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

13-52] INFINITE INTEGRALS

427

The condition R (v) > - I may now be replaced by the less stringent
condition R (v) >-, by analytic continuation.
An integral which may be evaluated in the form of a series by this method is
sinh rax
Hi sinh ax J, (bx) v + I d
which is a generalisation of Neumann's integral described in ~ 13-2; it is supposed that
R (a) +I (b)I <r and R(7)> -1.
I sinh az
By takingaz () (bz) zv 1 dz
'27ri sinh 7r z
round the contour used in this section, we find that the definite integral is 7ri times the
sun of the residues of
sinh az H (1) (bZ) ZV +
sinh rrz
at the points i, 2i, 3i,.... It follows that
Ys inh ax 2
(12) sinh x (bx) x + 1 dx- ( ) -1 n+l sin na. (nb).
Jo sinh rrx rr n=l
The series converges rapidly if b is at all large.
An integral expressible as a similar series was investigated by Riemann, Ann. der Phlysik
und Chemie, (2) xcv. (1855), pp. 132-135.
13'52. The generalisation of Hankel's integral.
Let is next consider the integral
1 Zf P-1H,') (as) dz
27ri (z2 + k2)+1 '
This differs from Hankel's integral in containing the (complex) number /[
in place of the integer mn. The conditions for convergence (with the second
type of contour specified in ~ 13'5) are*
>o, I R (v) < R(p) < 2R ()+-.
The contour is chosen with a loop to exclude the point ik, as shewn in
Fig. 31, and then there are no poles inside the contour; and the integral round
the large semicircle tends to zero as the radius tends to infinity. Hence
1 1x) xp-i dx
2wi j [HV(1) (ax) - e(P-2WL)i H,,() (axx ei)](X + 2)d+l
_ f(ik+)zP- Hv() (az)dz
- 27r Jo (Z2 + k))+
Now 1 f(ik-) zP dz ~1, e t R( -2a R(k)
*v Asi int ~13- 5-1 e a - (e) ri 2 -N~ow _27ri ( 2 + k)_L+i=- ~eQP-/ (2p - A) IF(p- + l)z
*As in ~ 13-51, we take R (k)> 0.