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

10-14]

ASSOCIATED FUNCTIONS

315

so that
sinh vt 1 (u+,1/u+,l+) dv d'
sinh 2t 27i J ( - 1)2 4sinh2
l f(u+, 1/+'I +) rp- 22r n sinh2m t 22P ' sinh2P t d,
-27rri Lo0 (-_ 1)2M+2 + - 1)p {(- 1)2 - 4sinh2 t}
whence it follows that, if we take p so large that R (p + 1 + v) > 0, then
sinh vt sin vr l (_)m F (m 1 (+2 s t+)(m +1 )
cosh - ~L~m=0 (2m + 1)!
02 (-)p r (2 + 1.) 1- (2 sinh t)2p+1]
+02 () J(p + (2p+.)!
On integrating these results, it follows that
osht. _in COSoT ' (-)mF(m+ I+v)P(M+-,v)
Jo 2w- ~,=0 (IZ)2m+1'
0 2v 2cos t- dt 2'/ 2 - f -m
sinh Vt e-zsinht dt, (_)m r (m + 1 + 2) r ( -l + 1- r)
~,,,h ul.,,.U""lp- sin v (2- v2) (I2 - v2) (32 _ V) 1
rn =0 J())2M+2
If v is real and z is positive, these asymptotic expansions possess the
property that the remainder after p terms is of the same sign as, and is numerically less than, the (p + 1)th term when p is so large that R (p + 1 + Iv) > 0.
It follows from ~ 10'13 (2) and (3) combined with ~ 10'11 (6) that
() (z- (z)_ n v 2-v 2 (12 -(2) (32 _ ) ]
- in Pv77- 7*2J
(2) E (Z)-Y(Z)- 1 + -cos L - -...]
- cosin 7r _v v (22 - v2) v (22 - v2) (42 - v'2) 1
7TZ Z Z'q Z5 '
Cylinderfunktionen (Leipzig, 1904), p. 228. The proof of this section does not seem to have
been given previously.
Since the only singularities of cosh vt/cosht and sinh vt/cosht, qua functions
of sinh t, are at sinh t = + i, it is possible to change the contours of integration
into curves in the t-plane on which arg (sinh t) is a positive or negative acute
angle; and then we deduce in the usual manner (cf. ~ 61) that the formulae
(1) and (2) are valid over the sector I arg z I < r.