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

176

THEORY OF BESSEL FUNCTIONS

[CHAP. VI

and so we have at once
( I Z)v (0+) Z2
J(1) ()= _( t-v-o exp {t — dt.
This result, which was discovered by Schlafli, was rediscovered by Sonine;
and the latter writer was the first to point out its importance.
When I arg z I < T, we may swing round the contour about the origin until
it passes to infinity in a direction making an angle arg z with the negative
real axis.
On writing t = lzu, we then find that, when I arg z \ < 7r,
1 (0+)/ 1\I
(2) J (2)= 27 u- - exp {z u-) - du.
This form was given in Sonine's earlier paper (p. 335).
Again, writing u = ew", we have
00 + 7ri
(3) Jv (z) = =2 ezsinhw-vv w,
valid when arg z < 17r. This is the first of the results obtained by Schlafli.
In this formula take the contour to consist of three sides of a rectangle, as
in Fig. 8, with vertices at oo - ri, - 7i, 7ri and x + vri.
-TTi
Fig. 8.
If we write t T 7ii for w on the sides parallel to the real axis and + iO for w
on the lines joining 0 to + Tri, we get Schlafli's generalisation of Bessel's integral
(4) J (2)=1 cos ( - - zsin 0) dO - e-vt-zsinht dt,
77 J0 TT. 0
valid when ] arg z I< 17r.
If we make arg z -+ 1~ r, the first integral on the right is continuous and,
if R (v) > 0, so also is the second, and J, (z) is known to be continuous. So (4)
is still true when z is a pure imaginary if R (v) is positive.
The integrals just discussed were examined methodically by Sonine in his
second memoir; in that memoir he obtained numerous definite integrals by
appropriate modifications of the contour. For example, if. be an acute angle
(positive or negative) and if
2- 7 + < arg z < ~r - w,