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

562

THEORY OF BESSEL FUNCTIONS

[CHAP. XVII

In order to establish these formulae, it is first convenient to effect the
generalisation to complex variables of the expansion of the reciprocal of the
radius vector given by ~ 17'21 (6). That is to say, we take the expansion
1 + 2 Z Jn (nz)cos nu,
n=1
which we denote by the symbol S(z, O), and proceed to sum it by Kapteyn's
method (explained in ~ 17'3), on the hypotheses that q is a real variable and
that z lies in the domain K. We define a complex variable r by the equation
b = r-z sin.
The singularities of J, qua function of b, are given by cos 1 = l/z, that is
= arc sec z - /(z2 - 1)
None of these values of q is real* if z lies in the domain K; and, as Sb increases from 0 to oo through real values, 4 describes an undulating curve
which can be reconciled with the real axis in the +-plane without passing
over any singular points.
It follows that if, for brevity, we write
U t-l exp {Lz (t- 1/t)},
then
1-fU2 dt
(z, ) 27rJ(r+) 1 - 2 Ucos t + U2 t
with the notation of ~ 17'3. By the methods of that section we have
i2-~f 1-_U2 dt
2S(z, 27i)Jr+,,)l - 2Ucos + 2 t '
and so 2S (z, b) is equal to the sum of the residues of the integrand at those of
its poles which lie inside the annulus bounded by r and y.
We shall now shew that there are only two poles inside the annulus, and,
having proved this, we then notice that these poles are obviously t = e~i.
By Cauchy's theorem, the number of poles is equal to
1 f dlog(1-2Ucos+ U2)dt
27ri (r+,y-) dt
1 f d log( - 2 Ucos ( + U2)
7Ti (r+) dt
+I f d log [t2 exp {-z (t - /t)}] dt
cit tdt + 2
92f r 1~ n dU
__ 2iJr[LE US cos(n+1)+) I dt+2
7r (r+) =O dt
=2,
* It is easy to shew that such values of 0 satisfy the equation
ef~ _ z exp,%/(1 - 2)
l+/(-z th)
so that I e-i| < 1.