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

620 THEORY OF BESSEL FUNCTIONS [CHAP. XIX
and change the order of the integrations. We deduce that
( rd- 2x 'r 'r
2- (x sin 0) dO -f(O0) = ' (x sin 8 sin i) sin 0 d dO, r 0 7T o JO
71-.Jo/('8m^^7(Jo^
2w fP r Sin sin 0 cos X dX dO
2^.f.. cs0(xsin. X)
= 2 Jo7 (xsin -s- in cos x)
2x s sin O cos dO d
w (CoS2 X - cOS2 0)
0 f' (X sin X) Kare sin cos ) cos X dX
= f'(xsinx) cos XdX
J o
=f () -f(0),
and so, when g (x) is defined by (4), g (x) is a solution of (3).
Now it is easy to verify from (4) that, when f' (x) is a continuous function
with limited total fluctuation in the interval (0, 7r), so also is, g (x); and
therefore, by Fourier's theorem, g(x) is expansible in the form
00
()= ao + X a cos rmx,
w=l
where
a,= - g (u ) cos mudu
=- j /(O) + /' (u sin cb) d] cos mudu,
and this series for g (x) converges uniformly throughout the interval (0, 7r).
Hence term-by-term integrations are permissible, and so we have
Xf() -J0 g (x sin O).dO
/i(~)==2o y^gsmd2
=2 f {Ia0 + o a, cos (mx sin 0)} dO
7 Jo m=1 )
= ao+ i aC.Jo(mx),
m=l
and this is the expansion to be established. It is easy to verify that the
values obtained for the coefficients am are the same as those given by
equation (2).
When the restriction concerning the limited total fluctuation of f'(x)
is removed, the Fourier series associated with g(x.) is no longer necessarily
convergent, though the continuity, of f' (x) ensures that the Fourier series