A treatise on the theory of Bessel functions, by G. N. Watson.

162

THEORY OF BESSEL FUNCTIONS

[CHAP. VI

It is to be observed that, when R (z) > 0, both integrals are convergent,
and differentiationrs under the integral sign are permissible. Also, both
integrals are analytic functions of v for all values of v.
Ini order to express the first integral in terms of Bessel functions, we
expand the integrand in powers of z, the resulting series being uniformly
convergent with respect to t on the contour. It follows that
/^(1+, -1-) oo m ^,v-+-m /(l+, -1 —)
z" eizt (t2 - )- dt tm (t2 - )V-a dt
f(1+JA,- =O mi JA
Now tm (t2 - 1)Y- is an even or an odd function of t according as m is even
or odd; and so, taking the contour to be symmetrical with respect to the origin,
we see that the alternate terms of the series on the right vanish, and we are
then left with the equation
r(l+,-l-) _m+2 (+)
v1 eizt (t2 -1)-t dt = 2 v v; - tOn (t2 - 1)v dt
J A mn =O (2m)~ Jo
-X (_)m v+2mz (1+)
X-= 2 2~ u, o-, (t- 1u - a ndu,
mn=o (2n)! Jo
on writing t= /,u; in the last integral the phases of u and u- 1 vanish when
u is on the real axis on the right of u = 1.
To evaluate the integrals on the right, we assume temporarily that
1R (v + -) > 0; the contour may then be deformed into the straight line
from 0 to 1 taken twice; on the first part, going from 0 to 1, we have
- 1 = (1 - ) e-, and on the second part, returning from 1 to 0, we have
u - 1 = (1 - ) e+wi, where, in each case, the phase of 1 - u is zero.
We thds get
r(l+) I
f Ul- (u - 1)v- dlu= {e- ( — ). e(v-.)i}J um — (1 _,)v- dui
r (., + - ) r (v + I)
= 2icos 7r (r V1
r(m+^ +1)
Now both sides of the equation
fO u+ -* (u - 1)"- du = 2i cos 7 r(r- ~ (V -)
0 o r (m+ + v - 1)
are analytic functions of v for all values of v; and so, by the general theory of
analytic continuation*, this result, which has been proved when R (v + ~) > 0,
persists for all values of r.
7 Modern Analysis, ~ 5-5. The reader will also find it possible to obtain the result, when
/ (v + 1) < 0, by repeatedly using the recurrence formula
f<0 _?n- n+ v+ n'+ 1 J(l+)ni- (uL- )"ir-+du,
[(l+)nu-~ (ul - l)V+' —2- du = v+^ - ( + d
which is obtained by integrating the formula
d {u"+~i (u-1)YV+T+h} =(os+ v +n+l) urm — (u-l)V+'L+i + (v+ +~) um- (t_ — l)"+v- ~
the integral is then expressed in terms of an integral of the same type in which the exponent of
u - 1 has a positive real part.