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

180

THEORY OF BESSEL FUNCTIONS

[CHAP. VI

Modifications of (2) and (3) are obtained by replacing w by w ~+ ri; it is
thus found that*
(8) H(1) (z) - i i eiz coshw-vwdw
2e-iv2i oo0+Ti&
= -- - eizcoshw cosh v. dw,
rtz Jo
e~vni [' c- ~ ~ri
(9 H() e() - iz_ 2 -coshw- vw dw
H ( ) -O.~ _O +i
Ii I 7 — j7r
=p2e-r e t = -. e-iz coshw cosh vw. dw,
Tri Jo
provided that i arg z I < 7r.
Formulae of special interest arise by taking z positive (= x) in (6) and (7)
and -1 < R (v) < 1. A double application of Jordan's lemma (to circles of
large and small radius respectively) shews that, in such circumstances, we may
take wo = tr in (6) and w =- 17r in (7). It is thus clear, if u be replaced by
+ iet, that
e-vti FoO h 2e^ 'riO fX
(10) HfPj) (x) = - eeosh t- tdt = J ex cosh cosh vt. dt,
elvi f 2eliVm CO
(11) H(2) (x) () =- e-ixcosh t-vt dt = - - e-ixcosht cosh vt dt,
7r J - o( rtz J o
and hence, when x> 0 and - 1 < R (v) < 1, we have
(12) Jv (x) = J sin (x cosh t - ~ Tr). cosh vt. dt,
2 ry
(13) Yv (x)=- - cos (x cosh t - vrr). cosh vt. dt;
and, in particular (cf. ~ 613),
(14) ~ (l
~~~~(14) Jo (x) = 2 r sin xt. dt
(15) Y0 (x) =- 2 f cos xt. dt
(15 1\/(t2V )
when we replace cosh t by t.
The last two formulae are due to Mehler, Mlath. Ann. v. (1872), p. 142, and Sonine,
Math. Ann. xvi. (1880), p. 39, respectively; and they have also been discussed by Basset,
Proc. Camb. Phil. Soc. viii. (1895), pp. 122-128.
A slightly different form of (14) has been given by Hardy, Quarterly Journal, xxxnII
(1901), pp. 369-384; if in (14) we write x=2 \/(ab), xt=au+b/u, we find that
(16) si-= iJ(au + 0 (2 J(ab)}.
Jo \ uJ u
NOTE. The reader will find it instructive to obtain (14) from the formula
P2 0 T sin (n +1) a,
Pn r J C{2(co 0 O-cos )}
combined with the formula ~ 5-71 (1). This was Mehler's original method.
* Cf. Coates, Quarterly Journal, xxI. (1886), pp. 183-192.