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

13-55] INFINITE INTEGRALS 433
and thus we have
]0 fo ei(x+k) Xp —1
2wi*= 2 *j0r epriH (lx) (xer) - e-PriH^(l1 (xe-i)} dx
] (f ei(x+k) xp-i
= ( +1)+ [Jf (x) Isin (p + v) 7r + 2i cos v7r cos p7r}
+ i Y (x) sin (p - v) 7r] dx.
In particular, taking m = 0, we get
1 ei(x+k) Xp-i
(2) kP-H() (k) = I [r ei+k Xp- [Jv (x) {sin (p + v) 7r + 2i cos v7r cos prr}.TT o xr lk
+ i Y (x) sin (p - v)7r] dx.
If we consider the integral
1 er -i(z+k)
2'-Li J (- z)p- H,(2) (- z) dz,
we find that
1 r 00 e~- ($+k),CP-1
(3) kCp-1H(2) (k) = e-(+ [Jv (x) tsin (p + r) r - 2i cos vT7 cos p7r}
-iYv (x) sin (p - v) 7r] dx.
If we take p = 1, v = 0, we get
(4) Jo (k) = 2 si ( + k) () dx,
7. o x + ko
2 fxCos x+c)
(5) Yo (k-) = - - co (x) dx.
7r Jo x -+ k(
The last two results are due to Sonine*.
More generally, taking p = v + 1 and - I < R (v) < -, we get
1 f o' e~i(x+k) Xv ev~
(6) + J, (x) dx = 2 cos v (1) ~ i Yv(k)},
77iJo x+l 2i cos vw
a result also due to Sonine.
cos (x 1 k1 '. \
By writing cos(- in (x + k) dt,
x+lk x+ k lJ
and using the formulae (6) and (7) of ~ 13'42, Sonine deduced from (5) that
(7) 2 C'" J(x) 2 i'h'r
(7) (7) 2 - o Jo (x) dx + - sin (k cos 0) dO,
V/ Jo x+k VrJo
and hence from ~~ 356 (2), 9-11 (2),
(8) Yn (k) =- 2 0, (x + k) Jo (x) dx + - sin (c cos 0 - n7r) cos nOdO.
* See also Lerch, Monatshefte fur Math. und Phys. i. (1890), pp. 105-112.

W. B. F.

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