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

474

THEORY OF BESSEL FUNCTIONS

[CHAP. XIV

Hence it follows that, if the limit on the right exists, then
udu& ~ F(R, ~))g(uR)R(dRdn~)
~ 3 30 (dRd-D)1
= lim F(R, ()G(XR)(dd).
DAo J - tr R
14'63. The proof of Neumann's integral theorem.
We are now in a position to prove without difficulty the theorem due to
Neumann stated in ~ 14'6. We first take an arbitrarily small positive number
e and then choose the sectors in which the convergence of fl (R, <>) to zero is
uniform, in such a way that the sum of their angles exceeds 27r- e. We then
choose 8 so small that 1f (R, p) < e in these sectors whenever R < 8; and we
take the upper bounds of
n(R, )+ F(R, ) and / G (t) du
to be B and C.
We then apply the second mean-value theorem. We have
o xi (R, )G) G (R)
= x (+ 0, O ) G (xR) R+ ~{x (8, D) - XI (+, >)} G (R) dR,
where 0 8.
NdR As due
Now G (X) = G (t) < 2C.
Hence
fF(li,4)G( xi) dl7 / s du_
F (R, <) G (xR) - = F (+ 0, ~) G (u)-+ +,
where [ is less than 2eC inside the sectors in which convergence is uniform,
and is less than 2BC in the exceptional sectors.
Hence it follows that
f F(R, m) G (XR) (dl d) -27rflF(+o0,i ) G () du
< 27r. 2e+ e. 2BC
= 2eC 2- + B}.
Hence, for large values of X,
f' fo F(R, qi) G (XR) (d 2d w F(+ 0, cD) o (du
< 2eC(27 + B) + o (1),
that is to say
lim f 7r o )G R(dRd d:) du I
lim | F(R, () G (XR) (d ) - 27rwlF(+O 0, () G (u)2eC (27r + B).