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

436

THEORY OF BESSEL FUNCTIONS

[CHAP. XIII

Finally, as an example suggested by ~ 13'55, we shall consider
x xP-1 Jv (ax) dx
o0 (x + k)+1 '
in which a > 0 and I arg lk < yr. It is first to be supposed that
R(v)>0, R(+l+1)>R(p+v)>O.
The integral is equal to
1 f rc r (- S) P-k1 x) sd+'S
27-i. o J-. (v + s+) (x + k)+1 dsdx
2i z r (-s) r(p + + 2s) r (+ 1 - - p -\ 2s) +2 p+v+2s —l d
27r-i j r(V + +1)r ( + ) a2 /
cP - -1 r F = (-)m( Ct/)v+ r F(p + V + 2m)
sin (p + v - u) 7r. r (Iu + 1) o m-om! Pr (y + m+ 1) r (p + v - + 2m)
Eo ( ak)+-+l-P+m r (mt + m + 1) sin I (p + v -/ - m) r 1
o~=0 m! r(' + 1v -1p + 1m+ ) r( I- - Ip+ m+ )J *
The first series reduces to J, (ak) when / = 0, and the second series is then
expressible by Lommel's functions (cf. ~ 10'7). In particular we have
(7) f XV J. (ax) dx 2rk" [H_ v(ak) - Y_ (ak)]
x + k 2 cos v\provided that - < R (v) < <.
The reader will find that a large number of the integrals discussed in this
chapter may be evaluated by this method.
13*61. Integrals involving products of Bessel functions.
If an integral involves the product of two Bessel functions of the same
argument (but not necessarily of the same order), it is likely that the integral
is capable of being evaluated either by replacing the product by Neumann's
integral (~ 5'43) and using the method just described, or else by replacing the
product J, (x) Jv (x) by
27 - OOr(,+ sl) r(p+s+ l) r(/ +^+ ) ds
in which the poles of r (- s) are on the right of the contour while those of
r (o/ + v + 2s + 1) are on the left; this expression is easily derived from ~ 5'41
by using the method of obtaining ~ 6'5.
The reader may find it interesting to evaluate
f xp-1 J. (ax) Jv (ax) dx
Jo (x2+- k2)x+l
by these methods. The result is a combination of two functions of the type 3F4, and the
final element in each function is a2k2.