.Ml
J^^S^^.% Sri!
FRANK BOWMAN
'III ll o
BESSEL
FUNCTIONS

INTRODUCTION
TO
BESSEL
FUNCTIONS
By Frank Bowman
Dover Publications Inc.
New York

This new Dover edition first published in 1958, is an
unabridged and unaltered republication of the first
edition.
Standard Book Number: 486-60462-4
Library of Congress Catalog Card Number: 58-11271
Manufactured in the United States of America
Dover Publications, Inc.
180 Varick Street
New York, N.Y. 10014

PREFACE
This small book has grown out of lectures given
from time to time in the College of Technology,
Manchester. It is hoped that it may serve as
an introduction to the larger treatises on Bessel
functions and their applications.
The author is indebted to Dr. W. N. Bailey for
reading the manuscript, and Mr. H. Tilsley for
drawing the figures, and particularly to Dr. S.
Verblunsky for reading the manuscript and proofs
and for many helpful suggestions.
F. B.
V

CONTENTS
CHAPTER
I. Bessel Functions of Zero Order
PAGE
II. Applications
III. Modified Bessel Functions
IV. Definite Integrals .
V. Asymptotic Expansions
VI. Bessel Functions of any Real Order
VII. Applications .
Index
VII

TABLE I
-■■■■-- ——
X.
0
1
2
2-405
3
3-832
4
5
5-520
6
7
7-016
8
8-654
9
10
10173
11
11-792
12
13
13-324
14
14-931
Jo(aO-
+ 1
+
+
-—
—
—
—■
<
+
+
+
+
—
.— .
— ■
— •
+ '
+ '
+ '
+ '
•7652
•2239
0
•2601
•4028
•3971
■1776
0
■1506
3001
3001
1717
0
0903
2459
2497
1712
0
0477
2069
2184
1711
0
Ji(s).
0
+
+
+
+
—
—
—
—
— .
+
+
+
+ ■
— .
— ■
— .
— .
i
+ •
•4401
•5767
•5191
•3391
0
•0660
■3276
•3403
•2767
0047
0
2346
2715
2453
0435
0
1768
2325
2234
0703
0
1334
+ -2065
ix

TABLE II
Roots of the Equation Jn(x) = 0
1
2
3
4
5
Jo-0.
2-405
5-520
8-654
11-792
14-931
Ji = 0.
3-832
7-016
10-173
13-324
16-471
J2 =0.
5-136
8-417
11-620
14-796
17-960
J3 = 0.
6-380
9-761
13-015
16-223
19-409
,^ = 0.
7-588
11-065
14-373
17-616
20-827
J5-o.
8-771
12-339
15-700
18-980
22-218
A few of the more modern books on Bessel Functions are given
below; a comprehensive bibliography will be found at the end of
Watson's " Theory of Bessel Functions ".
Watson, " Theory of Bessel Functions " (Cambridge, 1922).
Gray and Mathews, " Bessel Functions " (2nd edn., London, 1922).
Nielsen, " Handbuch d. Theorie d. Cylinderfunktionen " (Leipzig,
1904).
Schafheitlin, " Die Theorie d. Besselschen Funktionen " (Leipzig,
1908).
McLachlan, " Bessel Functions for Engineers " (Oxford, 1934).
Weyrich, " Die Zylinderfunktionon und ihre Anwendungen " (Leipzig
1937).
x

CHAPTER I
BESSEL FUNCTIONS OF ZERO ORDER
§ 1. BesseVs function of zero order.
The function known as Bessel's function of zero order,
and denoted by J0(x), may be defined by the infinite power-
series
J0(x) = 1-- + W~& ~~ 22 . 42 . 62 + ' ' " (1,1)
If uT denotes the rth term of this series, we have
ur (2r)2
which -> 0 when r -> oo, whatever the value of x.
Consequently, the series converges for all values of x, and since
it is a power-series, the function J0(x) and all its derivatives
are continuous for all values of x, real or complex.
§ 2. BesseVs function of order n, when n is a positive
integer.
The function Jn(x), known as Bessel's function of order
n, may be defined, when n is a positive integer, by the
infinite power-series
Xn / £C2 X^ \
J«(x) = 2^TlV ~~ 2 . 2w+2 + 2 . 4 . 2w + 2 . 2w + 4 — * ' 7
(1.2)
which converges for all values of x, real or complex.
In particular, when n = 1 we have
Jl(x) = 2 ~ 2^71+ 2^. 4^. 6 ~ ¥7¥TWTs + ' ' ' (1,3)

INTRODUCTION TO BESSEL FUNCTIONS
and when n == 2
X'
X'
X
6
X
8
J20*0 = 27i ~~ 22. 4. 6 + ¥. 42. 6 . 8 22. 42. 62. 8 . 10
+ . . . (1.4)
We note that Jn(x) is an even function of x when n is
even, odd when n is odd.
The graphs of JQ(x), Ji(#) are indicated in Fig. 1.
10
•8
•6
•4
•2
0
.-2
-•4
1
1 i
^ J
2\
.1 /V)
3 \
4
5/
«0»-
6 >
7
8 \
9
X
10
Fig. 1.
Extensive tables of values of Jn(x), especially of J0(#)
and J^x), have been calculated on account of their
applications to physical problems.*
§ 3. BesseVs equation of zero order.
By difiEerentiating the series for J0(x) and comparing
the result with the series for Jt(x), we find f
dJ0(x)
dx
= — J^x).
(1.5)
((
((
* See Watson : " Theory of Bessel Functions " ; Gray and Mathews :
Bessel Functions " ; Jahnke und Emde : " Funktionentafeln " ; Dale :
Five-Figure Tables."
d
f Cf. — (cos x) = — sin x.
dx

BESSEL FUNCTIONS OP ZERO ORDER 3
Again, after multiplying the series for Jt(x) by x and
differentiating, we find
d
fa{xJi(x)} = xJo(x)- • • • (1-6)
Using (1.5), we can write (1.6) in the form
d / dJJx)\ T , v ,, „.
/^ + ^ + ^^) = 0. . (1.8)
Thus y — J0(x) satisfies the linear differential equation
of the second order
or ^+11+^ = 0, • • (L10)
which is known as BesseVs equation of zero order.
§ 4. Bessel functions of the second kind of zero order.
A solution of Bessel's equation which is not a numerical
multiple of J0(x) is called a Bessel function of the second kind.
Let u be such a function, and let v — J0(x) ; then, by (1.10),
xu" + u' + xu = 0,
xv" + v' + %v = 0.
Multiplying the first of these equations by v and the
second by u and subtracting, we have
x[u"v — uv") + m'v — mv' = 0,
d
which, since u"v — uv" = -r-(u'v — uv'),
d
can be written -r-{x(u'v — uv')} = 0.

4 INTRODUCTION TO BESSEL FUNCTIONS
Hence x(u'v — uv') = B,
where B is a constant. Dividing by xv2, we have
u'v — uv' B
v2 xv2
that is, "t~(- ) —
dx\v/ xv2
and hence, by integration,
- = A + B —.
v J aw2
Consequently, since 0 = Jo(#)>
u = AJ0(a:) + BJoO*)]"^^ . (1.12)
where A, B are constants, and B =)= 0 since u is not a
constant multiple of J0(x), by definition.
§ 5. If, in the last integral, J0(x) is replaced by its series,
and the integrand expanded in ascending powers of x, we
find for the first few terms
xJ02(x) ~ x + 2 + "32 + ' ' '
and therefore
3»{x)\Jh)=Jo(a;)(log x + xi+ws + --:.
1 - 2"2 + • • ■) (l + 128 + • * •
^.2 g^.4
= Jo(») log » + ~l - Y28 + " * "
Consequently, if we put
r0(») = J0(x) lo§ ^ + J - 128 + • • • (L13)
then F0(#) is a particular Bessel function of the second
kind ; it is called Neumann's Bessel function of the second
kind of zero order; the general term in its expansion can be
obtained by other methods (§ 8).

BESSEL FUNCTIONS OF ZERO ORDER 5
Since J0(x) -> 1 when x -> 0, it follows from (1.13)
that Y0(x) behaves like log x when x is small, and hence
that Y0(x) -> — oo when x -> + 0.
§ 6. It follows from (1.12) that every Bessel function
of the second kind of zero order can be written in the form
AJ0(a:) + BY0(x).
The one that has been most extensively tabulated is
Weber's,* which is denoted by Y0(x) and is obtained by
putting
A=-?(log2-y), B = -
and hence
Y0(tf) = -{r0(s) - (log 2 - y)Jo(tf)}, . (1.14)
TT
where y denotes Euler's constant, defined by
y = lim(l +1+^ + . . . +- —log n)= 0-5772... (1.15)
n^oo\ 2 3 n ° /
We note that, when a? is small,
Y0(s) = -{log x - (log 2 - y) . . .} . (1.16)
TT
the remaining terms being small in comparison with unity.
As far as applications are concerned, it is usually
sufficient to bear in mind that Y0(x) is a Bessel function of the
second kind whose values have been tabulated ; that x
must be positive for Y0(x) to be real, on account of the term
involving log x in (1.13); and that Y0(#)-> — oo when
x -> + 0.
The graphs of J0{x) and Y0(x) are shown together in
Fig. 2.
§ 7. General solution of BesseVs equation of zero order.
Since J0(x) and Y0(x) are independent solutions of the
equation
^ + 1^ + 2, = 0,
(J. Jb JC CvJb
* Watson, § 3.54.

6 INTRODUCTION TO BESSEL FUNCTIONS
the general solution can be written
y==AJ0(x) + BY0(x), . . (1.17)
where A, B are arbitrary constants, and x > 0 for Y0(x) to
be real.
If we replace x by kx, where h is a constant, the equation
becomes
1 d2y , 1 dy ,
&2 dx* ^ kx kdx ^y
10
•8
•6
•4
•2
0
^o<*>
1
v/ /~ I
Yo
\
1
\
\
2\
1-«/
V
\
\
\
3 4
V 5
6 lj
8 \
9
10
Fig. 2.
Multiplying by &2, we deduce that the general solution of
the equation
d2y , 1 dy , 7Q
do;2 ' x dx ^ u
can be written
(1.18)
(1.19)
y = AJ0(te) + BY0(fe)
where k > 0 for Y0(&#) to be real when a; > 0.
§ 8. 2¾ general solution by Frobenius's method.
Bessel's equation belongs to the type to which Frobenius's
method of solution in series can be applied. Put
d2 . d
^3 (^ + 35+*)* '
(1.20)

RESSEL FUNCTIONS OF ZERO ORDER 7
and make the substitution
y = afi + cxxp+* + c2xP+* + czxp+* + . . . (1.21)
We obtain, after collecting like terms,
Li/ = p^-1 + (P + l)2c^ + {(P + 2)2c2 + 1)^+1
+ {(p + 3)2c3 + cJ^+2 + . . . (1.22)
Now let c19 c2, c8, . . . be chosen to satisfy the equations
(p + l)2Ci = 0,
(p + 2)*c2 + 1 = 0,
(p + 3)2c8 + cx = 0, . . .
Then, unless p is a negative integer,
Cj :==1 C3 == Cg === C7 = ... == u,
X
c« =
c, = —
(P + 2)*'
c2
(p + 4)2 (p + 2)»(p + 4)2
Substituting these values in (1.21) and (1.22), we deduce that, if
'-"b-T^+u+inrnr-'"} (L23)
and if p is not a negative integer, then
*S+S+^=p2a5e_i- • • • (i-24)
Putting p = 0 in (1.23) and (1.24) we see again that
y = J0(x) =1-- + __ - ___ 4- . . .
is a solution of Bessel's equation
Further, differentiating (1.24) partially with respect to p, we
get
£®+£@D+-(¾)-*-+"->••*
and hence, when p = 0,
^VM + 1/^, W^l -0
^"Up/o ^IWi \ Wo ~
from which it follows that (^y/^p)p^o is a second solution. Now,
from (1.23),
Xdx*
x

8 INTRODUCTION TO BESSEL FUNCTIONS
JJ-zMogz|l (p + 2)2+(p + 2)2(p + 4)2 • • .}
p ( 2x* __1 2a;4 / 1 1 \
+ * \(p + 2)2 P + 2 (P + 2)2(P + 4)Ap + 2 + M^J
2a° / 1 1 1 \ |
+ (p + 2)2(P + 4)2(P + 6)2lp + 2 + p + 4 + p + 6j --7-
Hence, putting p = 0 and Y0(#) = (½^p)p=o we obtain the second
solution
#2 x* ,, . ,v . #6
Fo(») = Jo(^)loga; + 2i - WT&k1 + *) + 22 . 42 . 62^ + * + *)""
. . . (1.25)
which is Neumann's Bessel function of the second kind of zero order,
in a form which indicates the general term (§5).
It follows that the general solution of the equation can be written
y = AJ0(z) + BY0(x),
which is equivalent to (1.17).
§ 9. To examine the convergence of the series that follows
J0(x) log x in (1.25), we can put, by (1.15),
1 + 3 + \ + • • • + \ = log n + y + €n' ' (L26)
where en -> 0 when n -+ oo. Hence if ur denote the rth term of
the series, we have
^~i ~ ~ W? log(r- 1) + y + €r-19
which -> 0 when r -> oo, whatever the value of x. Consequently,
the series converges absolutely for all values of x, real or complex.
§ 10. Integrals.
We notice next certain integrals involving Bessel functions
in their integrands. Firstly, from (1.5) and (1.6) we have
I3x(x)dx = — J0(ff), • • (1^27)
$xJ0(x)dx = xJ^x). . . (1.28)
Secondly, we note that the indefinite integral
SJ0(x)dx .... (1.29)
cannot be expressed in a simpler form, but on account of
its importance the value of the definite integral

BESSEL FUNCTIONS OF ZERO ORDER 9
[XJ0(t)dt . . . (1.30)
Jo
has been tabulated.*
Thirdly, we shall obtain a reduction formula for the
integral
$xnJ0(x)dx. . . . (1.31)
Put un = $xnJ0(x)dx = jxn"1d{xJ1(x)},
by (1.6). Then, integrating by parts, we have
un = xn~x . xJt(x) — jxJ^x) . (n — l)xn"2dx
= xn3x(x) + (n — l^-HJ^x),
by (1.5) ; and on integrating by parts again,
un = xnJt(x) + (n — 1)^-^0(^) — (n — l)2Sxn~2J0(x)dx,
that is,
un = xnJx(x) + (n — 1)^-^0(2:) — (n — l)X-2 (1-32)
which is the reduction formula required.
It follows that, if n is a positive integer, the integral
(1.31) can be made to depend upon (1.28) if n is odd, or
upon (1.29) if n is even.
§ 11. LommeVs integrals.
Put u = J0(olx), v — J0(jix), where a, j8 are constants ;
then, by § 7, writing u' = du/dx, u" = d2uldx2, . . . we have
xu" + u' + ol2xu = 0, . . (1.33)
xv" + v' + fi2xv = 0. . . (1.34)
Multiplying the first of these equations by v and the
second by u and subtracting, we get
x(u"v — uv") + (u'v — uv') = (/32 — ol2)xuv,
d
that is, -j-{x(u'v — uv')} = (/32 — ol2)xuv,
and hence, by integration,
(/32 — oi2)lxuvdx = x(u'v — uv'),
* Watson, p. 752.

10 INTRODUCTION TO BESSEL FUNCTIONS
and therefore
(/32 — a2)JaJo(a#)J'0(fix)dz
= x{aJ0'(ux)J0(fix) - pJ0'(px)J0(Kx)}, (1.35)
since u' = olJq(olx), v' = j8J0'(j8:r).
Again, multiplying (1.33) throughout by 2xu\ we have
d
2xu'-r-{xu') + 2oL2x2uuf = 0,
7
or -r-(x2u'2 + a2A2) — 2ol2xu2 = 0.
Integrating, we get
x2u'2 + a2#2^2 = 2oL2$xu2dx,
and hence
2<x2SxJ02(a.x)dx = ol2x2J 02(olx) -f x2ol2J0'2(olx),
and therefore
SxJ02(xx)dx = %x2{J02(olx) + J^kx)}. . (1.36)
In particular, when we integrate between the limits
0 and 1, we find from (1.35) and (1.36), respectively,
(jS2 - a2)[ xJ0{aux)J0(fix)dx
= aJ0'(a)Jo(i3) ~ j8J0Wo(a), (1-37)
f xJ02(oLx)dx = l{J02(a) + Ji2(a)} . (1.38)
J o
Corollary 1. If a, /? (/32 4= a2) are two roots of the
equation J0(x) == 0, then
I xJ0(olx)J0(j3x)dx = 0.
J o
This follows immediately from (1.37).
Corollary 2. If a, /3 (/32 4= a2) are two roots of the
equation
xJ0'(x) + HJ0(#) = 0,
where H is a constant, then
1 xJq(olx)J0(fix)dx = 0.
J o
This also follows at once from (1.37).

BESSEL FUNCTIONS OF ZERO ORDER 11
Examples I
1. Show that
(i) $x2J0(x)dx = x2Jx(x) + xJ0(x) — jJ0(x)dx9
(ii) Sx^J^dx = x(x2 - 4)Ji(a) -f 2x*J0(x),
(iii) ]x*J0(x)dx = x*(x* - 9)J1(x) + Sx(x2 - 3)J0(z) + 9JJ0(a;)^.
(iv) fx \ogxJ0(x)dx = J0(#) + # logo; Ji(t»).
2. Show that
r1 1
(i) I xJ0(oix)dx = — Ji(oc),
J o oc
f1 11 If*
a2J0(aa;)da; = -J^a) + -^(a) 5l J\(t)dt,
q OC OC OC J q
JI a2 4 2
a3J0(aa;)da; = —Jx(a) + —J0(oc),
0 OC OC
(iv) [*(:
4 2
a2)J0(aa;)da; = —^(oc) sJ0(a).
a3 AX ' a2
3. If a is any root of the equation J0(x) = 0, show that
r1 1
(i) I J1(oLx)dx = —,
J0 a
Jot
J±(x)dx = 1,
0
J 00
Jx(x)dx = 1.
o
If a ( + 0) is a root of the equation J±(x) = 0, show that
(iv) f xJ0(atx)dx = 0.
J o
4. If J0 == J0(x), Jx == Ji(#), show that
(i) fijjdx = - iJ02,
(ii) ^xJtiJ1dx = — J#J02 + JJJ02dr,
(iii) JattaJjdx = ^jy.
5. Show that
(i) Ja;J02<to = i*2(J02 + Ji2),
(ii) ]xJx2dx = Ja;2(J02 + Ji2) - xJ0Jl9
(iii) 2Sx*J0*dx = ^3(J02 + Ji2) + i*2J0Ji + i^Jo2 - iJJo2^
(iv) 2!x*JSdx = J^W + Ji2) - f*2J0Ji - I^Jo2 + IPV^,
(v) 3Ja3J02<fo = lx*(J0* + Jx2) + xsJ0J1 - x2Jt29
(vi) SfaPJSdx = itf4(J02 + Ji2) - 2a?3J0J1 + 2:^2.
6. If u = fxJ0(<zx)J0(f}x)dx, v = $xJ\(<xx)J\(fix)dx, show that
ecu — Pv = xJ ^clx)! 0(fix)9
fiu — OLV = #J0(a#) J !(/?#).

12 INTRODUCTION TO BESSEL FUNCTIONS
Deduce the values of u and v, and by differentiating partially
with regard to j3, and then putting j9 = a, deduce the values of
$xJ 02(oLx)dx, jxJ12(ccx)dx.
7. If f(x) is any Bessel function of zero order, show that
dx J0(x) xJ02(x)
where B is a constant; and hence that, if a, j3 are two consecutive
positive roots of the equation J0(x) = 0, the fraction f(x)/J0(x)
increases steadily from — oo to + oo (or decreases steadily from + oo
to — oo) when x increases from a to 0. Deduce that the equation
f(x) — 0 has one root between a and 0.
§ 12. Behaviour of Bessel functions when x is large.
If in Bessel's equation
Xdx^&x^Xy-">
we make the substitution
u = y^x . . . (1.39)
we find that u satisfies the equation
d2u
0+=
u.
dx2 \ 4:X2,
Now, when x is large enough, l/4#2 is as small as we
please compared with 1, and then we have approximately
d2u _
dx2
of which the general solution is u = C cos (x — A), and we
infer, by (1.39), that every solution of Bessel's equation
behaves like
C cos (x — A)
when x is large, where C, A are constants.
In fact, it will be shown in § 80 that, if x > 0,
J1°{X) = Qx)\°°* (X ~ i) + P{X))■> ■ (L40)
Y°{x)=GDVk* -1)+q{x)}' • (L4i)
where p(x) -> 0 and q(x) ~> 0 when x -> + oo.

BESSEL FUNCTIONS OF ZERO ORDER 13
Hence any solution f(x) of Bessel's equation of zero
order may be written in the form
f(x) = AJ0(tf) + BY0(z)
(A2 + B2)* {cos (x - A) + r(x)}, (1.42)
where A = \tt + tan~1(B/A), and r(x) -> 0 when x -> + °°-
Corollary. Every solution tends to zero when x -> + oo.
§ 13. Roots of the equation f(x) = 0, where f(x) denotes
any Bessel function of zero order.
Iff(x) denotes any Bessel function of zero order, it follows
from (1.42) that, for large values of x, the roots of the
equation f(x) — 0 are approximately those of cos (x — A) = 0.
We infer that the equation f(x) = 0 has an infinite number
of real roots, and that the large roots are A + (s — |)7r,
approximately, where s is any large positive integer.
In particular, the large positive roots of the equation
J0(x) = 0 are (s — £)77-, those of Y0(x) = 0 are (s — f)77-,
approximately, where s is any large positive integer.
§ 14. None of these roots can be a repeated root.
Proof. Let a be a root, and note, firstly, that a =j= 0>
since no Bessel function of zero order vanishes when x = 0.
Secondly, suppose that a could be a repeated root ;
then /(a) = 0, /'(a) = 0, and by substituting x — a in the
differential equation
xr(x)+f(x) + xf(x) = 0, . . (1.43)
it would follow that a/"(a) = 0, and hence /"(a) = 0, since
a #= 0.
Moreover, by differentiating the differential equation
and putting x = a again, it would follow that /'"(a) = 0,
and, by repeating this process, that all the derivatives of
f(x) would vanish when x = a. Consequently, from Taylor's
series
= (-)
\ttxJ

14 INTRODUCTION TO BESSEL FUNCTIONS
we should have f(x) s 0. Hence, x = a cannot be a
repeated root.
§ 15. Roots of the equations J0(x) = 0, Jx(x) = 0.
I. The equation J0(x) = 0 has an infinite number of
real roots, all simple.
This follows from §§ 13, 14 as a particular case. Another
proof of this theorem will be given later (§ 96, II).
II. The equation J0(x) = 0 has no purely imaginary roots.
Proof. Put x = ip, (/3 4= 0), in (1.1). Then
Jo(^) = ! + 22 + 22 . 42 ^~ 22 . 42 . 62 ^ ' * '
which cannot vanish, since all the terms on the right are
positive. Hence x = if} cannot be a root of J0(x) == 0.
III. The equation J0(x) = 0 has no complex roots.
Proof. Suppose that a + ib could be a root (a 4= 0,
b 4= 0). Then the conjugate a — ib would also be a root,
because the coefficients in the series for J0(x) are all real;
and, since (a + ib)2 4= (a — ib)2, it would follow from
§ 11, Cor. 1, that
I xJ0{(a + ib)x}J0{(a — ib)x}dx == 0.
But this is impossible, because the integrand is positive
throughout the range of integration, being the product
of x and a conjugate pair of complex numbers. Hence,
a + ib cannot be a root.
IV. The equation J0'(x) = 0 has an infinite number of
real roots.
Proof. By Rolle's theorem, since J0(tf) and J0'(^) are
continuous, the equation J0'(^) = 0 has at least one root
between every pair of roots of J0(x) = 0, and hence, by I,
has an infinite number of real roots.
V. The equation J^x) = 0 has an infinite number of
real roots.
This follows from IV, since J^x) = — J0'(^).

BESSEL FUNCTIONS OP ZERO ORDER 15
VI. The equations J0(x) = 0, 3x(x) = 0 have no common
root.
Proof. By I, J0(x), Jo'M have no common root. But
Ji(x) = ~~ Jo'Wj therefore J0(x), Ji(#) have no common
root.
VII. The equation
xJ0'(x) + HJ0(a) = 0, . . (1.44)
where H is any real constant, has an infinite number of real
roots.
Proof. Put
</>{x) s xJ0'{x) + HJ0(a)> . . (1.45)
and let a, |3 be a pair of consecutive positive roots of the
equation J0(x) = 0. Then
<£(a) - aJ0'(a), fl|8) = j8J0'(|8). . (1.46)
Now, since a, jS are simple roots of J0(x) = 0, neither
J0'(a) nor J0'(/3) can be zero ; and since J0(x) is continuous,
J0'(a) and J0'(j8) must be of opposite sign. Since a and jS
are positive, it follows from (1.46) that </>(<*) and </>(/3) are
of opposite sign, and hence, since <j)(x) is continuous, that
the equation <f>(x) = 0 has a root between a and /3.
Consequently, the equation <j)(x) = 0 has an infinite number
of real roots, at least one between every pair of
consecutive roots of J0(x) = 0.
§ 16. A few of the smaller positive roots of the equations
J0(x) = 0, Sx(x) = 0, along with those of the equations
in(x) = 0, (n = 2, 3, 4, 5) are given in a table at the beginning of
the book.
§ 17. Fourier-Bessel expansion of zero order.
Let a1? a2, a3 . . . denote the positive roots of the
equation J0(x) = 0, arranged in ascending order of
magnitude.
In general (see § 99), any ordinary function of
mathematical physics f(x), arbitrarily defined in the interval

16 INTRODUCTION TO BESSEL FUNCTIONS
0 < x < 1, can be represented over this interval by an
infinite series of the form
f(x) = AxJofcai) + A2J0(a;a2) + A3J0(#a3) + . . . (1.47)
which is called the Fourier-Bessel expansion of f(x) of zero
order.
For the expansion to hold good up to x = 1, a necessary
condition is plainly /(1) = 0, since every term of the
expansion vanishes when x = 1.
§ 18. Assuming the expansion to hold good, and that
term-by-term integration can be justified, the coefficients
A1? A2, A3 . . . can be determined with the aid of Lommel's
integrals (§ 11).
Multiply (1.47) throughout by xJ0(xois)dx and integrate
between the limits 0 and 1 ; then the general term on the
right-hand side will be
•l
xJ 0(xaLr)J 0(xa.s)dx
o
which vanishes when r 4= «5, by § 11, Cor. 1. Consequently,
every term on the right vanishes except the one in which
r = s, and we get
I xf(x)J0(xaLs)dx = As\ xJ02(xo(.s)dx = ^ASJ^(oLg),
Jo Jo
by (1.38), since J0(as) = 0. Hence
2 f1
As= xf(x)J0(xa.8)dx. . . (1.48)
The simplest cases in which this integral can be
evaluated in terms of tabulated functions are
(i) f(x) = a polynomial in x (see § 10), or log x;
(ii) f(x) = J0(Jcx), where k is a constant (see § 11);
(iii) f(x) = (1 — x*)*, where p > — 1 (see § 91).
Ex. Find the Fourier-Bessel expansion oif(x) = 1 — x2.
In this case we have, by (1.48),
I;
2 f1
A« == T 2/ \ X(l — X2)J0(XQLa)dxt

BESSEL FUNCTIONS OF ZERO ORDER 17
and hence, replacing a by as in Exs. I, 2 (iv), and putting J0(as) = 0,
Consequently,
I _ X2 ^ S f Jo(g«l) , JoQe«i) , Jpfotta) _L A
la^J^ax) a23Ji(a2) as'J^aa) " " "J'
which may also be written
1-^=87-¾¾ . . . (1.49)
where the summation extends over the positive roots of the equation
Jo(s) = 0.
This expansion holds good over the range — 1 < x < 1, since
1 — x2 is even.
We may gain an idea of the numerical values of the coefficients
from the table at the beginning of the book; thus we find approximately
1 - x2 = MO&Tofca!) - -140J0(ira2)
+ -045J0(a;as) - -02lJ0(ira4) + -012J0(ira6) — . . .
§ 19. Dini expansion of zero order.
An expansion similar to (1.47), but based upon the roots
of the equation
xJ0'(x) + HJ0(x) = 0, . . (1.50)
is called * the Dini expansion of f(x) of zero order. Three
cases may be distinguished, depending upon the values of
the constant H.
I. If H > 0, the Dini expansion has exactly the same
form as (1.47), viz.
f{x) = AXJ^xolj) + A2J0(xcl2) + A3J0(a:a3) + . . .
where 0Cj, ^2j ^3j ... are the positive roots of (1.50), and
0 <x < 1.
The coefficients are determined in the same manner as
before. We multiply both sides by xj^xu^dx, and integrate
between the limits 0 and 1 ; then the general term on the
right-hand side will be
xJ 0(xoLr)J 0(xoLs)dx
I
0
* Watson, p. 580.

18 INTRODUCTION TO BESSEL FUNCTIONS
which vanishes when r =}= s, by § 11, Cor. 2. Consequently
every term on the right vanishes except the one in which
r = s, and we get
xf{x)J 0(xoL8)dx = As xJ02(xoL3)dx = lAs{J02(as) + JxV,)},
Jo Jo
by (1.38), which determines the coefficient As.
II. If H — 0, equation (1.50) becomes
xJ0'(x) = 0 or xj±(x) — 0,
which has a double root x = 0, and in this case the series
has an initial constant term, thus
f(x) = A0 + AiJoOraO + A2J0(aa2) + . . . (1.51)
which may be regarded as an expansion based upon the
roots, 0, of the equation J±(x) — 0. The
constant A0 may be obtained by multiplying throughout
by xdx and integrating from 0 to 1. Thus we get, using
Exs. I, 3 (iv),
f1 f1
1 xjixjCbX '•—'■ -ci-0 I x ax — o'Aq
Jo Jo
which gives A0.
Since Ji(as) = 0, the constants As (s =t= 0) are given
now by
•1
xf(x)J 0(xx3)dx = lAsJ02(as).
o
III. If H< 0, the equation (1.50) has two purely
imaginary roots, and the Dini expansion involves an initial
term depending on them (see § 98, III).
Examples II
1. If a is a typical positive root of J0(x) = 0, obtain the
following expansions :—
J0{0LX)
i;
4
(ii) x2 = 2> 3T . ,J0(ax).

BESSEL FUNCTIONS OF ZERO ORDER 19
(iii)Jo(M = 2Jo(^(g8lJffi(g).
(-) * = 22{^T) - FJ?(=jI/*)*}^«)-
(v) log I = 2^
Jo(okb)
2. If a denotes a typical positive root of Ji(ck) = 0, obtain the
following expansions :—
(^ = 1 + ^2¾¾
2 ^-a2J0(a)
<fi>(1 - *2>2 = i - "S^g-
3. In an expansion of the form
f(x) = A0 + 2 AJ0(aa;), (0 < x < 1),
a
where Ji(a) = 0, show that JA0 is the average value of xf(x) over
the interval 0 < x < 1.
4. If a is a typical positive root of J0(x) = 0, and — a < r < a,
show that
a2 _ r2 = 8a2J 1_T (f?\
^-a3J Ja) \a/
5. If f(x) is defined arbitrarily in the interval 0 < x < h, and
a is a typical positive root of J0(x) = 0, show that
2J (—)

CHAPTER II
APPLICATIONS
§ 20. Uniformly stretched uniform membranes.
Bessel functions find their simplest applications in
certain ideal problems of mathematical physics, e.g. the
vibrations of a uniformly stretched uniform circular
membrane.
It is assumed that the ideal membrane is perfectly
flexible, i.e. that the stress across any line drawn on the
Fig. 3.
membrane is a tension perpendicular to the line at every
point and in the tangent plane to the membrane.
Consider such a membrane, plane in its equilibrium
position, under the action of a uniform tension T per unit
length over its boundary, and let T' be the tension per unit
length across a line element AB (Fig. 3). Take rectangular
axes in the plane of the membrane, with Oy parallel to AB,
and let AB = dy. Through A, B draw lines parallel to Ox,
meeting the boundary in C, D respectively. Let CD = ds
20

APPLICATIONS 21
and let the normal to the element CD make an angle j/r
with Ox. Consider the equilibrium of the element of
membrane ACDB. Resolving in the direction of the #-axis,
and remembering that the stress is everywhere perpendicular
to the boundary, so that the stresses across the edges AC,
BD have no components in the direction of the #-axis,
we have
Tdy = Tds . cos ifs = T . ds cos if, = Tdy,
and hence, dividing by dy,
T' = T;
thus the tension per unit length across any line drawn in
the membrane is constant and equal to that over the
boundary.
§ 21. Differential equation of the small vibrations of such
a membrane.
Suppose the plane of the membrane horizontal and the
effect of gravity negligible, and consider a motion in which
the displacement of every
point is small and the
gradient is everywhere small. Let
ABCD (Fig. 4) be arectangular
element of edges dx, dy, with
its centre at the point (x, y)
and its edges parallel to the
co-ordinate axes ; and let z -pio. 4.
be the displacement of the
centre from its equilibrium position.
The motion of the element is caused by the tensions
acting across its boundary and by possible external forces.
Now the tension across a line element of length dy passing
through the centre parallel to the edges AD, BC is Tdy,
and its component perpendicular to the xy plane is
approximately
Tdy . ^, = T-dy,
u lx9 ^x *'
y
0
1
!
D, C
Aj dx
i
i
dy
B
i
i

22 INTRODUCTION TO BESSEL FUNCTIONS
since the gradient is small. The corresponding components
across the edges BC, AD are respectively
Ix u lx\ to V 2
These act in opposite directions on the elementary rectangle,
and together they contribute
d22
to the forces causing its motion. A similar contribution
d22
T—j:. dx dy
is supplied by the tensions across the edges AB, DC. If
we suppose that in addition there is an external force Z
per unit area, and that the mass per unit area is cr, the
equation of motion of the element is
d22 /d22 <)22\
(Tfc^.-^2=-T^—+ —J .dxdy + Z.dxdy + . . .
the dots at the end indicating unknown terms of higher
order of smallness in dx, dy. Hence, dividing by dxdy and
putting c2 = T/cr, we obtain, in the Hmit, the differential
equation
u2 Vto2^^2/ <*' ' '
§ 22. Polar co-ordinates.
In polar co-ordinates, if we
consider the motion of the
element ABCD (Fig. 5) bounded
by the circles r ± \dr and the
radius vectors d ± ?dd, the ten-
FlG 5 sions across the edges AD, BC
contribute a component
1( T^. rdd) . dr, = TfS + lpf.rdr dO,

APPLICATIONS 23
to the forces causing the motion of the element, and the
contribution from the edges AB, DC is
~ T-
. dr\ ,dd, = r(~^\ .rdrdd.
i0\ no
The equation of motion of the element now leads to the
differential equation
^ = C.(^ + I^ + I^+? (22)
M2 W2 ^ r7>r^ r* WV ^ a K ]
This could also be deduced analytically from (2.1) by
changing the independent variables from x, y to r, 9.
§ 23. Special cases.
I. If the membrane is vibrating freely, i.e. if there is
no external force, then Z = 0. If, in addition, the motion
is symmetrical about the origin, i.e. z is independent of 0,
we have <te/d0 = 0, and under these conditions equation
(2.2) reduces to
II. If z is the same at every point along any line parallel
to the y-axis, so that z is independent of y, then 'dz/'dy = 0
and equation (2.1) reduces to
^ - c2— (2 4)
The general solution of this equation is known to be
z^f(x — ct) + ¥(x + ct), . . (2-5)
where /, F are arbitrary functions, and by considering either term
of this solution we are led to a physical meaning of the constant c.
Thus, if we consider the first term and put
zx — f(x — ct), .... (2.6)
then zx represents a displacement in which, at time t = 0,
*!=/(*) (2-7)
At time t' later, taking a new origin at the point x = ct', zx = 0,
and putting x — x' + ct', we find
*1 = /(*')•

24 INTRODUCTION TO BESSEL FUNCTIONS
Comparing this with (2.7), we infer that (2.6) represents a
displacement which is travelling in the positive direction of the #-axis,
unchanged in shape, and covering a distance ct' in time t', so that
the displacement is travelling with uniform speed c.
Similarly, if we put
z2 — F(x + ct)
we should find that z2 represents a displacement travelling in the
negative direction of the a>axis with uniform speed c.
Thus, c is the speed with which a one-dimensional displacement
could travel, unchanging in shape, across the membrane. It
maybe verified that c = V(T/a), has the dimensions of velocity.
§ 24. Normal modes of vibration.
It is known that, when any mechanical system is
vibrating freely about a position of stable equilibrium, it has
normal modes of vibration.
A normal mode of vibration is one in which all the
particles of the system vibrate with the same period and
pass through their mean positions simultaneously.
§ 25. Symmetrical normal modes of vibration of a circular
membrane, with the circumference fixed.
Let a be the radius of the membrane. The motion being
symmetrical about the centre, the equation to be satisfied
by the displacement z is (2.3), viz.
d2z fbh 1 lz\ /0 Qv
H* = C\*i + r»)- ' ' (2>8)
For the normal modes of vibration, z must be of the form
z = R cos (cut — e), . . (2.9)
where R depends upon r only ; and since the circumference
is fixed, the solution must satisfy the condition
z = 0 when r = a, for all values of t. . (2.10)
Substituting (2.9) in (2.8) and dividing throughout by
cos (cot — e), we find that R must satisfy the equation
dm l <m co2 ^
dr2 r dr c2

APPLICATIONS 25
and hence, by § 7, that R must be of the form
R = «.(2) + BY.(=).
Now, in the present problem, 2 is small at the centre
r = 0. But the Bessel function of the second kind Y0(cor/c)
becomes infinite at r = 0 (§6). Consequently, the
constant B must be zero, and hence
Z = AJ0(—J cos (cot — e). . . (2.11)
It remains to satisfy the boundary condition (2.10).
Substituting z — 0, r = a, we get
Jo(^)=0. . . . (2.12)
This equation determines the possible values of co and gives,
since co must be positive,
wa _
0^1) ^2j 35 • • • O^o • • •
C
where oc-i, &9' • • • are the positive roots of the equation
J0(x) = 0. It follows that, for the ideal membrane, there
are an infinite number of normal modes of vibration, whose
periods 277-/0^, 27r/co2J . . . 27r/cos, . . . are determined by
Cal Ca2 Cas /o i Q\
COi = , (Oo = , . . . CO* = , .... IZi.lOJ
a a a
If we distinguish the successive normal modes by the
suffixes 1, 2, 3, . . . we can write
zx = dJo^) cos (coxt - €l),
Z2 = C2J°(^r) C°S W "~ ^
Z3 = C3J0( —-J COS (co3£ — £3), . . .

26 INTRODUCTION TO BESSEL FUNCTIONS
where C1? C2, C3, . . . denote arbitrary constants which
must, however, be small compared with a, so as to keep
the displacement small at every point of the membrane.
In the first normal mode a radial section through the
membrane at any instant has the shape of the graph of
y = J0(x) from x = 0 to x = a1? since r varies from 0 to a
(Fig. 6.1).
In the second mode the shape is that of y = J0(x) from
x = 0 to x = a2. There is a nodal circle at r = aoL1joL2
(Fig. 6.2).
Fig. 6.1. Fig. 6.2. Fig. 6.3.
In the third mode the shape is that of y = J0(x) from
x = 0 to x = a3. There are two nodal circles at r = aa1/a3,
r == aoL2/oLz, respectively (Fig. 6.3) ; and so on.
§ 26. General initial conditions.
The most general motion of the membrane can be
represented as a sum of arbitrary multiples of the normal
modes, thus
^ = 2 CSJ0(^) cos M - es), . (2.14)
or, what amounts to the same thing,
*>, /roL\
z== Z Jo(-^)(As cos <*J + Bs sin cost). (2.15)

APPLICATIONS 27
Every term of this expression satisfies the differential
equation (2.8) and the condition that z = 0 at r = a; and
the coefficients As, Bs can be chosen so as to satisfy arbitrary
initial conditions. For, let these be
z = <f>(r), 0 < r < a, t = 0 ; . (i)
lz/U = ifs(r), 0 < r < a, t = 0. . (ii)
Then, putting t = 0 in (2.15) and using (i), we get
*<r> = AlJo(^) + A2J0(^) + A3J0(^) + . . .
Also, differentiating (2.15) with respect to t and then
putting t = 0, and using (ii), we get
The coefficients As, Bs have to be chosen so that these
equations are satisfied by values of r between r = 0 and
r = a. If we put x = rja, the equations become
(f)(ax) = AXJ ^xolj) + A2J0(#a2) + A3J0{xol3) + . . .
which have to be satisfied between x = 0 and a; = 1.
Consequently, by (1.48),
2 f1
As = y-r/—vl #<£(&#) J0(#as)cfcr, . . (2.16)
Ji (as)Jo
2 fi
Bs = —TTi—\l %Max)J0(xaL8)dx. . (2.17)
^8^1 (as)Jo
With these values of the coefficients, the value of z is
given by (2.15).
Examples III
1. If z = C(a2 - r2) and cte/<tf = 0 when t = 0, show that
r /TOLA COLxt (rOL2\ CK2t
hi—l) cos—- Jo(—-J cos—-
\ a J a , \ a ) a
«i8Ji(«i) oci'Ji(ai)
2. A uniformly stretched circular membrane, with the
circumference fixed, is acted upon by a force Z = F cos pt per unit area,

28 INTRODUCTION TO BESSEL FUNCTIONS
where F is a constant. Show that, in the notation used above, the
forced oscillation thus caused is given by
s = op>Zpa/c)Mi;) - j»(t)} cos^
provided 2w/p is not a natural period of vibration.
3. If the membrane has the form of a circular annulus of radii
a, b, the condition z = 0 being satisfied round both the circles r = a,
r = b, show that the periods of the normal modes of vibration are
2tt/(x) where a> satisfies the equation
'•(?)y-(t) - '•(?)*•(?;
0.
If b > a, show that the large roots of this equation are given by
co(b — a)/c = S7T, approximately, where s is a large positive integer.
[Use (1.40), (1.41).]
§ 27. Small oscillations of a uniform flexible hanging
chain, in a vertical plane.
Take the origin 0 at the equilibrium position of the
lower end of the chain (Fig. 7). Let I be its length, A its
mass per unit length.
Fig. 7.
Consider the motion of an element PQ of length dx,
with its midpoint at a height x above 0. Let T be the

APPLICATIONS 29
tension at the middle point of the element. The horizontal
component of this tension is approximately — T~byJ~dx,
and the corresponding components at the ends P, Q are
These act in opposite directions on the element and cause
its motion ; the equation of motion is therefore
Xdx.^ = l(T^).dx+ . . .
M2 7)x\ 7>x/
the dots at the end indicating unknown terms of higher
order of smallness.
Now T = gXx approximately, the vertical motion being
ignored ; hence
Xdx . —?= — (gXx—)dx 4- . . .
U2 Dxv lx/
Dividing by Xdx, since A is constant, we have, in the
limit, the differential equation
3 - £(■&• • • ■ <»•>
§ 28. To find the normal modes of vibration, we make
the substitution
y = X cos (wt — €), . . (2.19)
where X is a function of x only, and after dividing by
cos (out — e) we find that X must satisfy the equation
d2X. . dX. , a)2 v A /0 rtAX
^+^ + Tx = 0- • ' (2'20)-
This, as it stands, is not a Bessel equation, but the
substitution *
x^lgr2 . . . (2.21)
* See Lamb : ** Higher Mechanics," p. 219, for the physical meaning
of the new independent variable r.

30 INTRODUCTION TO BESSEL FUNCTIONS
transforms it into the Bessel equation
cZ2X , 1 dK ,
^, + 7^ + ^ = 0, • • (2.22)
from which follows
X = AJo(cot) + BYo(o)t). . . (2.23)
The constant B must be zero, for the same kind of reason
as in § 25, and hence
y = AJ0(o)t) cos (cot — e)
= AJ</2w J-\ cos (cot — e). . (2.24)
The condition that y — 0 when x = I gives the equation
J0(2^) = °> ' " • (2'25)
which determines the values of co, and hence the periods of
the normal modes of vibration ; thus
2coaJ- = al5 a2, . . . a„ . . . . (2.26)
where al5 a2, . . . are the positive roots of J0(x) = 0, and
the corresponding periods can be written
2 o I1 2 o I1 2 o I1
— x 27Ta/-, — X 27TA/-, . . . — X 2ffJ-, . . .
or 0-832 X 2ttJ--, 0-363 X 2ttJ--, . . .
The corresponding normal modes are of the form
Vi = CiJo(<*i\J-i) ^s {oht — ei)>
y2 = Cj^JJc^J-jJ cos (w2£ — £2), . . .
In the second normal mode (Fig. 8) there is a node given
by a2\/(^/^) = al5 or
£=(^y ==0-190.

APPLICATIONS 31
The part of the chain below the node vibrates in its own
first normal mode, i.e. the period of the second mode for
the whole chain is that of the first mode for the part below
the node.
In the third mode there are two nodes, in the fourth
three nodes, and so on.
Examples IV
1. If the initial conditions are
y = m{l — x), 0 < x < I, t = 0,
ly/lt = 0, 0 < x < I, t = 0,
show that
o ifr / lx\ C0S ^l* . t ( lx\ cos ^2^ , 1
y = g^J.^-j-j^ + J.^-j-j^ + . . .),
where u>s = \oLs\/(gjl),
2. Show how to satisfy the general initial conditions
y = <f>(x), 0 < x < I, t = 0,
(ty/dj = ^(x), 0 < x < I, t = 0.
[Cf. § 26.]
3. A uniform flexible chain of length I, suspended from one end,
rotates about a vertical axis through that end, in relative equilibrium.
Show that the possible values of <*), the angular velocity, are given by
a)X = icdVigfi)* w2 = i*2V(g/i)> w3 = l^V(gli)> . . .
where cc19 a2, a3, . . . are the positive roots of J0(x) = 0; and hence
that the periods of rotation are the same as the periods of the
normal modes of vibration in a vertical plane.
§ 29. Conduction of heat in an isotropic solid.
Suppose that one face of a uniform plate, of thickness d,
is maintained at temperature u and the opposite face at
temperature ux. Also, suppose the plate to be intersected
at right angles by a cylindrical surface of cross-sectional
area S. Then the quantity of heat Q that flows
perpendicularly to the faces from the first to the second in time t
within the cylindrical surface varies directly as u — ul9 S
and t and inversely as d, and we write
Q = K u-^ g ^ ^ ^ (2>27)

32 INTRODUCTION TO BESSEL FUNCTIONS
where K is a constant, called the coefficient of thermal
conductivity of the material of which the plate is made. This
is the law, founded on experiment, upon which the classical
theory of the conduction of heat is based.
§ 30. In order to adapt the law to an isotropic body
bounded by any surface, we imagine the body dissected by
isothermal surfaces. An isothermal surface is one at every
point of which the temperature is momentarily the same. The
direction of flow of heat at any point is normal to the
isothermal surface through the point, because there is no
temperature-gradient in any direction tangential to the surface.
Let u and u + du be the temperatures of two
neighbouring isothermal surfaces, and dS an element of the surface
u at a point where dn is the normal distance between it and
the surface u + du. Then the quantity of heat dQ that
passes in time dt across dS in the direction from the surface
u to the surface u + du is given by
dQ= -K^dSdt. . . (2.28)
^n
Next, let dS be any element of surface at any point of
the body, and let the normal to dS make an angle 0 with
the normal to the isothermal through
that point (Fig. 9).
Let u be the temperature of the
isothermal, and let dn be the element
of the normal to the isothermal
between it and the neighbouring iso-
. thermal u + du. Also, let dnt be
u+du ' ' 2
the element of the normal to dS
between the same two isothermals.
Then, since the projection of dS on
the isothermal surface is dS cos 0, the quantity of heat dQ
that flows across d$ in time dt is given by
dQ = — K— . dS cos 9 . dt.
?>n

APPLICATIONS
33
But dn = dnx cos 0, and therefore
dQ = -
K^S dt.
'dn-
(2.29)
p
Vd
dy
dz
rdx
Hence, since ^uj^n1 is the temperature gradient
perpendicular to dS, the law (2.28) holds good for any element
of surface dS, whether it is part of an isothermal or not.
§ 31. Differential equation of the conduction of heat.
To find the differential equation satisfied by the
temperature at any point in the interior of an isotropic body,
we begin by considering
a small rectangular
parallelepiped with its centre
at the point ¥(x, y, z) and
its edges, of lengths dx,
dy, dz, parallel to a
convenient set of Cartesian
axes (Fig. 10).
Let u be the
temperature at P, and let
'buj'bx, 'bu/'by, Tsuj'bZ, 'duj'dt
be its rates of change
with respect to the space co-ordinates x9 y, z and the time t.
We shall find two expressions for the increase in the quantity
of heat contained within the parallelepiped in time dt, and
equate them.
Firstly, the quantity of heat that flows across a section
through P parallel to the yz plane is, by (2.28),
— K — .dydz . dt.
?)X *
The quantities that flow across the two faces parallel to this
section are therefore
^2u dx\
Fig. 10.
(
-k£±k.'
^X d#a
- J . dydz . dt.

34 INTRODUCTION TO BESSEL FUNCTIONS
The upper sign corresponds to an inflow across the face
nearer to the yz plane, the lower to an outflow across the
opposite face, and by subtraction the nett inflow across the
two faces is
"dX'
'Kr^^dxdydzdt.
There are similar contributions to the inflow across the
other two pairs of opposite faces, and by adding it follows
that a first expression for the increment dQ in the quantity
of heat contained within the parallelepiped is given by
dQ==K(^2 + ^ + ^)dxdydzdt. . (2.30)
Secondly, let p be the density of the material and s its
specific heat. Then the mass of the parallelepiped is
pdxdydz, and since the increase in temperature during the
interval dt is "buJU . dt, a second expression for the
increment dQ is given by
dQ = s . p dxdy dz . —dt. . . (2.31)
Equating these two expressions for cZQ, and dividing
by sp dx dy dz dt, we find the equation satisfied by the
temperature u, viz.
lu fb2u 7i2U 7i2U\
where K = —. . . . (2.33)
sp
§ 32. Special cases.
I. If the flow is two-dimensional and parallel to the xy
plane, so that u is independent of z and dw/dz = 0, the
equation reduces to

APPLICATIONS 35
In polar co-ordinates (cf. § 22), the same equation reads
~bu _ /3¾ 1 7>u 1 3%\ ,„ 0_.
and if the flow is radial, so that u is independent of 6 and
'duj'dd = 0, it reads
U=K\lfi + r»)- ' ' (2'36)
II. If the flow is one-dimensional and so takes place in
one direction, which we take to be that of the #-axis, then
'bu/'by = 0 and bujbz = 0, and the equation further reduces
to
U=K^ ' " " (2'37)
III. Steady flow.—The flow is said to be steady when
the temperature at every point is constant, so that u is
independent of the time t and is a function of the space
co-ordinates only. The equation satisfied by u is then
found by putting 'bu/M = 0 in the appropriate equation
above.
§ 33. Boundary conditions.
Besides satisfying the differential equation in the
interior of the body, the temperature u must generally satisfy
certain equations over the surface, usually called the
boundary conditions. Three cases will be mentioned here :—
I. The surface may be maintained at a constant
temperature, say u0, perhaps by means of liquid at temperature
u0 flowing round it. In this case the function obtained
for u in the interior of the body, a function of x, y, z and t,
must reduce to w0 when the space co-ordinates refer to a
point on the surface.
More generally, the temperature over the surface may
be any given function of position and time.
II. The surface of the body may be impervious to heat,
in which case the condition to be satisfied over the surface

36 INTRODUCTION TO BESSEL FUNCTIONS
will be TiujTtn = 0, i.e. the temperature gradient will be
zero in a direction normal to the surface.
III. The body may be surrounded by a gas into which
heat is radiating from the surface. In this case, if u is the
surface temperature of the body, and u0 that of the
surrounding gas, we make the assumption that the body loses
heat at a rate proportional to u — u0 and we put the rate
equal to H(u ■— u0) per unit area per unit time, where
H is a constant, called the coefficient of emissivity or exterior
conductivity.
With this assumption, consider the quantity of heat
dQ gained in time dt by a coin-shaped element, of thickness
e, with one of its flat faces of area dS in
the surface (Fig. 11).
The quantity of heat that flows across
the interior flat face by conduction is
K . 7>uj7>n . dSdt, by (2.28), where 'du/'dn
is the temperature gradient in the direction
of the inward normal; while the quantity
Fig. 11 l°s^ ky radiation from the other flat face
is H(u — u0)dSdt; consequently, ignoring
the flow across the narrow cylindrical surface of the element,
we have
dQ = K—cZS dt — TL(u — un)dS dt.
7>n
But dQ is also given, as in (2.31), by
dQ = s . pedS . —dt.
The first of these expressions for dQ is proportional to
the area of the element, the second to its volume ; on
equating them, dividing by dSdt, and making e tend to
zero, we get
K— = H(u — Un),
7m
or :r- = Hu — Uq), • • • (2.38)
7>n

APPLICATIONS 37
where h = H/K. This is the condition that must be
satisfied when the space co-ordinates in u refer to a point on
the surface.
§ 34. Cooling of long circular cylinder.
To take an example, consider the cooling of a long
circular cylinder, initially heated to a uniform temperature ul9
when its surface is maintained at a constant temperature
u0. Let the radius of the cylinder be a, and suppose it
so long that its length may be theoretically regarded as
infinite. The problem is then a two-dimensional one, in
which the flow of heat is radial, and the equation to be
satisfied by the temperature u in the interior of the cylinder
is (2.36), viz.
<m _ /¾¼ 1 7m\
Hi "~ KYbr*+r Tr)'
The temperature at the surface is supposed to adjust
itself instantaneously to the value u0, so that the boundary
condition to be satisfied is
u = u0, r = a, 0 < t < oo.
The initial condition is
u = uL, 0 < r < a, t = 0.
Further, u will approximate to u0 as t increases, and
will always be finite throughout the cylinder and in
particular at r = 0.
The problem is a little simplified by first making the
substitution
v = u — u0 . . . (2.39)
Then v must satisfy the following equation and conditions :
¥=Kb+7W; • • • (2-40)
v = 0, r == a, 0 < t < oo ; . . (i)
v = ux — uQ, 0 < r < a, t = 0 ; . (ii)
v -> 0 when t -> oo ; . . . (iii)
v is finite, 0<r<a30<£<oo. . (iv)

38 INTRODUCTION TO BESSEL FUNCTIONS
Using a standard method of seeking particular solutions
of partial differential equations, we make the substitution
v = UT . . . (2.41)
in the equation, where R denotes a function of r only, and
T a function of t only ; the result can be written
*T dt ~R,\dr* + r dr)' ' ' ( ]
On the face of it, one #ide of this equation is a function
of r only, the other a function of t only, and since r and t
are independent variables, an equation of such a type is
impossible, unless each side is equal to the same constant.
Accordingly, put each side equal to a constant A ; then
T and R must satisfy the separate equations
^ = AkT, . . . (2.43)
d2R ldB,
■& + 7Te=XR' • • {2M)
The solution of (2.43) is of the form T = Ae*Kt, where
A is an arbitrary constant. This form, however, is only
possible if A is negative, on account of condition (iii) ;
accordingly, put A = — fx2, then
T = Ae-**
and (2.44) becomes
from which follows, by § 7,
R = BJ0(/*r) + CY0(/xr),
where B, C are constants, and jjl > 0 for Y0(//,r) to be real
when r > 0. But since Y0(//,r) -> — oo when r -> 0, we must
put C = 0 by condition (iv), and hence, merging A and
B into one constant,
v = Ae"^KtJ0(jir).

APPLICATIONS 39
Condition (i) will also be satisfied if
J0(/xa) = 0,
i.e. if pa = ol, or n = cx.ja, where a is a typical positive root
of the equation J0(x) = 0. Substituting this value of jtz,
we now have the solution
0 = Ae-^\T0(^),
which satisfies all the conditions except (ii). But, since
(2.40) is a linear equation, the sum of any number of
solutions is also a solution. Consequently we can write
down the more general solution
v = 2 Ae~Mla2J0(^j, . . (2.45)
(X-
where the summation extends over the positive roots of
the equation J0(x) = 0. This solution satisfies every
condition but (ii), since this is true of every term. But it
can be made to satisfy (ii) also ; for, putting t = 0, we have
only to choose the constants A so that
ii, — uv
<x
0 = 2AJo(^)> (0<r<a),
or, if x = r/a,
Ui — u0 = 2 AJ0(olx), (0 < x < 1).
»
By (1.48) and Exs. I, 2, (i), we find
A = 2(u1 — u0)
aJx(a)
and substituting this value of A in (2*45) and the resulting
value of v in (2.39), we obtain the final solution
u = u0 + 2(Ul - -o)I-j^e-^Jo(?), (2.46)
where the summation extends oVer the positive roots of
the equation J0(x) = 0.

40 INTRODUCTION TO BESSEL FUNCTIONS
Examples V
1. If the initial condition is u — u0 + ^(f)> (0 < r < a), and
the boundary condition is u = u09 (0 < t < oo), show that the
solution is
/ar\
u = w0 + 2Ae-*2«WJ— J,
a
where the summation extends over the positive roots of J0(x) = 0,
and
2 r1
A = T I a^(a#)J0(a#)cfo.
2. If the initial condition is u — u0 + ^(r), (0 < r < a), and
the surface is impervious to heat, so that the boundary condition
is 'bu/'br = 0, (r = a, 0 < t < oo), show that the solution is
= u0 + A0 + 2Ae"*2,"/o!Jo(3>
where the summation extends over the positive roots of J^x) = 0,
and
r1 2 r1
A0 = 21 x<t>(ax)dx, A = 1 x<f>(ax)J 0(oix)dx.
Jo Jo(a)Jo
3. If the cylinder is at a uniform temperature ux initially, and
at time t = 0 is placed in a gas at temperature u0, show that the
solution is
u = u„ + 2(Ml - Mo)2-^^> e-«WJ0(f),
where the summation extends over the positive roots of the equation
I
xJ q'(x) + ahJ0(x) = 0,
and h has the same meaning as in (2.38).

CHAPTER III
MODIFIED BESSEL FUNCTIONS
§ 35. Modified Bessel functions of zero order.
If we put k = i = \/("™ 1) ^1 (1-18), we obtain the
equation
d2y , 1 dy ^ /0 ,x
which is called the modified Bessel equation of zero order;
it can also be written
s(Bs)=ay- • • • (3,2)
A solution of this equation, and the only one, except
for a constant factor, that remains finite when x = 0, is
denoted by I0(x) and is given by *
loM = Jo(«0 = ! + 22 + 22742 + 22 42 . 62 + ' " " (3'3)
which is called the modified Bessel function of the first kind
of zero order.
§ 36. A modified Bessel function of the second kind can
now be defined as any solution of (3.1) which is not a
constant multiple of I0(x), an(l can be expressed in the form
AI0(*) + BI0(,){^,
(cf. § 4), where A, B are any constants (B 4= 0).
When x is small this solution (cf. § 5) behaves like
AI0(&) + b|i0(») log x - ~ - . . .J
* Cf. cosh x = cos ix.
41

42 INTRODUCTION TO BESSEL FUNCTIONS
In particular, if we put
A = log 2 — y, B = — 1,
where y denotes Euler's constant (§6), we obtain a
particular modified Bessel function of the second kind which
is denoted by K0(#), thus
K0(x) = (log 2 - y)I0(x) - \ I0(x) log x
X'
. (3.4)
We note that, when x is small,
K0(x) = (log 2 — y) — log x . . . . (3.5)
the remaining terms being small in comparison with unity ;
so that K0(#) -> + oo when x -> + 0.
The graphs of I0(#), K0(#) are shown together in Pig. 12.
Ex. 1. Show that the equation I0(x) = 0 has no real root; and
that the equation I0'(#) = 0 has no real root except x = 0.
Ex. 2. Show that
tr%.
Ko(a-) = -gftl'o(^) + iY0(ix)}f

MODIFIED BESSEL FUNCTIONS 43
provided that log (ix), which occurs in Yn(ix), has the value
\iti + log x when x is positive.
§ 37. The general solution of (3.1) can now be written
y = Alo(a) + BKo(a), . . (3.6)
where A, B are arbitrary constants.
Corollary. The general solution of the equation
*!L + I ^ _ khj = 0,
Ct££ ££ CiX
where k is a constant, can be written
2/ = AI0(fc») + BK0(Jh;), . . (3.7)
where k > 0 for K0(£#) to be real when x > 0.
§ 38, Since I0(#) is a solution of (3.2), we have
=(-¾ - *«■>• • • (3-8»
and inversely
Jxl0(x)dx = xl0'(x). . . . (3.9)
Replacing x by ax, we have further
jxl0((xx)dx = -I0'(o#), . . . (3.10)
oc
and, as in § 11, we find, if a, j8 are constants,
(/32 — a.2)$xl0(a.x)l0(l3x)dx
= x{pi0'(px)I0(*x) - (rio'MIoCto)}, (3-11)
JzI02(oaOfc = |~{IoaM - I0'2(aa;)}. . (3.12)
§ 39. Laplace's equation in cylindrical co-ordinates, when
the dependent variable is independent of 6.
When Laplace's equation
3+£+3- • ■ <«•>
is expressed in terms of cylindrical co-ordinates r, 0, z, it
reads
^ + l^.I^ + ^! = o (314)
Dr2 ^ r Dr^r2 Id2 + lz2 * " l '

44 INTRODUCTION TO BESSEL FUNCTIONS
and if u is independent of 6, reduces to
^+1^+ >!j = 0. . . (3.15)
7)r2 r dr <)z2 v
We shall now find solutions of this equation of the form
u = RZ,
where R is a function of r only, and Z a function of z only.
When we substitute u = ^SJL in (3.15), we find that the
resulting equation can be written
1 /<Z2R 1 dR\ _ 1 <WZ
R,\dr2 + r~dr / Z dz2'
each side of which must be equal to the same constant,
for the same kind of reason as in § 34. Firstly, putting
each side equal to — /z2, we find solutions of the form
u = (AJ0(/xr) + BY0(/xr)}(C sinh pz + D cosh /zz), (3.16)
or the equivalent form
u = (AJ0(/xr) + BY0(jur)}(Ce-'l» + De^). (3.17)
Secondly, putting each side equal to + /x2, we find
u = (AI0(/xr) + BK0(/xr)}(C sin pz + T> cos pz). (3.18)
Thirdly, putting each side equal to zero, we find
u = (A + B log r)(Cz + D). . . (3.19)
In each case, A, B, C, D denote arbitrary constants.
We must suppose /x > 0 for Y0(/xr) or K0(/xr), to be real
when r > 0.
§ 40. Steady flow of heat in a finite cylinder.
Consider the flow of heat in a finite cylinder of radius
a and length Z. Suppose that over one end the temperature
is a given function of the distance from the centre of that
end, and that the other end and the curved surface are
maintained at a constant temperature, which we take as the
zero temperature. Further, suppose that these conditions
have persisted for some time, so that a steady state has
been reached in which ^u/Tit = 0 at every point (§ 32, III).

MODIFIED BESSEL FUNCTIONS 45
Then, putting ~du/U = 0 in (2.32), we see that u satisfies
Laplace's equation. Cylindrical co-ordinates are appropriate
to the present problem, with the pole at the centre of one
of the ends and the 2-axis along the axis of the cylinder.
Then, as u does not depend on 6, the equation satisfied by
u is (3.15), viz.
^ + 1^ + ^ = 0. . . (3.20)
The boundary conditions we take to be
u = 0, (z = 0, 0 < r < a), . . (i)
u = 0, (r = a, 0 < z < I), . . . (ii)
u == c/)(r), (z = Z, 0 < r < a), . . (iii)
u is finite, (0 < r < a, 0 < z < Z). . (iv)
With these conditions in mind, we select from (3.16),
(3.18), (3.19) the types of solution likely to be suitable.
Firstly, having regard to (iv), we must put B = 0 in each
case, since Y0(/xr), K0(/xr), and log r all become infinite at
r = 0.
Next, having regard to (i), we put D = 0 in each case.
We then have possible solutions of the form
J0(/xr) sinh /z2, I0(/^) sin fiz, z.
Of these three solutions, the first will satisfy (ii) if
J0(/xa) = 0,
that is, if jjua = a, or jjl = aija, where a is any positive root
of the equation J0(#) = 0. The second cannot satisfy (ii)
because the equation I0(x) == 0 has no real root (§ 36, Ex. 1) ;
and the third obviously cannot satisfy (ii). Hence we select
the solution
u = AJ0(—} sinh —
\a J a
which satisfies all the conditions but (iii). The same is
true of the more general solution
u
(X
2AJ0(^)sinhf, . . (3.21)

46 INTRODUCTION TO BESSEL FUNCTIONS
where the summation extends over all the positive roots
of J0(x) — 0. Further, this more general solution will also
satisfy (iii) provided the constants A are determined so that
(putting z = I)
4>{r) = Z AJo(^) sinh p (0 < r < a),
or, if r = ax,
<f>(ax) = 2 AJ0(a#) sinh —, (0 < x < 1).
a a
Hence, by (1.48),
ad 2 f1
A sinh— = _ 0/ , I x6(ax)J Jo:x)dx . (3.22)
a Ji2(a)J0
When the value of A, thus found, is substituted in
(3.21), we obtain the final solution.
Examples VI
1. If 4>(r) = u0, a constant, show that the solution is
J0 — smh —
u = 2^2. jj»
« aJi(a) sinh —
a
where J0(a) = 0.
2. If the boundary conditions are u = u09 (z = 0, 0 < r < a) ;
u = 0, (r = a, 0 < 2 < I); m = 0, (2 = Z, 0 < r < a) ; show that
the solution is
a(l - z)
u
= 2m»z— —
« aJi(a) sinh —
a
where J0(a) = 0.
3. If the boundary conditions are u = 0, (z = 0, 0 < r < a) ;
w = 0, (z == Z, 0 < r < a) ; u = u0, (r = a, 0 < z < I) ; show
that the solution is
T fmiTr\ . m-nz
W = - > _____ ,
where ra = 2s — 1.

MODIFIED BESSEL FUNCTIONS 47
4. Show how to solve the problem when the boundary conditions
are u = «/»(r), (z = 0, 0 < r < a) ; u = <f>(r), (z = I, 0 < r < a) ;
w == f(z), (r = a, 0 < z < I).
5. If the boundary conditions are u = 0, (2 = 0, 0 < r < a) ;
M = ff>(r), (z = 1, 0 <r < a); "bufbr = 0, (r = a, 0 < z < I), so
that the curved surface is impervious to heat, show that the
solution is
u — A0z + 2 AJ0 ( — ) sinh —,
where the summation extends over the positive roots of the equation
Jx(x) = 0, and the coefficients A0, A are given by
^1A0 = I x<j>(ax)dx,
Jo
£AJ02(a) sinh—= I x<f>(ax)J0(oLx)dx.
d J 0
Verify that, if <f>(r) = uQ, a constant, this solution reduces to
u = u0z/l, as is obvious from physical considerations.
§ 41. Large values of x.
It is plain from the series (3.3) that I0(x) tends to +00
when x is large. In what follows we shall need an
approximation to I0(x) when x is large and positive. To obtain such
an approximation, we first put
u = y\/x . . . (3.23)
in (3.1), and find that u satisfies the equation
which, when x is large compared with 1, takes the
approximate form u" = u, of which the general solution is
u = Aex + ~Be~x, where A, B are arbitrary constants.
This suggests that any solution of (3.24), which tends to
00 when x ->■ + 00, will approximate to Aex when x is
large and positive.
§ 42. To attempt to improve upon this approximation,
we make the substitution
u = ve
X

48 INTRODUCTION TO BESSEL FUNCTIONS
in (3.24), and find that v satisfies the equation
£+£+£-<»• • ■ <«)
Assuming that this equation can be satisfied by a series
of the form
"=1+ci+&+! + --- • <3-26>
l*/ \AJ \AJ
we substitute this series for v in the equation and obtain,
after collecting like terms,
&> - (¾¾+{». *. - © v
X3
+ J2 . 3c3 - ( • ) c2
■5\2
1
1- = 0
x*
By equating the several coefficients to zero, it follows
that the equation is formally satisfied by the series, provided
the coefficients are given by
1
= 8'
32 12. 32
<>i
c, =
2.81 2 ! 82'
52 12. 32 . 52
Co — ——— Co —
it O O <2
3 . 8 * 3! 83
and hence
12 12 Q2 12 02 K2
*=1 + 8^+2T(8^+T!W+--- (3-27)
We are thus led to the expansion
t^ Aey, , 1' , l'.3» 1».3».5» \
^ = 7^ + 8^ + 21(8^ + 3I(to)» +•••> (3-28)
where A is some positive constant; it will be seen later
(§ 81) that A = 1/V(2tt).
When A has this value, (3.28) is an asymptotic expansion
(§ 78) of I0(x) ; the series on the right is divergent, but it

MODIFIED BESSEL FUNCTIONS 49
has the property that the sum of the first n terms gives an
approximation to I0(#) when x is large enough, with a
percentage error as small as we please.
Ex. 1. Show that, when x is small,
xl0(x) _ #2_ l/a^V , }f^\
2I0'(aO + 8 3V 8/ +6U/
Ex. 2. Show that, when x is large,
^) = 1 + 2+ 3
I0'(x) ' 2x " 8a;2 '
§ 43. Application to alternating current in a wire of
circular cross section.
The differential equations of the electromagnetic field
are based upon two laws which are sometimes distinguished
by the names of Ampere and Faraday, viz.
Ampere's law.—The line integral of magnetic force round
a closed circuit is equal to 47r x (the integral of electric
current through the circuit).
Faraday's law.—The line integral of electric force round
a closed circuit is equal to — — (magnetic induction through
at
the circuit).
§ 44. To apply these laws to determine the current
density at radius r in a wire of circular cross-section, through
which alternating current is flowing, let
a be the radius of the wire, p its specific
resistance, and fx its permeability; let
x be the current density and H the
magnetic intensity at radius r and
time t.
Firstly, consider a closed circuit
which is a circle of radius r, with its pIG# 13.
axis along the axis of the wire (Fig. 13).
Applying Ampere's law to this circuit, we have
27rrH = 4«r[rx . 2irrdr, . . (3.29)
Jo

50 INTRODUCTION TO BESSEL FUNCTIONS
and hence, after differentiating with regard to r,
1 ^ / TTX
r ^(rH) = 47r*' • • • (3-30)
Secondly, consider a closed circuit
which is a rectangle with one of its
sides, of length I, along the axis of the
Fig. 14. cylinder and the two perpendicular
sides of length r (Fig. 14). Applying
Faraday's law, we have
2) fr
px0l — Pxl= — -I /juHldr,
otj o
where #0 denotes the value of x when r = 0 ; and by
differentiation with regard to r,
^ = ^- • • ■ (3.31)
To eUminate H, multiply (3.31) by r, differentiate with
regard to r, and use (3.30); this leads to the equation
from which x can be found, and then H is given by (3.29).
§ 45. Let the total current through the wire be C cos cot,
an alternating current of period 2tt/o). It is convenient
to regard this current as the real part of the complex
number Cei(t}t, and correspondingly to regard x as the real
part of a complex number z that satisfies the equation
r lr\ dr/ ~J~ u' ' ' ^3,33)
Accordingly, we seek a solution of this equation of the
form
z = V{r)&*9 . . . (3.34)
remembering that when F(r) has been found, the real part
of z will be the actual current density.

MODIFIED BESSEL FUNCTIONS 51
Substituting (3.34) in (3.33), we find that F(r) must
satisfy the equation
;*(•§)-^- • • ^
d2F 1 d¥ _
dr2 r dr "™ '
where F denotes F(r), and
4. = ^, t = i +»(*!22£)\ . (3.36)
Hence, by § 37,
P(r) = Alo(fcr) + BK0(fcr).
The constant B must be zero, since z is finite at r = 0,
and therefore
z = AI0(kr)ei(ot. . . . (3.37)
The constant A can be found in terms of C. For, since
Gei0>t is the total current, we have
Ceiwt = \"z. 27rrdr
Jo
and therefore
C = 27rA\\l0(kr)dr = 2^^¾¾
by (3.10). This gives A in terms of C, and hence
«= dW0^- • ' (3-38)
Ex. Show that H is the real part of
§ 46. Equivalent resistance and internal self-inductance of
a length I of the wire.
The electromotive force along a length I of the surface
of the wire, where r = a, is the real part of
kCl0(ka) ioa

52 INTRODUCTION TO RESSEL FUNCTIONS
If we equate this to (R + io)h)Ceia)t, then R is called
the equivalent resistance and L the internal self-inductance
of length I of the wire ; this gives
27ral0 (to)
Further, if R0 is the resistance of length I for steady
current,
Pi
Ro = —h
ira2
and, by division.
R + toL __ ka I0(ka)
R0 " 2 I0'(te)' • • (6mM)
§ 47. Low frequency.
Now, if we put (3.36) in the form
k2a2 __ 7TjjLcoa2. __ . 2 __ /7r/zcoa2\*
^s" ~ ~^r* ~ •*j * - v~2^;'
we find, from (3.39), using § 42, Ex. 1, when the frequency is
small (co small, k small),
R + io)h . , . „ , /c4 i/c6
and hence, by equating real and imaginary parts,
— — i±!L4 <aL_ 2_k6
R0~ 3 " * "' R0~" 6---
or, since R0 = plj-na^, and a> = 2p/c2/7r/xa2,
R
= R0(l + |. ...), L = |(l-^- . . .). (3.40)
§ 48. 13¾¾ frequency.
Again, if the frequency is large (w large, k large) we have,
from (3.39), using § 42, Ex. 2,
R + iayL _ ka 1 3
R0 2 ' 4 ' 16to ' • •
= (1 + %)K + ] +
4 ' 32(1 + i)K

MODIFIED BESSEL FUNCTIONS 53
and hence, by equating real and imaginary parts,
R 13 coL 3
R0 ' 4 ' 64* • • •' R0 64* " " "
or
R = R0(, + i + ^. ..), L = ^(1-^...)(3.41)
§ 49. Verification of the value found for the equivalent
high-frequency resistance.
As a further example, we may verify the formula just
obtained for the resistance R, when the frequency is high,
by showing that the heat Q, generated per unit time in
length I, is ^RC2. We have, in fact, x being the real current
density,
Q = PlC f%2 . 2irrdrdt = 2ttP1 f [^-—Vrdrdt,
where z denotes the complex number conjugate to z. Now
for rapidly alternating current, the average value of z2
or z2 with respect to t, over an interval of a second, is
practically zero, and hence
Q = TTpl \l I"
J 0 J 0
zzr dr dt,
that is, substituting the value of z from (3.38), and writing
k for the conjugate of k,
477¾210 (ka)I0 (ka) J0 Jo
Using (3.11), and putting q = ka, q = ka, we can write the
result of the integration
pZC* (qqf / I„(g) _ I0(g) \
H 47«.« 32 - qAqI0'(q) ql0'(q))>
or, since g = ka = 2(1 + i)K,
plCW/ I.(g) _ I„(g) \
V 77«* Vjlo'te) W'

54 INTRODUCTION TO BESSEL FUNCTIONS
and therefore, by § 42, Ex. 2, since k is large for rapidly
alternating current,
0:-,W i 1 i 3 _I__L__L \
* Tra2 \g ^ If ^ Sq* * ' ' q 2g2 8ga' ' 7
7ra2\2 ' 8 ' 128/c * "
= -2^(- + 1+6^ -..)=^,
which is the result we set out to verify.
§ 50. The skin effect.
To conclude this application to the flow of alternating
current in a cylindrical wire, we shall verify that, for
sufficiently high frequencies, the current flowing through
a coaxial cylinder of radius r is small compared with the
total current, even when r is nearly equal to a, thus showing
that most of the current flows through a thin layer at the
surface—the well-known " skin effect."
By (3.38) the whole current flowing through a coaxial
cylinder of radius r is the real part of
27Tal0(ka))0 uv ' al0 (lea)
The ratio of this to the total current Ceio)t is
rlo'(fr)
al 0'(ka,y
and, by (3.28), when k is large, this is approximately equal to
r _^_ V(ka) _ /t\* *.--f)
a V(fcr) e*« ' W
which, for any fixed value of r, however small a — r may
be, is as small as we please when k is sufficiently large,
i.e. when the frequency is high enough.

MODIFIED BESSEL FUNCTIONS 55
§ 51. Kelvin's ber and bei functions.
The functions named ber x and bei x by Lord Kelvin
may be defined by
iJ—jn~x) — ber x + i bei x.
Now, since {(1 + i)x/\/2}2 = ix2, we have, by (3.3),
t /1 + L\ __ i i ^! . (^2)2 i (^2)3 ,
H V2 / ~~ + 22 ■+" 22. 42"+" 22. 42. 62 "^
and hence, by equating real and imaginary parts,
^ a = X - 2T~P + 2'. 4»°! 6» . 8'~ (3-42)
/yi2 /y»6 /ytlO
tel x = 22 - 2a. 42. 62 + 22. 42 . 62. 82. 102 ~~ ' ' * (3'43)
§ 52. If we now write (3.36) in the form
fc=L+_V m=(^)*, . . (3.44)
the results of the application begun in § 44 can be written
in terms of the ber and bei functions. Thus (3.38) will
be found to be expressible in the form
imC ber mr + i bei mr ., /0 AS.
2ira ber ma + & bei ma
and (3.39) in the form
R + iuyL _ ima ber ma + i bei ma . .
R0 ~" 2 ber' ma + i bei' ma'
In these forms the real and imaginary parts are in
evidence. Numerical calculation can be made with the
aid of the tables published in Kelvin : " Math, and Phys.
Papers," III, p. 493 ; Jahnke und Emde : " Funktionenta-
feln " ; McLachlan : " Bessel Functions for Engineers," etc.
§ 53. The reader will realise that a complete familiarity with
the behaviour of the function J0(#) for all values ot.x, real or complex,

56 INTRODUCTION TO BESSEL FUNCTIONS
would render it unnecessary to give separate names to the functions
I0(x), ber x, bei x, etc.
Ex. 1. Given that y — I0( —jirx) *s a solution of the equation
ld_/ dy\ _ .
xdx\dx) ~~ ly'
by putting y — u + iv in this equation, separating real and
imaginary parts, and eliminating u and v in turn, show that ber#
and bei x both satisfy the equation
xdxL dx\xdx\ dx/ J J
Ex. 2. If a, j3 are constants, show that
j8I0'(j9)J0(a) - aJ0'(a)I0(j3) ,
JxJq(olx)Io(Px) dx —
o
a2 + j32
and obtain the Fourier-Bessel expansion
a
where J0(a) = 0, 0 < a; < 1.
Deduce the Fourier-Bessel expansions of ber kx and bei lex.

CHAPTER IV
DEFINITE INTEGKALS
§ 54. BesseVs integral for J0(x).
If we expand the function eixsind in ascending powers
of x, we get
^sma _ i , fa sin e ■ (ix sin fl)2 (½ sin fl)3
Now integrate both sides between the limits 0 = 0,
6 = 27T, and use the formulae
i
2ji
sinn0d» = 0, (nodd),
0
(» — l)ln — 3) . . . 3 . 1 0 , ,
— v /v ' - . 27T, (weven);
»(» — 2) . . . 4 . 2
this gives
/*2jl / /yt2 /yt4 \
j^ eixmed9 = ^ _ |. + _5_ _...). (4.1)
and hence, since the series in brackets is J0(x),
1 c2n
J0{x) = ±\ (f***'d6. . . (4.2)
Equivalent forms are given by
n
J0(x) = - f2 eixsin ed6 = - [ Vcos dd6, . (4.3)
77 J n 77 J o
2
or, on separating the real parts,
n n
J0(#) = - 1 cos (x sin 0)d0 == - I cos (x cos 0)d9. (4.4)
77 Jo 77 Jo
57

58 INTRODUCTION TO BESSEL FUNCTIONS
Any one of the above definite-integral forms of J0(x)
may be called BesseVs integral for J0(tf),* being a particular
case (when n = 0) of Bessel's integral for Sn(x), (§ 86).
§ 55. Lipschitz's integral.
If, in the well-known integral
j
e~ax cos bxdx = 9 , L9, (a > 0),
a1 -j- b*
we replace b by b cos 9, we get
I e~ax cos (bx cos d)dx = 9 , ,» ^.
J 0 v ' a2 + 62 cos2 0
Since the infinite integral on the left is uniformly
convergent with respect to 9, we may integrate under the
integral sign with respect to 0 from 0 to \tt ; we thus find
f°° 7 f§ /l m 7fl P adfl
e~aa! da cos (bx cos u)dd = , r-^,
Jo Jo Jo a2+ 62 cos2 0
and hence by (4.4), if a > 0,
»00 ]_
oe-»J.(fa)*c=v(fl, + y) • • (4.5)
which is known as Lipschitz's integral.^
Corollary. When a -> 0 we get, if b > 0,
r
i
oo x
J0(bx)dx =-7. . . (4.6)
o b
and in particular, if 6 = 1,
J0(x)dx = 1. . . . (4.7)
i
00
0
§ 56. Weber's discontinuous integrals.
Interchanging a and b in (4.5) gives, if b > 0,
i
co I
J0(ax) . e~hxdx = -—-— . (4.8)
* Also called Parseval's integral, for historical reasons (Watson, p. 21).
t Watson, p. 384.

DEFINITE INTEGRALS 59
Since both sides of this equation are analytic functions
of b when the real part of b is positive, it follows that, if
b >0,
J"Max) . r<*♦«.>.& = v{a2 + \b+ -c)2} (4.9)
and hence, if b > 0,
\yo(ax) . «r<» +*»■& = ^^ = X2 + Y* (O0)
where X + iY = yV + 62 - c2 + 2i6c). . (4.11)
By equating real and imaginary parts in (4.10), we
deduce that
/»00
1 J0(ax) .
Jo
►00 ]£
e~bx cos cxdx = ya V2, . (4.12)
f00 Y
I J0(a#) . e~6ajsinc#efcr = y2 j, va* " (**^)
From (4.11) we have
X2-Y2 = a2 + 62-c2, . . (4.14)
XY = 6c, . . . . (4.15)
and hence, by eliminating Y and X in turn, we find that
X2 and — Y2 are the two roots of the equation in 6
* +b-z = l, . . . (4.16)
c2 + e ■ e
and that
2X2 = a2 + 62 - c2 + V{(a2 + 62 - c2)2 + 462c2}, . (4.17)
2Y2 = - (a2 + 62 - c2) + -v/{(a2 + &2-c2)2 + 462c2}. (4.18)
Suppose a > 0, c > 0. Then, by following the
continuous change in X + *Y" when c increases from 0, we see
from (4.11) that X and Y are both positive.
Now let b -> 0 ; then, if a > c, X -> yV - c*)>
Y -> 0 ; but, if a < c, X -> 0, Y -> V(c2 - «")• Hence
we find from (4.12), (4.13) respectively

60 INTRODUCTION TO BESSEL FUNCTIONS
I
00
JQ(ax) . cos cxdx = 0, (a < c), (4.19)
o
(a > c) ; (4.20)
i
oo i
JJax) . sin c# dx = —rm ^, (a < c), (4.21)
0 V (c ~ a2) '
= 0, (a > c), (4.22)
We have supposed a > 0, c > 0. If a < 0 or c < 0,
we need only note that both integrals are even functions
of a, and that the first is an even function of c and the
second an odd function of c.
§ 57. We shall next prove that
i
J0(ax) . e"hx . dx = tan-1^ . (4.23)
Q X J\.
where X is given by (4.17).
Proof. Since the infinite integral on the left of (4.12)
is uniformly convergent with respect to c, we may integrate
with respect to c under the integral sign from 0 to c. We
thus find
f°° T / v h~ sinc#7 fc Xdc IA ft.x
] J0(ax) . e-*» . —^-dx = ]qX2 + Y2- <4'24)
Now from (4.14), (4.15),
XdX - YdY + cdc = 0,
YdX + XdY - bdc = 0,
and hence
dc dX Xdc — cdX
X2 + Y2 bY-cX X(X2 + c2)' '
from which follows
(4.25)
f Xdc (Xdc — cdX A - c .. n„.
Jg + T'^J x* + c* ■ = ton~1x; • (4-26)
hence and from (4.24) follows (4.23), which was to be proved.

DEFINITE INTEGRALS 61
§ 58. Now suppose a > 0, c > 0, in (4.23), and let
6-^0. Then, as above, if a > c, X ->■ \^(a2 — c2), and
tan-^c/X) -> tan-^c/yV — c2)} = sin^c/a);
but, if a < c, X -> 0, and tan_1(c/X) ->■ 7r/2.
Accordingly, wo have
i
oo sin. Cic 7T
J0(ax) dx — -~, (a< c) . (4.27)
= 8111^-, (a>c). . (4.28)
If c = 0 the integral vanishes. If a < 0, or c < 0, we
need only note that the integral is an even function of a,
but an odd function of c.
Ex. Deduce the following well-known integrals as particular
cases of the integrals in §§ 56-58 :—
J 00 l) pOO c
e~bx cos ex dx = j-—i— | e~bx sin ex dx = r—-i—-,
0 o2-fc Jo b2 + c2
i:
00 , sin ex 7 , , c
e-6x fix = tan_17,
o x o
00 sin ex , tt . 7T _. .
a# = -,0, — -5, according as c > , =, < 0.
X 2i £
§ 59. Electrostatic potential of an electrified disc.
It was shown by Weber * that the potential of the
electrostatic field caused by an electrified circular disc
could be expressed as an integral of the same form as (4.23).
Let the disc be situated in the xy plane, with its centre at
the origin and its axis along the 2-axis ; let c be the radius of
the disc, and Q the total charge of electricity upon it. Let V
be the potential at any point due to the charge on the disc.
The differential equation satisfied by V in free space
is Laplace's equation
2)2V 2)2V 2)2V
*? + ** +1* = °' • • (4-29)
* Cf. Riemann-Weber: " Die Partiellen Differential-Gleichungen
d. Math. Physik," I, 6th edn., 1919, p. 342.

62 INTRODUCTION TO BESSEL FUNCTIONS
which, in cylindrical co-ordinates r, 9, z, when V is
independent of 0, as in the present problem, becomes
2)2V 1 7>V 2)2V
<)r2 r ?>r <>z2 v '
This equation, by (3.17), has a solution of the form
V == e~"z J0(ar), where a is any constant, and it follows,
by differentiation under the integral sign, that if z > 0
/•00
V = A e--J 0{<xr)f{a)daL . . (4.31)
Jo
is also a solution, where A denotes any constant, and /(a)
any function of a.
Now the charge distributes itself so as to make V = const,
over the disc but not beyond it, and this condition is satisfied
•
if we put/(a) = , for we then have
V = AfV"J0(ar) ^L^a) . . (4.32)
which, by (4.27), (4.28), reduces when z -> 0 to V = |ttA
if r < c, V = A sin"1 (c/r) if r > c.
It remains to determine the constant A in terms of the
charge Q. Now, if a is the surface density on the upper
face of the disc, we have
*"" = ~ fe),_. == A]0 J°(ar) Sin Cada = V(«2 - r>)
if r < c, by (4.21) ; and hence
A . . (4.33)
47r<v/(c2 - r2)'
Taking into account both faces of the disc, we have
therefore
re re n> Jo*
Q = 2joa.2Br*-Aji7F-^ = Ac. (4.34)

DEFINITE INTEGRALS 63
Hence A = Q/c, and by (4.33), (4.32).
Q
4:7tc<\/(c2 — r2)'
a =
(4.35)
tt Q f °° ^t/ xsinca7 ,4 nn\
V = - e-az J0(ar) da, . (4.36)
the constant potential over the disc being 7rQ/2c.
This gives V for positive values of z. For negative
values of z we need only note that V is an even function of z.
§ 60. Further, by (4.23) it follows that
V^-tan-1^, • • • (4.37)
where
2X2 = r2 + z2 - c2 + <s/{(r* + ^2- c2)2 + 4c2z2} (4.38)
Formulae (4.33), (4.37) can be found by other methods
(cf. Jeans : " Electricity and Magnetism," § 288).
Ex. Verify from (4.37) that V^ ,, ^—-r when +/(r2 + z2)
J ' ' V(r2 + z2)
is large compared with c.
Examples VII
1. Show that
00
Jou
o
a; — 1.
2. Show that J0'(#) -> 0 when a; -> <x>, and by integrating the
equation
J ."(as) + ±J.'(x) + J0(x) = 0
Ju
and using (4.7), show that
00 Ji(*)
j
da; — 1.
0 *^
3. If w, v, w denote the integrals
c^_xJ^x)dx_ r00 J0'(x)dx r00 a;J0"(a;)da;
Jo V(«2 + »»)' Jo V(«2 + x2)9 Jo V(«2 + x*Y

64 INTRODUCTION TO BESSEL FUNCTIONS
respectively, show that
u + v + w = 0,
gj + av + 1 = 0,
dv
a- w = 0.
aa
Deduce that, if a > 0,
00 #J0(#)dr r°° J^a;)^ 1 — e~a
Jo V(a2 + *2) ' Jo V(<*2
+ a;2) a
4. Show that
(i) f f 008(-008^).^^^0 = 2^^^(-).
(ii) fa f2"log^ -cos(^ cos ¢) .rdrdd = 27rc2|l - J0(-)}.
(iii) p f %2 - r2)cos^cos ¢) .rdrdO = ^W,/'-).
(iv) f J I0( r) cos (-cos fl) . rdrdO =
[As to the form of the result in (iii), see Exs. XI, 2, (i).]
5. If r, p, z denote the distances of a point from the origin, the
z -axis, and the xy-pl&ne respectively, show that if z > 0
100 1
e-'tJ0(pt)dt = -.
Also, if R, R' denote the distances of the point from (0, 0, c),
(0, 0, — c) respectively, show that if z > c
R' — R f °°(z$ + 1) sinh ct — ct cosh ct ,T , ,
—2—= ]0 p . e-rtj0(/rf)<ft.
6. Show that
1 p2w l r2w
I0(a) = — e-^ sin 0 ^^ = e* sin 0 dO.
ATT J o ^77" J o
§ 61. The Gamma-Function.
In the next two chapters we shall need the elementary properties
of the Gamma-function T(n), which may be defined in the first place
when n is real and positive by the integral
f00
r(n) = I e-xxn-xdx9 (n > 0); . . (4.39)
J 0

DEFINITE INTEGRALS 65
the condition n > 0 being necessary for the convergence of the
integral at the lower limit.
In particular, when n — 1 we have
{00
e—dx = 1. . . . (4.40)
0
Again, integrating by parts, we have
p -1 00 poo
r(n) = — e-*xn-x + (n — 1) \ e-axn~Hx,
and therefore, if n>l,
r(n) = (n - l)r(n - 1), . . . (4-41)
and hence, when n is replaced by n + 1,
r(n + 1) = nr(n). . . . (4.42)
It follows by repeated application of this formula that, if n is
a positive integer,
r(n +1) = n(n - l)(n - 2) . . . 3. 2 . 1 T(l),
that is, by (4.40),
r(n + 1) = n ! . . . (4.43)
If we substitute x2 for x in (4.39) we have also
T(n) = 2f e-^x%n-xdx, . . (4.44)
§ 62. The integral
r
J 0
cosTO 9 sinn 6d8
can be expressed in terms of Gamma-functions.
We consider the double integral
J 00 f 00
e-^-y^m-lyZn-ldx dy
0 ^0
in two ways. Firstly we have, by (4.44),
J oo poo
e-xzX2m-idx \ e-**y*n-xdy = £r(ra)r(w).
o J 0
Secondly, by transforming to polar co-ordinates,
n
'2 f°0
U
J 2 f °°
I e-f2(r cos fl)2"1-1^ sin 6)^-^drdd
o Jo
{oo <»2
e-f2r2m+2n-i^r cos2™-^ sin8"-1^ dO
0 ^0
n
'2
= \r(m + n) f cos2"1-1^ sin2n"l0 d0,
Jo

66 INTRODUCTION TO BESSEL FUNCTIONS
by (4.44). Equating the two values of u thus found, we have,
if m > 0, n > 0,
n
f2 cos2—i 0 sin2-* 9dd = ^m)r(n)x. . (4.45)
Jo '2r(m + n) v '
It follows that, if m > — 1, n > — 1,
' (^)r Onr)
l cos™ 0sin» 0d0 =—- -—\ \ T . . (4.46)
In particular, putting m == 0, n = 0, we have
f- r^,_{Hi)}2_{nt)}2
2"" Jo ~ 2r(l) ~ 2 '
and hence r(^) = Vw. . . . . (4.47)
Further, from (4.41), we have
r{i) = ir(i) = w>
r(i) = %r(i)= ^Vir,
r(i) = ir{i) = ^^V«,
and so on.
§ 63. Again, from (4.42), we have
) = Qi±i>, . . . (4.48)
Hn
n
from which it follows that r(ri) -> + <x> when n -> + 0.
For our present purpose we may now suppose that (4.48) defines
r(n) firstly for values of n between — 1 and 0, then for values
between — 2 and — 1, then for values between — 3 and — 2, and
so on ; the Gamma-function will then have been denned for all
real values of n ; for example, by (4.48) and (4.47), we shall have
n- i) = zri= -2V*,
l\~ s) — _ ii = j—3^"'
and so on.

DEFINITE INTEGRALS
67
The graph of r(n) for real values of n is indicated in Fig. 15.
Note that r(n) -> ± ooasw approaches a negative integer or zero.
§ 64. The Beta-function.
The Beta-function, B(m, n), may be defined, in the first place
for positive values of m and n, by the integral
B(m, n) = \ a;m-1(l — x)n-xdx.
(4.49)
The conditions m > 0, n > 0 are necessary to ensure convergence
of the integral at the lower and upper limits respectively.
The Beta-function can be expressed in terms of the Gamma-
function. For, if we put x = cos2 0, we find
B(m, n) = 2 [2 cos2™-1 0 sin2"-1 0 dO, .
and hence, by (4.45),
r(m)r(n)
B(m, n) = — ;—r. .
v r(m + n)
(4.50)
(4.51)

68 INTRODUCTION TO BESSEL FUNCTIONS
Examples VIII
1. Show that
co r(n)
{«-> 1(71)
e-axx*-Hx = —~9 (n>0, a > 0).
(ii) V° e~x*dx = ^-.
Jo ^
(iii) [^ e*—*dx = ^
eZax~x2dx = -V-e*2.
a
°o 1 /m + 1'
J00 i /m -4- i\
ar»e-«nda; = - W—I—j, (m > - 1, n > 0).
2. Show that, if n > — 1,
re Jt
f2 sin" 9d9 = f2 cosn Odd =
Jo Jo
vvr^
n + 1
2rf^'
3. Show that, if — 1 < n < 1,
jr
^.^,.-,(1 + -),(1-:).
4. By evaluating the integral
'2
J" sin2n-l ^COS2"-1 0d0,
J o
in two ways, show that
r(n)r(n + i) = Zi-^VirrVn).
5. Evaluate Lipschitz's integral
Jco
e~axJ 0(bx)dx,
0
when 0 < b < a, by using the expansion of J0(bx), and Ex. 1, (i).
6. Prove that, if a > 0,
JCO 1
e~axJ0(b Vx)dx = ~~e-b2l*a.
0 ^
7. Show that, if m > 0, n > 0,
B(m, n) = l ——^— du = \ ——— dv.
Jj ww+n J0 (1 + v)m+n
§ 65. Euler's constant in an integral form.
As a preliminary to the next paragraph, we shall now obtain
one form in which Euler's constant y (§6) can be expressed by
definite integrals. We have

DEFINITE INTEGRALS 69
y = lim (1 + ~ + o + • • • + - — log n)
n->oo z ° n
= lim (sn — log n),
n->oo
where ^=1 + - + - + ... + -.
Now we can write sn in the form
?„ = f * (1 + a + *« + . • • + *«-i)e&? = f * ^
J o J o ■*•
Put 1 — # = —, #=1 , dx = ;
n n n
then
5,
- tt'-('-3"}!
-a:+D{'-('-3"}?
and hence
Proceeding to the limit * when n -> oo, we obtain
. dt- f — (ft. . . (4.52)
0 v J J t
J oo g-*
—eft.
This integral will be required in § 75. We have, if x > 0,
00 p—t pi p—t t*COp—t
= pi-(1 - ^ + r g*
J1 1 — e~* r00
.—*—* + ],
--(1:-0^-+1;
00 e>,
* See Bromwich : " Infinite Series," p. 459 ; Whittaker and Watson :
" Modern Analysis," § 12.2.

70 INTRODUCTION TO BESSEL FUNCTIONS
and hence, by (4.52),
> oo Q-t /•* i — e~t
—dt = -logo; - y + ]
J2
(ft
J* / £ £2
.^-21+31---
^-logx-y + a-^ + ^L-... (4.63)
In particular, when # is small,
ooe-«
—-eft == — log # — y. . . . (4.54)
Examples IX
1. Show that
y = 1 + (i + log i) + (J + log J) + (i + logf) + • • •
2. By integrating both terms in (4.52) by parts, show that
100
er% log t dt.
0
3. Show that
i"(l) = - y.
4. Show that
,00 e-xv x2 x?
—dv= -logx-y + X~2T2] +
2.2! ' 3.3!

CHAPTER V
ASYMPTOTIC EXPANSIONS
§ 67. HankeVs definite integral for J0(x).
Returning to the integral (4.3), viz.
n
J0(a;) = H2 e«" «"»•«»,
7TJ _n
2
and making the substitution
dt
t = sin 0, dd =
V(i - n'
we obtain
2 ri £ixt
•^^Lva-*»)**• • • (5-1)
The real part of the integrand is an even function of t,
the imaginary part is an odd function ; hence we have also
J°(x)=If, v(i~t2f- • • (5-2)
§ 68. We can at once verify that the integral on the
right of (5.1) is a solution of Bessel's equation, by
differentiating under the sign of integration. For, put
then
d2y dy /d2y \ dy
+£+x^x{d + y) +
dx2 ' dx ' * \dx2 u J { dx
dt
71
C1 C ifpixt -\

72 INTRODUCTION TO BESSEL FUNCTIONS
i
1 w~iv/(1 _ t2)eixt)dt
= [- »V(1 - t2)e
as was to be verified.
l
= 0,
-1
§ 69. HankeVs contour integral.
Now consider the integral
y = \a^r=i?f • ■ ■ (5-3)
as a contour integral in the plane of the complex variable
t, assuming for simplicity that x is positive. Then we find,
as above,
*2+l+**=[-.vu - **-l
and hence, if we put a = ± 1, b — irj, and let rj -> -f oo,
Xdx* + dx+Xy~{)'
More generally, it is easy to see that we should still get
this result if we put a = ± 1, b = R(cos [5 -\- i sin /3), and
let R -> oo, provided sin /3 is positive, that is, provided
0 < /3 < 77.
It follows that y is a solution of Bessel's equation if the
integration is carried out along any path joining either of
the points t = 1, t = — 1, to an infinitely distant point in
the upper half of the £-plane. Thus, the path might be
any straight line drawn from either of the points t = 1,
t = — 1, to infinity in the upper half of the £-plane.
(Actually, in the limit, this line may coincide with the real
axis from 1 to + °°> or from — 1 to — oo, but this cannot
be inferred by differentiation under the sign of integration,
because the integrals obtained in this way are not then
convergent.)

ASYMPTOTIC EXPANSIONS 73
§ 70. Again, the function
V(i - *2)
is a regular function of t at all points in the £-plane, except
the branch points £ = 1, t = — 1, of the denominator.
Consequently, by Cauchy's theory of contour integration,
the value of the integral
r gi xt
Jvo^)** • • • (5A)
taken round any simple closed contour which does not enclose
either of those points, is zero.
§ 71. It is sometimes convenient to choose a contour
part of which consists of an arc of a circle of infinite radius
with its centre at the origin. The value of the integral
(5.4), taken along any such arc in the upper half of the
£-plane, vanishes when x is positive, by a well-known theorem
in contour integration.*
§ 72. When the integral (5.4) is to be evaluated along
any contour, it is necessary to pay attention to the
continuity of y^l — t2), the denominator of the integrand.
We shall assume that x is positive, and confine ourselves
to contours in the upper half of the £-plane ; accordingly,
for the sake of following the way in which y"(l — t2) varies,
we indicate in Pigs. 16-1, 16-2 the correspondence between
the upper half of the £-plane and a plane on which y^(l — t2)
is represented, the branch points £ = 1, £=-1, in the
first of these planes being marked by indentations. The
value of <y/(l — t2) at t = 0 is taken to be + 1-
§ 73. Use of the contour HBDPH. Hankel functions.
Pirst take the integral (5.4) round the closed contour
HBDPH. The part of the integral corresponding to each
* Whittaker and Watson, § 6.222.

74 INTRODUCTION TO BESSEL FUNCTIONS
of the infinitesimal circular quadrants at H and B vanishes,
and so does that corresponding to the infinitely distant
arc DF of an infinite circle (§ 71). Consequently, since
the value of the integral taken round this closed contour is
zero (§ 70),
)ixt
pf*=0- . (5.5)
,1 p+1„ ,-1 V ^
VJ-i Ji J -l+.w VX1 —
Now, by (5.1), we have
ttJ0{x) = J
iixt
V(i - n
dt;
(5.6)
+ / OO
fa*-
'G
1
K
lA^JLB
OO
-1 r O
Fig. 16*1. t-plane.
C!
B\
+oo
^.
.Q~>-'
+oo -/oo
Fig. 16-2.
V(l — J2)-plane.
and, in accordance with an accepted notation, we put
77 ri+ioo
\ K0U(x) = J
>iz*
V(i - t«)
cZtf, .
n
dt;.
(5.7)
(5.8)
then, from (5.5), we have
Jo(*) = i{Hoft>(a:)+H,<«>(a)}. . . (5.9)
§ 74. The functions H^ar), H0<2>(a:) which are also
solutions of Bessel's equation (§ 69), are called Hankel

ASYMPTOTIC EXPANSIONS 75
functions after Hankel, who wrote an outstanding memoir
on Bessel functions.*
We shall next express these functions as integrals in
which the range of integration is real. Along BD put
£ = 1 + irj, dt = idrj ;
then, paying attention to the continuity of \/(l — t2),
(§ 72), we must write
V(l ~ t*) = V(~ 2wj + 772) = V2e~^ V(t7 + frq*)9
and hence, by (5.7),
77 f00 e^e^^idri
from which follows
^) = ^^)^-77^^^ . (5.10)
^ Jo V(v + ¥1)
or further, putting u = xrj,
In the same kind of way we find from (5.8)
^-(=)^50-^ <6-12>
Thus, when x is positive, H0(1)(a), H0(2)(tf) are a
conjugate pair of complex numbers,f and J0(x) is the real part
of either, by (5.9). We shall prove that their imaginary
parts are iY0(x), — iY0(x) respectively.
§ 75. Use of the contour ABDEA.
Take the integral (5.4) round the contour ABDEA.
This gives
al fl+ioo pO \ fiixt
.+J, +L)w^=0- • (613)
Along AE we put
t = irj, dt = idrj ;
V(i - <2) = V(i + v2);
* Math. Annalen, I, 1869.
t H0(l)(a?), Hq(2)(#) are not conjugate when x is not real.
H,

76 INTRODUCTION TO BESSEL FUNCTIONS
then, having regard to § 67 and (5.7), we can write (5.13)
in the form
Putting
Ho^)^) = J0(x) + <Y(»), . . (5.14)
and separating the imaginary parts, we find
77v f00 er*n f1 sin art ,,
which we may also write, with rj = sinh u, t = sin 6, v — eM,
n
-Y(a:) = — I 6-* sinh u gu + gin (^. gin 0)^0
^ Jo Jo
n
= — I e 2V *; 1_ I sin (a: sin 0)d0.
Ji ^ Jo
The first integral on the right may be expanded in the
form
2 ^
dv.
i;
+ 2»+2!V2t;/ +' ' *
v
Now, suppose that x is small. Then, by Exs. IX, 4,
the first term of this expansion is approximately equal to
!og (ix) + 7> or lo§ x — (log 2 — y)-
The remaining terms are small in comparison; for
example, since e~*xv < 1, the numerical value of the second
term is less than
i
00 x , a?
x 2<A = 2-
Again, since sin (x sin 6) < a? sin 0, the value of the
second integral in the above expression for \ttY(x) is less
than
j
n
2
a? sin 9 dO, — #..
0

ASYMPTOTIC EXPANSIONS 77
It follows that, when x is small,
Y(x) = -{log x - (log 2 - y) . . .}. . (5.15)
77
Now, since J0(x) and H0(1)(#) are solutions of Bessel's
equation (§ 69), it follows from (5.14) that Y(x) is also a
solution, and hence that
Y(x) = AJ0(z) + BY0(aO,
where A, B are certain constants. Then, by (5.15) and
(1.16), when x is small,
2
-{log x - (log 2 - y) . . .}
77
= A(l — . . .) + B-{log x - (log 2 - y) . . .}.
77
Equating coefficients of log x, and then the terms
independent of x, we find B = 1, A = 0, and hence
Y(x) - Y0(x).
Consequently, by (5.9) and (5.14), we have now shown
that
H0(1)M = J0(a) + iY0(x), . . (5.16)
H0<2>Oz) - J0(a) - iY0(»). . . (5.17)
Also
J0(x) = HH0(1)(tf) + W*)}, . . (5.18)
Y0(tf) = - HH0<i>(:r) - HoW(a:)}. . (5.19)
§ 76. Use of the contour DBCD.
If x is positive, the integral (5.4), taken along the part
of an infinite circle that bounds the positive quadrant,
vanishes (§ 71). Hence, if we evaluate the integral round
the closed contour DBCD, part of which consists of an
infinite circular quadrant, we get
/f1 f°°\ e*xt
(L.+1 )7iH^s* = °'
that is, by (5.7)

78 INTRODUCTION TO BESSEL FUNCTIONS
Now, along BC (see Fig. 16-2),
V(i -*») = - W(t2 - l),
and therefore
77- poo pixt
^B^(x)^-i\iW-jf, . (5.20)
from which follows, on equating real and imaginary parts,
Ex. 1. Obtain Mehler's integral forms :
2 f oo
J0(x) = - \ sin (x cosh u)du,
7rJ0
2 f°o
Y0(sc) = 1 cos (x cosh w)dw.
ttJo
Ex. 2. Struve's function of zero order, H 0(x)9 may be defined by
Show that
77-.. . , # #3 #5
2h.(») - p - pr-gs + x.. 3.. 6i - • • • • (5-23)
Also, by taking the integral
Jpixt
V(i - *2)
round the contour ABDEA show that
^Jo v(l + T)
§ 77. Asymptotic power-series and asymptotic expansions.
We shall digress to explain the meaning of an asymptotic
power-series and an asymptotic expansion.
Definition 1.—An expansion of a function f(x) in the form
\AJ \AJ \AJ

ASYMPTOTIC EXPANSIONS 79
will be called an asymptotic power-series for f(x), if it has the
property that, for every fixed value of n,
f(x) = a0 + - + -2+. . - + -^iH rs ' (5-25)
where en(x) -> 0 when x -» oo.
It follows from the definition that:—
I. A convergent power-series of the form (5.24) is a
particular case of an asymptotic power-series.
II. The sum Sn(x) of the first n terms of an asymptotic
power-series for f(x) is an approximation to f(x), in the sense
that the difference f(x) — Sn(x) is as small as we please
compared with l/#w-1 when x is large enough.
Proof. Put
*,(*) = /(*) - S.{x) = a" +Jn(x);
then
Rn(x) _ an + en(x)
1/x*-1 X
which is as small as we please when x is large enough, since €n(x) -> 0
by the definition.
COR. The difference f(x) — Sn(x) is as small as we please
compared with the last non-zero term in Sn(x), when x is large enough.
III. The sum of the asymptotic power-series of two
functions f(x), g(x) is the asymptotic power-series of their
sum f(x) + g(x).
IV. A given function f(x) cannot have two different
asymptotic power-series for the same range of values of x.
To prove this, suppose two different series possible, and consider
their difference.
V. On the other hand, two different functions can have
the same asymptotic power-series.
For example, it follows from the definition, and by remembering
that lim (xne~x) = 0, that the asymptotic power-series for e~* is
X-* 00
0 + 0 + 0 + . . . , (x > 0). Consequently, f(x) -\- e~x has the same
asymptotic power-series as f(x), for x > 0.

80 INTRODUCTION TO BESSEL FUNCTIONS
VI. A function may have no asymptotic power-series.
Such a function is e*, (x > 0), since e* -* + <x> when x -> + <x>.
VII. The product of the asymptotic power-series of two
functions f(x), g(x) is the asymptotic power-series of their
product f(x)g(x).
In particular, since the asymptotic power-series of e~* is
0 + 0 + 0 + . . . , it follows that, if f(x) has an asymptotic power-
series, then the asymptotic power-series of the product e~x .f(x) is
0 + 0 + 0 + . . . , (x > 0).
VIII. Let f(x) be a function whose derivatives of all orders
exist at x = 0. Then Maclaurin's series
/(*) =/(0) + a/'(0) + |V"(0) + |V"'(0) + . . .
whether it converges or not, has the property that, for
every fixed value of n,
/(*) =/(0) + */'(0) + ... + (^iriyi/^Ho)
+ ^{/(n)(0)+ *?„(*)}
where rjn(x) -> 0 when x -> 0.
By writing 1/x for x it follows that, if /(n)(0) exists for
all values for n, the function /(1/x) is represented
asymptotically by the power-series
for large values of x.
§ 78. Definition 2.—If the asymptotic power-series of the
quotient f(x)/g(x) is given by
g(x) x x2 x*
we shall say that
/(x)-^)(a. + 5+§ + 5 + ...)
is an asymptotic expansion of f(x).

ASYMPTOTIC EXPANSIONS 81
We shall also say that the sum of the asymptotic
expansions of two functions is an asymptotic expansion of their
sum.
§ 79. Asymptotic expansions of the Hankel functions.
Now write (5.11) in the form
2\*e*(*-i»> /V
where
(Z \* 0H*-*WJ / l\
/(j) = )D7i1+T) du'
Differentiating n times under the integral sign and
putting x = 0, we find
13 5 2n- 1 £
Jo
and hence, by §§ 61, 62,
/<«>(o) = (-r\ll ■ ■ • ^r1 ■ U?^un~idu'
/<">(o) = (-)»l-*-5--; {2n~1]. £ • Wn + I)
2" 2'
= (-)» 12. 32. 5a . . . (2n - 1)2. ^¾^.
O
Thus, /(n)(0) exists for all values of w, and it follows, by
(5.26), that f(l/x) is represented for large values of x by the
asymptotic power-series
, f i 12. 32 a2 12. 32. 52 a3 , 1
V77\ 8a: + 2! (8z)2 3! (8a:)3 + ' ' 'J
Substituting this series for /(1/a:) in (5.27), we obtain
Ho(1)(*} = Qx)'^X " ^(1 " ^ + P " 32 ^
Sx ' 2! (8a:)2
12. 32 . 52 a3
~~ 3! (8a;)
3 + • • -j J

82 INTRODUCTION TO BESSEL FUNCTIONS
and, replacing i by — i, we have also
/ 2 \ * f 7 1 2 02 9*2
ttb/ L 8a: ' 2! (8a;)2
12 . 32 . 52 i3
H 3l (&z0~3 + ' ' '.
These are asymptotic expansions of the Hankel functions
in accordance with the definition of § 78. They have been
derived on the assumption that x is positive (§ 69), but it
can be proved * that they remain true when x is replaced by
z, = | z | eie, provided that — tt < 9 < 2tt in the case of
H.0{1)(z)> and that - 2tt < 6 < tt in the case of H0<2>(z).
Note that the series in brackets do not converge for any
value of x.
§ 80. Asymptotic expansions of J0(x) and Y0(x).
If we now put
12 32 1 12. 32 52. 72 1
P = 1 ~ ~TT (tej"«+ 4! ' (to)*-- • * (5'28)
1 12 32 52 1
Q = te TT~ (top + • • * ' * • • (5,29)
we can write the asymptotic expansions of the Hankel
functions in the form
Ho«(a!) = (—) V"-*">(P - *Q), . . (5.30)
\ttx/
Ho(2)(*0 = (-)^-^^(2 + *Q), • (5.31)
and hence, by (5.16),
l°{x) = GD*{P cos (* ~ i) + Q "K* ~ l)}' (5-32)
JJx) =
/ 2\*
Y°{x) = (™) {p sin (* ~ i)_ Q cos (* ~ i)} (5-33)
which are the asymptotic expansions of the standard Bessel
functions of zero order, in accordance with the second
definition in § 78.
* Watson, p 198.

ASYMPTOTIC EXPANSIONS 83
The following deductions may be made :—
I. J0(x) = (—J lcos(x — ^j +2>(*)1, • (5-34)
Y°{X) = (^)\Sin(X ~ l) + q{x)}' ' (5'35)
Jo'{x)=~d)i{Sin(X~1l)+r{x)}' • (5>36)
where p(x), q(x), r(x) all -> 0, when x ->■ + °°»*
II. The positive roots of the equation J0(x) = 0 are
given approximately by
as = (s — \)tt. . . . (5.37)
This approximation is fairly good even for s = 1, 2, 3, . . .
§ 81. Asymptotic expansion of I0{x) and K0(#).
The asymptotic expansions of the modified Bessel
functions of zero order, I0(x), K0(#), may be deduced from
the formulae
I0(x) = 30(ix) = i{RoM(ix) + Ho<»(«e)}, . (5.38)
K0(x) = ^HoW(te), . . (5.39)
the second of which will be proved below (§ 82).
Assuming that the asymptotic expansions of TLq^^x),
H0(2)(#) hold good when x is replaced by ix (§ 79), we have
HoU>M =-<(A)V.{l - "L + ^f ^- • • •} (5.40)
*^ = &**{l + 8^+^(8^ + ---} • <8-41>
From (5.40) and § 77, VII, it follows that, if x > 0, the
asymptotic power-series of ^/xS.^(ix) is 0 + 0 + 0 + • • •
and hence further that
e~* <>/xTL0W(ix) = 0 + 0 + 0 + ...; . (5.42)
* As regards J0'(#), an asymptotic power-series deduced from a
Maclaurin series, as in § 78, VIII, can always be differentiated.

84 INTRODUCTION TO BESSEL FUNCTIONS
also, from (5.41), e~x'yfxH.^\ix) has the asymptotic power-
series,
e-v*h.«(«> = Q\i + h+lrf(k>+---} (5-43)
Remembering § 77, III, we have by addition and (5.38)
the asymptotic power-series
2e-V*Io(*)=(|)i{l +i + ^f(8^+ • • i
and hence the asymptotic expansion
l + Sx + ^T(Sxf+ • • '}• (5*44)
Again, from (5.39) and (5.40), the asymptotic expansion
of K0(x) is given by
/7T\* f 1 12 32 1 1
Ko(*> = (¾) 6~T - 8-, + -2! - (teji ~ " • J- <5-45)
Note.—From the integral (5.20) we conclude that H0{1)(ix) is a
pure imaginary, and with this conclusion (5.40) is consistent. But
(5.41) appears to imply that H0(2)(£e) is real, and that this is not
true follows from (5.38). Actually, from (5.41) we can only deduce
that, in the asymptotic power-series (5.43) of e-xA/xH.0{2)(ix), when
x > 0, the imaginary part is of the form 0 + 0 + 0+. . ., and
this is confirmed by (5.42), since the imaginary part of H0{2)(ix) is
— H0(1)(£»), as we see from (5.38).
§ 82. It remains to prove (5.39).
Proof. Since *KQW(x) is a solution of Bessel's equation,
it follows that T3.0^(ix) is a solution of the modified equation
(3.1), and hence, by (3.6), that
•
-T^^ix) = AI0(x) + BK0(x), . . (5.46)
where A, B are certain constants.
IoM =
>«
a/(2ttx)

ASYMPTOTIC EXPANSIONS 85
Now, from (5.20) we have
Jo
,0VI (•+*)*>
j;
i *>
and it follows, as in § 75, that when x is small
^0W(ix) = - log x + (log 2 - y) . . .
Substituting in (5.46), and using (3.5), we have, when
x is small,
— log x + (log 2 — y) . . .
= A(l + . . .) + B{- log x + (log 2 - y) . . .}.
Equating coefficients of log #, and then the terms
independent of x, we find B = 1, A = 0. Consequently, putting
these values in (5.46), we have
Ko(x) = ^HoW(«r), . . (5.48)
as was to be proved.
Ex. 1. Obtain the asymptotic expansion of K0(#) directly
from the integral (5.47), which, with (5.39), gives
Joo e~xt
Begin by putting £=1+-.1
Ex. 2. Calculate J0(#) from (5.32) when x = 10, and compare
the result with that tabulated at the end of the book.
Ex. 3. From (5.44) obtain the asymptotic expansions of ber x
and bei x (see § 51) :—
ber* = tW){Pcos (72 - i) + Qsin (72 - I)}'
bei x = Vo£)(Psin (72 -1) ~ Qcos (72 -1)}'

86 INTRODUCTION TO BESSEL FUNCTIONS
where
P=l + a00Bi+ __00BT + . ..
~ 1 . 77 , P. 32 . 2tT ,
Q = ~ sin T + sin — + . . .
8a; 4 2 !(8#)2 4
Ex. 4. Show that the asymptotic expansion of Struve's function
H 0(x) can be written
/ x at / x . 2 /, 1 , P. 32 P. 32. 52 . \
H o(«) = Y0(aO H 1 . H t ; +...).
x ' 7ra?V as8 a4 a;6 /
[See § 76, Ex. 2.]
Ex. 5. Show that (see e.g. Hardy: " Pure Mathematics," 7th
edn., § 167)
(l + />)-* = Sn + Rn,
where
S„ = 1 - \h + *-^2 - f-1!1!^ + • • • to n terms,
If the real part of h is positive or zero, show further that
13 5 /„ _ i\
ir„k 2'2'2 'n;( g>i/*i-•
Deduce that, in the asymptotic power-series for/(1/a?) in § 79, the
modulus of the difference between f(l/x) and the sum of the first
n terms of the series is less than the modulus of the (n + l)th term
for every positive value of x.

CHAPTER VI
BESSEL FUNCTIONS OF ANY REAL ORDER
§ 83. Bessel functions of any order.
We can define Jn(x), when n is any integer, to be the
coefficient of tn in the expansion of the function
e^ ", . . . . (6.1)
in ascending and descending powers of t. Now,
xt
xt , 1 /xt\2 , 1 /xt\3
Consequently, picking out the coefficient of tn in the product
of these two series, we find by the above definition, if n is
a positive integer, that
Jn(#) = ^1(1 -
2nnl\ 2.2n + 2
+ 2.4.2n + 2.2n + ± -•••)» (G-2>
and that
J-n(#) = o' . (1 -
+ 2.4.2^ + 2.2^ + 4 • " 'J' (6'3)
2»nl \ 2 .2n + 2
2.4.2½ +~2 . 2» + 4
The first of these agrees with the original definition of
Jn(x) in § 2 ; the second can be written
J-n(») = (-)nJn(»). • • (6.4)
87

88 INTRODUCTION TO BESSEL FUNCTIONS
The graphs of J0(x), J^x) have been given in Fig. 1.
Those of J2(tf), J*ix)> JeOz) are shown in Fig. 17.
§ 84. If n is not an integer, Jn(x) can be defined by the
series
Jn(x) =
& /
n + 1)\
2nr(n + 1)
1
X'
+
2 . 2n + 2
x*
2 . 4 . 2n + 2 . 2½ + 4
. . .), (6-5)
where 71 denotes the Gamma-function (§ 61).
•6
0
1
2
, Jo
c^z
3
4 6
|itJ
6
7\
s^6
9
X
10
Fig. 17.
In particular, by putting n = |, n = — \ in turn, we
find
Ji(a;) = Gi)*sin *•
J-*(a;)=w) cosa:-
. (6.6)
• (6.7)
§. 85. The modified Bessel function In(x) is defined by
the series
.2
lnW - 2nri<n + ^(1 + 2 .
X'
2n + 2
+
x'
2 . 4 . 2n + 2 . 2½ + 4
[cf. (3.3)]. We note that
+ . . .) (6.8)
. (6.9)

BESSEL FUNCTIONS OF ANY REAL ORDER 89
In particular, we find
I4(a;) = (—J sinha?, . . (6.10)
l_,(x) = (—) cosh a;. . . (6.11)
\7TX/
§ 86. BesseVs integral for Jn(x).
In accordance with the definition in § 83,
+ J0(x) + J^t + J2(x)t2 + . . . (6.12)
which, by (6.4), may be written
cH'-l) = J0(a.) + J^ft -1) + J2(x)(t* + £
+ JB(x)(tfi-±) +. . . (6.13)
Making the substitution t = eie, we get
eix*[ne = J0(x) + lAS^x) sin 0 + 2J2(a:) cos 20 + . . . (6.14)
and hence, by equating real and imaginary parts,
cos (x sin 0) = J0(#) + 2J2(#) cos 20
+ 2J4(a:) cos 40 + . . . (6.15)
sin (x sin 6) = iJ^x) sin 0 + 2J3(a:) sin 30 + . . . (6.16)
These are Fourier expansions, and from them it follows,
by the ordinary rule for finding the coefficients in a Fourier
series, that, if n is even,
If*
J„(#) = - cos (x sin 6) cos n0d0, . (6.17)
ttJo
0 = - sin (x sin 0) sin n0d0 ; . (6.18)
ttJo
while, if w is odd,
1 f*
0 = - cos (x sin 0) cos w0 d0, . (6.19)
If71
Jn(x) = -1 sin (# sin 0) sin w0 d0. . (6.20)

90 INTRODUCTION TO BESSEL FUNCTIONS
By adding we find, if n is any positive integer,
1 f *
Jn(x) = - cos (nd — x sin 0)d0, (6.21)
77 J o
which is known as BesseVs integral for Jn(x).
Examples X
1. By considering the expansion of the product
j('-f) JH)
e . e
in ascending and descending powers of t, in two ways, obtain the
Addition formula for J0(x), viz.
(i) Jo(* + 2/) = J0(s)Jo(y) - 2Ji(*)Ji(y) + 2J2(x)J2(y) - . . .
By putting i/ = — x, y = x in turn, show that
(ii) 1 = J0*(x) + 2JV(a) + 2J22(z) + • • • •
(iii) J0(2a;) = J0*(x) - 2J^{x) + 2J22(a) - . . .
Show also that
(iv) J1(2«) = 2J0(a)J1(aj) - 2J1(a)J8(a) + 2J2(a)J3(a;) - . . .
[Compare the trigonometric formulae 1 = cos2 x + sin2 x,
cos 2# = cos2 x — sin2 x, sin 2# = 2 sin x cos #.]
2. Show that
Jo(* + W) = J0(a)Io(y) - 2J2(^)I2(2/) + 2J4(z)I4(2/)
- *{2J1(aj)I1(y) - 2J,(aj)I8(y) + . . .}
3. Show that
00
eixcosO = JQ(X) _|_ 2^inJn(x) cos nd.
n=l
By considering the integral
J2tt
eix cos 0 . e-iy cos (0-«)d0
o
in two ways, obtain the generalised Addition formula for JQ(x), viz. :—
00
Jo{\/(a2 + y2 — %XV °os a)} = J0(x)J0(y) + 2^JB(a;)J„(t/) cos na.
4. Show that, if n is an integer,
00
J„(* + 2/) = 2 J*(*)J«-*>(2/)-
p = — CO

BESSEL FUNCTIONS OF ANY REAL ORDER 91
5. By expanding cos nd, sin nd on the right-hand side of (6.14) in
ascending powers of sin 0, and then arranging both sides in ascending
powers of sin 0, and equating coefficients, show that
1 = J0(x) + 2J2(z) + 2J4(a;) + 2J6(z) + . . .
x = 2{J1(a) + 3J3(*) + 5J5(z) + 1J7(x) + . . •}
x* = 8{J2(x) + 4J4(a) + 9J6(a) + 16J8(aO + . . .}
6. Obtain the expansions in the last example also by
differentiating (6.14) with respect to 0 repeatedly and putting 6=0.
7. By expanding Jn(bx) and integrating term by term, show that,
if n >—1, a> 0,
8. Show from Ex. 1, (ii), that, if x is real and n a positive integer
[Jo(*)|<l, |J„(*)| <1/V2.
9. Prove from the series for J„(z) that, if n > 0, and r = | z |,
1 Jw(2) I < 2«r(n + I)" '
Deduce that J'„(z) -> 0 when n -> 4" oo , for all values of z.
10. If z — a; + iy, (2/ > 0), and if 0 < t < 1, prove that
| cos zt | < ev.
Hence, using Hankel's integral (§ 93) for J„(z), prove that
rnev
I Jn{z) I < 2T(n + 1)*
where r = \z\.
§ 87. BesseVs differential equation of order n.
If we write
y = xn~ 2 . 2w + 2 + 2 . 4 . 2w + 2 . 2w + 4 ~ ' ' " (6'22)
we find at once, on differentiating term by term, that
d ( dy\ _ 8 „ _ (n + 2)V+ii
xdxVdx) ~nX 2.2n + 2
(n + 4)V>+ *
+
2 . 4 . 2ra + 2 . 2w + 4

92 INTRODUCTION TO BESSEL FUNCTIONS
Also
^2/^tn+2 ,™2/vt7i+4
n2y n2xn 2 . 2n + 2 + 2 . ± . 2n + 2 . 2n + 4
By subtraction,
d/Jy\ _ _+2 , ^+4
a / dy\ . ,. , 5
dx\ dx/ J ' 2 .
2n + 2
= — x2y,
and hence xJ~(xt) + (x* " w% = °> ■ • (6*23)
or ^ + ^ + (^-.^)^ = 0, . (6.24)
which is known as BesseVs equation of order n.
Since y = 2nr(n + l)Jw(ff), it follows that
^¾^ + "^r +(*2 - n2)J«w = °; (6-25>
in other words, y — Jn{x) is one solution of Bessel's equation
of order n. The general solution of this equation will be
referred to later.
§ 88. Bessel's equation may also be written
d2y , 1 dy , /_ n2\ _ ... ft/ix
Replacing x by fcr, we find that y = Jn(kx) is one solution
of the equation
S+MS+(*•-£)»-«>■■ • <6-2"
§ 89. Recurrence formula.
If we write down the series for Jn-i(a?) and Jn+iO*0>
add and subtract them, and compare the sum with the
series for Jn(x), and the difference with the series for the
derivative Jn'(x), we find the following two formulae,
which, together with other forms in which they can be

BESSEL FUNCTIONS OF ANY REAL ORDER 93
expressed, are known as the recurrence formulce of the Bessel
functions :—
2n
-;Jn(x) = Jn-^X) + Jn+1{x), . . (6.28)
2Jtl» - J^x) - Jn+iO*0- • • (6.29)
By adding and subtracting and dividing by 2, we obtain
the two formulae
n
Jn-i(x) = -;J«(z) + Jn'(x), ■ • (6.30)
n
Jn+lO*0 = - JnW - Jn», • • (6.31)
which express Jw_i(#), Jn+i(#) in terms of Jn(x), Jn'(x).
The last two formulae can also be written
d^XnJn{x)} - Xnj^ix), . . (6.32)
dx
d f Jnfo)) = __ J«+i(g)
dx[ xn ] xn
In particular,
(6.33)
fa{xJi(x)} = ^Jo0*0> • • (6.34)
^J0(x) = - J^x), . . (6.35)
results which have already been noticed in § 3.
Inversely, we have
\xn3n_x{x)dx = xn3n{x), . . (6.36)
JJ^fa =_££!>; . . (6.37)
and hence also, if a is a constant (4= 0),
[xn3n.1{<^)dx = ^^, . . (G.38)
\Jn+^)dx = _ W^ (639)
J xn a.xn v '

94 INTRODUCTION TO BESSEL FUNCTIONS
§ 90. By differentiating (6.31) repeatedly, and after each
differentiation substituting for 3n"{x) from the differential
equation (6.25), it follows that Jn+1(#) and all its derivatives
can be expressed in the form
VJn{x) + QJn'(s),
where P, Q are polynomials in 1/x.
Moreover, Jn_i(#) and all its derivatives can be expressed
in this form also, as we see by treating formula (6.30) in
the same way.
Corollary 1. Jn{x) and all its derivatives can be
expressed in the form
PJ» + QJ*'(s),
where P, Q are polynomials in 1/x, and n — p is any integer,
positive or negative, or zero.
Corollary 2. If n is an integer, Jn(x) and all its
derivatives can be expressed in the form
PJ0(aO + QJo'(s),
where P, Q are polynomials in 1/x.
Examples XI
1. Show that
(i) J,"(a) = - J0(x) - i J.'(ar),
(ii) J „'"(*) = i J,(ar) + (| - l) J.'(*).
2. Show that
2
(i) Jt(x) =-Ji(a;) - J0(tf),
Ju
(ii) J3(X) = (| - l)^) - |J„(Z),
("0 J«(*) = (f - 3 J^» " (S - O"1^-
3. Show that
(i) J,'(«) = IU*) + (J - l) Jo'(*),
(ii) J,'(«) = (3 " l)j.(«) + (^-j)-1"^)-

BESSEL FUNCTIONS OF ANY REAL ORDER 95
4. Show that
(i) J2(X) = JQ(x) + 2J0"(a),
(ii) Jz(x) = 3^) + 4J/»,
(iii) Jn+2(z) = J2n + 1 - 2n{n2x~ ^}^) + 2(n + l)Jn"(tf).
5. Show that
/ 2 \ * /sin x \
J#> = fe j ("IT ~ cos V'
( 2 \ \ ( . cos x\
J-#>=U)'(-Sin*
x
and express J:>(#), J- &(x) similarly.
6. Show that
(i) Jf+ifo) = __ i A ^-(^)
' £n+1 a; rfx a;"
Deduce that
^ ' xn+T \x dx) Xn '
7. Show that
(i) 2JB'(a>) = Jn-t(x) - Jn+1(x),
(ii) 4J/'(a?) = J„_2(x) - 2Jn(z) + J„+2(*),
(iii) 8Jn'"(a>) = Jn_3(z) - SJ^x) + 3Jn+1(aj) - J,+ ,(aj)f
8. Show that
(i) zJ0(*) = 2{Jx(a;) - 3J3(*) + 5J6(z) -...},
(ii) $JQ(x)dx = 2(^(^) + J3(x) + J5(a;) + ...},
(iii) JxJ02(x)cto = 2{Jx2(x) + 3J32(a) + 5J52(x) + • • •}
Obtain corresponding results when J0{x) is replaced by Jn(x) on
the left-hand sides.
9. Show that
n
Jo(a?)Ji(») ~ J«(a)J*+i(aO = 2 ^ Jr(a;)J/(aj).
r = l
Deduce that
Joo
Jn(t)Jn+1(t)dt = iJ02(tf) + Jl2M + ... + J«*(a0,
a:
^i;^ = i J0*(x) + Jx2(^) + ... + JViM + WW-
X

96 INTRODUCTION TO BESSEL FUNCTIONS
10. Show that
[xnJn(x)dx = (2rc — l){xn-1Jn-1(x)dx — xnJn-x(x).
Deduce that, if n is a positive integer,
I xnJn(x)dx depends upon fj0(#)cfo.
Evaluate
11. Show that if n > 0
{oo poo
0 Jo
Deduce that, if n is a positive integer,
J 00
Jn(x)dx = 1
0
[for n — 0, seo (4.7)] ; and also that
r
00 J«(z)^ 1
-dx — -.
x n
Hence, if b > 0, show that
f°°T tu w l f°°J»N,
1 J„(o.r)aa; = T, I v cfa
Jo 6 J0 a;
1
n'
12. Show that
2w
—■!,,(») = I„-i(«) - I»+i(«),
2rtt(^) = in-iW + i„+iW.
13. Show that
/ 2 \ I / . sinh a;
I«(a;) == — cosh x ,
*v \irxj \ X J
cosh #\
^= (i)*(sinha;-
a;
14. Show that y — ln(kx) is one solution of the equation
d*y , ldy fi., , ™2
dx2 x dx
(*- + J>-«-
(When & = 1, this equation is called the modified Bessel equation
of order n.)

BESSEL FUNCTIONS OF ANY REAL ORDER 97
15. If n is an integer, show that Jn(x) can be expressed in the form
Jn(x) = J2(x)An_2 + Jx(a;)Jn_3
where ^„-2> ^„_3 are polynomials in l/x; express them as determinants
of order n — 2, n — 3 respectively.
§ 91. Sonine's integral.
Consider the integral
Jn, v = f V - x*)p • *"* ^ni^dx, . (6.40)
Jo
where n > — 1 and a is a constant.
When p = 0, we find at once, using (6.38),
^,0 = ^1^. • - • (6.41)
Next, suppose that p is a positive integer. Then,
by (6.38), we have
and hence, after integrating by parts,
j -¾
u n. 33 un+li J>-1
Applying this reduction formula repeatedly, we get
T -2"P!T
that is, by (6.40) and (6.41),
fl(l - &)• . **ijn{«x)dx = 2PF{1 + 1}J„+P+1(«). (6.42)
Jo a17
Changing the notation by putting x for a, and £ for #,
we can write this result in the form
Jn+„+1(*) = 2*r(T+ i)l/wM) • *n+1(1~ *')f*' (6,43)
It holds good if n > — 1, f> > — 1. For, although we
have so far supposed p to be a positive integer, it is easy

98 INTRODUCTION TO BESSEL FUNCTIONS
to verify that this restriction is unnecessary, by expanding
Jn(xt) and integrating term by term, using § 64. The
conditions n > — 1, p > — l9 are necessary to ensure
convergence at the lower and upper limits respectively.
Putting ni = n + p + 1, we obtain another form of the
last result, viz.
J»(«) = 2nJr{m - n)\0J"{xt) ■tn+1(l ~ t%)m-n-ldt (6-44>
where m > n > — 1.
§ 92. The integral on the left of (6.42) or on the right
of (6.43), (6.44) will be called here Sonine's integral* It is
one of several integrals with which his name is associated
in the theory of Bessel functions.
Sonine's integral enables us to express a Bessel function
Jm{x) as an integral involving Jn(xt), of lower order, provided
that m > n > — 1.
§ 93. Deductions from Sonine's integral.
Put n = ■— J in (6.44) ; then, since
J-i{xt) = (Js)*cos "*'
by (6.7), we find, if m > — J,
xm r1
Jm{x) = 2"»-V^(m + l)j „ °°8 ** • (1 - t')m-'dt, (6.45)
which may be called HankeVs integral for Jm(x).
Putting t = sin 0, we have, further, if m > — J,
tc
Xm f2
J«(a) - 2m.ly^m + X)J0COS (:r sin 0) cos2W 0 rfl9' (6,46)
which is known f as Poisson's integral for Jm(#), or BesseVs
second integral for Jw(#).
* " Sonine's First Finite Integral," Watson, p. 373.
| Watson, p. 24.

BESSEL FUNCTIONS OF ANY REAL ORDER 99
Examples XII
1. Show that, if n > — 1,
J«+i(a)
>
a
(i) \ xn+1Jn(czx)dx
J o
(ii) f V+Hl - x*)Jn(*x)dx = 2Jw+22(oc),
(iii) rV+vi - x*)*jn{oix)dx = 8Jw+3b(qc).
2. By writing
x*» = {1 — (1 — a2)}*
= 1 - »d(l - a*) + *<78(1 - a;2)2 - . . .
show that, if p is a positive integer,
2f x^+iJ0(<xx)di
J 0
In a similar way, when n > — 1 find the value of the integral
f xn^^+1Jn{aix)dx.
J 0
3. Show that, if n > — 1,
a) rv+^a^z = **±iW - 2J-^(a).
Jo a a2
(ii) (V+'J,,^)^ = ^^ - ^u#) + !£-±J<£L>.
J 0 a a-1 a*5
4. Show that
f« a?J0(igy) rf _ si
5. Show that
sin ay
n
sin x
(i) f J0(# sin 6) sin 0 dO =
Jo
71
(ii) f2 J^a sin 0) sin2 0 dB =
J n
sin x — x cos #
o x
JX
2
(iii) (2JM(xSine)sin«+iflde= (£j J^i(as).

100 INTRODUCTION TO BESSEL FUNCTIONS
6. Show that, if n > 0,
Deduce that there is a number f such that
J-w = s^nri? • • • (6-47)
where 0 < f < x.
7. By writing (6.44) in the form
m\ / 2m-nr(m — n))0 nK ' v ' v '
and using the mean value theorem for integrals, show that there is a
number £ such that
^W^'W), • (0.48)
where 0<£< x, m>n> — 1.
8. By putting m = ^, x = it in (6.48), show that, if — 1 < n < J,
the equation Jn(%) = 0 has a root between 0 and it.
Also, by putting m = — ±, x = \tt, show that if — 1 < n < — J,
the equation Jn(%) = 0 has a root between 0 and \tt.
9. If p is a positive integer, and n > | + 2p, show that
T M - XP+1 f °° Jn(**)(*8 ~ 1)P,7,
Deduce that
.00
{00
Jn(atf)*-n+2p+1cft
1
1f 2 /2\2 /2\s 1
10. Assuming the first result in the last example to hold good *
when p > — l,n> ^ + 2p, deduce that, if — }2 < m < -J-,
T f — 2m+1 f°° sma^
* It can be deduced from Watson, p. 417, (5), The condition p > — 1
is necessary for the convergence of the integral at the lower limit ;
n > J + 2p is necessary for convergence at the upper limit, since Jn(xt)
C cos (xt — A)
behaves like when t is large (see § 95).
Vixt)

BESSEL FUNCTIONS OF ANY REAL ORDER 101
11. Making the same assumption as in the last example, show
that
00 xJ0(xy)dx cos ay
i
a v(*2 - «2) y
§ 94. LommeVs integrals.
Put u = Jn(ax), v = Jn(fix) ; then, by (6.27), using
dashes to denote differentiation with respect to x, we can
write the equations satisfied by u and v in the forms
XU" + u> + (a* _ ^xu = o, . (6.49)
xt," + v' + ()82 - ^xv = 0, . (6.50)
from which we find, by the method of § 11, that
(j82 — a2)J#Jn((x.x)Jn(f3x)dx
= x{aJ nr (cae)J n(fix) — pJn'(f}x)Jn(a.x)}. (6.51)
Again, multiplying (6.49) throughout by 2xu', we have
d
2xu'-rr(xu') 4- 2(ol2x2 — n2)uur = 0,
ax
d
or -t-{x2u'2 + (&2x2 — n2)u2} = 2ol2xu2.
Integrating, we get
x2un -f- (a2#2 — n2)u2 = 2a.2$xu2dxi
and hence
l#Jn2(oc#)<&r = — x2Jn'2(a.x) -f [x2 ^\Jn2(o(.x) -, (6.52)
which may also be written, with the aid of (6.30), (6.31),
f x2
\X3 n2(0LX)dx = -rt{Jn2(aa:) — Jn-l(aaOJn+l(atf)}. (6.53)
In particular, when we integrate between the limits
0 and 1, we find from (6.51) and (6.53), provided n > — 1
to ensure convergence at the lower limit,

102 INTRODUCTION TO BESSEL FUNCTIONS
(/?2 — a2) I xJ n(a.x)J n(fix)di
J o
\
x
iff \ i tit \| i
= «Jn'(a)J„(j3) - ySJ»'0S)Jn(a), (6.54)
1
aJn2(aa:)6Za; == |{Jn2(a) — Jn-i(a)Jn+i(a)}. (6.55)
o
Examples XIII
1. If a, 0, (jS2 * a2), are two roots of the equation Jn(x) — 0,
show that, if n > — 1,
{xJn(cLx)Jn(f$x)dx — 0.
o
Show also that the same result holds good if a, /?, (j32 * a2),
are two roots of the equation
xJn'(x) + HJn(z) = 0,
where H is a constant.
2. Show that (6.54) can be written in the form
(jS2- <X.*)^xJn(KX)Jn(Px)dx= ^$n-MJn+AP) - Jn+l(*)Jn-l(P)}.
3. Derive (6.52) from (6.51) by differentiating partially with
respect to /?, and then putting j3 = a.
4. Write down the differential equations satisfied by Jm(x),
Jn(x), and show that, if m + n > Q>
0
5. Show that
(m« - n')fJ-(i8y"(a!)«fa = a{Jm'(a)Jn(a) - J.(a)J.'(«».
J£C2
aJn'Mda = ^{J«2(a:) - J„-i(a:)Jn+i(«)}-
(ii) J x*»+*Jn(x)Jn+1(x)dx = hx*»+*Jn+1*(x).
Jx2n+2
x*»^Jn*(x)dx = ^--^{Jn^x) + J„+1»(*)}.
^2n+1J«-l(^)Jn+l(^)^ = 4n _L 2^-1^^^1^) + J«MJ« + >(»)}•
6. If n > — 1, show that
CxIn*(Kx)dx = £{In2(a) + I.-i(a)I.+ 1(a».
J 0
7. Show that (cf. Exs. XI, 8)
i*2{J02(x) + Ji2(*» = Ji20*0 + 3J32(a^) + 5J52(z) + • • •

BESSEL FUNCTIONS OF ANY REAL ORDER 103
Show also that
lx*{Jn\x) - Jfl^1{x)Jn+1{x)}
- (n + l)Jn+12(x) + (n + 3)Jn+32(*) + (n + 5)Jn+52(a;) + . . .
Deduce that
-r = j!>(aj) + 3J32(a) + 5Js2(a) + IJr^x) + . . .
^r 2 * * *
55-^ = J^a) - 3J32(a) + 5J52(a) - 7J72(.r) + . . .
7JT "5" 2 2 2
§ 95. Large values of x.
If we make the substitution
u = y^Jx . . . (6.56)
(cf. § 12) in Bessel's equation (6.24), we find that the
equation satisfied by u is
£--(1-^1--- • <6-5"
2
Now, when x is large enough, the term (n2 — 1)/%2 is as
small as we please compared with 1, and then we have
approximately
d2u
dx2
= — u.
of which the general solution can be written u = C cos (#—A),
and we infer that every solution of Bessel's equation of
order n can be written in the form
y == -^{cos (x-\) + r{x)}9 . . (6.58)
where C, A are constants, and r(x) -> 0 when x -> + °°-
We deduce that
I. If x > 0, y <\/x is bounded.
II. If y = f(x) is any solution of Bessel's equation, the
equation f(x) = 0 has an infinite number of real roots, and
consecutive large roots differ by tt approximately.
We shall now examine the roots of the particular
equation Jn(x) = 0 more closely.

104 INTRODUCTION TO BESSEL FUNCTIONS
§ 96. Roots of the equation Jn(x) = 0, when n is real.
If a is a root of the equation Jn(x) = 0, it is plain from
the series for Jn(x) that — a is also a root. It will therefore
suffice to confine our attention to positive roots.
I. If k > 1, the equation Jn(kx) = 0 has at least one root
between a and a -\- tt, when a is sufficiently large.
Proof. Put u = u(x) — \/(kx)Jn(kx) ; then, replacing
x by kx in (6.57), we find
£ = - (*'- *^)». ■ • (C-59)
Also, put v = sin (x — a) ; then
g = -*>. . . . (6.60)
Multiplying the first of these equations by v and the
second by u and subtracting, we get
d ( dv du\ /72 n2 — l\
t uj ih- = ' fc2 -— 1 z-^ )uv.
dx\ dx dxl \ x2 I
Now integrate between the limits x = a, x = a + tt.
Then, at the lower limit, v = 0, dv/dx = 1 ; at the upper
limit v == 0, dv/dx = — 1 ; hence
'a+n' n2 — Is
ra-rn /
— u(a + tt) — u(a) = I (¾2 — 1 —
x2
uvdx.
In the integrand on the right, u is continuous throughout
the range of integration if a > 0; v is positive ; and the rest
of the integrand is positive if a is large enough.
Consequently, by the mean value theorem for integrals, we have,
for sufficiently large values of a,
[a+n / ™2 1\
_ u{a + 7T)- U{a) = «(|)J ^ (¾2 - 1 ^j-lj
vdx,
where a < £ < a + tt.
Since the integral on the right is now necessarily positive,
it follows that u(a), u(£), u(a + tt) cannot all be of the same
sign, and hence that the equation u(x) = 0 has at least

BESSEL FUNCTIONS OP ANY REAL ORDER 105
one root between a and a -\- tt. The theorem to be proved
follows at once.
Corollary. The equation Jn(x) — 0 has at least one
root between a and a + kn, where k is any number greater
than 1, and a is any sufficiently large number.
II. The equation Jn(x) = 0 has an infinite number of
real roots, all simple.
Proof. From I, Cor., it follows that, if k > 1, the
equation 3n(x) = 0 has at least one root between every
consecutive pair of terms of the sequence
a, a + fen, a + 2ibr, a + 3Jbr, . . .
provided a is sufficiently large. The equation Jn(x) = 0 has
therefore an infinite number of positive real roots.
That all these roots are simple follows from the
differential equation, as in § 14. (See also Exs. XIV, 3.)
III. If n > — 1, the equation Jn(x) = 0 has no purely
imaginary root.
Proof. This follows from the series for Jn{x), by the
same kind of reasoning as in § 15, II.
IV. If n > — 1, the equation 5n{x) = 0 has no complex
roots.
Proof. This follows from Exs. XIII, 1, by the same
kind of reasoning as in § 15, III.
Corollary. From II, III, IV it follows that, if »> —1,
the equation Jn(x) = 0 has an infinite number of simple
real roots and no others, except possibly x = 0.
V. If n is any real number, the equation Jn(x) = 0 has
no root in common with either of the equations Jn-i(#) = 0>
Jn+i(x) = 0 (except possibly x = 0).
Proof. From the recurrence formulae (6.30), (6.31), it
would follow that a common root of Jn(x) — 0 and either
of the equations Jn_x(^) = 0, Jn+1(x) = 0, would be also
a root of Jn'(x) = 0, which is impossible, since all the roots
of Jn(x) = 0 are simple (except possibly x = 0). (See also
Exs. XIV, 3.)

106 INTRODUCTION TO BESSEL FUNCTIONS
VI. If n is any real number, the equation J„+i(#) = 0
has at least one root between every pair of positive roots of
Jn{x) = 0.
Proof. Since Jn(x)/xn and its derivative are continuous,
it follows from the recurrence formula
dx
Jn(»)l Jn+lfc)
by Rolle's theorem, that the equation 3n¥1(x) = 0 has at
least one root between every pair of roots of Jn(x) = 0.
VII. If n is any real number, the equation Jn-^x) = 0
has at least one root between every pair of positive roots of
Jn(x) = 0.
Proof. This follows in the same kind of way from the
recurrence formula
^{xnJn(x)} = aPJ^x)
and Rolle's theorem.
VIII. If n is any real number, the positive roots of
Jn(x) = 0, Jw + x(x) = 0, interlace.
Proof. Prom VI, the equation Jn + x(x) = 0 has at least
one root between every adjacent pair of positive roots of
Jn(x) = 0 ; and from VII, Jn(x) = 0 has at least one root
between every adjacent pair of positive roots of Jn + ±(x) = 0.
It follows that between every adjacent pair of positive roots
of either equation there lies one and only one root of the
other, i.e. the roots of the two equations interlace.
IX. If n is any real number, the equation
xJnf(x) + JLJn(x) = 0, . . (6.61)
where H is a real constant, has an infinite number of real roots.
This follows by the same kind of proof as in § 15, VII.
Note.—Equation (6.61) includes as particular cases
Jn'(x) = 0, (H = 0);
J«-i(a) = 0, (H = »);
Jn+iO*0 = 0« (H = — n).

BESSEL FUNCTIONS OP ANY REAL ORDER 107
Examples XIV
1. If a is a root of Jn(x) = 0, show that
(i) Jn'(a) = Jn_i(a) = — JB+1(a).
/,-,-\ Jn+2(^) _ %Jn+iM __ _ 2Jn-i(*) _ _ Jn-2(a)
in) ; z~ — — — 1 •
n+1 a a n — 1
2. If a is a root of Jn'(#) = 0, show that
= J„_i(a) = J„+i(a).
a
3. If a is a positive root of Jn(x) = 0, (n > — 1), show that
J o
Deduce that the equation Jn(x) — 0 has no root in common
with any of the equations Jn'(x) = 0, J„+i(#) = 0, J„_i(#) = 0,
xJn'(x) + HJn(sc) = 0, except possibly a; = 0.
4. If a is a positive root of Jn'(x) = 0, (r* > — 1), show that
j" ^^(0^)^=°6 2~,W Jn'(«) = - iJJoOOY'ta).
Deduce that, if a is the least positive root of J„'(#) = 0, and
jS the least positive root of Jn{x) — 0, then j3 > a > n.
Deduce also that the maximum values of Jn(x) are all positive,
and that the minimum values are all negative.
5. If a is a positive root of Jn{x) = 0, (n > — 1), show that
f xJn2(ax)dx — — JaJn'(a)Jn"(a).
J 0
Deduce that J„'(a), J„"(a) are of opposite sign, and interpret
this on the graph of Jn(x).
6. Show that the graphs of Jn+1(x), Jn-i(x) intersect at a point
below each maximum point, and at a point above each minimum
point on the graph of Jn(x).
7. Show from (6.48) that the smallest positive root of the equation
Jn(x) = 0 increases as n increases, if n > — 1.
8. If a is a root of Jn(x) = 0, deduce from the result in Exs.
XIII, 4, by differentiating partially with regard to m, and then
putting m = n, that
L ~bn Jx=a aJn'(a)J0 x

108 INTRODUCTION TO BESSEL FUNCTIONS
Hence show, by differentiating the equation Jn(a) = 0, that
da. 2n c*Jn2(x)
clol _ 'in Caon*[x)
dn aJn+12(a)J0 x
I
Deduce that, if n > 0, the positive roots of Jn(x) = 0 increase
as n increases.
9. If Jn(a) = 0, (n > 0), show that
JxJn-x2(ax)dx = I xJn2(oix)dx = \ xJn+l2(ccx)dx.
0 ^0 Jo
10. If y = f(x) is any solution of Bessel's equation of order n,
show that the equation f(x) = 0 has one root between every
consecutive pair of positive roots of the equation J„(#) = 0. [See Exs. I, 7.]
§ 97. Fourier-Bessel expansion of order n.
Let a1? a2, a3 . . . denote the positive roots of the
equation Jn(x) = 0, (n > — 1), arranged in ascending order
of magnitude. Then it follows from (6.54) that, if r 4= s,
xJ n(xoLr)J n(xoLs)dx = 0, . (6.62)
o
and from Exs. XIV, 3, that
xJn2(xoLs)dx = |Jn+i2(as) . . (6.63)
o
It can be proved (see § 99) that, if n > — J, a function
f(x) which is arbitrarily defined in the interval 0 < x < 1,
subject to certain conditions of integrability, can be
expanded in an infinite series of the form
f(x) = A1Jn(a:a1) + A.2Jn(aa2) + A3Jn(^a3) + . . . (6.64)
The coefficients As can be formally determined by
multiplying throughout by xJn(xa.s)dx and integrating between
the limits 0 and 1. For, we then have
I xf(x)Jn(xa.s)dx = JaJ xJn(xar)Jn(xa.s)dx.
Jo rsml Jo
By (6.62) every term on the right vanishes except the
one in which r — s, and hence, by (6.63),
xf(x)Jn(xcLs)dx = iAsJn+12(as),
J o
f

BESSEL FUNCTIONS OF ANY REAL ORDER 109
from which
2 f1
^8= t 17—\ xf(x)Jn(xaL8)dx. . (6.65)
Jn+i (as)J0
The expansion (6.64) is called the Fourier-Bessel expansion
oif(x) of order n.
§ 98. Dini expansion of order n.
An expansion similar to the Fourier-Bessel expansion,
but based upon the roots of the equation
x~n{xJn'{x) + HJn(a:)} = 0, . . (6.66)
is called the Dini expansion off(x) of order n.
Three cases may be distinguished, depending upon the
values of the constant H :—
I. If H >— n, the Dini expansion has exactly the same
form as the Fourier-Bessel expansion, viz.
f(x) = A1Jn(^a1) + A2Jn(#a2) + A3Jn(aa3) + . . .
where oc^, 0^25 ^3? • • • are the positive roots of (6.66).
By the same method as before, the determination of
the constants As depends upon the integrals
i
l
xJn(xoir)Jn(xoLs)dx = 0, (r 4= s),
0
(see Exs. XIII, 1), and
xJn2(xa.s)dx = i{Jn2(as) —Jn-i(as)Jn+1(as)},
J 0
which follows from (6.55). Hence we find
2| xf(x)Jn(xoLs)dx
Jw2(as) — Jn-i(as)Jn+i(as)'
II. If H =— n, equation (6.66) becomes
x~n{xjn'(x) — nJn(x)} = 0,
that is, %~n+1Jn+i(x) = 0>

110 INTRODUCTION TO BESSEL FUNCTIONS
which has a double root x — 0, and in this case the Dini
expansion has an initial term of the form A0xn, so that now
f(x) = A0Xn + A±J ^^) + A2Jn(X0L2) + ...,
which may be regarded as an expansion based upon the
positive roots (including zero) 0, a1? a2, a3, . . . of the
equation Jn+1(x) — 0.
The constant A0 may be found by multiplying
throughout by xn+1dx and integrating from 0 to 1 ; thus,
i w^=A°\r+id*=2(^ftj'
which gives A0, the remaining terms on the right vanishing
in virtue of
i
V+1J„(a;as)da; = J"+l(oCs) = 0, (a, #= 0).
0 as
The constants As(s 4= 0) are now given by
1
1
xf{x)Jn(xc(.&)dx = lAsJn2(as).
0
III. If H < — n, equation (6.66) has two purely
imaginary roots, ± ia0, and the Dini expansion begins with a term
depending on them ; * it is of the form
f(x) = A0ln{xoL0) + AJnixxJ + A2Jn(aa2) + . . .
The coefficients As (5 4= 0) are found as in I, and A0 is
found by multiplying throughout by xln(xoi0)dx and
integrating from 0 to 1. [See Exs. XV, 8.]
§ 99. Validity of the expansions.
We shall not attempt here to establish the validity of the Fourier-
Bessel and Dini expansions, but refer the reader to Watson, " Theory
of Bessel Functions," Chap. XVIII, where the expansions are proved
to be valid in the open interval 0 < x < 1 when n > — J, provided
that f(x) has bounded variation in every closed interval contained in
the open interval 0 < x < 1, and that the integral f | f(t) \ ^/1 dt
0
exist?.
It must be pointed out that these are not necessary conditions
* Watson, p. 597.

BESSEL FUNCTIONS OF ANY REAL ORDER 111
for the expansions to be valid : they are sufficient conditions on
which a proof of the validity of the expansions can be based.
We may add that, if f(x) is continuous, the Dini expansion
converges uniformly to f(x) in any interval 0 < a < x < 1. The
Fourier-Bessel expansion converges uniformly in any such interval
if and only if the condition/(1) = 0 is satisfied, this condition being
plainly necessary because the Fourier-Bessel series at x = 1 is
0 + 0 + 0 + . . . .; if this condition is not satisfied, the interval
of uniform convergence does not extend to the end point x = 1.
For the end point x = 0, see Watson, § 18.55.
§ 100. Example.
One function that can be represented by a Fourier -
Bessel or Dini expansion is xn(l — x2)v, (n > — J, p > — 1).
For, by § 97 and (6.42), we find the Fourier-Bessel expansion
r(p+l) 4 a*-*Jn+1»(a) '
where the summation extends over the positive roots of
Jn(x) = 0.
Again, by § 98, II, and (6.42), we find the Dini
expansion
an(l-s2)P= r(n + 2)x" 0ff1 ~ Jw+P+1(a)J«(aaQ
r(p +1) r(n + p + 2) ^ " 4 a^Jn2(a) '
where the summation extends over the positive roots of
Examples XV
1. Obtain the following (Fourier-Bessel) expansions, in which the
summations extend over the positive roots of Jn(x) = 0.
i x« = 2 > nK '
~ aJn+1(oc)
(ii) z«(l - *■) = 8(n + 1)X J7/?!-
(hi) a«(i - »■)■ = i6TJw\'ttt)J;/(flf).
~ asJw+12(a)
(iv) a"+2* = T ^1^ where
^- Jn+i2(a)
2 /2\2 /2\3
A = aJw+l(a) ~~ PW) J"+2(a) +p(p ~ l\~) J«+s(a) - • • •
and p is a positive integer [see Exs. XII, 2].

112 INTRODUCTION TO BESSEL FUNCTIONS
WW^,(^(a,_^iW.
Off
2. Show that, if p > ■— 1,
(1 - a*)v = l + 2*+*r(v + i)YJ^l{a)Jo{ax)
U } p + 1 + (P+1,i a*+ij0t(a)
where the summation extends over tho positive roots of J^x) = 0.
3. Show that, if n > — |,
-(1 - ^ = (n+1;:+3) -32(w+2) 2 ss?)
whore tho summation extends over the positive roots of the equation
J«+i(«) = 0.
4. Show that, if — 1 < x < 1,
00
V(l - x2) = "
(1) ./n-^ = ff 2 Jl(fi7r) sin S7TX;
8 = 1
00
(ii) W(l -x>) = ff 2 Jl(r7r) Sin r7rX'
where r = s — J;
... x (3x ^ Ji(a) sin oca;)
(1U) V(l - *2) " "\T + ^ sin* a J
where the summation extends over the positive roots of the equation
x = tan x.
5. Show that, if — 1 < x < 1,
1 5.
(i) V(l -a2) = "" ^- J°(r7r) C°S ^
s = 1
where r = s — \;
1 °°
(") V(1 __ ^) = i + w 2 Jo(57r) C0S S7r*'
» = 1
G. If <? > 0, — 1 < # < 1, and J0(a) = 0, show that
JQ(CC)J0(0LX)
Deduce that
1 _ 9 ^ sin a J0(a#)
1 - x2) ~ Z* ~~^~ J?(a)'
V(
a

BESSEL FUNCTIONS OF ANY REAL ORDER 113
Multiply throughout by xdx, and by integration deduce that
1 — \/(l — x2) _ 9^r sin a Jx(ax)
x ~ 2L a2 ^1( a) *
ex
7. Deduce the Fourier sine and cosine series as particular cases
of the Fourier-Bessel expansion.
8. Assuming the validity of the Dini expansion in III, § 98, show
that the coefficient A0 is given by
21 xf(x)In(xa0)dx
A — J0
In2(a0) + In-xCaoJIn+^ao)
§ 101. The Fourier-Bessel double integral.
The following argument * shows the plausibility of
expressing an arbitrary function in the form of a double
integral analogous to the Fourier double integral.
Let a function f(x) be defined from x = 0 to x = h, by
f(x) = <f>(x), 0 < x < a, . . (i)
f(x) — 0, a < x < h. . . (ii)
Then by Exs. II, 5, we have
/m=Ip^>j.(¥)j.(?H
where the summation extends over the positive roots of
the equation J0(x) = 0, and 0 < x < h. It follows from
(ii) that, if 0 < x < a <h,
#»=zpjy>>j.(¥) *®*
Now every term of this series tends to zero when h -> oo.
Consequently, when A is large, a finite number of terms
at the beginning of the series can be neglected. Hence,
if for ol8 in the 5th term we make the substitution (5.37)
as = (s — £)77-,
the error that we make by this substitution in the first few
terms of the series will vanish when h -> oo.
* Cf. Riemann-Weber, I, p. 199 (6th edn.).

114 INTRODUCTION TO BESSEL FUNCTIONS
Again, by (5.36), with this substitution,
Ji2(as) = sin2 (a, — ~) =
7T0LS \ 4/ 7T0LS
and hence
00
^w=i9J>wjo(x)j»(t)^
Now put
2/s = J, oy = y,+1 -ys = ~,
oo -a
then </>(a) == 2 2/*82/ </>(t)Jo(^ys)Mtys)tdt,
8=1 J°
and we infer that, in the limit when h ->■ oo,
Ioo pa
J0(xy)ydy <j>{t)J<>{yt)tdt. . (6.68)
o Jo
If the function </>(#) is suitable (see Note below), we can
put a = oo, and we then have a function, defined from 0
to oo, expressed as a double integral, thus
TOO TOO
#r)= 30{xy)ydy\ <f>(t)J0(yt)tdt . (6.69)
Jo Jo
This formula and the more general formula
Ioo poo
3n{xy)ydy\ <j>(t)Jn(yt)tdt . (6.70)
o Jo
can be proved rigorously under certain conditions.
Note.—Sufficient conditions on which a proof of the validity of
(6.70), when x > 0, can be based are : n > J, 4>(t) has bounded varia-
Ioo
I ¢{^) I V* dt
o
exists. [See Watson, § 14.4.]
Examples XVI
1. Verify (6.70) when
(i) +(t) = j ; (ii) <j>(t) = t«e-"<\
[Use Exs. XI, 11 ; Exs. X, 7.]

BESSEL FUNCTIONS OF ANY REAL ORDER 115
2. Show from (6.70) that if
{00
<f>(x)Jn(xy)xdx = 4>(y),
0
{00
ip(x)Jn(xy)xdx = <f>(y).
o
(i) By putting n = 0, j>{x) = e~amfx9 verify that, if a > 0, 6 > 0,
I
J 0
00 xJ0(bx) e~ab
dx —
Via2 + x2) b '
[See (4.5) and Exs. VII, 3.]
(ii) By putting n = 0, <j>(x) = smax9 verify that
x
xJ0(xy) sin ay
dr =
0 V(«2 - *2) 2/ '
[See (4.21) and Exs. XII, 4.]
(iii) By putting n — 0, <f>(x) = , verify that
x
i
00 xJ0(xy)dx cos ay
V(*2 - «2) y '
[See (4.19) and Exs. XII, 11.]
§ 102. General solution of BesseVs equation.
We have seen (§ 87) that y = Jn(x) is a solution of
Bessel's differential equation,
d2y 1 dy
dx2
+^2+0-^-^-- • <6-71»
Since this equation remains unaltered when n is replaced
by — n, a second solution is y = J_n(#), and if n is not
an integer the general solution can be written
y = A3n{x) + BJ_n(z). . . (6.72)
But, if n is an integer, J_n(#) = ( — )nJn(x), and (6.72)
is no longer the general solution. In this case, by the
same method as in § 4, the general solution can be written
in the form
y = AJn(z) + BJn(x)\^^y . . (6.73)
If we substitute for Jn(x) its series in the last integral,
and expand the integrand in ascending powers of x, the

116 INTRODUCTION TO BESSEL FUNCTIONS
integral takes the form
\(J&i + . • •+^ + ^ + ^+ b** + b*fi + ...)dx
= ~~ 2^ ~~ * • • ~ 2^ + a» iog *
+ b0 + b-f + b-f + . . .
where b0 is a constant of integration. When this expression
is multiplied by Jn(x) and the product substituted in the
second term of (6.73), we obtain a second solution of the
form
Jn(x)(a0 log x + b0) + — (c0 + ctx2 -f c2x* + . . .)
Here
o? o? i> 2? • . • are definite constants ; the
constant b0, however, is arbitrary, and its value can be
chosen so as to give the most convenient form to the second
solution. The form which is now generally accepted as
the standard one is known as Weber's, and is denoted by
Yn(x). In terms of Jn(x) and Yn(#), the general solution
takes the form
y = AJn(a) + BYn(a). • . (6.74)
There is no need here to determine the actual values of
the coefficients in the expansion of Yn(x). It is sufficient
to remember that any second solution of the equation,
when n is an integer, behaves like l/xn when x is small, and
involves log x.
§ 103. If we replace x by kx in (6.71), we obtain the
equation
3+=2+(- -£»-•• • <"5>
of which the general solution may be written
y = AJn(kx) + BYn(kx), . . (6.76)
or y = AJn(kx) + BJ_n(kx), . . (6.77)
according as n is an integer or not.

BESSEL FUNCTIONS OF ANY REAL ORDER 117
§ 104. Transformations of BesseVs equation.
A number of transformations of Bessel's equation,
together with their solutions, can be obtained by finding
the equation satisfied by
y = x«Jn(pxr) . . . (6.78)
where a, j8, y are constants. If we put
v = L £ = fr?> • • • (6-79)
(6.78) gives rj = Jn{£), and hence
t>% + (% + «• - »■), = 0,
which may be written
Now £
by (6.79), and hence
>d_/td7i\ _l±(rd7A
*d£Vd£)~y* dx\ dx)'
Again, from (6.79) we find
di) _ y' <xy
dx cc"_1 xa'
and further,
r d /. dv\ _ y" (2« - W ■ «V
dx\ dxj x*~* «"-i "^ a;"'
Hence the equation satisfied by ?/ is
1 / 2/" (2a - 1)2/' , oA/
^-+3) + (^-^=0,
y2 Va*-2 of"
or
d2y 2 a — 1 dy , ^02 2/vt2v_2 , a2 — w2y2>
cZ#2 a: dx
+ (pyx*?-* + a Jly)y = 0. (6.80)

118 INTRODUCTION TO BESSEL FUNCTIONS
The general solution of this equation is
y = x*{A3n{fr?) + BYB(j8a?)}, . . (6.81)
or y = x«{AJn(Pxv) + BJ.n(]3^)}, . (6.82)
by §103, according as n is an integer or not.
§ 105. Particular cases of equations whose solutions can
at once be written down in terms of Bessel functions are
obtained by giving particular values to the constants
a, /3, y, n. For example,
(i) a = 0, y = 1, gives
dx* ^xdx^\p xVy '
(ii) a = \ gives
(iii) a = \, /3=1, y = 1, gives
d2y ( n2 — iN
da:2
+ (i-^r-;
,. . ! „ 2¾ m + 2 1
(1V) a = *> ^ = ^+2« y = -^r> n = ^+2> ^lves
(v) a. = n, ]8 = 1, y = 1, gives
d2w 2¾ — 1 dy ,
— + 2/ = 0 ;
Cvu!/ X CLX
(vi) a = — n, jS = 1, y = 1, gives
cZ2v , 2n 4- 1 dy
Examples XVII
1. Show that the general solution of the equation
4^+9^ = 0
can be written
y = V*{AJ_i(a;f) + BJi(a#)}.
Also solve the equation in series.

BESSEL FUNCTIONS OF ANY REAL ORDER 119
2. Show that the general solution of the equation
^ + 4^=0
is y = V^{AJ_i(x2) + BJi(x2)}.
Also solve the equation in series.
3. If y satisfies the equation
d2y
dx2 u
and if dy/dx = a when x = 0, show that y can be written in either
of the forms
/ x6 x12 \
y = ^ + B(i-0+gaflallal2-..-j,
y = ax + C V^^Lj (%) >
where B, or C, is an arbitrary constant.
4. Show that Riccati's equation
y + by2 = cxm
dx ^ u
is transformed into
^-r — bcxmu — 0
dx2
by the substitution by = --=-.
u ax
Hence show how the solution of Riccati's equation can be
expressed in terms of Bessel functions.
5. Use the last example to show that the general solution of the
equation
-1 = x2 + y2
dx ' ^
can be written
AJ_g(jx2) + Jf(^2)
y~Xm J^dx2) - AJidx2)
where A is an arbitrary constant.
From this solution, or by solving the equation in series, verify
that, if y = a when x — 0,
y = a + a2x + o,zx2 -f (a4 + i)#3 + • • •
If y — 0 when x — 0, show that
x3 x1 2xxl
y = 3" + 63 + 2079 + ' ' '

120 INTRODUCTION TO BESSEL FUNCTIONS
§ 106. Contour integral for Jn(x).
It follows from the definition of Jn{x) in § 83 that,
when n is an integer, Jn(x) is the residue of the function
e2\ t)
tn+l .... (6.83)
at t = 0, and hence that Jn(x) can be expressed as a
contour integral in the form
jm=M/H)&- • (6-84)
where C denotes any simple contour surrounding the origin.
Moreover, by a simple modification, Jn(x) can be
expressed as a contour integral which can be regarded as
defining Jn(x) for all values of n, real or complex.* It is,
however, beyond the scope of this book to pursue the study
of Bessel functions further from this point of view.
§ 107. HankeVs contour integral.
From (6.45) it follows that
J-<*> = w^k+T)Leixt{1 - ****• (6-85)
We can verify that this expression satisfies Bessel's
differential equation ; for if we put
y = xn\ eixt(l — t2)n~Ut,
we find by differentiation
^2¾ + 4^ + (a2 - n2)y
dx2 dx v w
= — ixn+1 i {ixeixt(\ — t2)n+* — (2n + l)teixl(l — t2)n~l}dt
= - ixn+1[X ^xt(l - t2)n+*}dt
= — fa«n+ip««(l — t2)n+*
= 0, if n > — |.
* Whittaker and Watson : " Modern Analysis," § 17.2.

BESSEL FUNCTIONS OF ANY REAL ORDER 121
§ 108. More generally, if we put
y = xn\ eixt(l — t2)n~Wt,
J a
we find that
,2d2y . Jy
X'
dx'
+ x-y- + (x2 — n2)y = — ixn+1\eixt{\ — t2)n+i
a
and hence, if x > 0, that y is a solution of BesseFs equation
of order n if the integral is taken along any contour from
either of the points t = ± 1 to an infinitely distant point
in the upper half of the £-plane (cf. § 69).
Accordingly, we can define further solutions (Hankel
functions) when n > — \, x > 0, by the formulae
- H""^) = 2-V.i> + «1. e"'(1 - ^-
- H"M<*> = 2-V^(» + nil,/"" " '"'""*
These functions of order w correspond to H0(1)(#),
TL0(2)(x) in the theory of Bessel functions of zero order
(§ 73). From them we can, for example, develop the
asymptotic expansions of order n.

CHAPTER VII
^---—■
/ ^*"^ 1 f
/ ^r 1 f
1 ^ 1 f
I s I f y
1 x I*/
I/ 1 f /
If lf/
\t ii/
It If/
■ L'/
Q
\ xM
p\ /\
jC *Xv. \
/\ s ^*v \
\ \ \ \
\ \ X \
\ * \ \
\ * \\
\\ \\
\\ V
x 1
c
Fig. 18.
S A
APPLICATIONS
§ 109. Kepler's problem.
In the ideal problem of planetary motion a planet P
moves in an ellipse under the gravitational attraction of
a sun S situated in one of the foci, the area swept out by
the radius vector SP
during any interval of
time being proportional
to that interval (Fig.
18).
Let A'A be the
major axis of the
ellipse, and let a line
drawn through P
perpendicular to A'A meet the circle described on A'A as
diameter in the point Q. Draw CP, CQ, SQ. In the usual
notation of the ellipse, let 2a be the length of the major axis
A'A, 26 that of the minor axis, and e the eccentricity.
Further, let a point M describe the circle AQA' at such
a constant speed that it coincides with P at A and A'. Let
the time t be measured from an instant when P is passing
through perihelion at A, and let r = SP, 0 = angle ASP,
<j) = angle ACQ, i/» = angle ACM, all measured at time t.
In the terminology of astronomy, 9 is called the true
anomaly, <f> the eccentric anomaly, and ip, which is
proportional to t, the mean anomaly.
Kepler's problem was to express such variables as r,
6, $ explicitly in terms of the time t, or, what comes to the
same thing, in terms of the mean anomaly ifs.
122

APPLICATIONS 123
§ 110. BesseVs solution.
Consider, for example, the expression of </> in terms of i/j.
Since i/j and the area ASP are both proportional to t,
we have, if T is the period in which the complete ellipse is
described,
t _ ip __ area ASP _ area ASQ
T ~~ 2^ ~ Mb — ^2 *
Now area ASQ = sector ACQ — ASCQ
= \a2<j> — \ae . a sin </>,
and hence ifj = <f> — e sin </>. . . (7.1)
The problem is now to solve this equation for </>, so as
to express <j> explicitly in terms of i/j.
It is evident that </> — ip is an odd periodic function of (p,
with period 27r, and may therefore be expanded in a Fourier
sine-series. Consequently, we put
cj> _ tfj = Bx sin ifj + B2 sin 2i/j + B3 sin 3^ + . . . (7.2)
The coefficients Bn are now given, in accordance with
the usual rule, by
77 Cn
2Bn = 1 {</> — $) sin nipdifj
L At Jo Jo rb
In the first term on the right, </> — i/j vanishes at both
limits, and in the second term the integral of cos rnfjdip
between the limits 0 and 7r, is zero ; consequently
2 f*
Bn = — J cos nip dcp
TTTIJq
2 f»
= — I cos (n<f> — ne sin <f))d(f),
7rnj0
by (7.1), and hence, by (6.21),
2
Bw = -J«(»e). . . . (7.3)

124 INTRODUCTION TO BESSEL FUNCTIONS
Putting n = 1, 2, 3, . . . and substituting the values
of B1? B2, B3, . . . in (7.2), we obtain
sin 2ifj
/ / . of t / xsin 0 sin '
¢ = ^ + 2^(6)-^ + J2(2e)—^
+ J3(3e)8i^ + . . .} (7.4)
which is the expression required.*
Next, it is evident that r is an even periodic function of
ifj, with period 2tt, so that r can be developed into a Fourier
cosine-series of the form
r = ^A0 + Ax cos ifj + A2 cos 20 + A3 cos 3^ + . . . .
We leave it to the reader to show that
r = a(l — e cos </>),
and to deduce that
r- = 1 + *■ - 2e{j/(e)2£ii + J2'(2e)^ + . . j (7>5)
Again, 0 — ip is evidently an odd function of ifj, with
period 27T, so that there is an expansion of the form
g _ tfj = Cx sin ip + C2 sin 2ifj + C3 sin Sip + . . .
For the determination of the coefficients in this case,
see Watson, " Bessel Functions," p. 554.
Ex. Show that
1
1 — e cos <j>
Deduce that
1
= 1 + 2{Jx(e) cos 0 + J2(2e) cos 20 + J3(3e) cos 3«/» + . . .}
= 1 + 2{Jx»(6) + J22(2e) + J32(3e) + ...}.
§ 111. Critical length of a vertical rod.
When a thin uniform elastic rod has its lower end clamped
vertically, the vertical position of equilibrium is stable if
00
* Series of the type /.A8Jw+8{(w + s)x}, where As is independent of x,
s = 0
are called Kapteyn series (Watson, Ch. XVII).

APPLICATIONS
125
the length of the rod is less than a certain critical length.
But for a rod of this critical length, the vertical position is
one of neutral equilibrium only, so that, if the upper end
is slightly displaced and held fast until the rod is at rest,
it will remain in the displaced position when released.*
This will appear in what follows.
§ 112. Let I be the length of the rod, a the radius of
its cross-section, w the weight per unit length. Put I=|7ra4,
and let E be Young's modulus for the
material of which the rod is made. ~~*t i ^
Suppose the rod to be in equilibrium in
a position deviating slightly from the
vertical (Fig. 19). Take the origin 0 at the
upper end of the rod in the vertical position,
the #-axis vertically downwards, and the
2/-axis in the plane of the rod. Let P be a
point (x, y) on the rod, and Q a point
(£, 77) above P.
Consider the equilibrium of the part of
the rod above P. The moment about P
of the weight of an element wd£ at Q is
wd£(rj — y), and by integration we obtain
the moment about P of the weight of the
part of the rod above P. Again, by the usual theory of
elastic rods, the moment of the elastic forces about P is
EI d2y/dx2. Hence, since the part above P is in equilibrium,
Fig. 19.
EI
d2y
dx2
= W(7]
Jo
y)d£.
By differentiation with respect to x, we get
w-^-dt;
J Q CLX
mdi = [w{r} ~y)]^
* dy
= 0 — w-r-x,
dx
* Greenhill: Proc. Camb. Phil Soc, IV, 1881.

126 INTRODUCTION TO BESSEL FUNCTIONS
that is, EI;ri + wxlf — °- • • • (7-6)
Put k2 = wfEl; then
Comparing this equation with (iv), § 105, we deduce that
i=v^m+BJi(^)}, . „8)
which can also be written, by expanding the Bessel functions,
£-0-178 + ---)+^-5^ + --)- <7-9'
where A, B or a, b are arbitrary constants.
Two conditions that must be satisfied by the particular
solution required are :
(i) d2y/dx2 = 0 at x = 0, since there is no bending
moment at the upper end ;
(ii) dy/dx = 0 at x — Z.
Condition (i) gives 6 = 0. Condition (ii) can only be
satisfied by a = 0, unless I satisfies the equation
h2 73 JU76 £679
- 2.3^2.3.5.6 2.3.5. 6. 8. 9^" " K }
that is, the equation
J-*<cr) = 0- ■ • ■ (7-n)
Now * the least root of the equation J-^{x) = 0 is
x = 1-8663. Hence the rod cannot bend from the vertical
until the length I is given by
, 3
9kT*
~ = 1-8663,
and hence
Z3 = 7-84EI/W. . . . (7.12)
* Gray and Mathews : " Bessel Functions," p. 317.

APPLICATIONS 127
This gives the critical height of the rod. For example, for
a steel rod of diameter 0-1 inch, density 0-28 lb. per cu. in.,
E = 13,000 tons per sq. in., we find I = 80 in.,
approximately.
§ 113. From a practical point of view, it is perhaps easier
to solve equation (7.7) in series, and to solve (7.10) by trial.
The solution is given here as an example on Bessel functions
of fractional order.
Ex. In the problem of the small vibrations of a flexible string
of length I with its ends fixed, the displacement y satisfies the
differential equation
Vy _ T a»y
where T is the tension and p the line density.
Show that the normal modes of vibration are given by
y = X cos (cot — e),
where X is a function of x which satisfies the equation
and vanishes at x = 0 and x = I.
Hence, if P = Po M + y J = Po£,
show that g + ^X = 0, (*«=*gf),
and deduce that the periods 2ir/a> of the normal modes are given by
V&2T
where fi is a root of the equation
Ji(/x)J_j(A^) =- J_j(At)Jj(A/x),
and A - V(l + k)\
§ 114. Circular membrane with the circumference fixed.
Normal modes of vibration.
We return to the problem of the vibrations of a circular
membrane with the circumference fixed, no longer assuming

128 INTRODUCTION TO BESSEL FUNCTIONS
that the vibrations are independent of 6. With the same
notation as in § 25, the differential equation to be satisfied is
^-^+1^.1¾ /713)
To find the normal modes of vibration, we try a solution
of the form
z = R® cos (cot — e),
where R, 0 are functions of r, 6 only. The result of the
substitution can be written in the form
dr2 + r dr + c2 /R ~~ 0 d0*' ' ( }
which is only possible if each side of this equation is equal
to the same constant, since the variables r, 6 are
independent. Putting both sides equal to n2, we deduce that
0 and R respectively satisfy the equations
d20
_ = -*«©, . . . (7.15)
d2R . ldR . (to* n\^ n „_._,
Prom (7.15) we then have
0 = C sin (nd - j8),
where C, jS are arbitrary constants. Now, if the membrane
is subject to no constraining force except that at the
circumference, z must be a single-valued function of position
and so must be of period 2tt in 0. Hence n must be an
integer, which, without loss of generality, we may take to
be a positive integer, or zero.
The general solution of (7.16) can then be written
B = AJ.(^) + BY.(2),
and as the particular solution required for the present
problem is plainly one that remains finite when r -> 0,
we must put B = 0, since Yn(x) -* oo when x -> 0.

APPLICATIONS 129
Hence, merging A and C into one constant, we have
the solution
z = AJn(y) sin (rc0 - j8) cos M - c), . (7.17)
where A, /J, e are arbitrary constants, and n is any positive
integer, or zero.
Further, if z = 0 at r = a, for all values of 0 and t, the
equation
j.(t) = ° • • • (7J8)
must be satisfied by o>. Accordingly,
coa
—■ — ocj, a2, a3, . . .
c
where, since co is positive, a1? a2, a3, . . . denote the positive
roots of the equation Jn(x) = 0. Consequently, the normal
modes of vibration are given by
z - AJB(^) . sin (nd - j3) . cos (^ - c), (7.19)
where a is any positive root of Jn(%) = 0.
§ 115. The normal modes when n = 0 have been
discussed in § 25. For the discussion of any other single
mode, the constants /J, e are of no importance, as they
depend only upon the initial line from which 6 is measured,
and the instant from which t is measured. Consequently,
any normal mode for which n 4= 0 may be written in the
form
z = AJn( —) . sin nd . cos —. . . (7.20)
\a/ a
This represents a doubly infinite system of normal modes,
for there is an infinite number of values of n, and an infinite
number of values of a for each value of n.
When n = 1 we have
ctcc8
a
z == AJxf—^J . sin 0 . cos

130 INTRODUCTION TO BESSEL FUNCTIONS
where Jx(as) = 0, (s = 1, 2, 3, . . .)• In each mode of
this set, the radii 0 = 0, 0 = tt together form a nodal
diameter. For s = 2, there is one nodal circle, for 5 = 3
two nodal circles, and so on (Figs. 20.1, 20.2, 20.3).
5=1
Fig. 20.1.
When n = 2 we have
z = A J2
5=2
Fig. 20.2.
5=3
Fig. 20.3.
roLs\ . nn cta.s
M . sin 20 . cos —-,
a
a
where J2(as) = 0, (s = 1, 2, 3, . . .). In each of these
modes the radii 0 = 0, 9 = tt together form one nodal
5-1
Fig. 21.1.
5=2
Fig. 21.2.
5=3
Fig. 21.3.
diameter, and the radii 9 = \n, 6 = ftt form another
(Figs. 21.1, 21.2, 21.3).
§ 116. General initial conditions.
The most general solution that can be obtained by adding
together arbitrary multiples of the normal modes may be
written
* = f i ^n(-) . sin (nd - £) . cos (—' - e), (7.21)

APPLICATIONS 131
where qls denotes the 5th positive root of Jn(x) = 0. The
suffixes n, s might have been appended to the arbitrary
constants A, j8, e, but have been omitted to lighten the
appearance of the formula.
These arbitrary constants can be chosen to satisfy
general initial conditions such as
z = </>(r, 0), t = 0, . . (7.22)
7>z/7)t = i/t{r9 0), t = 0. . . (7.23)
§ 117. In particular, if the membrane is started from
rest, with an initial displacement given by z = <j)(r, 0), we
have to satisfy the conditions
z = 0(r, 0), J = 0, . . . (i)
dz/M -= 0, $ = 0. . . (ii)
Condition (ii) is satisfied by putting e = 0 in every
term, and we may then write the solution, with a slight
change of notation, in the form
Z = 2 2 J»( ^ ) (A". s c°3 ^0 + Bn, a sin ^) COS
Putting t = 0, we see that, in order to satisfy (i), the
coefficients A, B must be determined from the expansion
OO 00
fa, ")=Z I Jn(~)(A- » C0S nd + B*. . Sil1 ^)-
To find AWj s, multiply both sides by
Jnf—-] . cos nd . rdr dO,
and integrate over the membrane. Then every term on
the right vanishes except the one that involves An, s and
we get
f f*r#(r, 0) jj^cos nOdrdO
J o J 0
"■ a
~ K' 8J o J
rJ.X^s)cos2w0cZrd0

132 INTRODUCTION TO BESSEL FUNCTIONS
= 7rAn> sl ax . Jw2(#as) . adx
Jo
as in (6.63). This determines Ant8. To find Bw>s we first
multiply the expansion throughout by
nV a
. sin n6 . rdr dO,
and integrate over the membrane as before.
Examples XVIII
1. Show how to find the periods of the normal modes of vibration
of a membrane in the form of a sector of a circle of radius a and
angle w/m, the condition z = 0 being satisfied at every point of the
boundary.
Also show how to find the solution which satisfies the arbitrary
initial conditions
z = 4>(r, 0), (0 < r < a, 0 < 0 < irfm),
Izfbt =■ 0(r, 0), (0 < r < a, 0 < 0 < n/m).
2. Investigate the normal modes of vibration of a complete
circular membrane of radius a, with the condition z — 0 satisfied
all round the circumference and also along the radii 0 = 0, 0 = 2n
(a sector of angle 2n). Show that in the simplest case, in which
there are no nodal radii,
A . S-rrr . 0 Siret
z = —7- sin — sin - cos -,
\r a 2 a
where s is an integer.
[Note that in this case the nodal circles divide any radius into
equal parts.]
3. Show that the normal modes of vibration of a sector of a
circle of radius a and angle 120°, with the condition z = 0 satisfied
at every point of the boundary, in the simplest case, in which there
are no nodal radii, are given by
A / a . roc roc\ . 30 eta.
— sin cos — sin -=- cos
■\/r\roL a a J 2 a
where a denotes a typical positive root of the equation x = tan x.

APPLICATIONS 133
4. Discuss the normal modes of vibration of a membrane in the
form of a circular annulus bounded by concentric circles of radii
a, b, with the condition z = 0 satisfied at every point of both circles.
5. In the problem of the circular membrane of radius a (§ 26),
vibrating with circular symmetry and satisfying the condition z = 0
at r = a, if the initial conditions are
-°('-S)'- S-*
where p > 0, show that
, - C . 2«/XP + DZa-^o © cos f.
where J0(a) = 0.
6. In the problem of the uniform flexible hanging chain of length
I, making small oscillations in a vertical plane (§ 27), if the initial
conditions are
,_.(,-■)+,(,-■)•, *_„.
show that
„ = a V a2(X ~
a6Ji(a) "° V"t"V«y """ V2\Z
0 «r- a*a - 2(a2 — 8)6 T ( \x\ /at lg\
a
where J0(°0 = 0.
7. Show that a solution of the equation
~b2v 17iv v ~b2v
Sr* r 7yr ~ r2 dz2 ~~ >
which satisfies the boundary conditions,
v = 0 when z = 0 and when r = a,
Vr(a2 — r2) ,
?; = when z = I,
a3
is
v
J, f ^) sinh ^
= icvj u; 2.,
* a3J2(a) sinh -
where Ji(a) = 0.
8. Laplace's Equation (§ 39) in cylindrical co-ordinates, r, 0? z is
l)r2 ~^~ r Tr ~*~ r* Yd2 ~*~ ^z2 "

134 INTRODUCTION TO BESSEL FUNCTIONS
Show that the solutions of the form u — f{z)g(r)h(0), which are
everywhere finite and single-valued functions of position, are given
by
u = (Ae-w -f- Be^)Jn(/xr)(C sin nO + D cos nO),
u = (A sin fiz + B cos fj,z)In(fxr)(C sin nO -f- D cos nO),
u = (Az + B)rn(C sin nO + D cos n0),
where n is a positive integer or zero.
9. If U, V are two solutions of Laplace's equation, both
continuous and one-valued throughout a region bounded by a simple
closed surface, it is known from Green's theorem that
where ~b/~bv denotes differentiation in the direction of the outward
(or inward) normal to the surface, and the integral is taken over
the surface.
Prove (6.54) when n is a positive integer, by putting
U = e-"z Jn(ar) cos nO, V = e~~P* J„(j8r) cos nO,
integrating over the surface of the cylinder bounded by r = 1,
z = 0, z = I, and making I ->• + <x>.
10. Solve the problem of § 40 when the boundary conditions
are u = 0, (z = 0, 0 < r < a) ; u = 0, (r = a, 0 < z < I) ;
u = <f>(r, 0), (z = I, 0 < r < a).
11. Solve the problem of § 40 when the boundary conditions
are u = 0, (z — 0, 0 < r < a) ; u — 0, (z = Z, 0 < r < a) ;
u = f(z, 0), (r = a, 0 < z < I).
[This requires a double Fourier series.]
12. Solve the problem of § 40 when the boundary conditions
are u =. \ji(r, 0), (z = 0, 0 < r < a) ; u— <f>(r, 0), (z = I, 0 < r < a);
it = f(z, 0), (r — at 0 < z < /).

Addition theorem for J0(x), 90.
generalised, 90.
fpr Jn(x)9 90.
Alternating current in wire, 49.
Ampere, 49.
Asymptotic power-series, 78.
expansions, 48, 78, 81, 85.
Ber x, bei x, 55, 85.
Bessel functions, 1, 87, 120.
second kind, 3, 5, 116.
Bessel's equation, 3, 91, 115.
first integral, 58, 89.
second integral, 98.
Beta-function, 67, 68.
Chain, vibrations of, 28, 31, 133.
Conduction of heat, 31, 40, 44,
134.
Dini expansion, 17, 109.
Disc, electrified, 61.
Euler's constant, 5, 69.
Faraday, 49.
Fourier-Bessel expansion, 15,
108.
double integral, 113.
Frobenius, 6.
UAMM A -function, 64.
Green's theorem, 134.
Kapteyn series, 124.
Kelvin, 55.
Kepler's problem, 122.
Laplace's equation, 43, 133.
Lipschitz's integral, 58, 68.
LommePs integrals, 9, 101.
Mehler, 78.
Membrane, annular, 28, 133.
circular, 24, 127, 133.
circular sector, 132.
Modified Bessel functions, 41, 88,
96.
Neumann, 4, 8.
Normal modes of vibration, 24.
Parseval, 58.
Poisson, 98.
Potential of electrified disc, 61.
Recurrence formulae, 92.
Riccati's equation, 119.
Riemann-Weber, 61, 113.
Roots, 12, 13, 42, 100, 104, 107.
SKiN-effect, 54.
Sonine's integral, 97.
Struve's function, 78, 86.
Transformation of Bessel's
equation, 117.
Vertical rod, 124.
Weber, 5, 61, 116.
discontinuous integrals, 58.
Hankel functions, 74, 121.
Hankel's integrals, 71, 72, 98, 120.
Heat conduction, 31, 40, 44, 134.