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

148 THEORY OF BESSEL FUNCTIONS [CHAP. V
This obvious mode of procedure does not seem to have been roticed by any of the
earlier writers; it was given by Nielsen, Math. Ann. LII. (1899), p. 228.
The series for J (z) cos z and J0 (z) sin z were obtained by Bessel, Berliner Abh. 1824,
[1826], pp. 38-39, and the corresponding results for Jv (z) cos z and Jv (z) sin z were
deduced from Poisson's integral by Lommel, Studien iiber die Besselschen Functionen
(Leipzig, 1868), pp. 16-18. Some deductions concerning the functions ber and bei have
been made by Whitehead, Quarterly Journal, XLII. (1911), p. 342.
More generally, if we multiply the series for J,^ (az) and J, (bz), we obtain
an expansion in which the coefficient of (-)m albv (lz)I+v+2m is
2m
mw a2m-2n V2I
n=On\ (vF + n + l).(m - n)! r t + m-n+1)
a2m 2F (-m, - t-m; v+l; b2/a2)
mn! r (+ m +1) r(v+1)
and so
(2) J, (az) J(bz) = (az)' (2b)
x (-)m (I2)lm 2F, (-m,- L- n; + 1; b2/C2)
m=O! P (r + f+ 1)
and this result can be simplified whenever the hypergeometric series is
expressible in a compact form.
One case of reduction is the case b = a, which has already been discussed;
another is the case b = ia, provided that I2 = 2.
In this case we use the formula*
F(a,; a- +1; - 1) =2 r( + )r(a- + l)
2o, p+ Q / 2r a+)r(la-3+ 1)
and then we see that
()rn ( az)2v+2M CoS jm1r
(3) J, (az) L (az) 2 2
(3) J z) J (a) = r (-m ~ i+ 1) r ( + -m + 1) 7 (z + m + 1)
o>d _ (-)m ( Iaz)2v+4m
n~=0om! F(v-+ i + 1) r (v + 2m + 1)'
(4) J, (az) Iv (az)= (-) (az)'m cs ( 2 - m)7r
m=O0 1n! r (2J V "m + + ) r (- + + 1)'
(5) J (az) l(az)= ( (- (iaz)m cosv n)
=o ~ r (IV + 2n + l) r (-1V + 2 + ].)
If we take a = ei-i in (3) we find that
(6) ber 2 (z) + bei,2 (z)= E 2(-)2+4m., =0,! 2 r (v +mr, + 1) F (v + 2n + ~1)'
an expansion of which the leading terms were given in ~ 3-8.
* Cf. Kummer, Journalfilr Math. xv. (1836), p. 78, formula (53).