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

13.73] INFINITE INTEGRALS 445
Now, T being positive, we have
(2U + 7i) exp {- 2X (U+ 7)2} e2i sinh Tsinh U d
00
= 2 f0 (v + n7r) exp {- 2X (v + 7r)2} -2xsinh Tcoshvdv
00
and, since ve-2xsinhTcoshv is an odd function of v, it may be proved that the
last integral is
ro
27i e-2x sinh Tcosh dv + 0 (X),
_ 00
where the constant implied in the symbol 0 (X) is a function of T such that
its integral with respect to T from 0 to o is convergent.
In like manner,
f (2 U + 77t) exp {- 2X (U 4+ 1 7r)2} e-2i sinh Tsinh U d U
j -00
00
= J 2v exp (- 2Xv2) e-2xsinh Tcoshv d = 0.
- co
Hence it follows that
J^(X) I Y(x ) aJ ( x) _2 e T-2xsinh cosh v-2vTdvdT
av a - jJ e
87 jV rrjo J -oo
=- oK0 (2x sinh T) e-2T dT..7ro
The extension to the case in which the argument of the Bessel functions is
complex with a positive real part is made as in (1).
It should be mentioned that formula (2) is of importance in the discussion
of descriptive properties of zeros of Bessel functions.
The reader may find it interesting to prove that
v[() - a Y ( ) a (Z)
2Z2[V (Z) av > () Yv (z) ]-(- (Z) aY. (z () (z]
and hence that
(3) J,,(z) a^ ) - Yv (z) a ( — 4z) f (z2 cosh 2T- v2) Ko (2x sinh T) e-2^TTdT.
0 V aV 7rZ2
Other formulae which may be established by the methods of this section are
(4) J, (z) JV (z) YL (z) Y (Z)
=- K- (2z sinh t). {e(+ v)t cos ( - v) r + e - (+) t} dt,
4 sin (Mt- v) 7r
(5) J. (Z) Y, (z) - y (Z) Y (Z) sin2 0 Kv7z (2z sinh t) e(+ v) t dt;
these are valid when R (z)>O and IR(/ - v) l <; they do not appear to have been
previously published.