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

458

THEORY OF BESSEL FUNCTIONS

[CHAP. XIV

By taking X sufficiently large (n remaining fixed after e has been chosen) the
last expression can be made less than 2Ae//VX, and so the original integral
is o (1/VX).
(II) If the upper limit is infinite, choose c so that
I \F(R) V/R.dR <,
and use the inequality
| F(R) J, (\R) Rd.R F (R)J (\R) RdR + F I (R) I R. dR;
then, proceeding as in case (I), we get
C~FIR\IXRRI 72BnK 2Ae
F (R ) J~ (\R) R dR <..3 + 27.
fa A2' V^X- +/X
The choice of n now depends on e through the choice of c as well as by the
mode of subdivision of the range of integration (a, c); but the choice of' n is
still independent of X, and so we can infer that the integral (with upper limit
infinite) is still o (1//X).
(III) If F(R) \/R is unbounded*, we may enclose the points at which it is
unbounded in a number p of intervals 8 such that
tf F (R) VR.dR<e.
By applying the arguments of (I) and (II) to the parts of (a, b) outside these
intervals, we get
bi! F(R)Jv <2Bn (p +1)K 3Ae
[bF(R) J(XR)RdR ^< + ~/'
Ia - V-X
where K is now the upper bound of | F (R) ],/R outside the intervals 3. The
choices of both K and n now depend on e, but are still independent of X, so
that we can still infer that the integral is o (1//X).
14'42. The inversion of Hankel's repeated integral.
We shall next prove that, when v > - a, anrd F(R) V/R. dR exists and
is absolutely convergent, then
fudu F (R) Jv (uR) Jv (ur) RdR
0o 0o
lim F (R) { (,R) J^ (ur) hudn RdR,
— oo J 0o
provided that the limit on the right exists.
* Cf. Modern Analysis, ~ 9-41.