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

17-31]

KAPTEYN SERIES

563

the integral of each term of the uniformly convergent series vanishing, just
as in ~ 17'3.
Now the residues of
' 1 U2 1
1- 2Ucos + U" t
at t = e~i are both equal to 1/(1 - z cos *); and therefore we have proved that
1 X
(3) 1 ---, =1+2 E J (nz) cos no,
1 - cos n_=1
in the circumstances postulated; and the series on the right is a periodic
function of b which converges uniformly in the unbounded range of real
values of b.
Hence, when R () > 0, we may multiply by e-~ and integrate thus:
roo oo roo _ Ao e-~
e-0 d~ + 2 E Jn (nz) e- cos n dO = de.
n=l 1 - COS#
That is to say,
(4) f e-(-z sin )O d= + =2 E r.,
n=1 n+-2
where the path of integration is the undulatory curve in the +-plane which
corresponds to the real axis in the +-plane; and, by Cauchy's theorem, this
undulatory curve may be reconciled with the real axis.
Now, when the path of integration is the real axis, the integral on the left
in (4) is an integral function of z; and this function may be expanded in the
form
X zm eac o m d
Z -i- r e sinq~d~.
2m = 0. Jo
By changing the sign of z throughout the work we infer the two formulae
(2 J2 (2nz) - z2m fm-1 e in f d,
<5> 2 2 ^ -. S d e - sin d,
()7=1 4n+ 2 + na=l (2m)!
0(6) 2 21 (2n 1\) 00 2mr-1 2m-2 f
(6) 2 2 ^ ^"r1^1 = 2 -- e-q sin2'~-' + '
62 =1 - lm=1+ (2 X m- 1)! j
which are now established on the hypotheses that z lies in the domain K and
that R (') > 0.
By dividing the paths of integration into the intervals (0, 7r), (wr, 2r),...
and writing 2 7r + 0, 2 r + 0,... for ~ in the respective intervals, we infer that
fJe- sin2 Md = sinh I —f cosh ~0. cos2M 0d0
{2 + 22} { 42 + 2... t2 + 42}'
36-2