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

178

THEORY OF BESSEL FUNCTIONS

[CHAP. VI

Replace 0 by wr -0 in the second integral on the right, and it is found on reduction that
(1) Y (z) = - sin (z sin 0-) - ) dO (evt + e-t cos vr) e-sint dt,
Y )jo 'f7eJo
a formula, practically discovered by Schlafli (who actually gave the corresponding formula for Neumann's function), which is valid when arg z < -7r.
By means of this result we can evaluate
f riezsinhw-vw dw,
when i arg z I< -rT; for we take the contour to be rectilinear, as in Fig. 9, and
lri
\
0
— o
Fig. 9.
write - t, iO, t + 7i for w on the three parts of the contour; we then see that
the expression is equal to
- evt-zsinhtdt + ei(zsinO-vO) d + e-tzsinh dt
7rz o 77Jo 07J J7T
and this is equal to J, (z) + i Y, (z) from formula (1) combined with ~ 6'2 (4).
Hence, when arg z I < ar, we have
J ro+7ri
(2) H('e () = - ez sinh w-Vw dw,
(3) H_(2) (z)- -1. ez sinh w -w dw.
Formulae equivalent to these were discovered by Sommerfeld, Math.. Ann. XLVII. (1896),
pp. 327-357. The only difference between these formulae and Sommerfeld's is a rotation
of the contours through a right angle, with a corresponding change ill the parametric
variable; see also Hopf and Sommerfeld, Archiv der Math. und PAys. (3) xvIII. (1911),
pp. 1-16.
By an obvious change of variable we may write (2) and (3) in the forms
1 0o exp 7fi 1\
(4) HV() (z) = u —' exp u - - du,
1 rKo exp(- 7i) P 1\)d;
(5) H ).(z) = - -1 I ep- z u du;
H~(')~ 7 (z ) ui