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

11-1-11-3]

ADDITION THEOREMS

359

We take the formula (Parseval's integral)
Jo ()- &1.r &Cos Od9 1 | et cOS(-os L)d
7r j eicosod= 2 -TTJ
which is valid for all (complex) values of wr and a, the integrand being a periodic
analytic function of 0 with period 27r. We next suppose that a is defined by
the equations
sina=Z-z cos b, cosa = z sin b,
and it is then apparent that
J. (v) = 2- exp {i (Z - z cos c>) sin 0+ iz sin (b cos 0} dO
- 7rJ v { Jen(Z) emi} eiZ sin ( - 0)d
27r?=-o -T
2 1 X Jrm (Z) ei (0+) -izsin 0d
= s Jm (Z) en (z) emi4,
m= — o
the interchange of the order of summation and integration following from
the uniformity of convergence of the series, and the next step following from
the periodicity of the integrand.
If we group the terms for which the values of in differ only in sign, we
immediately obtain Neumann's formula.
The corresponding formulae for Bessel functions of order + ~ were obtained by Clebsch,
Journal fur Math. LxI. (1863), pp. 224-227, four years before the publication of Neumann's
formula; see ~ 11'4.
11'3. Graf's generalisation of Neumann's formula.
Neumann's addition theorem has been extended to functions of arbitrary
order v in two different ways. The extension which seems to be of more
immediate importance in physical applications is due to Graf, whose
formula is
(1) J (i). - z- - if) = Jv+m (Z) Jm (z) eni*,
[ Z - ze m=-oo
and this formula is valid provided that both of the numbers zei. \ are less
than Z |.
* Math. Ann. XLIII. (1893), pp. 142-144 and Verhandlungen der Schweiz. Naturf. Ges. 1896,
pp. 59-61. A special case of the result has also been obtained by Nielsen, Math. Ann. LII. (1899),
p. 241.