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

144

THEORY OF BESSEL FUNCTIONS

[CHAP. V

It will first be shewn that the series on the right of (1) is a uniformly
convergent series of analytic functions of both z and t when
z < r, R <ltl^A,
where r, R, A are unequal positive numbers in ascending order of magnitude.
When m is large and positive, J_-m (t) J (z) is comparable with
Ir (n- )
sin vw. ( R)v. (r/) r" m
and the convergence of the series is comparable with that of the binomial
series for (1 - r/R)v. When m is large and negative (=-n), the general
term is comparable with
(_)n (IA)" (aAr)
r (v + n+ 1). n!
and the uniformity of the convergence follows for both sets of values of m by
the test of Weierstrass.
Term-by-term differentiation is consequently permissible*, so that
(a -) E Jv-m (t) Jm (z) -= S {Jv- (t) Jm (z)-Jv-m (t) J'm (z)}
at Z j 0 =_= -m=- o
2= {2Jv-m-1 (t) - J-M+l (t)} Jm (z)
2M= - Go
1 X
- J_- (t) J- J- (z) - Jm+l (z)},
m M= -00
and it is seen, on rearrangement, that all the terms on the right cancel, so
that
00
S Jv-m (t) Jm (Z)= 0.
Hence, when I z < t I, the series E J_, (t) Jm (z) is an analytic function
n= -- 0
of z and t which is expressible as a function of z + t only, since its derivates
with respect to z and t are identically equal. If this function be called
F (z + t), then
F (z + t)=- J (t) J,, (z).
If we put z = 0, we see that F (t)- J (t), and the truth of (1) becomes
evident.
Again, if the signs of v and rm in (1) be changed, we have
J-_ (z + t) = I (-)m J-V+m (t) Jm (),
n= - o
and when this result is combined with (1), we see that
(3) Yv (z +t)= E Yv-rn(t) i( z).
-n= — oo
* Cf. Moderl) Anlalysis, ~ 5-3.