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

16-4] NEUMANN SERIES 535
Hence
(3) X (4n + 2) J2l (x) J21 (t) dt
=f/(x)- J0 (x- ) f'(v)- t ) { f(t+v)+ f(t-)dt dv.
Now write
f' (v)- - J )f {(t + v) +f(t - v)} dt = F(v),
so that F(v) is a continuous function of v, since J, (t)/t has an absolutely
convergent integral.
If then we are to have S =f(x) when x has any value in such an interval
as (0, X), we must have
r0 (x - v) F(v) d 0,
throughout this interval; and, differentiating with respect to x,
F (x) J J(x - v) F (v) dv.
/o
Since J (x - v) j < 1/V2, it follows by induction from this equation, since
F(x) <_2o I F(v) dv,
that F(x) A
x I it2. n!,
where A is the upper bound of I F(x) i in the interval and n is any positive
integer.
If we make n -> so, it is clear that F (x) - 0, and so the necessity of equation
(2) is established.
The sufficiency of equation (2) for the truth of the expansion* is evident
from (3).
It has been pointed out by Kapteyn that the function sin (xcosec a) is
one for which equation (2) is not satisfied; and Bateman has consequently
endeavoured to determine general criteria for functions which satisfy equation
(2); but no simple criteria have, as yet, been discovered.
[NOTE. If f(x) is not an odd function, we expand the two odd functions
2 {f (X)-f(-X )}, (X {x(X) +f(-X }
separately; and then it is easy to prove, by rearranging the second expansion, that
=0
f(X)= E a.J (X),
where ao f (x) J1 (I x ) dx,
"an =f |x) J. (x) (n > 0),
provided that the appropriate integral equations are satisfied.]
* The sufficiency (but not the necessity) of the equation was proved by Kapteyn.