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

18-6]

FOURIER-BESSEL SERIES

617

To prove this theorem we observe that, when p = 0, 1, 2,..., we may write
t"+,p+-,= | Jmti J(jv(t),
= 0
where the coefficients,,, are determined by the formula
a -r-, t ( +2 J (jmt ) dt;
and the series on the right converges uniformly in (0, I -A) and oscillates
boundedly in (1 - A, 1). It is therefore permissible to multiply the expansion
by tif(t) and integrate term-by-term.
It follows that
t"+ip+~ f (t) dt- am t f (O J, (jnt) dt
=0.
Since all the integrals
tv+2p+f(t) dt (p 1, 2, 3,...)
are zero, it follows that tvf (t) is a null-function, by Lerch's theorem*, and
the theorem stated is proved for Fourier-Bessel series. The theorem for Dini
series can be proved in precisely the same way, and it is theoretically simpler
because the Dini series associated with tv+2P does not fail to converge uniformly in (1- A, 1).
It is possible to construct a theory of series of Bessel functions of the types.e aJv J(j, ), n bmiJv (m i ),
(where the coefficients am and bm are any constants) which resembles Riemann's
theory of trigonometrical series )t.
Such a theory is, however, more directly associated with Schlomilch's
series of Bessel functions, which will be discussed in Chapter xI; and it
seems convenient to defer the examination of the series
' 00 00
l2 aJ(jvx), X bm bJ (bmeX )
m==l m-1
by Riemann's methods to ~ 19 7, when the discussion of the series forms a simple
corollary to the discussion of Schlomilch series.
* Lerch, Acta Mathematica, xxvii. (1903), pp. 345-347; Young,: Messenger, xL. (1910),
pp. 37-43. Cf. ~12-22.
t Cf. Modern Analysis, ~~ 96 —9632.