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

192

THEORY OF BESSEL FUNCTIONS

[CHAP. VI

provided that arg iz I < rr; and
l(0 = ~ __i-.i
(6) =nretV+1)i H1 (z) I= 2 ri r (- - - s) r (- s) (- 1iZ)v+2S ds,
provided that ] arg (- iz) ] < l-r; and, in each integral, c is any positive number
exceeding R (v) and the path of integration is parallel to the imaginary axis.
There is an integral resembling these which represents the function of the
first kind of order v, but it converges only when R (v) > 0 and the argument
of the function is positive. The integral in question is
(7) J. () = 2ri t r (- )(x)v+2Sds;
2- c, ()i r (v +s~l)
and it is obtained in the same way as the preceding integrals; the reader
will notice that, when s I is large on the contour, the integrand is 0 ( s -v-l).
6*51. Barnes' representations offunctions of the third kind.
By using the duplication formula for the Gamma function we may write
the results just obtained in the form
(1) J(z) eZ = (2z) f(0+) (v + s + 1). (+ 2iz )sds
2ivJ7r r (s + ) r (2 + s + l)sin s7r
Consider now the integral
(2z)v 00i
(-2^ /- I r (-s) r (- 2 - s) r ( + + ).(2iz)ds,
2izvw'7 -*i
in which the integrand differs from the integrand in (1) by a factor which is
periodic in s. It is to be supposed temporarily that 2v is not an integer and
that the path of integration is so drawn that the sequences of poles 0, 1, 2,...;
- 2p, 1 - 2v, 2 - 2v,... lie on the right of the contour while the sequence of
poles -v -- - v -,... lies on the left of the contour. In the first
place, we shall shew that, if I arg iz < 32, the integral taken round a semicircle of radius p on the right of the imaginary axis tends to zero as p -- o;
for, if s = peti, we have
7T2 rf (V + s +~). (2i),)
s (- s) r (- 2 - s) r (F +s + ).( = r (s ) (2 +S-i-1) sin sr sin (2v + s) 7r
and, by Stirling's formula,
1r(~ +s+ ~).(si,)8
log F (s + 1) (2v + s + 1)
pei~ log (2iz) - (v + pei) (log p + iO) + p eO - log (2vr);
and the real part of this tends to - co when - I < 0 < -T, because the dominant,
term is - p cos 0 log p. When 0 is nearly equal to _+ r, I sin s7r I is comparable
with I exp [pvr sin 0 } and the dominant term in the real part of the logarithm
of s times the integrand is
p cos 0 log 2z- p sin 0. arg 2iz - p cos 0 log p + p0 sin 0 + p cos 0 - 2p Isin 0
and this tends to - o as p - oo if i arg iz < 37-r.