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

136 THEORY OF BESSEL FUNCTIONS [CHAP. V
512. Indefinite integrals containing two cylinder functions; Lonmel's
second method.
An alternative method has been given by Lommel* for evaluating some
of the integrals just discussed. By this method their values are obtained in a
form more suitable for numerical computation.
The method consists in adding the two results
d
dz {fZ" W (z) (, (z)} = - Z {t (z) ^+ (z) +,+ (z) t( (z)}
+ (p +~j + +) zP-lj (Z)V ( Z),
d
d- {ZP Lt+1 (z) V+, (z)} = ZP {X, (z) vi+l (z) + + (z) Zv (z)}
+ (p - - - 2) zP-1,+1 (z) '+1, (z),
so that
(p + + v) z-1 i, (z) W (z) dz + (p- - v - 2) Z P-1 +, (z) W +, (z) dz
= { (z) (V (z) + +1 (z) ( v+i(z)},
and then giving special values to p.
Thus we have
= - 2 (~ +~ +1) {) (Z) () + ( (Z) + ++ (z)+,
2(/ ~ v+ ) (z) ~ (z)+ +l () Yel~z)}.
rz _ zt+Y-2
(2). +y+1 W () W (z) d =2 _+ {I ( W, (Z + v,.+ (Z) V+ (z)}.
As special cases of these
frs Z^~~~,~-2Z
(3) Jir-pp-L ^-1+l (8) il ^ -24u + 2
rZ z2y+2
(4) Z+1 (z) dz = 4 + 2- {2 (z) + +1 (z)}.
Again, if p be made zero, it is found that
(u + v)j f, (z) 9v (2) dz - ( + v + 2) t (z) dV+l (z)
= 'e (Z) iv (Z) + A+1 (z) v+l (Z),
so that, by summing formulae of this type, we get
(5) (tz + v)f z (z) V - (z) z -( + v + 2n) W +n (z) i-v+? (Z) d
Z - z CZ
n-l
=, (Z) v (z) + 2: ct,+m (Z) Wv+,n (z) +,,+n (2) v+n (Z).
m==l

* Math. Ann. xiv. (1879), pp. 530-536.