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

560

THEORY OF BESSEL FUNCTIONS

[CHAP. XVII

we see that, if we can find a circle r with centre at the origin of such a radius
that on it the inequality
(2)X exp { (t-1l/t)} < 1
(2) < 1
t
is true, then
1 r l +t-exp z(t- 1/t)} dt
(3) ( )= 27z J (r+) 1 -t-' exp [z (t- 1/t)} t
To investigate (2), we recall the analysis of ~ 8'7. If z = peia, t = eu+i,
where p, u, a, 0 are all real (p and u being positive), then (2) is satisfied for
all values of 0 if
p \/(sinh2 zu + sin2 a) - u < 0;
and when u is chosen so that the last expression on the left has its least
value, this value is (~ 8'7)
log z exp V(1 - z2)
which is negative when z lies in the domain K. Hence, when z lies in the
domain K, we can find a positive value of u such that the inequality (2) is
satisfied when t = eu.
Again, if we write 1/t in place of t in (3) we find that
(4) ( 1 fI + texp {-~z(t- 1/t)} dt
Z) = 27 J(y+) 1 - t exp {- (t- l/t)} t
where y is the circle I t ee-.
When we combine (3) and (4) we find that
If1 t + exp {1z (t- l/t)} at.2S (z)= 2
() =) 27rJ(r+,-) t- exp {1 (t- l/t)} t
and so 2S (z) is the sum of the residues of the integrand at its poles which lie
inside the anculus bounded by F and y.
We next prove that there is only one pole inside the annulus*, and, having
proved this, we notice that this pole is obviously t = 1.
For the number of poles is equal to
1 _ d log [1 - t- exp {~z (t - 1/t)}]
27ri r+,y-) dt
1 d log [1 - t- exp {~z (t - t)] ct
I cit
1'r+ )dlog [1- t exp {-l-(- I-/t))] dt
27ri (r+) dt
1 d log [1 - t expt- ~z (t - l/t)}]
r't (r+) cdt
+ 1 f d log [- t exp {- 2z (t - 1/t)}] dt
27rz (r+) dt
* The corresponding part of Kapteyn's investigation does not seem to be quite so convincing
as the investigation given in the text.