f_T
r\T{n +r H
r=0
^1 1
'!Г(и +r + 1):
)
(since the factor r in the numerator makes the term with
r = 0 vanish; we remember that 0! = 1)
00 * л
2/ IV+i (? 4- 1W«(«+i)-i
(where we have set s = r 1)

106 BESSEL FUNCTIONS CH. 4
^—> .г!Г(я -\- s -4- 2) I
5, I /x\2s+n+l
^-w ^11 I я -p J -p Z) \Z/
(on use of equation D.6)).
(iii) From result (i) above, by carrying out the differentiation of the
product on the left-hand side, we have
which, on dividing throughout by xn, gives
п-Ш +Л(х) =Л-1(«),
and hence J'H(x) =/n_1(x) - -Jn{x).
X
(iv) We carry out the differentiation in result (ii) above to obtain
-tuf^-*Jn(x) + x-"J'n(x) = -x'-*Jn+1(x)
which, on multiplying throughout by xn, gives
Thus /;c*)=^/.(*)-/•«(*).
(v) Add results (iii) and (iv) and the result follows immediately.
(vi) Similarly, if we subtract result (iv) from result (iii) we obtain the
required relationship.
Theorem 4.9
All the results of theorem 4.8 remain true when the Bessel functions of the
first kind are replaced by the corresponding Bessel functions of the second
kind.
Proof
We shall prove that result (i) remains true for Yn(x); a similar method
will prove result (ii), and then results (iii-vi) follow in the same way as they
did in theorem 4.8.

§4.5 HANKEL FUNCTIONS 107
Thus what we have to prove here is that (d/dx){xnYn(x)} = xnYn_1(x).
We must consider separately the cases of integral and non-integral n.
(a) Non-integral n.
Here we may write
Y (лд = cosnjtjn(x) -J^{x)
sin rm
so that
А
sin rm
(where we have used theorem 4.8 (i) for the first derivative and
theorem 4.8 (ii) for the second)
[OSW7T J'„_!(*) +/-(«-1H*0]
sin rm
1
sin {(и — \)л + л}
1
7l}Jn^{x) +/-(«-1H)]
—sin (и — 1
cos (и —
xn[~cos{n - \)njn_x{x)
sin [n — Vpi
(b) Integral n.
Here we note that Yn(x) = lim Yr(x) and that by part (a)
Taking the limit of this result as v —> n gives the required result.
4.5 HANKEL FUNCTIONS
We define the Hankel functions (sometimes called Bessel functions of
the third kind) by
Hn^(x)=Jn(x)+iYn(x)
Hn^(x)=Jn(x) \Yn{x).

108 BESSEL FUNCTIONS CH. 4
These are obviously independent solutions of Bessel's equation. Both
are, of course, infinite at x = 0; their usefulness is connected with their
behaviour for large values of x, which we shall investigate in a later section.
Theorem 4.10
All the recurrence relations of theorem 4.8 remain true when Jn(x) is
replaced by either Я„A)(х) or Hni2)(x).
Proof
Again we prove relation (i) only, the remainder following as before.
But we know relation (i) to hold for both/„(л:) and Yn(x), so that we have
d
d*1
and ^{*ПВД} =
Hence
d .d
so that
which, on remembering the definitions of H?\x) and H^\x), gives
d*1
(taking the plus sign)
and —(
(taking the minus sign).
4.6 EQUATIONS REDUCIBLE TO BESSEL'S EQUATION
Theorem 4.11
The general solution of
is AJn(Xx) f BYn(kx).

§4.6 EQUATIONS REDUCIBLE TO BESSEb's EQUATION 109
Proof
Make the substitution t = he
dy dy
so that .Z = Д-Z
dx dt
d2y d2y
and ~A = x-jU-
dx2 dt2
Then the given equation becomes
t2w+tdi + {t2-n2)y==0
which is Bessel's equation of order n, so that the general solution is
у = AJn(t) + BYn{t)
and hence, replacing t by 2.x, the general solution of the original equation is
у = AJn(hc) + BYnB.x).
Theorem 4.12
The general solution of
is
Proof
We first make the change of variable у = xaz.
Then ^ = Xх ^ + ax^z
dx ax
and _? = *« у 2^-х — + a(a -
dx2 dx2 dx
so that the given equation becomes
лА+2 Т-, + 2«^+1 -r + a(oc - 1)Л + A 2)^ *'1
dx2 dx
+ {fj2y2x2v + (a2 - и2г)}Л = О
which, on collecting terms in d2^/dx2, dz/dx and z becomes
-b Ma - 1) -I- A - 2a)a •) ft*y2x2Y I (a2 n2y-)}z - 0

110 BESSEL FUNCTIONS CH. 4
which simplifies to
- d*2 + % + {(}2у2х& ~ nY]z = 0> DЛ5)
Now change the independent variable x by setting t = xy.
_, dz dz dt dz
Then -г— == -г- — = y# —
ax dt ax at
dzz , л. „dz , , d dz
and -г—- = y(y — 1 r-' \- yx7'1 -r--—
dx2 v ' dt dxdt
I 14 v ad* , vi v . d2^
= МУ — 1)Ж'"а -г- + УЛ7.!'*7
so that equation D.15) becomes
y*x2V d? + y{y ~1)xV af+ yxV tt + ^Y*2y ~ n*y2]z = 0>
which, on collecting like terms and cancelling a common factor of уг, gives
But this is just the equation of theorem 4.11 with solution
я = AJtft) + BYJtft).
Hence the solution of the original equation is obtained by substituting
back the relationships у = x?z and t — xY:
у =
4.7 MODIFIED BESSEL FUNCTIONS
Consider the differential equation
X% ~ (*2 + ni)y = °- DЛ6)
This has the form of the equation in theorem 4.11 with Я2 = — 1.
Thus it has the general solution
у = AJJix) + BYn(ix).
Now, the solutions Jn(ix) and Yn(ix) have the disadvantage of not

§4.7 MODIFIED BESSEL FUNCTIONS 111
necessarily being real when x is real. However, we may take the constant
multiple of JJix) defined by
J»W — * Jn\}X)
and use it as one of the independent solutions of equation D.16). It is easy
to show that it is a real function of x:
In(x) = i-nJn(ix)
+7+T) {2)
1
~ U 'Щп+r
= Z НГ(я + r + f) U/ • DЛ7)
/„(») so defined is called the modified Bessel function of the first kind,
and equation D.16) is called Bessel's modified equation.
Theorem 4.13
// n is integral, /_„(») = /„(*).
Proof
(by theorem 4.1)
= IJx).
We may obtain the second independent solution to Bessel's modified
equation by considering yn(i#); or alternatively we may employ a method
similar to that used for the definition of Yn(x).
In(x) and I-Jx) are both solutions of equation D.16). When я is non-
integral they are independent solutions (since JJix) and /_„(Ьс) are in-
independent solutions). However, when я is integral /_„(#) = /«(#). Define
now
K^~2 sTn^ ' DЛ8)
When n is non-integral, this is well defined, and /,,(.v) and Kn(x) together

112 BESSEL FUNCTIONS CH. 4
provide independent solutions of equation D.16). When я is integral,
Kn(x) is indeterminate as it stands and has to be defined by
Sin V71
2 jr[cos vn]v=n
(by L'Hopital's rule)
(-1)-ГЭ Э
As in theorem 4.2, it may be shown that this provides a second in-
independent solution of Bessel's modified equation for integral values of n.
The explicit series for Kn(x) may be obtained from those for Jn(x) and
Yn(x) by use of the following theorem.
Theorem 4.14
Kn{x) = ?i«-»{/n(«) + ПЭД} = |
Proof
n I-Jjc) — In(x)
Kn{x) =
2 sin пл
(by definition D.18))
it
2 sin nn
But from the definition of Yn(x) in theorem 4.2 we have that
J-Jix) = cos rnijjix) — sin пл YJix).
Hence
л i™ cos пл]п{\х) — i-nJn(ix) — i" sin яте ^„(i^)
2 sin яте
But
— i cos пл — i-2"-1 = —i cos nn + i.i~2n
= —i cos пл + i(em/2)~2"
(writing i = em/2)

§4.8 RELATIONS FOR MODIFIED BESSEL FUNCTIONS 113
= —i cos nn + i e~m71
= — i cos tin + i(cos tin — i sin rm)
= sin тт.
Thus, using equation D.19) we have
Kn(x) =-2i"
4.8 RECURRENCE RELATIONS FOR THE MODIFIED BESSEL
FUNCTIONS
The modified Bessel functions satisfy recurrence relations similar to,
but not identical with, those of theorem 4.8. Also, in contrast to the func-
functions Jn{x) and Yn(x), which satisfy the same relations as one another,
In(x) and Kn(x) satisfy different relations from one another.
Theorem 4.15
(i) ^{
(ii) ^x{=c
In
(vi) /»_!(*) - /,«(*) = —/«(ж).
Proof
We shall prove results (i) and (ii). The remaining results follow from (i)
and (ii) exactly as in theorem 4.8.
(i) From theorem 4.8 (i) we have
d
Replacing x by ix gives

114 BESSEL FUNCTIONS CH. 4
which, on using the fact that In{x) = i~nJn(ix), becomes
l_d .n/
i dx
and this equation, on cancelling a factor of i2" throughout, gives
d
"J~\X -ln\x)f — X ln-l\X).
ax
(ii) From theorem 4.8 (ii) we have
~{x-Jn{x)} = - x-»Jn+l(x),
which, on replacing x by \x, becomes
jLr-* -»7 r _ _¦-» -«г rlx)
Thus
r -7-{i-nx-ninIn(x)} = — i-nx-nin+1ln+1(x)
and hence
1 d
so that
Theorem 4.16
(ii) ±{x-»Kn(x)} = -x-"K
(iii) K(x) = -ЛГп^С*) -
(iv) K(x) =-Kn(x)
X
(vi) л:,^*) - кп+1(х) = - ~

§4.8 relations for modified bessel functions 115
Proof
Again, we prove only results (i) and (ii). Also, in the same way as in
theorem 4.9, we prove the results for non-integral n and appeal to con-
continuity for a proof in the integral case.
(i) From theorem 4.14 we have
JVjjl Л J ——* i x Yi \ /
and by theorem 4.10 we know that H\}\x) satisfies the first recurrence
relation of theorem 4.8.
Hence —
ax
which, on replacing x by ix, gives
Thus
so that
and hence
(ii) Similarly H^\x), by theorem 4.10, satisfies the second recurrence
relation of theorem 4.8 so that
- ¦T-{inx"-i-n-1KJx)} = {"х^
1 ах л л
{x
which, on replacing x by ix, gives
and hence, using theorem 4.14,
. \ {i пх-п-Л-п-1К„(х)} = -i-nx~n-\-п-гКпп(х).
1 d.V Л 71

i 16 besse'l functions ch. 4
Therefore
from which a common factor of i~2»-2 cancels to give
~{x-"Kn(x)}=-Knn(x).
4.9 INTEGRAL REPRESENTATIONS FOR THE MODIFIED BESSEL
FUNCTIONS
Theorem 4.17
1 /x\n Г1
(i) In(x) = ___-__y j ^ e-*(l - <2)(«-5) dt
(n > —I).
(я > -\, x > 0).
Proof
(i) By equation D.17) we have
/„(*) = i-nJn(ix)
and by theorem 4.7 we have
so that
'•(>t) = '"¦¦ (vsyrbrI"©" Г, <• - '¦)-
(ii) We first show that the integral
P = я" f" e-^f2 - I)" df (я > ~\, x > 0) D.20)
satisfies Bessel's modified equation

§4.9 REPRESENTATIONS FOR MODIFIED BESSEL FUNCTIONS 117
We have
dP Г00 , Г00 <* . ,
— = пхп~х e~xt(t2 - 1)"-J dt + xn —te-x'(<2 — I)"-1 dt
dx Jl J l
and
d2P чч Г00 ч , ,
-— = n(n — \)хп~г t~x4t2 - ])"-' dt
ax* J i
so that
d2P
+ 2m;"-1 j -* e~x'(t2 - I)"
(•00
i2 е-ж((г2 - 1)"-! d*
J l
dP
х- (ж2 + я2)Р
ax2 ax
= xn {n(n - 1) - Inxt + *2*2 -Ь я - xt - x2 - n2} e-xt(t2 - l)nl dt
J l
= xn+1 Г {x(t2 - 1) - 2(я i- J)<} e-^^f2 - I)" dt
J
{xe-xt(t* - 1)"+J - (n + %){t2 - 1)"-*2*е-*"} dt
= -«"+1 f ^-{e-xt(t* - l)»+*}df
J l df
since the integrated part vanishes at the upper limit, t = oo, because of
the factor e~x* (remembering that we are considering here x > 0) and at
the lower limit, t = 1, because of the factor B2 — l)n+* (remembering
that я + \ > 0).
Thus P satisfies Bessel's modified equation and therefore must be of
the form
P = AIn(x) + BKn(x).
We show now that A = 0. We do this by considering the limit as
x—*¦ oo. From the series D.18) for In(x) we see that In(x) consists of a
power series with positive coefficients, and hence IJx) —>¦ oo as я: —>¦ oo.
Consider now P(x). We have, from the definition, P(x) > 0, and we shall
show that P(x) is less than some quantity which tends to zero as x tends
to infinity so that P(x) itself must tend to zero as x tends to infinity.
Since the asymptotic behaviour of an exponential dominates that of a

118 BESSEL FUNCTIONS CH. 4
power, we must have (i2 — l)n~* < e""/2 for x large enough (say for
x>X).
Then for x > X we have
P(x) <x» ["
(•CO
= Xя e~x
[2 Iе
e-xt/2
x Ji
t/2dt
= 2л;"-1 е-ж/2
which —>¦ 0 as x —>¦ oo, again using the fact that the exponential dominates
the power.
Thus we have shown that P(x) —> 0 as ж —> oo, whereas In(x) —> oo.
Hence P(x) cannot contain any multiple of /„(#) and we must have
P{x) = BKn(x).
To determine the constant В we examine the behaviour of both P{x)
and Kn{x) as * —> 0.
For Kn{x) only the lowest power of x is important in this limit, and we
may use the definition
n I_n{x) - /.(*)
4 ' 2 sin ил:
together with
to see that the lowest power of x in IJx) is
Г(и + 1)\2/
and hence in Kn(x) is
я 1 fx\ -"
2 sin итг Г(-я + 1)\2/
Hence as x —> 0, /Сп(лг) —>¦ r=^j r—; (-1 which, on making use
2Л. \\ — tij sinnsz\x/
of the result of theorem 2.12 that
Г(ж)ГA - x) = -
sin nx

§4.9 REPRESENTATIONS FOR MODIFIED BESSEL FUNCTIONS 119
may be written in the form
To study the behaviour of P(x) near x = 0 we make the change of
variable
x
so that dt = - dw.
Also, when t = 1, м = 0 and when t = со, и = со.
Thus, using equation D.20),
P(x) = xn \ e-"-"( — + — ) ' - du
w Jo \x x2/ x
1 Г00 / 2x\n~^
= - e-M e~u 1 + — M2"-xdM.
ж" Jo \ m/
For small values of x we may make the approximations e~x =2= 1 and
A 4 ) ^ 1 to obtain
1 f°°
Р(л:)->— е-"иа»-Чм
47 ж" J о
(by the definition of the gamma function).
Thus for small values of x the result P(x) = BKn(x) reduces to
glvmg 5
But by theorem 2.10 we have
xn
ГBи)
so that
2*»-Т(я)Г(я + I)
в =
2»Г(и Ч-

120 BESSEL FUNCTIONS CH. 4
Hence
giving
*¦<*>=щ)гтт,р<*>
and thus, using the definition of P(x),
4.10 KELVIN'S FUNCTIONS
Consider the differential equation
**;&+*&-(*'** +«*» = <>• D-21>
This is of the form of Bessel's modified equation
with A2 = i&2. Hence, since the general solution of Bessel's modified
equation is
у = АЩх) + BKn(hc),
the general solution of equation D.21) must be
у = Aljpkx) + BKJpkxtf
Also, since In(x) = i~nJn(ix), we may take the independent solutions of
equation D.21) as Jn(i3nkx) and ^„(i1'2^).
Of course, when x is real/„(i3/2a:) and A"n(it/2*) are not necessarily real;
we obtain real functions by the following definitions:
Ьег„ж = Re/n(i3/2a:)
/n(i3/2x) = bern x + i bein x. D.22)
kei« x = Im i~".Kn(i1/2*)
i-"-K:n(il/2a;) = кег„ х + i kein x. D.23)
f Strictly speaking, i* is a two-valued function with values еы/4 and ein/i. Wo
remove this ambiguity by taking the value e"*/4.

§4.11 SPHERICAL BESSEL FUNCTIONS 121
(If n = 0 the notation used is often ber x, etc.)
Thus the general solution of equation D.21) is given by
у = ^i(bern kx + i bein kx) + ^42(кег„ kx + i keire kx). D.24)
4.11 SPHERICAL BESSEL FUNCTIONS
Consider the equation
*2 P +
ax
This is of the form of the equation of theorem 4.12 with
1 -2a=2
. + lx4 + ikV lV + !)}У °- D-25)
* ax
Solving these equations for a, /5, у, п gives
«= ~h ?=&, У = 1, n =/+^
and hence by theorem 4.12 the general solution of the above equation is
given by
у = Ax-iJlH(kx) + Bx-*YlH(kx)
= AJ^kx) + A2yt(kx)
where we have defined the spherical Bessel functions j\(x) and yt{x) by
iM D-26)
and ^j, ^42 are new arbitrary constants related to А, ВЪу At = л/Bк/л)А,
A2 = VBk/n)B.
Spherical Hankel functions may also be defined in a way exactly anal-
analogous to the definition of Hankel functions:
(*) =jt(x) + iyi(x) D.28)
=№ - W(*)- D-29)
In applications it turns out that the spherical Bessel functions of most
interest are those of integral order. We shall show that such spherical

122 BESSEL FUNCTIONS CH. 4
Bessel functions may in fact be written in closed form in terms of ele-
elementary functions. First, however, we prove recurrence relations anal-
analogous to those in theorems 4.8, 4.9 and 4.10 for the Bessel functions.
Theorem 4.18
jx) is any ofjn(x), yn(x), Щ\х) or h%\x), then
(i) ~{х-Чп{х)} = х^Ч^х).
(ii) ^"f()} ЧЛ)
() a) „&) u)
X
(iv) Qx) = \(x) - f.+1(*).
(v) Be + 1)ВД = nf.^*) - in + l)fn+1(x).
(vi) f.-^*) + f«+1(*) = -"-^Ц*).
Proof
We first prove the results for/n(x).
(i) From theorem 4.8 (i) with n replaced by я + \ we have
which, on using definition D.26), becomes
^{«"+УB/яУ;«(
and this simplifies to
(ii) From theorem 4.8 (ii) with n replaced by я -\-\ we have
Again, use of definition D.26) gives

§4.11 SPHERICAL BESSEL FUNCTIONS 123
and hence
-{*-«/„(*)} - - x-%+1{x).
(iii) If we carry out the differentiation on the left-hand side of (i) above,
we obtain
xn+1j"n(x) +{n + l)x%(x)
On dividing throughout by xn+1, this gives
and thus
X
(iv) Carry out the differentiation on the left-hand side of (ii) above, and
we obtain
Cancelling a common factor of x~n then gives
;«(*) - -jn{x) = -y»+
and hence
(v) Multiply result (iii) by n, result (iv) by (n +1) and add, and we
obtain the required result.
(vi) Subtract result (iv) from result (iii), and we obtain the required
result.
The proofs for yn(x), h^ix), and ^2)(#) follow in the same way, since
these functions bear the same relationship to Yn+i(x), H^l^x) and
H^lx(x) that jn(x) does to Jn+i{x) and by theorems 4.9 and 4.10 the
various Bessel functions all satisfy the same recurrence relations.
Theorem 4.19
sin x
0) JO(X) = " ¦-.
X
() ()

124 BESSEL FUNCTIONS CH. A
(iii) /#>(*) = -i -.
ОС
(iv) /#>(*) =i—.
X
Proof
(i) We have, by equation D.26),
AM - VW
(from equation D.6));
but, by the corollary to theorem 2.10, we have
so that we now have
M*j Z ^ IJ r\Br
1)!
(on recognising the infinite series for sin x)
(ii) We have
(by definition D.27))
V \2ж
sin ^л
(by the definition of Yn(x) in theorem 4.2)

§4.11 SPHERICAL BESSEL FUNCTIONS 125
\2x) ? K Jr!r(-Hr + 1)\2
(by the series D.7)).
But again, by the corollary of theorem 2.10, we have
so that
r=0
-
Br)!
1
COS X,
X
(on recognising the infinite series for cos x).
(iii) h$\x) =jo(x) +iyo(x)
(by definition D.28))
1 . Л
= - sin x — i - cos x
X X
= (cos x + i sin x)
ОС
X
(iv) hf{x) =;„(*) - iyo(x)
1 . , Л
= - sin x -j-1- cos я:
X X
= - (cos x — i sin x)
X
X

126 BESSEL FUNCTIONS CH. 4
Theorem 4.20 (Rayleigh's Formulae)
If n is a non-negative integer, then
d
Proof
We give the proof for (i) only, the proofs for the other three results
being similar.
We shall use the method of induction; i.e., we shall assume the result
true for n = N, say, and then prove it true for n = N + 1. Then since
theorem 4.19 guarantees the result true for я = 0, it follows that it must
be true for all positive integral я.
Assuming the result true for n = N implies that
From theorem 4.18 (ii) we have
(using the fact that we are assuming the result to
be true for n = N)
which thus proves the result true for я = N -(- 1.

§4.12 BEHAVIOUR OF THE BESSEL FUNCTIONS 127
Hence, by the remarks above, the result must be true for all positive
integral n.
We may use the results of this theorem to write down explicitly the first
few spherical Bessel functions of integral order. We obtain in this way:
sin ж cos x
h{x) = T* IT1
= [~i ~ -) sm * - ^5 cos *,
= ё - йsin • - (^ - Эcosx' D-30)
cos x sin x
3 1\ 3 .
-(^i - -J cos ж - — sin x,
^*) = -(^ - il) cos *-(•- i
It is to be noted that because of the relationships D.26) and D.27) all
the above information concerning spherical Bessel functions of integral
order provides an equal amount of information concerning Bessel func-
functions of half-odd integral order.
4.12 BEHAVIOUR OF THE BESSEL FUNCTIONS FOR LARGE AND
SMALL VALUES OF THE ARGUMENT
Theorem 4.21
As x —> oo we havef
(ii) Yn{x) ~ (A)° sin \x - (n + i
(iii) Я5»»(*) ~ (~J" exp \i\x ~(n
(iv) m\x) ~ Q-) exp [-i{* - (n + i
f Here we use the symbol ~ rather loosely to mean 'behaves like'. A more
precise definition is: f(.v) ~ r(.y) as x -> a if lim (f(х)/к(х)) 1.

128 BESSEL FUNCTIONS CH. 4
(vii) ./„(*) ~-sin (*-yJ 5
(vin) yn(x) cos (ж - — I;
(ix) h$Xx) ~ - -exp [i{* -
(x) A<f>(*) ~ I exp [-i{* - (rm/2)}].
X
Proof
We shall first prove result (vi) and then deduce the remainder of the
results from it.
(vi) By theorem 4.17 (ii) we have
We now make the change of variable
x
so that dt = - 4m.
x
Also, when t — 1, м = 0 and when t = oo, м = oo.
Thus ад^^^};.-^;^)
»-1
= Vji (?) lt е- Г e"V-»fl + ~Y * du.
1 (n H- y)\w ж J о \ 2ж/
For large values of ж we have м/2ж small, so that we may take as an
approximation
1.

§4.12 BEHAVIOUR OF THE BESSEL FUNCTIONS 129
Then
(by definition B.1))
\2x) ¦
(iii) From theorem 4.14 we have
Kn(x) =^i^Hn^(ix)
where, if n is non-integral, we take the value of the many-valued function
in+1 given by e(fal/2)(n+ *\
Then
P{ix) = I exp {(-к/2)(я 4 1)} ВД
and hence, on replacing x by — ix,
= - exp {(-ш/2)(я + 1)} /C(-i
Thus Hi?'(*) ~ ^ exp {(-m/2)(n + 1)} (~^j*
^^ exp ft-
= f-^V е'л/4 exp {(-i7t/2)(« + 1)} c«*
(on writing —i = e~W2)
(-|)8 exp [i{« - (я
(iv) Since Я<,2>(л) =/„(*) - iYn(x)
and 7/<1»(x)=yn(x)+iFnM
we have H^(x) = {H?\x)}*
— ) exp \y\x -
exp [ - i{(w

130 BESSEL FUNCTIONS CH. 4
(i) We have
/„(*) = Re #?>(*)
~ Re (~K exp [i{* - (» + i)(^
(ii) Similarly
Yn(*) = I
~ Im r^V exp [i{* - (и + J
(v) By definition we have In(x) = i~nJn(vc) so that, by part (i) above,
we have
/2 V
i-»-*(—1 i[exp {-х-(п+^)(л/2)[} +ехр {x+(n + $(n/2)i
=[~n~lvkx)exp <*+(и+«Gr/2«
(keeping only the dominant term for ж —> oo)
j—»— ^
(writing exp (in/2) = i)
1
e*.
(vii) *.(*) = (?J/.+,(*)
(by definition D.26))
_ . . . cos-sa: —(n + l,_
.2*/ \яж/ • I v '2.
(by result (i) above)

§4.12 BEHAVIOUR OF THE BESSEL FUNCTIONS 131
1 . / яи\
= - sin I * I.
* V 2/
(viii)
(by definition D.27))
n
(by result (ii) above)
1 . / rm n\
= - sin I x —— — - I
x \ 2 2/
1 . (n ( nn\\
= sins- —1л; —— I >•
x U \ 2/J
1 / яя\
= cos [x .
* \ 2/
(ix) ^>(Ж) =;.(*) + iyn(x)
i exp [1{* - (яя/2)}].
(x) *<?>(*) = {*<?>(*)}•
Theorem 4.22
As x —> 0 то
Г(-,', 1H"

132 BESSEL FUNCTIONS CH. 4
-In* (я=0);
(iii) jjx) ~ ,-, (я integral) ;f
(iv) ^„(ж) ~ ПхП+1 (и integral).
Proof ^
(i) We consider the series D.6) for Jn{x). As * -> 0 only the lowest
power of x will be important. But this lowest power of x is just given by
the term with r = 0, so that we have
(ii) When n is not an integer we have
YJX) =
sin игт
and we pick out the lowest power of x occurring in this expression. This
will be contained 1п/_п(л;) and will be given by the result of (i) above, so
that we obtain
sin югЦ-п +1)\2,
n \x
using the result of theorem B.12) that
Г(н)ГA - n) = -/
sinmt
When n is an integer we pick out the dominant term from the series for
Yn(x) given by theorem 4.3. Remembering that a power of x dominates a
logarithm, we see that
Yn(x) ~ - In * (и = 0)
•)• By n\\ (n double factorial) we mean
5.3.1 if я is odd

§4.13 GRAPHS OF THE BESSEL FUNCTIONS 133
and Yn(x) ~ - \{n - 1)|(|)
(iii) We have from definition D.26)
V \2
(using repeatedly the fact that Г(и + 1) = яГ(и))
Bя + 1)Bя- 1). . .ЗЛ.д/гт
(where we have made use of the result of
theorem 2.6 that Щ) = у/л)
xn
Bя +1)!!'
(iv) yJL*)=
1 Bя-1)Bя-3)...3.1.-у/д
\/n xn+1
~ xn + l
4.13 GRAPHS OF THE BESSEL FUNCTIONS
We give here some graphs of various Bessel functions; they bring out
rather clearly the behaviour for large and small x discussed in the previous
section.

134
i-o
0-5
-0-5
-1-0
BESSEL FUNCTIONS
Fm. 4.1 /„(*), я = 0,1,2
cir. 4
-0-5 -
Fm. 4.2 Yn(x), n = 0, 1, 2

§4.13
GRAPHS OF THE BESSEL FUNCTIONS
135
I 2 3 X
Fig. 4.3 /„(ж), n = 0,1, 2
I 2 3
Fig. 4.4 Kn(x), n = 0, 1, 2

136
BESSEL FUNCTIONS
CH. 4
0-4 -
Fig. 4.5 Mx), n = 0,1, 2
Fie. 4.6 yn(x), n = 0,1, 2

§4.14
ORTHONORMALITY OF THE BESSEL FUNCTIONS
У (Graphs oscillate for large *)
у *ber x
137
Fig. 4.7 ber x, bei x
Fig. 4.8 ker x, kei x
4.14 ORTHONORMALITY OF THE BESSEL FUNCTIONS; BESSEL
SERIES
Theorem 4.23
Г а2
xjd? jX)JJt; x) dx — - •'/„ (?fl)JEv
Jo J 2 ™ ' >}
if St ami S, are roots of the equation ./„(f«) 0.

138 bessel functions ch. 4
Proof
We first show that if |s and |3- are distinct roots of the equation Jn(f a) — 0
then
Jn{?iX) &ndjn(!-3x), being Bessel functions, must satisfy the following
equations
and
which may be written in the form
and ^1^(^I + (||Ж2 _ „2)MjX) = o. D.33)
Multiplying equation D.32) by A /x)JJ?fX\ equation D.33) by
A /x)Jn(?{x) and subtracting, we obtain
Applying the result u(dv/dx) ~ (d/dx)(uv) — ^(dM/da1) to both first and
second terms gives
(| fWJLidMfc) = о
and thus
(if - ifkMf**Ш^) = о.

§4.14 ORTHONORMALITY OF THE BESSEL FUNCTIONS 139
Integrating from 0 to a, we obtain
+ (If - ф Г xJn^x)Jn(^x) dx = 0.
J 0
The first term will now vanish at the upper limit since
/«(?<«) =/«&«) =o,
and at the lower limit because it contains a factor x.
Accordingly (|? - |?) Р*Л
J 0
= 0
r
and hence х]п{^гх)]п{^х) dx = 0
J
х]п{^гх)]п{^х)
J о
provided |, ^ f,-.
To complete the proof we require to prove that
if MSa) = 0.
Now, if we denote Jn(?x) by z we have
x2z' + лг' + (I2*2 - я2)г = О
which, on multiplication by 2#', becomes
2x2z"z' + 2xz'2 + 2(Px2 - n2)zz' = 0
and this is readily seen to be equivalent to
— {x2z'2 - n2z2 + Px2z2} - 2?2xz2 = 0.
d»
Integrating this equation from 0 to a gives
[*»*'• - я2г2 + iVz2]a0 - 2|2 f"*»»d* = 0
J о
which, on replacing z byjn(?x), becomes
- 2I2 |%{/„(|х)}2 cb = 0
and hence, using the facts that Jn(?a) --- 0 and nJn@) -- 0,
г{Л(?г)}Ыг О.

140 BESSEL FUNCTIONS CH. 4
Hence
But, by theorem 4.8(iv), we have
which may be rewritten in the form
so that
Га кШЫJ dx =
1
(again using the fact that J«(?a) = 0)
{J(l)}2
Theorem 4.24
Ifi(x) is defined in the region 0 < x < a awrf caw ie expanded in the form
ciJ«{^ix) where the |4 are г/ге roo^ o/ f/ге equation Jn(!-a) = 0,
Г x?(x)Jn(te)
J 0
Proof
We have
CO
i
so that xi{x)JJ??c) =

§ 4.15 INTEGRALS INVOLVING BESSEL FUNCTIONS 141
and hence
Г xi(x)Jn(^x) d* = У ct ГlxU
Jo (-[ J о
t UZ#)Ute) dx
([ J о
(by theorem 4.23)
Thus
2
о
which completes the proof.
oa
The series/^ сг/п(|,ж), with ct as given above, is called the Bessel series
for i(x). The conditions under which i(x) may be expanded in this way are
given in the next theorem, which we quote without proof.
Theorem 4.25
ra
If i(x) is defined for 0 < x < a and (л/х){(х) dx is finite, then
J о
(with Ci as defined above) is convergent and has sum i(x) if i(x) is continuous
at x and i{f(x+) + Kx~~)} tf^{x) is discontinuous at x.
4.15 INTEGRALS INVOLVING BESSEL FUNCTIONS
Theorem 4.26
2(тс/2\п-т Г1
Ш = Цп^пТ) J оA " t2)n-m-4m+1M*t) 61
(n > m> —1).
Proof
Consider the integral
I [ A /2)"-m-1/mflJm(x0d/
J 0

142 BESSEL FUNCTIONS CH. 4
and substitute the series D.6) for/m to obtain
f1
(on making the substitution и = ?2).
But A — u)n~m-1um+Tdu
Jo
= B(n - m, m + r + 1)
(by the definition of the beta function, pro-
provided n — m > 0 and m + r ¦+ 1 > 0. This
last condition is equivalent to m > — 1,
since r takes on values between zero and
infinity.)
_ Г(и - m)Y{m + r + 1)
~ Г(п + r + 1) •
(by theorem 2.7).
Hence / = ?Г(я —m)?(—\)r - ~, ^———
— ^ (n - m)y-J ? (- ) Г|Г(И +r + 1)
~J»(*).
Thus
which is the required result.
Theorem 4.27
J 0

§ 4.15 integrals involving bessel functions 143
Proof
From theorem 4.6 we have
1 [n
Jo(x) = - c°s {x sin ф) d<f>.
71 J 0
Hence
fco Л со J (-я
e~axJo(bx) dx = e""* - cos (bx sin ф) йф dx
Jo Jo л J о
= - e""* Mexp (ifcc sin ф) + exp (—iia; sin ф)} dx dф
л J о L J о J
(interchanging the order of integration)
_ 1 Г" Гехр {-(a - \Ъ$тф)х) exp {-{a + ib sin ^жЛ c°d .
2л J о L — a+i6sin<? — a— iisin^ Jo
1 p/ 1 1 \
2л J о la — ib sin </> a + ii sin ^/
= g Г" l d,
л J о a2 + b2 sin2 ^ ^'
But this last integral may be evaluated by elementary means (e.g., by
the substitution и = cot ф) to give
Theorem 4.28
JJbx) dx = -- (if n is a non-negative integer).
Jo b
Proof
We first prove the result for n = 0 and n = 1, and then show that if the
result is true for я = N, it is also true for я = N + 2, thus proving the
result for all non-negative integral n.
For n — 0 we take the limit as a —>• 0 of the result of theorem 4.27,
obtaining
1
1X b

144 BESSEL FUNCTIONS CH. 4
For n — 1 we make use of theorem 4.8(ii), which says that
so that, by taking n = 0, we have
and replacing x by bx gives
с
which is equivalent to
±Ш = -/,(*),
^Jo(b) = -/,(**).
Hence Ji(bx) dx = — j
Jo о
__ 1
~v
since /0( со) = 0 and /0@) = 1, results which are, implied by theorems
4.21(i) and 4.22(i).
If we now use theorem 4.8(v) and integrate from 0 to oo, we obtain
1 Г00
2 J о
Remembering that n > 0 so that Jn(co) =/„@) = 0, we have
0 = if{/-i(*
/Jo
and thus
f7() [W
o Jo
Replacing n — 1 by ЛГ and л by йд; gives the result
Л OD Г CO
JN+i(bx) dx =
Jo Jo
f00 1 f00
Thus if JiAbx) dx — -, we also have
Jo 6 Jo
the proof is complete.
1
x) dx = -, and hence
о

§4.15 INTEGRALS INVOLVING BESSEL FUNCTIONS 145
Theorem 4.29
Г 2n+1T(n + ^) abn
(a > 0).
Proof
(i) From equation D.6) we have
1 Л2г+в
NT* 1 /ЙлЛ2
so that
e—d«
Jo
But ["e4"»"*!!»
Jo
0 ~ a2r+2n+lw
(writing t = ax)
1
+ 2я + 1),
(by definition B.1) of the gamma function),
so that
1 2Г(
> r\ Г(>-|-я) \2
(using the result of theorem 2.2 that Г(я | 1) -- «Г(я)).

146 BESSEL FUNCTIONS CH. 4
But by theorem 2.10 we have
so that
Г Jn(bx)x» c~a* dx
J о
2r+" 1
r=O
Г(г_+ и _+ -i)
a2r+2n+l yj
Г(г + н
r\
1 / *^\-"
2i
r + l)/i'V
W
+
r!
(by the binomial theorem)
nr
Г(я + ^) (" + i)
^b T(n+i)r\\ay'
on repeated use of the result xT(x) = T(x + 1) (theorem 2.2).
Hence |л(И*. е- d* = ^Г(
2иГ(и
(ii) This result follows immediately from result (i) if we differentiate
both sides with respect to a.

§4.15 INTEGRALS INVOLVING BESSEL FUNCTIONS 147
Theorem 4.30
(i) Jo Jn(bx)x^ exp (-a*2) d* =
(ii) ^Jn(bx)x"+* exp (-«*») d* =
(a > 0).
Proof
The proof is very similar to the proof of theorem 4.29; so we shall not
give it here.
Theorem 4.31
r f 0 (b>a)
(i) sin axjo(bx) dx = < 1
J ° Ц*2 - *2) {Ь К a)-
(ii) Pcos axjjbx) dx = \ V{b2 - a2) ^ > "^
J° I 0 (i<a).
Proof
We shall obtain these results by replacing the a occurring in theorem
4.27 by \a, although it should be noted that strictly speaking this is not
allowed, since the proof of theorem 4.27 depended on having the real part
of a positive. It is possible to give a procedure justifying taking the real
part of a zero, but we shall not describe it here.
Thus we have
-a2)
so that
cos axjo(bx) dx - i sin ax J0(bx) dx = —— -.
Jo Jo v\y — a )
lvquating imaginary parts of both sides gives
sin axjn(bx) dx 0 if a - ; b.

148 BESSEL FUNCTIONS CH. 4
Equating real parts of both sides gives
J ^ cos ax J0(bx) dx = _ if b > a
cos ax J0(bx) dx — 0 if b < a.
J о
4.16 EXAMPLES
Example 1
Use the generating function to prove that
We have from theorem 4.5 that
exp ji(* + y){t - jjj =
П— — CO
so that Jn(x + y) is the coefficient of tn in the expansion
of
But
exp |i(* + y)(t - -j| = exp
* + y)(t - -j| =
op op
= 2 J&y- 2
= 2 JrWs(y)r+s.
r,s— — oo
For a particular value of r we obtain f" by taking s = n — r, so that, for
this value of r, we obtain the coefficient of tn as Jr(x)Jn-r(y); hence the
total coefficient of tn is obtained by summing over all allowed values of r.
00
Accordingly the coefficient of tn is equal to 2, Jr(x)Jn_r(y) and thus

§4.16 EXAMPLES 149
Example 2
Г 00 T /~\ 1
Show that J-^^ dx = -.
Jo ж п
From theorem 4.28 we have
f°°
Jn(x) dx = 1.
J о
But from theorem 4.8(vi) we have
2ra
Г °° I (x) Г °° С °°
so that 2и ^^ dx - /„..(ж) dx + /n+l(x) dx
J 0 л •/ 0 JO
= 1+1,
and hence dx = -
J о ж n
Example 3
If Si ere the solutions of the equation /o(l) = 0,
oo T / {• \
show that ^ jfj Дуг2 = - J In ж @ < ж < 1).
We use theorem 4.25 to write
—iln x =
with
2 x( -1 In x)J0(?iX) dx
Jo
ж In ж J0{?ix)
Jo
'I'o evaluate the integral appearing in the numerator we use the series
D.6) for y0:

150 BESSEL FUNCTIONS CH. 4
(and we note that since/о^Л = 0 we shall have
= 0). D.34)
We shall also require the integral xn In x dx.
J о
Integrating by parts gives
fl 1 Л Г l +i 1 I' f1 1 „+1 l A
xn\nxax = \- тж"+11пдг — -xn+1.-dx
Jo \_n + 1 JoJow + l x
Jo« + 1
_ 1
Thus f дг1пдг/0(|4л;)сЬ
Jo
,x dx
+ 1
D.35)
1Y * (*')"! l \
r40 (M2/ I Br+2J/
(using equation D.35))
(using equation D.34)).
Thus we have
1

§4.16 EXAMPLES 151
and hence
— i \n.X = У у—^г
Z—/ <? • /
Example 4
SAo» that Jn{x) Y'n{x) - J'Jx) Yn(x) = -
X
where A is a constant, and by considering the behaviour for small x show that
A = 2/n.
We have
d
dx^nYn ~JnYn>
_ т у" л- T' Y' T" Y — V Y'
— T Y" — T"Y
(all functions having argument x);
but bothjTn(x) and Yn{x) satisfy the equation
x2y" + xy' + (x2 - n2)y = 0
so that
= U-xY'n - (x2 - я») Yn} - {-*/; - («» - n*)Jn}Yn
= ~~x{Jn.Yn —JnYn}-
Hence, writing JnY'n — J'nYn = z, we have
dz z
dx x
or
dz dx
-+- =0.
z x
When integrated this gives
In z + In x = constant
which is equivalent to
In zx = constant
and hence ~.v constant A, say.

152 BESSEL FUNCTIONS CH. 4
Thus z = — so that, by the definition of z,
X
j у' — T'Y — -
X
For small values of x, we have, by theorem 4.22,
/„(*)-1, Y,(*)~^ln*
so that, considering the relationship for n = 0, we have
Hence
giving
and thus
2
лх
A
2 1
Л X
2
лх'
А
2
л
-о.2
л
In х
Example 5
We shall show this by first proving that
and then integrating this relationship from 0 to x.
We have
)j

§4.16 EXAMPLES 153
t2
= t(Jn ~ Jn-ljn+l) + 2"B/n/n ~Jn-lJn+l ~~ Jn-lJn+V
(it being understood that the argument of all Bessel
functions here is t)
= КЛ -Jn-Jnn) +
* Jn-l Jnljn+l ~ Jn-l\Jn " Jn+llj
(using theorem 4.8(iii), (iv) and (v))
t2 2
n Jn-lJn+l) ~Ь л •"IJn-i/n+l
Integrating from 0 to x now gives
= №
Example 6
C/se the generating function to show that Jn(—x)
We have, from theorem 4.5,
so that J /»(-*)«" =
QO _
7(.-_ - 00
CO
- 2 (-i)n'"/
quatintf the coefficients of /" on both sides gives

154 BESSEL FUNCTIONS CH. 4
PROBLEMS
A) Show that
~k/»D/«+1(*)} = х{Д{х) - Л+1(*)}.
Гя/2 sin л;
B) Show that (i) J0(x cos в) cos в dd = ;
Jo яг
(ii) их cos в) de = ^-C0SA
JO X
C) Show that J'n{x)J_n{x) — Jn{x)J'_n(x) = A/x where Л is a constant;
by considering the behaviour for large values of x, show that
A = B sin пл)/л.
D) Find the solution of the differential equation
which is zero at л; =0.
CO
xr^ Xn
E) Show that 2 -xJn{a) =/o{V(«2 - 2a*)}.
F) Use the generating function to prove that
and deduce that \J0(x) \ < 1, \Jn(x) \ < 1/V2 (я = 1, 2, 3 . . .)•
G) show ла, j;-&^a d,=^r_f)«. > -}.
(8) Prove that 2 W*»I" = «P (f,(* - l)} 2 **'V.W
n= — oo n=—oo
and deduce that /„(я) =
(9) Show that exp S^t + 7)} =
A0) Show that /„(дг), YJx), KJx) and /„(x) all satisfy the differential
equation
2 d3j 2и2 + 1 d2y In* + 1 dy /и4 - 4и2 Л _
* + 1 dy /и4
jc5" dbc + \
if2 Sc2 H jc5 dbc + \ ж4

PROBLEMS 155
00
A1) Expand x2 in a series of the form 2. Crjotyrx) valid for the region
r = l
0 < x < a, where Ar are the roots of the equation /0(Aa) = 0.
A2) Use the differential equation satisfied by jn(x) to show that
{n{n + 1) - m(m + \)}jn{x)jjx) = — {x%(x)jm{x) - jn{x)j'm(x)}.
Hence show that
sin {(n - m)n/2}
dx =
if тгг ^ п, and use L'HopitaPs rule to deduce that
Г 00
{jrix)Y dx = ———
Determine also the values of jm(x)jn(x) dx (m ^ n) and
J — 00
Г {jn{x)Ydx.
J — 00
A3) Show that
(i) bern x =
A1 I ПР1 V ——
iTo Л!(и + Л)!
A4) Prove that
(i) fberfdf = xbei'x;
J о
rx
(ii) f bei t dt = —x ber' ж.
J о
A5) Show that, for large values of x,
(ii) bei .v ~
v'

5
HERMITE POLYNOMIALS
5.1 HERMITE'S EQUATION AND ITS SOLUTION
Hermite's equation is given by
g g 0, E.1)
and in applications we are required to find solutions which are finite for all
finite values of x and are such that exp (\x2) y(x) —> 0 as x —*¦ oo.
The methods of Chapter 1 are applicable, and if we try for a solution of
the form *-
z(x, s) =
we find that the indicial equation has roots * = 0 and s = 1, with ax in-
indeterminate when * = 0, so that * = 0 gives the two independent series
solutions. The recurrence relation for the coefficients then has the form
W = (r + l)(r+2)' E-2)
This recurrence relation may now be used to construct the two series
solutions. However, from these series it can be shown that both solutions
behave like exp (x2) for large values of x. Thus they cannot satisfy the
requirement that exp (%x2) y(x) —> 0 as x —> oo; this can only be satisfied
if the series terminate. From equation E.2) we see that this will be so if,
and only if, я is a non-negative integer, for then ап+г and all subsequent
coefficients in the corresponding series will vanish. We shall now write the

§ 5.2 GENERATING FUNCTION 157
series in descending powers of x. Using equation E.2), rewritten in the
form
2(я - г) г+"
we obtain the series
f п(я - 1) я(и - 1)(и - 2)(я - 3) .
"I 2.2 22.2.4 ' ' '
, , 1Vr4»-l).-¦(«-2г+ !)_.,
У 1у"(»-!)•••(»-2т+ 1)
2'.2.4...2r ^
Г if я is even
я'
[W= \i(,,-l) if» ie odd)
The standard solution is obtained by making the choice of 2" for the
arbitrary constant an; the solution is then denoted by Hn(x) and called the
Hermite polynomial of order n:
ВД = 1 (-1)' -i
5.2 GENERATING FUNCTION
Theorem 5.1
Proof
We wish to pick out the coefficient of tn in the power series expansion
of exp Btx - t2).
Now,
r=0 s-0
->ы"'=--

158 HERMITE POLYNOMIALS CH. 5
For a fixed value of * we obtain tn by taking r + 2s = n, i.e., r =n — 2s,
so that for this value of s the coefficient of tn is just given by
, 1V B*)"~2S
^ ' (fi-2*)!*!"
The total coefficient of /" is obtained by summing over all allowed
values of s, and, since r = n — 2s, this implies that we must have
я — 2s > 0, i.e., * < \n. Thus, if n is even, * goes from 0 to \n, while if n
is odd, s goes from 0 to \{n — 1); that is, in all cases, s goes from 0 to [\n]
with [|k] denned as above.
Thus we have:
№.] j
coefficient of <n = > ( — \)s ^тт-,Bж)"-2<>
^0 (я-2*)Ыч
= „>.(*>.
(by definition E.3)).
5.3 OTHER EXPRESSIONS FOR THE HERMITE POLYNOMIALS
Theorem 5.2
Proof
By use of the generating function of theorem 5.1 and Taylor's theorem,
which states that
_ y /d"F^ Г
we have
ldtn
But

§5.3 OTHER EXPRESSIONS FOR HERMITE POLYNOMIALS 159
so that
and we have
Theorem 5.3
Proof
We have
H (x\ — (~\\n егТ-—- е
dxn
da:"
/1 d
\2 dAr
2d*
dV
/1 d \
and hence -~) e2te = Г е2'*
e2te = Г е2
Thus K4^)p^J
„=o «i"\2d*;
Expanding both sides in powers of t, we have
exp -Tj- > —j—= > Я„(ж)-
(using the generating function property of theorem 5.1).
t The exponential of an operator is defined by its powerseries expansion. Thus, for ex-
/d\ у1/A\" v1 d" (Л V x d"f
ample, cxpf , J 2, „i( Ax) ' Zn\ d.v-sothet cxpL J fW Z «! d,-

160 HERMITE POLYNOMIALS CH. 5
Equating the coefficients of tn on both sides now gives
and hence
ВД= 2-{exp (-!
5.4 EXPLICIT EXPRESSIONS FOR, AND SPECIAL VALUES OF, THE
HERMITE POLYNOMIALS
We may use either the definition E.3), or theorem 5.2 or theorem 5.3 to
write down an explicit expression for the Hermite polynomial of any order
we choose. For the first few orders we obtain
H0(x) = 1
jj /x\ _ 2x
H2{x) = 4*2 - 2
Ha(x) = 8л;3 ~ 12л
Щх) = 16** - 48л;2 + 12
Нь(х) = Ъ2х* - 160л;3 + 120*.
Theorem 5.4
н2п@) = (-1)ге ^г; я2п+1@) = о.
Proof
From the generating function of theorem 5.1 with л; = О, we have
e"'1 =
which, on expanding the left-hand side in powers of t, reduces to
Equating coefficients of corresponding powers of t on both sides gives
Я»@) = О if я is odd
(which is equivalent to Я2и+1@) = 0 with n a non-negative integer) and
(-1)» _. — HJ&) j^-r-. (with n a non-negative integer)
n! (ля)!

§5.5 PROPERTIES OF HERMITE POLYNOMIALS 161
which yields
5.5 ORTHOGONALITY PROPERTIES OF THE HERMITE
POLYNOMIALS
Theorem 5.5
Proof
f 00
e-*eHn(x)Hm(x) dx = l«n!(Vn)dnm.
J — 00
We have e + = > Hn(x)~-
(-00
so that e~x' Hn(x)Hm(x) dx is the coefficient of (Лт)/(я!»г!) in
J — 00
Г00
the expansion of e"** e~*s+2fa: е""8* + 2ет dx.
J — 00
But
J_ooe C e + dx
fi " " I PVf^ < V** I , /(Q I , /|V> Л V
С I Сли i Л \^ Aj\ о j^ L 1Л t UA
J — 00
Г 00
= e~*a~-sa exp [~{x - (j + 0}2 + (* + 02] dx'
J — 00
= e2s( exp [ — {x — (s + ?)}2] dx
J -00
= e2"' [ exp(-M2)dw
J — 00
(changing the variable of integration to и — x — (s + t))
(by the corollary to theorem 2.6)
2 (><:;'"

162 HERMITE POLYNOMIALS CH. 5
Hence the coefficient of (insm)/(n\m\) is zero if m ^я and is (л/лJпп\
when m = n.
Г °° ( 0 if w, ^*? n
Hence e-*'Hn(x)Hm(x)dx = \
or, making use of the Kronecker delta,
Г e~*! Hn{x)Hm(x) dx = (V^J"n!<5mn.
J
5.6 RELATIONS BETWEEN HERMITE POLYNOMIALS AND THEIR
DERIVATIVES; RECURRENCE RELATIONS
Theorem 5.6
(i) H'n{x) = inH^x) (n > 1); Я;(Ж) = О.
(ii) Hn+1(x) = 2хЩх) - 2nHn^{x) (n > 1); Нг(х) = 2^Я0(ж).
Proof
(i) If we differentiate both sides of the generating function relationship
with respect to x, we obtain
= It exp Btx - t2)
=2* 2 вд-,
m = 0
м0
Equating coefficients of tn gives for n = 0
ВД = 0
and, for и > 1,
H'n(x)
=
и! (я - 1)!
which reduces to

§ 5.7 WEBER-HERMITE FUNCTIONS 163
(ii) Differentiating both sides of the generating function relationship
with respect to t gives
and thus, performing the differentiations,
B* - 2t) exp Btx - *») = ^
n=0
We now note that the term with я = 0 does not contribute to the
summation on the right-hand side (we remember that 0! = 1) and thus,
on use of theorem 5.1, the above equation becomes
„i""V*>-
which is equivalent to
2я V —HJx) -2 У - #n(x) = 7 —-— HJx)
n=0 ' n=0 ' n=l ^ ^'
which may be written in the form
2tn -^ tn v tn
jj (^л 2 у JJ (x) = > H (x\
n! ¦*—' f и — 1)! n ^—' я!
n=0 n=l v ' n=0
Thus, equating coefficients of tn for я > 1, we have
which on multiplying throughout by n! becomes
2xHn(x) - 2nHn_x{x) = Hn+1(x).
Similarly, equating coefficients of t° gives
2xH0(x) = H^x).
5.7 WEBER-HERMITE FUNCTIONS
An equation closely related to Hermite's equation is
-^ + (A - x2)y = 0. E.4)
If we make the substitution
у ;с 'Ч-

164 HERMITE POLYNOMIALS CH. 5
we find that the above equation reduces to
g2^-MA-!), = „. E.5)
This is Hermite's equation of order n with 2я = A — 1. If, as is usually
the case in applications, we are looking for solutions of equation E.4)
which are finite for all values of x, we must have a solution of equation E.5)
which does not tend to infinity any faster than exp (x2/2) as x tends to
infinity, and by the discussion of Section 5.1 this implies that \{X — 1)
must be integral. With n = \(X — 1) the solution of equation E.4) is then
given by
ВД = e~*V2 Hn(x).
fn(x) is called the Weber-Hermite function of order n.
5.8 EXAMPLES
Example 1
Prove that, if m < n,
A
From the generating function of theorem 5.1 we have that
/d»
\dxm/
is the coefficient of tn/n! in the expansion of
Am
But — exp Btx - t*) = Bt)m exp Btx - t*)
00
2t
n=0
1
= 2"
~ x^ 1
И (v\ir
setting r = m + n.
The coefficient of tn is therefore
От If (v\
(n — m)\

§ 5.8 EXAMPLES 165
and hence the coefficient of tn/n\ is
(n -и)!1
which proves the required result.
Example 2
Evaluate x e~*! Hn{x)Hm{x) dx.
J — 00
We have, from theorem 5.6(ii)
xHn(x) = пНп_у{х) -(
so that
P x e~*'Hn(x)Hm(x) dx
J — 00
= P e~
J — 00
,Hn-m(x),
dx
(by theorem 5.5)
= (V^"-^!^-!,™ + (
Example 3
Show tha
From equation E.3) we have
Show that Pn(x) = t-^v-. Г tn e"'! Я«(^) dL
y-\/7l)n\ J 0
so that
2 Г00
(л/я)я! J°
О f CO I
= —\ tne-'' У <-\Y '1L 2п-2гхп~2Чп-2г dt
(лУпЫ Jo /ii..
h (V«)r!(n-2r)! Jo
(by theorem 2.4)

166 HERMITE POLYNOMIALS CH. 5
Ш 2"-2r(-l)Txn-*r Bn-2r)\ /_
а./тг\г\(*1 — OvW ?2n-2r/« r\|
•y 71 Jf Iy7l — ?Jfy. ?* \n — Ту.
(by the corollary to theorem 2.10)
, Bn-2r)l
r=u k ; 2"r!(n - 2r)\(n - r)!
(by equation C.17)).
PROBLEMS
A) Show that f °° x2 e~xt {Hn(x)}* dx = {л/пJ»п\(п + J).
J —00
B) If f (x) is a polynomial of degree »я, show that f (x) may be written in
the form
f (*) = 2 сгЩх)
r=0
1 Г" _ •
where cr = e * f (x)Hr(x) dx.
i.Y\\7l J -oo
(-oo _
Deduce that e * f (x)Hn(x) dx = 0 if f(x) is a polynomial of de-
J —00
gree less than n.
9 г oo
C) Show that e~*' = -— e"'1 cos 2xt dt.
л/л J о
By differentiating this result 2я times with respect to x, show that
O2/n+lf 1\n g*' Г °°
Я/у\ — * L I #»~*' /2n pne 9r/ H/
Of^liA I ^~" I t ft LU9 4лЛ1г \XLj
y/тс J о
and obtain a similar expression for H2n+1(x).
Deduce that HJx) = ——-.
V л J -
D) Use the result of problem 3 to show that
= A - t*)-1'2
е-я+яй
n=0
and deduce that
-Г- = (l — Pj-1" exp
n='o *ni

PROBLEMS 167
E) Use the result of problem 4 to show that
(i) J^ {Hn(x)}* e-2*' dx = 2»-*Г(п + *);
(•¦ /1 - «»V> e--
~ J U + Ч lTl
(Ц) 2-я1 J -
(
1 + x* ~ J -oo U +
F) Prove that
(ii) 2xWn{x) - inW^x) = Wn+1(x);
(iii) W'Jx) =
G) Evaluate
(i) Г" fm(x)fn(x)dx;
J — oo
(ii) P ^(x)KWdx.
J —00

6
LAGUERRE POLYNOMIALS
6.1 LAGUERRE'S EQUATION AND ITS SOLUTION
Laguerre's equation is
and in applications we usually require a solution which is finite for all
finite values of x and which tends to infinity no faster than ex/2 as x tends
to infinity.
The methods of Chapter 1 apply, and if we look for a solution of the
form z(x, s) = 2. ar xs+r we obtain an indicial equation with double root
s=0, and the recurrence relation
= (s + r - n)
The two independent solutions are then given by
<*,0) and — .
L osjs=o
The second of these we know from Chapter 1 contains a term of the form
In x, and so is infinite when x = 0. Since we require a solution finite for
all finite x we can only obtain this from z(x, 0). In this case the recurrence
relation takes the form

§ 6.2 GENERATING FUNCTION 169
However, the infinite series obtained from this relation may be shown
to behave like e* for large values of x, so that, from our remarks above, it
does not behave well enough as x tends to infinity. The way round this
difficulty is to make the series terminate, and from equation F.2) we see
that this happens if я is a positive integer. For, in this case, an ^0 but
an+1 and all subsequent coefficients will vanish. In such a case the relation
F.2) is best written in the form
(n-r)
ar+i - -«> + 1J
and we obtain the solution
1 v ' (HJ
(the highest power of x being xn)
.. n(n — 1)...(»- r -f 1)
We define the standard solution as that for which a0 = 1; we shall call it
the Laguerre polynomial of order n, and denote it by Ln{x):
6.2 GENERATING FUNCTION
Theorem 6.1
exp {—xt/{\ -1)} _
* ' n = 0
Proof
We have
1 x^ 1 / xt V
1 1
cxp { #'/A — 0} ¦"¦
A /) A t
V -> С 1 У Yr/r
^ r! ' A /)r"

170 LAGUERRE POLYNOMIALS CH. 6
= 1 + (r + 1)* + {- ^U(Y^ *
2! ' 3!
(by the binomial theorem)
QO /
8=0
so that we now have
^- exp {-*/(l - f)} =
For a fixed value of r the coefficient of tn in this expansion is obtained by
taking r -\- s = n, i.e., s =n — r. Thus, for this value of r, the coefficient of
V is given by
( ^(г!)«(я-г)!*-
The total coefficient of f is obtained by summing over all allowed values of
r. Since s = n — r, and we require $ > 0, we must have r < и. Hence the
total coefficient of t" is given by
(by equation F.3)),
which proves the required result.
6.3 ALTERNATIVE EXPRESSION FOR THE LAGUERRE
POLYNOMIALS
Theorem 6.2
ex d"
LJx) = — — (xn e~x).
Proof
Leibniz's theorem for the nth derivative of a product states that
d" . ^ n\
7Zk(uv) =

§6.4 EXPLICIT EXPRESSIONS FOR LAGUERRE POLYNOMIALS 171
so that we have
e« dn, ч e'v n\ dn~r dr
и! dxn я! * (я — r)\r\ dxn-r dxr
But
J?> ?-,
and hence we obtain
еж dre, . e*
\r\ r\
= Ln(x)
(by definition F.3)).
6.4 EXPLIOT EXPRESSIONS FOR, AND SPECIAL VALUES OF, THE
LAGUERRE POLYNOMIALS
We have an explicit series for Ln(x) given by equation F.3). For the first
few Laguerre polynomials this gives
L0(x) = 1
Lj(x) = -x + 1
ВД = ^;(*2 - 4* + 2)
ВД = lf-x* + 9x> - 18* + 6)
Lt(x) = |y(x4 - 16л;3 + 72x2 - %x + 24).
Theorem 6.3
(i) M<>) l.

172 laguerre polynomials ch. 6
Proof
(i) Set x — 0 in the generating function of theorem 6.1 and we obtain
00 -
* 4 1 — t
00
-2
(using the binomial theorem)
so that equating coefficients of tn on both sides gives Ln@) = 1.
(ii) Ln(x) satisfies Laguerre's equation F.1), so that we have
d2 d
x—Ln{x) + A - x)-^Ln{x) + nLn(x) = 0.
Setting x = 0 in this equation and using part (i) above we obtain
L'n@) + n = 0
and hence L'n@) = —n.
6.5 ORTHOGONALITY PROPERTIES OF THE LAGUERRE
POLYNOMIALS
Theorem 6.4
f CO
e~x Ln(x)Lm(x)dx = dnm.
J о
Proof
From the generating function of theorem 6.1 we have
exp {~xt/(l — t)} X4 r / n „
и=0
exp {—xs/(\—s)\ •sr*
and = У Lm(x)sm,
I — s *—<
m=0
so that
2r , sr , ч exp{—x^/(l — t)} exp {—xV(l — *)'
*u mU " 1 — t ' \ — s
n,m — 0
Г»
Thus Q~xLn(x)Lm(x) dx is the coefficient of f"sm in the expansion of
J о
Jo
1=Г e_x exp {-xt/(l - t)} exp {-xs/(l - s)}
J ' 1 — t ' 1 — s

§6.6 LAGUERRE POLYNOMIALS AND THEIR DERIVATIVES 173
1 [™ { ( t s \\
A - 0A —s) Jo I V 1 - t 1 - */J
i Г i
A - 0A —s)l 1 + {*/(! - 0} + {*/(! - 0}'
1
(l - 0A - 0 + ti\ - 0 + *(i -')
1
1 -rf
и=0
Thus the coefficient of tnsm is 1 if n — m and 0 if я ^ m, i.e. it is dnm.
Hence e-xLn(x)Lm(x) dx = dnm.
J о
6.6 RELATIONS BETWEEN LAGUERRE POLYNOMIALS AND THEIR
DERIVATIVES: RECURRENCE RELATIONS
Theorem 6.5
(i) (n + l)Ln+1(x) = Bn + 1 - x)Ln(x) - nLn^{x).
(ii\ a;L;(»;) = nLn(x) - nL^x).
2
r=0
Proof
(i) If we differentiate the generating function with respect to t and use
the fact that , . 1
at 1
dtl-t A - 02
we obtain
which, on further use of theorem 5.1, becomes

174 LAGUERRE POLYNOMIALS CH. 6
Multiplying throughout by A — tJ gives
2
и = 0 и=0 ге=О
and hence
CO 00 00
2 Ln{x)nf-1 - 2 2 ад»*- + 2 ад*1-*1
n=0 n=0 n—0
со со
=2 ад*- - 2 ад*"+1 -
n=0 та = О та=О
Relabelling the summations so that the general power appears as tn in
each gives
2 Ln+I(x)(n + 1)*» - 2 2 Ln{x)nt« + 2 i.-i(*)(fl - l)i"
n=—1 и^О n=l
GO^ CO CO
= 2 ад*" - 2 L«-i(*)*n - * 2Ln(x)fl
n=0 n=l n—0
and then equating the coefficients of ?" on both sides of the above equation
gives
(n + l)Ln+1(x) - 2nLn(x) + (я - IJL.-xW
= L.(«) - !»,_!(«) - »;Ln(»;)
which reduces to
(и + l)Ln+1(x) = Bя + 1 - ж)Д.(«) - hL._i(*).
(ii) If we differentiate the generating function with respect to x we
obtain
L'(x)tn - L
^2.(*)f. F.4)
n=0
Thus
OO 00
Л >^ T'(x\tn — t >^ T (r\tn
/ ^_^ »\ / ^_, \ /
and hence
CO CO CO
2ТЧ \j* V T't 4^,4-1 V Г /
2 2
n=0 n=0

§ 6.6 LAGUERRE POLYNOMIALS AND THEIR DERIVATIVES 175
which, on relabelling, becomes
OO OO OO
2 V—4 XT*
L (x\tn — / L (x\tn = / L (x\tn
и=0 n=l n-l
Equating coefficients of t" on both sides of the above equation gives
К(х) - LU{x) = -?„_!(*) in > 1). F.5)
If we now differentiate result (i) above with respect to x we obtain
(я + l)L'n+1(x) = Bn + 1 — x)L'n(x) — Ln(x) — пЬ'л_г{х)
and if we use equation F.5) in the rewritten forms
L'n+i(x) = L'n(x) - Ln(x)
л«Л T (y\ __ T (ы\ j T (ы\
ALLKX. ^-*n lv / ~~~ П\ / ~T" П IV /)
we shall have
(n + \){L'n(x) - Ln(x)} = Bя + 1 - x)L'n{x) - Ln(x)
This simplifies to give
4t\ / n\ f "П—Х\ /
and hence
(iii) If, in equation F.4), we expand 1/A — t) in the form y^ V we
obtain
CO OO OO
?—1 4—1
и=0 r=0 s=0
r,s=O
so that L'n(x) is given by the coefficient of tn on the right-hand side of the
above equation. For a fixed value of s we obtain t" by taking r + s + 1 = я,
i.e., r = n — s — 1, and then the coefficient of i" is —La(x). We obtain the
total coefficient of t" by summing over all allowed values of s, which, since
we require r > 0, implies that я — s — 1 > 0, i.e., s < я — 1.
This means that we have
n-l
coefficient of tn = 2^ ~ L3(x),
n-l
and hence I''n(x) - = ¦ У^ I'S*).
1-0

176 LAGUERRE POLYNOMIALS CH. 6
6.7 ASSOCIATED LAGUERRE POLYNOMIALS
Laguerre's associated equation is
Theorem 6.6
If z is a solution of Laguerre's equation of order n + k then dkz/dxk
satisfies Laguerre's associated equation.
Proof
Since z is a solution of Laguerre's equation F.1) of order n + k, we have
d2z .. 4ds
+Ate+(+% °
Differentiating this equation k times and using Leibniz's rule for the
k\h derivative of a product gives
d*
+ * 1
+ (» + *)^i* = 0-
Thus
d* d** I /b , 1 Nd d** . d% П
which just states that d*#/d«* satisfies equation F.6).
From the above theorem and the fact that the Laguerre polynomials
Ln(x) satisfy Laguerre's equation, it follows that (dk/dxk)Ln+k(x) satisfies
Laguerre's associated equation. We define the associated Laguerre poly-
polynomials by this solution together with the constant factor ( — 1)*:
i?(*) = (-i)*^W«). F.7)
Theorem 6.7
Proof
From equation F.3) we have
n+k

§ 6.8 PROPERTIES OF ASSOCIATED LAGUERRE POLYNOMIALS 177
so that, by equation F.7),
ib n+ к / , т \ i
(я+*-г)!(г1)«
(since {dk/dxk) operating on powers of x less than k gives zero)
я-t к , ,. . .
•_iv lw + ^ —-х-"
1 (n+k-r)\(r\y(r-
(since (dk/dxk)xr = r(r — 1). . . (r — k -\
n
84o J (я+Л -A -
(changing the variable of summation to s = r — k)
— > f 1V
(n — s)l(k + s)[s\
which is the required result.
6.8 PROPERTIES OF THE ASSOCIATED LAGUERRE POLYNOMIALS
Theorem 6.8 (Generating function)
exp {—xt/(l — t)} °°
n rtfc+i
Proof
From theorem 6.1
t)}
n=0
Differentiating both sides k times with respect to x, we have
d* exp { xt/{\ - t)} dk
dxk' A t) dxk
v ' UK
(since /<,,(.v) is a polynomial of degree n and so, if n ¦ U, will give
zero vvlii'ii ilifferentiated l< times).

178 LAGUERRE POLYNOMIALS CH. 6
Carrying out the differentiation on the left-hand side, we obtain
1 — t dx* ?¦*
and thus, using equation F.7),
4 ' n=0
which, on cancelling a common factor of (—\)ktk, becomes
exp {-xt/{\ - t)}
к1 г;
Theorem 6.9
_ у
L*(x) = — 'e-*xn+k).
Proof
Since this is almost the same as that of theorem 6.2, we shall not give
it here.
Theorem 6.10 (Orthogonality property)
JO n m
Proof
Again this proof is very similar to a previous one (that of theorem 6.4)
and will be omitted.
Theorem 6.11 (Recurrence relations)
(i) iit-i(*) + Lkn-\x) = Lkn(x).
(ii) (я + 1)L*+1(«) = Bh + k + 1 - *)#(*) - (я
(iii) *LJ'(«) = «ВД " (« ВД
n-l
«¦ = 0

6.8 PROPERTIES OF ASSOCIATED LAGUERRE POLYNOMIALS 179
K)OF
(i) Using theorem 6.7 we have
||
\n-l~r)\{k+r)\r\ + ?Л
У IV (" + ^1)!
1 \n
(n -n)\(k - 1 +n)\n\
n — 1 /iT-i\i f ^ -f*4
(я+ft-l)! / 1 1 Ъ
V ' (и - r - 1)!(A + r - l)lrl \(й + г) (и - r)J
IV"—-
nV
^ iy (n+k-l)\ (n-r)+(k+ r)xr
¦ (я — r — 1)!(Л + r — l)!r! (й + r)(n — r)
n\
n~X .. (n+k)\ . „__x"
nl
iv x
? )(n-r)\(k+ry.rl
= L\(x) •
(again using theorem 6.7).
(ii) Differentiate k times the result of theorem 6.5(i) with я replaced by
t -f k, and we obtain
n -|- k + 1)~?«+*+i(«) = Bя + 2? + 1)^1,^*)
ind thus, usinf» Leibniz's theorem for the &th derivative of a product,

180 LAGUERRE POLYNOMIALS CH. 6
(n + k + i)^kLn+1+k(x) = Bh + 2k + 1)—LB+,.(*)
Making use of definition F.7) we now have
(n+* + l)(-l)*LJU*)
= Bя + 2k + i)(-i)*L*(«) - *(-i№) - лс-i)*-1^;!^)
-(я+Л)(-1)*^_г(*)
which, when we take account of part (i) above with n replaced by n -\- 1,
gives
(n + k + l)L*+l(x) = Bя + 2k + \)Lkn(x) - хЩх)
and this simplifies to
in + l)Ll+x{x) =Bn+k + l- х)Щх) - (ii
(iii) If we differentiate k times with respect to x the result of theorem
6.5(ii) with n replaced by n + k, we obtain
dk dk dk
{Lk(x)} = (n + k)-^Ln+k(x) ~ (я + k)j^kLn f*-iW
which, on use of Leibniz's theorem for the &th derivative of a product,
becomes
dk , dk dk
dk
~ {n + Щ—Ьп
Then, by equation F.7), we have
xL*[x) + Шп(х) = (n + k)Ll(x) - (n + ОДи
and thus
xLi'(x) = пЩх) ~(п
(iv) If we differentiate & times with respect to x the result of theorem
6.5(iii) with n replaced by n + k, we obtain

§6.8 PROPERTIES OF ASSOCIATED LAGUERRE POLYNOMIALS 181
H + jfc-1 j.
C|"| fjint / 1 \& 7" I *vA У T (v\
(since for r < k Lr(x) is a polynomial of degree less
than k and so when differentiated k times just
gives zero)
— 2
8 = 0
(changing the variable of summation to s — r — k)
8=0
n-1
Therefore L%(x) = - 2
r-0
(v) By theorem 6.7 we have
^ (я-г)!(А+г)!(г-1)!
(the term with r = 0 vanishing on differentiation)
n-l
(changing the variable of summation to ^ = r — 1)
{n-1 +k+ 1I
(from theorem 6.7).
(vi) Comparing results (iv) and (v) above, we have
M-l
which, when n is replaced by я + 1, gives

182 LAGUERRE POLYNOMIALS CH. 6
6.9 NOTATION
It is important to note that certain authors use different definitions for
the Laguerre and associated Laguerre polynomials from those given here.
Sometimes the Laguerre polynomials are defined such that the gen-
generating function has the form
exp {-**/(! - t)}
= 2ад?
м=0
so that -5?„(х) thus defined is equal to n\Ln(x).\
The associated Laguerre polynomials may be defined by
giving the relationship ?ehn(x) = {-\)кЬкп_к(х).
6.10 EXAMPLES
Example 1
n
Prove that Ц+е+\х + у) = ]Г Ц{х)Ь^п_г{у).
We have, from theorem 6.8,
Z* n )X+y)-t П _ Ла+Р + 2
so that L%+P+\x -\- y) is the coefficient of Vх m the power series expansion
of
-t)}
(
But
{(+)f/(l f)} = exp {-**/(! - f)} exp {-
O
r=0 8=0
(again using theorem 6.8)
f We use JS? to denote alternative definitions, in order to distinguish them from
ours, but it must be remembered that authors adopting these alternative defini-
definitions will use L.

§ 6.10 EXAMPLES 183
We obtain Г by taking r + s = n; so that for a fixed value of r we have
s = n — r,and hence, for this value of r, the coefficknt of tn isL^(x)L^_r(y).
The total coefficient of tn is obtained by summing over all allowed values
of r, which, since s — n — r and we require s > 0, implies that r < n.
Hence the coefficient of tn is /> L*(x)L^_r(
so that Ц+е+1(х + y) =
r=0
Example 2
¦<o L™(x)
*\hnvn ihnt T U^/fvfW — (* — t/vf\m/2 > n \ / ~
^ (и + ту.
(where Jm is the Bessel function of integral order m).
We shall prove the equivalent result that
*-i (n +1
n=0 v '
From equation D.6) we have
so that
l
- 2
,2 т^ттуы
We wish to show that the coefficient of P1 in this expansion is
!?(*)/(* + »)!.
To obtain the coefficient of tn we take r + s == n so that for a fixed
value of r we have s -- n ¦ - r, and for this value of r the coefficient is
v ; »!(w l г)!(я r)\

184 LAGUERRE POLYNOMIALS CH. 6
The total coefficient is obtained by summing over all allowed values of r,
which, since s = n — r and we require $ > 0, is given by я — r > 0, i.e.,
r < n.
Hence the coefficient of tn is
я ..
2 {~l)rA(m + r)\{n-r)
-iv
+т)\ r\(m + г)(я - г) I
Tm(\
(using theorem 6.7),
and the required result follows.
Example 3
S/гоя; <Aai P e-'L*@ df = e^{L*(x) - LJU(*)}-
J a;
Integrating by parts, we have
f" e-'L*@ d^ = f-e-'L^)]00 - f " -е-«.Ы(*) df
ji L Ax J x
4 = 0
(by theorem 6.11(iv)).
Thus
f " e-«L*(«) df +T f " e-'X,r*(«) At = е-ед
and hence
r=0 ^^

PROBLEMS 185
Therefore
= е-ВД - e-LJU*)
(by equation F.8) above)
PROBLEMS
A) Show that Zi'@) = ^я(я - 1).
B) If f(x) is a polynomial of degree m, show that f (x) may be expressed
in the form
/¦to
with cr = t~xLr{x)i{x) Ax.
J о
Deduce that JJ e-**Ln(*) dx = {°^Л< » ^ ^
C) Prove that
(x - trLJt) At = _^J_ д
Jo (от + я + 1)!
D) Show that
^f! di.
E) Prove that
F) Show that
f'e '.v*"№)]2'lv (" ',*)!Bn I Л I 1).
J ii и!

186 LAGUERRE POLYNOMIALS CH. 6
G) If L*(x) is denned for non-integral k by the generalisation of the
result of theorem 6.7
L4x) _ r(" + * + Ц
iy
U ] (n-r)\T{k+r
show that Щ\х) = (-1)» ^J^H^Wx)
and L-i/«(*) = (_1)« Л-HUVx).
(8) With !.*(«) defined for non-integral k as in problem 7, show that

7
CHEBYSHEV POLYNOMIALSt
7.1 DEFINITION OF CHEBYSHEV POLYNOMIALS; CHEBYSHEV'S
EQUATION
We define the Chebyshev polynomials of first kind, Tn{x), and second
kind, Un(x), by
Tn(x) = cos (n cos-1 x) G.1)
and UJx) = sin (я cos-1 x), J G.2)
for n a non-negative integer.
Theorem 7.1
(i) ВД = \[{x + iV(l - *2)}n + {* - iV(l - *2)}"].
(ii) Un{x) = -ii[{* + i V(l - *2)}n - {* - i V(l - *•)}"]•
Proof
(i) Let us write ж = cos в and we obtain
.TJx) = cos (я cos" cos в)
= cos я0
{ }
= K(ei9)n + (е-")"}
= ^{(cos в + i sin 6)" + (cos в — i sin
(ii) The proof is similar to that of (i), and so will not be given.
f The transliterations Tchebichcf, Tchebicheff and Tschebyscheff are also
found.
X Sometimes the Chebyshev polynomial of the second kind is defined by
Wn(x) sin {(я I IH-os-'aWO -*') {WO - x'))/l Лм i(.v).

188 CHEBYSHEV POLYNOMIALS CH. 7
Theorem 7.2
f\ TT t \ — V
(П) Un(X) - ?
r-0
Proof
(i) From theorem 7.1(i) we have
ВД = l
1
= 2
n!
2 "^""'OVCl - *2)}r + 2 "C^"-r{--iV(l - **)}r
r=0
(by the binomial theorem)
Now, when r is odd ( —l)r = —1, so that 1 + ( —l)r = 0, and when r
is even ( —l)r = 1, so that 1 + (-l)r = 2. Hence we have
2 "c^n"rA - x*y/2 [r 2-
But if r is even we may write r = 2s with s integral, and the requirement
r < я means that s < я/2. This, since s is an integer, is equivalent to
s < [и/2] with [я/2] as defined before, namely the greatest integer less
than or equal to я/2.
Thus,
(ii) The proof is similar to (i) above.
We may use theorem 7.2 to write down the first few Chebyshev poly-
polynomials :

§ 7.1 DEFINITION OF CHEBYSHEV POLYNOMIALS 189
ВД = 2x2 - 1, Щх) = V(l - хгJх;
Г,(ж) = 4x3 - 3*. Щх) - V(l - *2)D*2 - 1);
ВД = 8л;4 - 8ж* + 1, Щх) = VO - x2)(Sx3 - 4ж);
Гв(ж) = 16х5 - 20л;3 + 5ж, Щх) = V(l - **)A6*4 - 12л;2 + 1).
We note that Un{x) is not actually a polynomial, but is instead a poly-
polynomial multiplied by the factor \/(l — x2), whereas ^„(x) as defined above
is a polynomial of degree n.
Since Tn(cos 6) = cos пв and Un(cos в) = sin пв, we note that the
Chebyshev polynomials provide expansions of coswfl and sin n6 /sin в in
terms of powers of cos в.
Theorem 7.3
Tn(x) and Un(x) are independent solutions of Chebyshev's equation
Proof
We give the proof for Tn(x); the proof for Un(x) is similar.
We have, from definition G.1),
±Tn(x) = ±cos(ncos-iX)
= -sin (я cos-1 ж).я.
_
Vl-1 x )
(remembering that (d/'Ax) cos x = —l/-y/(l — л;2))
Also, ,
d2 „ . ,
d/ я ,\
= — "S —— . Sin (Я COS XJ >¦
d^Lv(l — ж ) J
иж я — я
—-r— sin (я cos-1 ж) -I- -¦ ~ ¦ cos (я cos-1 ж).,. м2>
C0S V
я2
¦ лЯ)«»(Я1Ч|* '.V).

190 CHEBYSHEV POLYNOMIALS CH. 7
Hence
A - ж2I'2
nx
sin (и cos x) — и2 cos (и cos-1 x)
sin (и cos x) + и2 cos (и cos-1 x)
A - x2I'"
= 0,
which proves the required result.
The fact that Un(x) and TJx) are independent solutions follows from
observing that Tn(l) = 1 while UJl) = 0, so that Un(x) cannot be a
constant multiple of Tn(x).
7.2 GENERATING FUNCTION
Theorem 7.4
V(l - x2) _ у
' n=0
Proof
Again we prove only result (i), the proof of (ii) being similar.
Let us write x = cos в = ^(e19 + e~le), so that we have
1 - t2 1 - t2
l-2tx+
1
0-
= A
= A
-(eP
1
-<«)
+ e~ie)i + f2
— t2
00 00
r=0 »=0
(by the binomial theorem)
00
r,»=0 r,e=O

§ 7.2 GENERATING FUNCTION 191
We wish to pick out the coefficient of tn in this summation and show
that it is T0(x) when и = 0 and 2TJx) otherwise.
We consider the cases n — 0 and n — 1 separately, since for these values
of n we obtain tn from the first summation only, whereas for n > 2 we
obtain tn from both summations.
я = 0 is obtained only by taking r = 0 and 5 = 0 in the first summation,
so that the coefficient of t° is
ei(o-oy> = ! = ад
n = 1 is obtained by taking either r — 1 and 5 = 0 or r = 0 and * = 1,
so that the coefficient of tx is
ei9 + e-'9 = 2 cos в
= 2ВД
(remembering that ТЦя) = Tn(cos в) == cos tiff).
For и > 2 we obtain the coefficient of t" by taking r + s ~ n (i.e.,
s = n — r) in the first summation and r -\- s -\-2 = n (i.e., s = n — r — 2)
in the second summation. The coefficient of tn is therefore
n n-2
V
2 =
г=0
:цг-(п-г))в "V |{r-(
~" 2-,е
г=0
п
= е- J е-
с 1 _
,-г-2))в
2г9 g-K"
ei29\K+l
_ ив
n-2
.-2)9 у ei
r=0
¦ e
12Г9
1 - (e»)-1
1 ^219
(summing to n + 1 and я — 1 terms, re-
respectively, the two geometric series which
both have the common ratio e21")
ae
и 1 - e
e2i9) ^ gl^j _ е2Ш)
~~ 1 p.2'9 ' 1 P219
X С X С
(on rearranging the terms)
= 2 cos nO
(again remembering that T,,(x) T,,(cos 0) - = cos nO)

192 CHEBYSHEV POLYNOMIALS
Theorem 7.5 (Special values of the Chebyshev polynomials)
@ Tji) = it
tjl-1) = (-i)«,
T2n@) = (-l)«,
^2«+l@) = 0.
(ii) Щ1) = 0,
Un(-\) = 0,
tf*.@) = o,
f/2«4l@)=(-l)".
Proof
Again we prove results (i) only.
Setting x = 1 in definition G.1) gives
Г„A) = cos (я cos-11) = cos и.О = cos 0 = 1.
Setting x = —1 gives
Tn(—1) = cos(n cos-1 -1) = cosnn = (-1)".
Setting x = 0 gives
Tn@) = cos (и cos-1 0) = cos я|
@ if n is odd
(—l)n/2 if n is even.
Thus all four results are proved.
7.3 ORTHOGONALITY PROPERTIES
Theorem 7.6
.. г1 здад, f°
Г1 Г/ (x\TJ (х) Г° от ;z?w
(ii) ГГ "(?d« = V1/2 ^ = «^0
J-iV(l-^2) [о « = н=0.
Proof
We prove result (i) only, the proof of (ii) being similar.
If we set x = cos в we have

§ 7.4 RECURRENCE RELATIONS 193
J - 1 V(l — X2) J n Sin 0
= I cos пв cos mO dO G.3)
J о
(by definition G.1))
= | ?{cos (и + mH + cos (и - mH} dO
j о
1Г 1 1 • 1Л
= о sin (я + mH sm (n — m)Q
2L В+Ш n — m Ju
(provided n — m ~A 0)
= 0.
If и - я = 0 we have from equation G.3)
J -l л/A — * ) Jo
= f" 1A + cos 2яб) dd
J о
1Г 1 1"
= L\0 sin2w0
2L 2n Jo
(provided я ^ 0)
If и — от = 0 we have from equation G.4)
Г адад^ rldfl
and thus the proof is complete.
7.4 RECURRENCE RELATIONS
Theorem 7.7
(i) Tn+1(«) - 2xTn(x) + Тп.г(х) = О.
(ii) A - *aO';(*) = -nxTn{x) -|- «Г».^).
(iii) f/n(l(x) 2*ад + гл,..^) - о.
(iv) (I x-)U',,(x) tixUn(x) I nUn ,(.v).
^aiii we provide proofs for tlic polynomials ol first kind only.

194 CHEBYSHEV POLYNOMIALS CH. 7
(i) By writing x = cos 0, the required result is
cos (и + 1N—2 cos б cos пв + cos (и — 1H = 0.
However,
cos (и + 1)9-2 cos 0 cos пв + cos (и - 1H
= cos пв cos в — sin пв sin в — 2 cos в cos «0
+ cos пв cos 0 + sin «0 sin 0
the result which was to be proved,
(ii) By writing x — cos в, the required result is
But
d
A — COS2 6)-T-. — COS И0 = —И COS 0 COS Я0 + И COS (« — 1H.
4 d(cos в) ч '
A - cos2 0)- cos я0 = sin2 0 ( - -— •— cos nO)
v d(cos 0) \ sin в dd )
= —sin 0(—я sin «0)
= и sin б sin пв
and —я cos б cos пв + я cos (я — 1H
= —и cos 0 cos 0 + «(cos я0 cos в + sin «0 sin 0)
= и sin пв sin 0,
so that the required result is proved.
7.5 EXAMPLES
Example 1
Show that V(l - x*)Tn{x) = Un+1{x) - xUn{x).
If we replace x by cos 0 and use the consequences of definitions G.1)
and G.2) that
Tn(cos 0) = cos пв
and ?/n(cos 0) = sin пв,
we find that the result we require to prove is just
sin 0 cos пв = sin (и + 1H — cos 0 sin пв.
But sin (n + 1H — cos 0 sin иб
= sin пв cos б + cos «0 sin в — cos 0 sin я0
= cos пв sin 0,
which proves the result.

§ 7.5 EXAMPLES 195
Example 2
Show that У T2r(x) = -\ 1 + -— ;M2n+1(x)\.
Again we write x = cos 0, so that we have
n
V т (^
n
= 2 r2r(cos0)
П
= V cos2j-6
r = 0
= R\ls
„ 1 -
(using the result for the sum to n +1 terms
of a geometric progression)
R A - e'^' + ^Xl - e-2i9)
6
{1 - cos {In + 2H - i sin Bи + 2H}{l - cos 2в + i sin 20}
- ei29 - e-i2
^ ^ ол [A - cos 20)A - cos Bи + 2)°} + sm 2e sin Bи
2 — 2 cos 2$
l{l ¦¦- cos Bn + 2H -|- |^f- sin Bn + 2)o}
If sin Bя + 2H cos 0 - cos Bn + 2H sin 0
.1Г sin {B« -i- 2H -fl}]
~2L ' sinO" J
1/ sinBw | 1H
211 ! sinO
If, .
21 !

196 CHEBYSHEV POLYNOMIALS CH. 7
PROBLEMS
A) Show that V(l - *»)ВД = «ВД - Tn+1(x).
B) Show that Tm+n(x) + Tm_n(x) = 2TJx)Tn{x).
C) Prove that T'n{x) "
D) Show that 2{Tn{x)Y = 1 + Tin{x).
E) Show that {Tn(x)? - Тп+1(х)Тп_г(х) = 1 - xK
F) Show that Tm{Tn(x)} = Tn{Tm(x)} = Tmn(x).
G) Show that {1/д/A — x2)}Un(x) satisfies the differential equation
(8) Use Chebyshev's differential equation and the equation in problem G)
above to show that
and
[H-D] (я_г_1)(

8
GEGENBAUER AND JACOBI
POLYNOMIALS
8.1 GEGENBAUER POLYNOMIALS
It is possible to define new sets of polynomials by generalizing some of
the results already proved for the Legendre, Hermite, Laguerre or Cheby-
shev polynomials. We give here only two particularly useful sets; those
obtained by generalizing in two different ways the generating function of
the Legendre polynomials, given in theorem 3.1. We shall define the
Gegenbauer polynomialJ* of degree n and order A, C\{x), as the coefficient
of tn in the expansion of
1
A - 2xt + ty
(Note that the Legendre polynomial Pn(x) is in fact equal to C|(x).)
Thus
1 Чг4 i
N СHx)tn (8 1)
A — 2xt b fI ,-frb
It may be shown that such a power series expansion is valid for | t | < 1,
|* | < 1 and A > - J.
We shall omit proofs of the following properties. All the results may be
obtained by methods similar to those used in preceding chapters.
I' Also sometimes called the ultnisphcricHl polynoMiiiil.

198 GEGENBAUER AND JACOBI POLYNOMIALS CH. 8
Theorem 8.1 (Power series expansion)
гл/„ч _ X^ / 1V Цп - г + X)
Theorem 8.2 (Orthogonality property)
^l x) C[x)L(x)uxZ л
Theorem 8.3 (Recurrence relations)
(i) (n + 2)Ci+2(x) = 2(A + n + l)*Ci+1(«) - BA + и)С^(х).
(ii) под =
(iii) (n + 2X)C
(iv) fiCSK») = (и - 1 + 2X)xC"n_1{x) - 2ЯA - x*)Cxnz\{x).
(v) CJ'(*) =
Theorem 8.4 (Differential equation)
C*(x) satisfies the differential equation
8.2 JACOBI POLYNOMIALS
We may generalize the Legendre polynomial generating function even
further. We define the Jacobi polynomial Р^'Р)(х) as the coefficient of t"
in the expansion of
2xt+t*y"{l ^
Р*'рХхУп- (8-2)
2
n=0
We note that the Legendre polynomial Pn(x) is in fact equal to
ру\х).
Again the following properties may be proved by methods similar to
those used in previous chapters.

§ 8.2 JACOBI POLYNOMIALS 199
Theorem 8.5 (Series expansions)
(i) !*¦»(*)
у
^Г(а+г+1)Г(я+/?-г+1Хя-г)!г!\ 2 A 2
? Г03+г+1)Г(и+а+/3 + 1)(и-г)!Н \2
Theorem 8.6 (Orthogonality property)
Г A -xf(\ +xfff-'i\x)P^P\x)dx
J -1
Bя + ос + 0 + 1)и!Г(и + ос + /3 + 1)
Theorem 8.7 (Recurrence relations)
(i) 2и(« +р+и)(а +/3+2и -
= («+/3+2и -l){a2-
- 2(а
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Theorem 8.8 (Differential equation)
P(n'P\x) satisfies the differential equation
A x^ -\ {fi а (а I (i [ 2)^ !• «(« | « |- /? 1- l)y = 0.
As well as being related to the Legendrc functions (recall that
/'„(.v) /T'"(v) C',1,-(.v)), tlie (;едепЬаиег ami Jacobi polynomials arc
related to eaili other and to the C'hebysbev polynomials. In I'aet, tlie

200 GEGENBAUER AND JACOBI POLYNOMIALS CH. 8
Gegenbauer polynomials are just a special case of the Jacobi polynomials
and the Chebyshev are just a special case of the Gegenbauer:
_ Щ + 1)Г(И + 2Я) _
Cn{X) ~ ГBА)Г(и + A + \f* W'
Tn(x) = ? lim ^ (И>1); (8.4)
^ А-Я) ^
Un(x) = V(l - ^2)C^_1(x). (8.5)
These results will be proved in the next chapter (Example 5, p. 215).
There also exist relationships with the Laguerre and Hermite poly-
polynomials, which we shall state but shall not prove:
Щх) = lim P^\l - 2x/p); (8.6)
Hn(x) = n\ lim A-«/2C^(x/VA). (8.7)
A-i-ao
8.3 EXAMPLES
Example 1
Shorn that -^(x) = 2"'
From theorem 8.3(v) we have
d „,, .
so that
dx2 dx
= 22Я(Я + \)С1±\{х),
and, if we repeat this process m times in all, we obviously obtain
Am
f>n{x) = 2™A(A + 1)(Д + 2) . . . (Я + w -

PROBLEMS 201
Example 2
Show that
Leibniz's theorem for the nth derivative of a product gives
x (-l)n""(+)() ( + )(
_ V и! f n,-, ППп+\Щ*+п+\)
- Zrl(n-r){ lj Г(^+И-г+1)Г(а+г + 1I1+^ V X) '
Hence we have
2"и!
1 М т ;
^0 1)Г(9+я-г+1)г!(я-г)!\, 2
(by theorem 8.5(i)).
PROBLEMS
A) Show that P<a'">( -*) = (- 1)«Р№«)(Я).
B)ShowthatP-.">(l) .^±^+1)
C) Show that
dv"'" (Л) ^ Г(« i a i II ¦ 1) '-"

202 GEGENBAUER AND JACOBI POLYNOMIALS
D) Use the method of induction to show that
V,f , nrv ч (» + 2A)CfrO - (я *
^(r+A)Cr(«)- 2A -x) "
E) Show that
F) Show that CitK*) =

9
HYPERGEOMETRIC
FUNCTIONS
9.1 DEFINITION OF HYPERGEOMETRIC FUNCTIONS
Let us define (a),, (the so-called Pochhammer symbol) by
(a), = a(a + 1) . . . (a + г - 1)
Г(ос + r) , ...
= ; (r a positive integer)
l(a)
(a), = 1.
Then we define the general hypergeometric function
mFJxu <xa, . . . am; f}u /?2, ...?„; л;)
by
ЛК «.', •••«»; ft- Л, ..¦/»-; *) = 2 l^v^^K' <9Л>
/Г \Pl)r\Pt)r ¦ ¦ ¦ \Рп)гГ1
/ц l)r\Pt)r
The notation
p Га1> а2> • • • am! "I
'" " \ ft Я R • \
\J>1> Pi) • • • Pn > J
is also often used.
We shall show that many of the special funetions encountered up to now
(and indeed many of the elementary functions) may be expressed in terms
of hypergeometric functions. We shall confine ourselves to the two separate
eases: m n 1 (in which case the function will he called the continent

204 HYPERGEOMETRIC FUNCTIONS CH. 9
hypergeometric function or Kummer function) and m = 2, n = 1 (in
which case we shall merely call it the hypergeometric function).f
Convergence of the series (9.1) needs to be considered. The following
results may be proved using the standard techniques of convergence
theory:
(i) The confluent hypergeometric series is convergent for all values of x.
(ii) The hypergeometric series is convergent if | x | < 1 and divergent if
| x | > 1. For x = 1 the series converges if /? > a.x + <x2, while for x = —1
it converges if /3 > xt + <x2 — 1.
The following theorem shows the intimate relationships which exist
between the hypergeometric functions and the special functions already
considered.
Theorem 9.1
(i) />„(*) = 2F^~w, n + 1; 1;
(iii) /„(*) = ~(?jxFx(n + I; In + 1; 2b).
(iv) #2n(*) = (-1)" (-^ 1F1(-k; i; *»).
(v) Hin+1(x) = (-1)"
(vi) А1(*) = 1л(-я;
(vii) ?*(*) = ^^Д
(viii) TJx) = iF^-n, n;^;
(ix) ад = V(l -^2)» л(-« + 1, я +
(x) С»*) = ^щ|^ Л(-», » + 2Д; Я
+ The confluent hypergeometric function i-Fi(ot; /?; л:) is often denoted by
M(a, /3, x), and the hypergeometric function *Fi(ai>asi P>x) is often denoted by
u a2, /8, ж).

§9.1 DEFINITION OF HYPERGEOMETRIC FUNCTIONS 205
(xi) pw>(*)
Proof
In each case the result may be proved by expanding the hypergeometric
function as a series, using definition (9.1), and comparing with a known
series for the given function. We shall illustrate this method by proving
result (i) only.
We have, from definition (9.1),
F I n n A- 1 • 1 • 1 ~ X\
^-«,« + 1,1,^
, AO rl.
We may take и to be a non-negative integer, since it is for only these
values of n that the Legendre polynomials are defined.
Then we have
(-n)r = (-«)(-* + 1)(-я + 2) . . . (-я + r - 1)
= (-\yn(n - l)(n - 2) . . . (n - r + 1)
= (-1)ги!/(и -r)\ iir<n.
If r > n + 1, then (— n)r = 0, for it will contain a zero factor.
Also
(n + 1)P = (n + 1)(« + 2) . . . (n + r)
= (я+г)!/я!
and
(l)r = 1.2.3...r
= r\
so that we now have
r=0
f=0 v ''v "/
To show that this is the same as Р„(х), it is easiest to write Pn(x) as a
power series in (x 1). We do not as yet have such a power scries, but we
use Taylor's theorem to write
where by /"^A) we mean the rth derivative of /'„(л) cv.ilii.itcd .it л I.

206 HYPERGEOMETRIC FUNCTIONS CH. 9
To calculate Pn(r)(l) we use the generating function for the Legendre
polynomials given in theorem 3.1:
1 = V p Шп
л/A - 2tx + t2) ^-t nVJ>
so that, by differentiating r times with respect to x, we have
Ofv j_ f2\ —1/2
= (— 2t)r(—J)(—J — 1)( — | — 2) ... (-^ — r + 1)A — 2f* +
= Vr\(\ + 1)(| + 2) . . . (i + r -
= rl.3.5 . ..Br -
= <ГУ^A -2f» +
Hence, setting x = 1, we have
Л trBr\'
- *W A _ л-1
-l-2r
Br + l)Br + 2)Br + 3)
+ 3!
(by the binomial theorem)
2rr\ f^Q Br)\s\
writing r +5 = и.
Equating coefficients of tn gives
2rr\(n - r)\
= 0 for n < r.

§ 9.2 PROPERTIES OF THE HYPERGEOMETRIC FUNCTION 207
Hence, from equation (9.3), we have
2VI(«-r)! r!
(by equation (9.2)).
9.2 PROPERTIES OF THE HYPERGEOMETRIC FUNCTION
Theorem 9.2
.jF^a, /3; y; x) = ^(/3, a; у; ж).
Proof
This follows immediately from definition (9.1):
r=o GO*
Theorem 9.3
The differential equation
d2y , dy
(the hypergeometric equation or Gauss's equation) has 2Fi(a.,ft; y; x) as a
solution. If у is not an integer a second independent solution is given by
a^-WCoc + 1 - y; ft + 1 - y; 2 - y, x).
Proof
These results may be proved by using the series solution method of
Chapter 1. It is found that the roots of the indicial equation are 0 and
1 - - y, so that they lead to independent solutions provided у is non-
integral. It is found that these independent series solutions are just the
hypergeometric functions given in the theorem.
Theorem 9.4 (Integral representation)
if У />' ¦<».

208 hypergeometric functions ch. 9
Proof
From definition (9.1) we have
?j Mr
Го Г(а)Г03)Г(у + г) г I
Г(а)Г(/ОГ(У-/о2Г(а+') Цу+r) г!
1
(by theorem 2.7)
а 4- »Л Г fl - /у-0-i/fl+r-i н/ -
(by definition B.2) of the beta function, which
is valid if у - ft > 0 and /3 + r > 0)
1 } ^ Г() !
г!
_ v
(using the binomial theorem).
Theorem 9.5
Each of the following twenty-four functions is a solution of the hyper-
geometric equation:
V\ = 8^l(a,/S;y;*)
V2 = A - x)'-"-Vifr - а, у - P; у; *)
V, = A - яГУ^У - «, /3; у; ^-
^ =Л(«,/';«-ь/'-|-1 -y;i -
F6 =ж«^Л(а И y;/J I - 1 у; a 1 fi {- 1 y; 1 -ж)

§ 9.2 PROPERTIES OF THE HYPERGEOMETRIC FUNCTION 209
F7 = X-\Fx(ol, a + 1 - y; « +0 + 1 - y; 1 - -^
V8 = *¦ Vi(/J + 1 - y, /?;«+/? + 1 - y; 1 - ^
, a + 1 - y; a + 1 - /?; ^
F10 = (-*y-y(l - *)*—V,(l - /3, у - /J; a
Fu = A - ж)^л(«. У - /3; a + 1 - /?; j-^
+ l - y, l -
- У, /3; /3 + 1 - a; ^
- a, у - a; 0 + 1 - a; ^
^5 = (l - ^-^(/J, у - «; /3 + i - «; j-^rx)
^io = (-*)l-y(l - л)"-"-1^ + 1 - у, 1 - a; /? + 1 - a;
F17 = «^"^(a + 1 - y,/J + 1 - y; 2 - y; *)
Vla = xl-v(l - xy—f^l - a, 1 - /?; 2 - y; x)
F20 = xx-\\ - xy-^\FU + 1 - y, 1 - cc; 2 - y; -=—
Vn = A - xy—'jFfa - а, у - /3; у + 1 - a - /?; 1 - x)
VM = хг-"A - *)"-«-"tF1(l - a, 1 - /3; у + 1 - a - 0; 1 - x)
- a, 1 — a; У Ь 1 — a - /5; 1 - -
KM .V1 '¦(! л-)'1 • y.',(y //, 1 /?; у i 1 a /I; 1

210 hypergeometric functions ch. 9
Proof
Each function may be verified to be a solution by means of change of
variable, direct substitution and use of theorem 9.3.
Since the hypergeometric equation is of second order, there must be
only two independent solutions, so that a linear relation must exist be-
between any three of the twenty-four functions given above. In fact, it may
be shown that
V-, = P2 = V3 = Vt,
v5 = p6 = v7 = vB,
v9 = v10 = pu = vlt,
Pl3 = P]4 = Pi 5 = Pl6l
Pit = PI8 = Pi9 = P2o,
and Vn = P22 = V23 = V2i.
For the remaining relations the reader is referred to Erdelyi et al.,
Higher Transcendental Functions, Vol. I, pp. 106-8.
The six hypergeometric functions JFx{a. ± 1, /3; у; x), JF^a., fl ± 1; у; x),
2F1(a, jS; у ± 1; x) are said to be contiguous to 2^i(<x> fi\ 71 x). It may
be shown that between 2Fx(a, Д; y; x) and any two functions which are
contiguous to it, there exists a linear relationship whose coefficients are
linear functions of x. Since we can choose 2 from 6 in 6C2 = 15 ways,
there will be 15 such relationships altogether. For details the reader is
referred to Erdelyi et al., Vol. I, pp. 103-4.
9.3 PROPERTIES OF THE CONFLUENT HYPERGEOMETRIC
FUNCTION
Theorem 9.6
The confluent hypergeometric function jF^a; Д; x) is a solution of the
equation
(the confluent hypergeometric equation or Kummer's equation). If p is not an
integer a second independent solution is given by
^-VXa - jff + 1; 2 - 0; x).
Proof
As for theorem 9.3 above.

§9.3 CONFLUENT HYPERGEOMETRIC FUNCTION 211
Theorem 9.7 (Integral representation)
/or /9 > a > 0.
Proof
Similar to theorem 9.4 above.
Theorem 9.8
Solutions of the equation
гаек fry
1
у = ж11' е-ж/22г,
where z is a solution of the confluent hypergeometric equation with
v. — \ — k — m and ft ~ 1 — 2m.
Proof
The result follows immediately on direct substitution and use of
theorem 9.6.
Corollary
The independent solutions of equation (9.4) are given by
Mk<m(x) = *»+•» е-*/у^A - к + m; 1 -f 2m; x)
and
MK_m(x) = x*-m e-^2^! - /e - m; 1 -- 2m; x).
Proof
Since the independent solutions of the confluent hypergeometric
equation are
i*"i(«; P> *)
and
we have, from the theorem, that two independent choices for z arc
t/<\(\ k m; 1 2m; x)
and
.v-'",/•',(,\ к ¦ w; 1 • 2m; .v),

212 HYPERGEOMETRIC FUNCTIONS CH. 9
so that the corresponding solutions are
xi-me-s/y^ -k-m;l-2m;x)
and
xi+me-v/^F^ — k + m; 1 + 2ти; ж).
Мкт(х) and M;. _m(x) are known as Whittaker's confluent hypergeo-
metric functions.
Theorem 9.9
Each of the following four functions is a solution of the confluent hyper-
geometric equation:
V1 = ^(a; Д; x)
F4=*1-"e«1JP1(l -a]2-p; - x).
Proof
By change of variable, direct substitution and use of theorem 9.6.
The four confluent hypergeometric functions iFx(a ± 1; 0; ж), i^\(a;
/б zb 1 > x) are said to be contiguous to iFJa.; E; x). It may be shown that
between 1F1(a; f}; x) and any two functions contiguous to it there exists a
linear relationship whose coefficients are linear functions of x. Since we
can choose 2 from 4 in 4C2 = 6 ways, there will be six such contiguous
relationships altogether. For details the reader is referred to Erdelyi et al,
Vol. I, p. 254 (see Bibliography).
9.4 EXAMPLES
Example 1
From theorem 9.4 we have
- fl Io "-
— В) J О
(by definition B.2) of the beta function)

§9.4 EXAMPLES 213
Г(у) Г(/?)Г(у - ос - fl
Щ)Т(у - /?) Г(у - а)
(by theorem 2.7)
Г(у-а)Г(у-/?)
Example 2
Show that
^(а, Д; у; ж) = A - *)~ Vi(«, у — j»; у; ^^
(i.e., ^ = F3, мяи^ г/ге notation of theorem 9.5).
If we set т = 1 — / in theorem 9.4 we have
¦ f (l-Ty-4
Jo
X-V
" dx
on further use of theorem 9.4.
Example 3
Prove the relation of contiguity
y{y - 1 - By - 1 - а - /?)x} 2^(а, Д; у; х)
(«. Д; r 4-1; *)
We prove this result by expanding the hypergeometrie functions in
power series. Since
we see that the coefficient of x" on the left-hand side above must be

214 I-IYPERGEOMETRIC FUNCTIONS СИ. 9
Ла+я)Г(/»+я)Г(у)
.Г(«+»-1)Г05+я-1)Г(у)
.Г(«+я-1)Г(Д+я-1)Г(у+1)
Г(а)ГС8)Г(у+я-2)(я-1)!
(у+я-2) (у+я-1)(у+я-2)
Г(сс+я-1)Г(/3+я-1)Г(у-1) 1_
Г(а)Г08)Г(у+я-2)(я-1)! я(у+я-1)(у+я-2)
+ я(у+я-1)(у+я-2)} Ч
Г(«+я-1)Г03+я-1)Г(у-1)у(у-1)
Г(а)ГAв)Г(у+я-2)(я-1)!я(у+я-1)(у+я-2)
.[у\-2п+п+п) +у{(а+
- 2(га —1)« - яа —п0 -
+ я(я—2) + я(я-1)} + я(я-1)(а+/3+1) + яа/
= 0.
Thus the relation is verified.

§ 9.4 EXAMPLES 215
Example 4
Show that ^(a; 0; x) = e* ^(/9 - a; /9; -ж).
From theorem 9.7 we have
If we make the change of variable t = 1 — т, we have
W I -Р-Я-Щ _ T\»-l pi(l-t) J
j^ A r) e dr
- а)Г(а) J о U >
/J - a; a; -*)
(using theorem 9.7 again).
Example 5
Show that
(iii) Щх) = V(l - x2) CU(x).
(i) From theorem 9.1(xi) we have
Г(Я-| 4)Г(и |2Я)
(ii) I'Vom theorem 9.1(x) we have
(by theorem 9.1(x))

216 HYPERGEOMETRIC FUNCTIONS CH. 9
The trouble at A = 0 comes from ГBА). Hence we have
n lim CM - n lim 1 .limr(w
2 - Т
2 ^o АЦ2Т) я^о „j ^\~n' n+U> Л+*' 2
J_.rW F( i.l^x\
BЯ) и! а Л И'Я'2'-2~У
lim
^-o 2АЦ2А)
Ц2А)'2 Л *'"'*' 2
_ lim L_
' '" 2
= 1.2^-я,я; 4;- 2~-
()
(by theorem 9.1(viii)).
(iii) From theorem 9.1(x) we have
'(-"+¦¦" -
/ 1 — ж\
= я^-я + 1; я + 1; f; —2~-J
(by theorem 9.1(ix)).
PROBLEMS
A) Show that
(i) A -xT* = ,*>,?;?;*);
(ii) ln(l -*) = -«,^A,1; 2; л);
(iii) e® = xFx(a; a; ж).
B) Show that ^(oc; y; x) = lim ^^a, ^; y; x/(l).
C) Show that
+ 0 - а)ГA + I/
^a^-./S -a + 1; -1) =
ГA + /?)ГA -|- -J/J - a)

PROBLEMS 217
and use Example 2 of Section 9.4 to deduce that
D) Show that
~ Л(а, /9; у; *) = - Л(а + l,
алг у
and deduce that
^n ^(a, /I; y; x) = (-^-я 2^(а + n, p + я; у + n; x).
E) Prove the contiguity relationships
(i) (*-р)А(*,Р;у;х)
(а + l, j8; у; *) - ДЛ(«. Д + 1; у; *);
(ii) (а - jSKF^a; ^ + 1; л) + j8(* + j8 - lj^a; Д; *)
-/j(J8-iIJp1(«;j8-i;*) = o.
F) Use the confluent hypergeometric function to show that
and
G) Evaluate the integral
Г e-^F^x; p; x) dx
Jo

10
OTHER SPECIAL FUNCTIONS
In this chapter we discuss briefly other special functions which the
reader is likely to encounter. Several of these are denned by integrals
which are impossible to express in terms of known functions, and of these
integrals several have no special properties of interest—all the useful
information is contained in a table of values of the function.
10.1 INCOMPLETE GAMMA FUNCTIONS
These are denned by
'"-Чг (ЮЛ)
Г(*,а)= Г
J a
and
y(x, а) = Г е-'**-1 dt. A0.2)
• °
From definition B.1) of the gamma function we see that
Г(*, а) + y(x, «) = Г(«). A0.3)
10.2 EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS
The exponential integrals are denned by
Щх)=\' ~dt A0.4)
J —CO I
and Ej.(x) = f — dt. A0.5)
J X t

§ 10.2 EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS 219
We note that
ВД = -Щ-х).
(Some authors use the notation ei(x) for Ej{x).)
The logarithmic integral is defined by
It is easy to see that
oin*
Н(ж) = Ei(lnx) = -.b\(-l
A0.6)
A0.7)
A0.8)
0-2X04 0-6 0-8 10 1-2 1-4 1-6
Fig. 10.1 Exponential integrals
The sine and cosine integrals are defined, respectively, by
.. . fx sin t
ei(«) = —- dt
J 00 t
I /•,./ ч f * COS ' •
and Ci(.v) d/;
(a^ain tlu- nailer is warned to beware of different notations).
A0.9)
A0.10)
A0.11)

OTHER SPECIAL FUNCTIONS
UU)
2-0 -
сн. 10
-1-0 -
-2-0 -
Fig. 10.2 The logarithmic integral
Fig. 10.3 Sine and cosine integrals

§ 10.3 THE ERROR FUNCTION AND RELATED FUNCTIONS 221
The following results may readily be verified:
si@) = - |, A0.12)
(see Example 1, p. 225)
{Ei(L») - Ei(-i*)}, A0.14)
d(x) = %{Щ™) + Ei(-Lc)}, A0.15)
Ei(±i«) = Ci(x) ± i si(x). A0.16)
The graphs of the above functions are shown in Figs. 10.1-10.3.
10.3 THE ERROR FUNCTION AND RELATED FUNCTIONS
The error function is denned by
erf x = -?- [%-''<!*. A0.17)
\/л J о
By the corollary to theorem 2.6 we see that
erf oo = l. A0.18)
As a generalisation of the error function we define the functions
'** <10Л9)
(N.B. Ex(x) given by this equation is not the same as the Ex(x) of the
previous section.)
We see'that
erf* = .E2(x). A0.20)
The Fresnel integrals are denned by
S(x) = f*sinj*ad* A0.21)
Jo 2
f* ж
Clx) = cos-*2d/. A0.22)
Jo 2
and
It may be shown that
.S'(./) f'(¦/•) J A0.2.1)

OTHER SPECIAL FUNCTIONS
CH. 10
10 -
0 8-
0-6 -
0 4 -
0 2-
i
10 2 0 30
Fig. 10.4 The error function
0-8 -
0-6 -
o-5 [ 1—Ц-А—,
0-2 -
1-0 2-0 3-0 4-0
Fig. 10.5 Fresncl integrals

§10.4
RIEMANN S ZETA FUNCTION
223
and that the Fresnel integrals are related to the error function by the rela-
relationship
C(x) + iS(x) = l-p erf
A0.24)
The graph of the error function is shown in Fig. 10.4 and the Fresnel
integrals in Fig. 10.5.
10.4 RIEMANN'S ZETA FUNCTION
Riemann's zeta function is defined by
A0.25)
This series is divergent for x < 1, convergent for x > 1.
It may be shown that
A0.26)
Vtn. 10.6 Hic-тапп'я zeta function. (The graph is of C(x) — 1. Note the logarith-
logarithmic: .scale on tin- vertical axis. Kor comparison, the dotted line i:< the Kniph of
2 '.)

224 OTHER SPECIAL FUNCTIONS CH. 10
and that in general ?Bra), with n integral, may be expressed in closed
form.
The graph of ?,{x) — 1 is given in Fig. 10.6. Note the logarithmic scale;
for comparison, the graph of 2-" is given by the dotted line.
10.5 DEBYE FUNCTIONS
These are denned by
ад = 1оеТ^~1аЛ (Ю.27)
It may be shown that these functions are related to the Riemann zeta
function by Dn( oo) = и!?(га + 1).
10.6 ELLIPTIC INTEGRALS
We define the elliptic integrals of the first, second and third kinds
respectively by
-Л
A0'28)
E(k, ф) = [ V(l - k* sin2 0) d6 @ < k < 1), A0.29)
J о
Some of the importance of elliptic integrals lies in the following theorem,
which we state without proof.
Theorem 10.1 %
If R(x, y) is a rational function of x and у and if P(x) is a polynomial of
degree at most four, with real coefficients, then $R[x, V{P(x)}] dx can be
expressed in terms of elliptic integrals.
If the upper limit in each of definitions A0.28), A0.29) and A0.30) is
л/2, then we have the definitions of the so-called complete elliptic integrals:
E(k) = f* * V(l - k* sin2 0) dfl, A0.32)
J о
n'% d6
0).
A033)

§ 10.7 EXAMPLES
10.7 EXAMPLES
Example 1
Show that si@) = -л/2.
From definition A0.9) we have
f°°sin* ,
Jo t
To evaluate this integral we consider the integral
е-0" sin* ,
so that si@) is given by —/@).
Now,
— = — e" sin t dt
da Jo
= —
J
= - Г е-«и?* - е-") dt
Jo L\
= -ъ Г {<*-'+I>t - e—v} dt
Zi J о
= _i_ Г 1 e(-a+l
2i L— a + i
= _!M M
2i\a — i a + i/
— a — i
Hence, by integrating with respect to a we obtain
/(a) = — tan a + constant.
But J(oo)=0,
so that we must have 0 = — tan oo + constant,
and hence 0 =——-(- constant,
л
constant ¦•

226 OTHER SPECIAL FUNCTIONS CH. 10
Thus /(a) = - tan-1 a + ^
so that /@) = — tan-1 0 + - = ^
and hence si@) = - 7@) = — ~
Example 2
f00 1
Show that e~st si(t) At = tan-1*.
Jo s
Integrating by parts, we have
(•со г j Iе0 Г00 1 d
e-s( si@ dt = \-- e~st si(t) - — e~st - si(t) At
Jo w Is wJo Jo j At K '
1 1 Г00 d
= - si@) + - e-s< -;- si(t) dt
s s J о At
1/ яЛ If00 tsint
= "I-o) + - e
Д 2j jJo t
(using equation A0.12) and definition A0.9))
ж 1,. .
-2s + S^
(with / as defined in Example 1 above)
л 1/ ,
+t
(as shown in Example 1 above)
tan-1 j
Example 3
Show that (i) erf (—ж) = — erf x;
(ii) | erf ж | < 1.
(i) From definition A0.17) we have
erf* = 4- e-fld*
л/я: J о

§ 10.7 EXAMPLES
so that
erf (-*) = — e-(i dt = -4- e""*. -du
(on changing the variable to и — —t)
Vж J о
= — erf ж.
(ii) First suppose x > 0. Then, since e~i! > 0, we have erf x > 0.
Also
erf x < -4-f Г е-" df + f " e^'2 dt)
Vjr\ Jo J ж /
2 Г°°
= 4" е-Л
уя J о
= erf оо
= 1
(by equation A0.18)).
A similar argument holds if x < 0.
Example 4
that —— = —AFl /~>z:)-F[ /->-) >¦
J о VB - cos x) V31 \V 3 2/ \V 3 4/J
We have
Г/2 dx _ f"/" -dy
J о \/B — cos ж) J n \/{2 — cos (те — j)}
(making the change of variable у = тг — x)
dy
= Г
J я/2 \/B Г COSj)
_ Г» _dy
~~ J я/2 л/{2 Ч- 1 -2sin2(j/2)}
я/4 VC 2 sin2«)
(making the change of variable z v/2)
2 Г- d»

228 OTHER SPECIAL FUNCTIONS CH. 10
2 /Г/2 d* j^4 dz
J
Jo
(by definition A0.28)).
PROBLEMS
A) Show that
Г00 . Г°° . . л
(i) cos ж Сл(ж) dx = sin ж si(a:) da; = —-;
JO JO T"
(ii)
Г °° гоч /
B) If Да) = A - е-а() Г-~ dt, show that
Jo ^
d/
da ~a2 + 1
and deduce that /(a) = | In A + a2).
Prove that e~s' Ci(<) dt = — In A + ^a).
i 2x
C) Show that erf * = — y(i, x2) = —- ^ft; |; -
Y7f утг
|;
D) Show that T(x, t)=C erf {x/\/D-kt)} satisfies the following conditions:
(i) 7X0,0 = 0;
(ii) 7X*,0) = C;
..... dT J*T
A11) Tt = * aT--
E) Show that (i) f C(t) dt = жС(л;) - - sin ^ж2;
Jo 7Г 2
(ii) f% S(t) dt = *S(*) + - cos ^ж2 - -.
Jo л 2 ж
F) Show that —^-. = —^Ц = (V2)^Do
J n \/(sinж) Jo \/(cos ж) \-y/2

PROBLEMS
G) By making the substitution x = 2 sin в, show that
J о
(8) Show that
J о V{0 + *2H + 2лг2)} V2l \V2/ \V2 4/J

APPENDICES
1 CONVERGENCE OF LEGENDRE SERIES
We wish to investigate the convergence of the series obtained in Section
3.1 as solutions to Legendre's equation.
These series are of the form
_ **¦" (Al.l)
GO
on И > /7 v2tt + l
where in both cases we have, from equation C.7),
We shall discuss only the series (Al .1), since the discussion of the other
series is similar in all respects.
First we note that equation (A1.3) implies that all the terms in the series
(Al.l) for n > I have the same sign, so that we may apply tests for series
of positive terms.
If x =? 1 we may apply d'Alembert's ratio test:
if ^ un is a series of positive terms and if lim un/un+l = a, the series is
divergent if a < 1, convergent if a > 1 and the test provides no informa-
information if a = 1.
Here un = a2nx2n, so that we have
so that
Bи +
Bn - /)
lim
И-КГ
+ 2
1)Bя +
Bи + /
М»Н1
¦2)
+ 1)
я2
1
X

euler's constant 231
and the series is hence divergent if x2 > 1 and convergent if x2 < 1; that
is, it is divergent if | x | > 1 and convergent if | x | < 1.
If x = ±1 we must use Gauss's ratio test:
if ^ un is a series of positive terms, and if the ratio un/un+l can, for
7S=0
n > some fixed N, be expressed in the form
Мя+1 П
with p > 1 (where by 0A /nv) we mean some function f(n) such that
lim {{(n)/(l/np)} is finite), then ^ M« is convergent if /u > 1 and
divergent if /* < 1.
Here, since we are considering * = ± 1, we have
Mn Bw + l)Bw+2)
м/^ Bи
and it is a matter of simple algebra to prove that
я ^ {4и2 + 2и - /(/ + 1)}я
Now, we see that the last term is 0A /я2), since we have
lim Г W + 1X1 + ») /11
»-«> L{4w2 + 2n- 1A + 1)}я/ w2J
= lim /(/
{4n2 + 2я 1A + \)}n
which is finite.
Hence the conditions of Gauss's ratio test are satisfied; we have /u, = 1
and thus the series is divergent. This means that we have shown that the
Legendre series is divergent for x = ± 1.
2 EULER'S CONSTANT
Assuming f(x) to be a continuous, positive and decreasing function, let
us consider the sequence
1B)

232
APPENDICES
We first show that un > 0. This follows immediately from Fig. A2.1,
since f(l) + fB) + . . . + {(n) is the area of the rectangles shown, while
yUx)
I 2 3
Fig. A2.1
rn
i{x) dx is the shaded area underneath the curve, which is obviously less
J 1
than the area of the rectangles, thus making un > 0.
We now show that un+1 < un. For we have
fB) + • • ¦ + f(n)
f (ж) d*
= f(n
and from Fig. A2.2 we see immediately that this is negative.
\
Fig. A2.2
Hence «n+1 — un < 0, i.e., wnfl < м„.

DIFFERENTIAL EQUATIONS 233
Thus the un constitute a decreasing sequence, every member of which is
positive, and thus by a general principle the sequence must possess a
limit as и tends to infinity. Hence lim м„ exists.
The choice of i(x) = l/x will satisfy the conditions that f(#) be con-
continuous, positive and decreasing.
Then
= 1
= 1
1 1.
2 + 3
1
I - (
n ) \ x
H In n.
n
So we know that lim {1 + A/2) + A/3) + . . . + (l/и) - In n) exists.
It is called Euler's constant and is usually denoted by y. It may be shown
that, correct to four decimal places, у = 0-5772.
3 PIFFERENTIAL EQUATIONS
Equation
A - x2)y" - Ixy' + 1A + l)y = 0
A - x*)y" - 2xy'
= 0
хгу" + ху' + (x2 - n*)y = 0
x2y" + ху' — (хг + пг)у = 0
хгу" i A — 2
I №V** I" («2 - nV)}v 0
Solutions
Pi(x) Legendre polynomials
Qi(x) Legendre functions of the
second kind
P^(x) Associated Legendre
polynomials
Q"!(x) Associated Legendre functions
of the second kind
x\v" I 2.yv'
/(/ i l)}.v О
Jn(x) Bessel functions of the first kind
Yn(x) Bessel functions of the second
kind
ттB)/ i j-Hankel functions
In(x) ~)
„ , . >-Modined Bessel functions
Js.n(x)J
iicsstl fuiu'tions
y.(.v)/ Г

234
APPENDICES
Equation
у" - Ixy' + Iny = 0
у' + (A - x*)y = 0
xy" + A - x)y' + ny = 0
*•/' + (A + 1 - x)y' +• иу = 0
A - ж2)/' - xy' -I- я'у =» 0
A - л;2)^" - BA + \)xy'
+ n(n + 2l)y = 0
*A - x)y" + {y - (a + p + ОДУ
- *Py = 0
x%y" + (P — x)y' — <xy =0
Solutions
Hn(x) Hermite pol>Tiomials
Weber-Hermite functions
Ln(x) Laguerre polynomials
Lkn{x~) Associated Laguerre polynomials
jj . ' > Chebyshev polynomials
C^(x) Gegenbauer polynomials
Рп(ъР)(х) Jacobi polynomials
Hypergeometric function
a^-V^a + l -y; p + 1 -y; 2-y; x)
4*
=
Confluent hypergeometric function
\;2-P;x)
М*,т(ж) "I Whittaker's confluent
hypergeometric functions
^_,и(л:)/
4 ORTHOGONALITY RELATIONS
Function
Relation
Legendre polynomials
Associated Legendre
polynomials
f1 P,(x)Pm(x) = ~~i
J -1 II + \
w)!
I f1 P?(x)P?(x
1-х2
; J -l L x
-dx =
— m)\

ORTHOGONALITY RELATIONS
235
Function
Bessel functions
Spherical Bessel functions
Hermite polynomials
Weber-Hermite functions
Laguerre polynomials
Associated Laguerre
polynomials
Chebyshev polynomials
Gegenbauer polynomials
Jacobi polynomials
г
J о
Relation
= -{
where fa and Xj are roots of /n(Aa) = 0
J — 00
j:
J —00
)dx =
In + 1
e-*'Hn(x)Hm(x) dx =
x)dx =
Г e-'Lm(x)Ln(x) dx =дт
J о
Jo
dx
n!
Tm(x)Tn(x)dx
г.,
{0 m ^ и
я/2 m = и =? О
О т = и = О
fХ A -
dx
2А)
Г A -
Г(я + « + 1)Г(я + ;8 Ч-

236 APPENDICES
5 GENERATING FUNCTIONS
Function
Legend re polynomial
Associated Legendre
polynomials
Bessel functions
Hermite polynomials
Laguerre polynomials
Associated Laguerre
polynomials
Chebyshev polynomials
Gegenbauer polynomials
Jacobi polynomials
Generating function
(R = A - 2xt + <2I/2)
1 \r*
- = 2_, tnpAx)
n=0
Bm)!(l -*2)»'2 V-n-
2mm\Rm+
exp Btx - t1
exp{-**/(!
exp{-*i/(l
A - *)*+
R*
Я2
CO
ra=0
r=0
t)}= 2 r
00
n=0
CO
-t)} V .„7
~t)} 2rL
n=0
OD
re=0
0ad + « + Ry
Mx)
00
n=0

HINTS AND SOLUTIONS TO
PROBLEMS
CHAPTER 1
A) The general solution is Ayx(x) + By2{x) in each case, where A and 1
are arbitrary constants and yx{x) and уг(х) are given by
2
l
,11.5.9
valid for all x;
(ii) yi(x) = e-*,
_,_S?f UnU, 1 1
Уъ\х) — In ж. e / \ 1) IV 9 i
valid for all ж;
n=3
j2(x) = jj(x) In x + 1 4
_9
1 113
и - 2 н - 1 n 2J
valid for all x;
2
/ 1 1 1
+ 2 + 3 + ' ' ' + iT^
= A - - *)-*,
8.11.14. . . (Зи I 5) J
10.13.16 ... (Зи i 7)'V"
v;iliii for | .v | • 1 ;

238 HINTS AND SOLUTIONS TO PROBLEMS
valid for | x | < 1;
(vi) yi(x) = x\
y2(x) = х2 In х + хг
n=l
valid for all *;
(vii) У1(х) = x,
valid for | ж | < 1;
n-2
y2(x) = y^x) In x + x-1 +- %x
„3n -1
-\n ~2)\n\
»-1: «
valid for all x;
(ix) yi(x) = 1 + -|xs + &**
valid for all x.
B) (i) Impossible; (ii) impossible; (iii) possible.
C) The general solution is Ауг(х) + Вуг(х) where A and В are arbitrary
constants, and yi(x) and уг{х) are given by:
у 2.7.16... B»'-n I 1) 1
A) У^Х) = X + Z ,!1.3.5...B.-1Г *'
„!3.5...Bn-|

HINTS AND SOLUTIONS TO PROBLEMS 239
CHAPTER 2
A) Use definition B.1) of the gamma function with a suitable change of
the variable of integration.
B) Use theorem 2.5.
C) Use theorem 2.5.
D) (ОЖЫ)^
(ii) T(a). (Make the change of variable \/x = e».)
(iii) (b — a)m+n-lB{m, я). (Make the change of variable
у = (ж - «)/(? - a).)
(iv) -Б( ->/> + 1 )• (Make the change of variable у = xn.)
Й +F(Make the d-* ot:atife
(vi) B(\, |) = я. (Make the change of variable и = 1/A + t).)
F) Г(-« = -2V*; Г(-1) =^V«-
G) Use theorem 2.12.
CHAPTER 3
A) Use theorems 3.8(ii) and 3.5.
B) Use theorems 3.8(vii), 3.8(ii) and 3.5.
C) Consider (л;2 - II as the product (ar — 1 )'.(* + II.
f(-l )<'-«/« (r + l)(r-l)l . if, is odd
D)cr= П(г + 1)/2}1{(г - 1)/2}|
[О if »¦ is even.
Use the result of Example 2.
E) Differentiate the generating function for the Legendre polynomials
m times and use definition C.36).
F) Use theorems 3.8(ii) and 3.5.
B/я if n is even
0 if n is mid.

240 HINTS AND SOLUTIONS TO PROBLEMS
G) Use theorem 3.8(i).
(8) Use theorem 3.8(ix).
A0) Use theorems 3.8(viii) and 3.16 and Example 5.
A1) Use definitions C.61) and C.36).
A2) Use theorems 3.2 and 3.15 and integrate by parts / times.
CHAPTER 4
A) Use theorem 4.8(iii) and (iv).
B) Use the infinite series for/0 a
C) Compare with Example 4.
D) Axl'2J2/s(±(л/А)*2'3) (with .4 an arbitrary constant). Use theorem 4.12.
E) Use the infinite series for/,.
F) Use the fact that 1 = exp |>{*-(l/*)}].exp [-x{t—(l/t)}] and pick
out the coefficient of x° on the right-hand side.
G) Use theorems 4.8(ii), 4.21 (i) and 4.22(i).
(8) Use the generating function.
(9) Use the generating function for the /„.
A0) Show that both Bessel's equation and Bessel's modified equation may,
by further differentiations, be written in the given form.
j:
jm(x)jn(x) dx = 0 (m ^ n).
A3) Take real and imaginary parts of the infinite series for Jn(i3/2x).
rx
A4) Take real and imaginary parts of the integral tjg(i3/2t) dt; use
J о
theorem 4.8(i) to help evaluate the integral.
A5) Take real and imaginary parts of the asymptotic form of/0(i3/2#),
given by theorem 4.21(i).
CHAPTER 5
A) Use theorem 5.6(ii) and 5.5.
r oo r oo r oo
C) Use the fact that 2 e~*' cos 2xt = e~1' cos 2xt = Re
JO J — oo J — ai
exp (-f2 -\-2ixt)At.

HINTS AND SOLUTIONS TO PROBLEMS 241
#2n fl(«) --- ( 1)" 22"- e*' f e-'7»»+1 sin 2*/ d*.
Vл J о
D) Using the result of problem 3 for both Hn(x) and Hn(y) gives
У Н^Щп = 1 exp (** + y) П P ехр (-и2 - a2 + 2iux
n—0
+ 2iery — 2иг>2) du dv.
Performing the integrations gives the required result.
G) (i) 2-n\Wn)dnn.
(ii) 2"-1я!(уЪ)<»т,п_х -2-(я + 1)!(л/я>»«..+1.
CHAPTER 6
C) Use equation F.3) and integrate term by term.
D) Use equation E.3) and integrate term by term.
E) Use theorem 6.9.
F) JJse theorems 6.11(ii) and 6.10.
G) Use equation E.3).
CHAPTER 7
A-6) Make the substitution x = cos в.
(8) Use the series solution method of Chapter 1, and remember that n is
integral.
CHAPTER 8
A) Use definition (8.2) or theorem 8.5.
B) Use definition (8.2).
C) Use theorem 8.5(ii).
D) Use theorem 8.3(i).
E) Use definition (8.2).
F) Use theorem 8.1 and equation C.17).
CHAPTER 9
C) Use theorem 9.4.
(f>) Use theorem 9.1.
G) (\/s),l'\(y., 1 ; /(; s). I 'so theorem 4.7.

242 HINTS AND SOLUTIONS TO PROBLEMS
CHAPTER 10
A) Integrate by parts.
f °°
B) Write e~'* Ci (t) At as a double integral and change the order of
J о
the integration.
E) Integrate by parts.
F) First prove the equality of the two integrals by making the substitution
у = (л/2) — x in the first. Then consider the second integral, and make
the substitution cos x = cos2 u.
(8) Make the substitution x = tan 0.

BIBLIOGRAPHY
Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions,
Dover, New York A965).
Artin, E. The Gamma Function, Holt, Rinehart and Winston, New York A964).
Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. Higher
Transcendental Functions (Bateman Manuscript Project), Vols. 1-3, McGraw-
Hill, New York A953).
Fletcher, A., Miller, J. С P., Rosenhead, L., and Comrie, L. J. An Index of
Mathematical Tables, Vols. 1 and 2, 2nd edn., Addison-Wesley, Reading,
Mass. A962).
Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics, Cambridge
vUniversity Press, London A931).
Hochstadt, H. Special Functions of Mathematical Physics, Holt, Rinehart and
Winston, New York A961).
Jahnke, E., Emde, F., and Losch, F. Tables of Higher Functions, 6th edn.,
McGraw-Hill, New York A960).
Lebedev, A. V., and Fedorova, R. M. A Guide to Mathematical Tables, Per-
gamon Press, Oxford A969).
Lebedev, N. N. Special Functions and their Applications, Prentice-Hall, Engle-
wood Cliffs, N.J. A965).
MacRobert, T. M. Spherical Harmonics, 2nd edn., Methuen, London A947).
Magnus, W., Oberhettinger, F., and Soni, R. P. Formulas and Theorems for
the Functions of Mathematical Physics, 3rd edn., Springer-Verlag, New York
A966).
McLachlan, N. W. Bessel Functions for Engineers, 2nd edn., Oxford University
Press, London, A955).
Rainville, E. D. Special Functions, Macmillan, New York A960).
Relton, F. E. Applied Bessel Functions, Blackie, London A946).
Slater, L. J. Confluent Hyper geometric Functions, Cambridge University Press,
London A960).
Sneddon, I. N. Special Functions of Mathematical Physics and Chemistry, 2nd
edn., Oliver and Boyd, Edinburgh A961).
Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd edn., Cam-
Cambridge University Press, London A931).

INDEX
Associated Laguerre polynomials
176
definition of 176
generating function for 177
notation for 182
orthogonality properties of 178
recurrence relations for 178
Associated Legendre equation 62
Associated Legendre functions 62
definition of 62
orthogonality relations for 66
properties of 65
recurrence relations for 68
Bessel functions 92
of the first kind 94
asymptotic behaviour of 127
definition of 94
generating function for 99
graph of 134
integral representations for 101
orthogonality properties of 137
recurrence relations for 104
of the second kind 97
asymptotic behaviour of 127
definition of 97
explicit expression for 98
graph of 134
recurrence relations for 106
of the third kind (see Hankel functions)
Bessel series 141
Bessel's equation 92
Beta function 23
definition of 23
Chebyshcv polynomials 187
definition of 187
generating function for 190
orthogonality properties of 192
recurrence relations for 193
series for 188
special values of 192
Chebyshev's equation 189
Confluent hypergeometric equation
210, 212
Confluent hypergeometric function
203
contiguity relations for 212
definition of 203
integral representation of 211
Cosine integral 219
Debye functions 224
Differential equations
satisfied by various special functions
233
solution of by Frobenius' method 1
Elliptic integrals 224
Error function 221
Euler's constant 231
Exponential integral 218
Fresnel integrals 221
Frobenius' method 1
summary of 7
Gamma function 23
ck-finition of 2.1, 30
Kiapli ol 31

246
INDEX
Gauss's equation 207
Gegenbauer polynomials 197
definition of 197
differential equation satisfied by 198
generating function for 197
orthogonality properties of 198
power series for 198
recurrence relations for 198
Generating function 236
for associated Laguerre polynomials
177
for Bessel functions of the first kind
99
for Chebyshev polynomials 190
for Gegenbauer polynomials 197
for Hermite polynomials 157
for Jacobi polynomials 198
for Laguerre polynomials 169
for Legendre polynomials 46
Hankel functions 107
asymptotic behaviour of 127
definition of 107
recurrence relations for 108
Hermite polynomials 156
definition of 157
generating function for 157
orthogonality properties of 161
recurrence relations for 162
special values of 160
Hermite's equation 156
Hypergeometric equation 207, 208
Hypergeometric functions 203
contiguity relations for 210
definition of 203
integral representation of 207
relationship with other special func-
functions 204
Incomplete gamma functions 218
Indicial equation 3
Integral representation
of Bessel functions 101
of confluent hypergeometric function
211
of hypergeometric function 207
of Legendre polynomials 49
of modified Bessel functions 116
Integrals involving Bessel functions
141
Jacobi polynomials 198
definition of 198
differential equation satisfied by 199
generating function for 198
orthogonality properties of 199
recurrence relations for 199
series for 199
Kelvin's functions 120
graphs of 137
Kummer function 204
Kummer's equation 210
Laguerre polynomials 168
definition of 169
generating function for 169
notation for 182
orthogonality properties of 172
recurrence relations for 173
special values of 171
Laguerre's equation 168
Laplace's integral representation 49
Legendre duplication formula 29
Legendre functions of the second kind
70
graphs of 84, 85
properties of 77
Legendre polynomials 42
definition of 46
generating function for 46
graphs of 82, 83, 84
Laplace's integral representation for
49
orthogonality properties of 52
recurrence relations for 58
Rodrigues' formula for 48
special values of 50
Legendre series 55
Legendre's associated equation 62
Legendre's equation 42
Leibniz's theorem 63
L'Hopital's rule 39, 97
Logarithmic integral 219

INDEX
247
Modified Bcsscl functions 110
asymptotic behaviour of 128
graphs of 135
integral representations for 116
recurrence relations for 113
Neumann functions (see Bessel func-
functions of the second kind)
Neumann's formula 78
Orthogonality properties 234
of associated Laguerre polynomials
178
of associated Legendre functions 66
of Bessel functions 137
of Chebyshev polynomials 192
of Gegenbauer polynomials 198
of Hermite polynomials 161
of Jacobi polynomials 199
of Laguerre polynomials 172
of Legendre polynomials 52
of spherical harmonics 80
for Bessel functions of the second
kind 106
for Chebyshev polynomials 193
for Gegenbauer polynomials 198
for Hankel functions 108
for Hermite polynomials 162
for Jacobi polynomials 199
for Laguerre polynomials 173
for Legendre polynomials 58
for modified Bessel functions 113
for spherical Bessel functions 122
Riemann's zeta function 223
Rodrigues' formula 48
Sine integral 219
Spherical Bessel functions 121
asymptotic behaviour of 128
graphs of 136
Rayleigh's formulae for 126
recurrence relations for 122
Spherical harmonics 78
orthogonality properties of 80
Pochhammer symbol 203
Rayleigh's formulae 126
Recurrence relations
for associated Laguerre polynomials
178
for associated Legendre functions
68
for Bessel functions of the first kind
104
Ultraspherical polynomials (see
Gegenbauer polynomials)
Weber-Hermite functions 163
Whittaker's confluent hypergeometric
functions 212
Zeta function 223