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

48

THEORY OF BESSEL FUNCTIONS

[CHAP. III

f1
n = 0, namely t -(1 -t)-dt, which is convergent, we find that, when
o
R (v)>,
I (' -0' -q I | (_y M 2
1 (2r
f () = ( + 1) r ()m im-2= (2) 2 t)n- d
whence the result stated follows by making the substitution t= sin2 0 and
using the fact that the integrand is unaffected by writing r - 0 in place of 0.
When -- < R (v) < ~, the analysis necessary to establish the last equation is a little
more elaborate. The simplest procedure seems to be to take the series with the first two
terms omitted and integrate by parts, thus
o ( )m z2?m 1 m 1?1 o2; t t l
2 t - (1 - t)-' dt= () t (1 - t) -- dt
m=2 (2m)! Jo m=2 V + (2m)! o0
|= J {o+ ' ()(2n2)! (1 - t) dt
1t v + 2 (2 i ' ( t)
f 1 t- d o ( _ ) 1t 2 - I dt
= o;'~ — dt l m=2 (2min)!
on integrating by parts a second time. The interchange of the order of summation and
integration in the second line of analysis is permissible on account of the uniformity of
convergence of the series. On adding the integrals corresponding to the terms m= 0, m= 1
(which are convergent), we obtain the desired result.
It follows that, when R (v) > -, then
Jv J = r (v + 1) r (1) 1tv- ( - t)- cos z (1 - t)} dt.
Obvious transformations of this result, in addition to (1), are the following:
2 (z) cs (zt) d,
(2) Jv1 ) N V fr (Z) 1 - t2)v,- Cos (zt dt,
(3) JV () = r(. j + ) r () _S 1 - t2)^- cos (zt) dt
(4) Jv (Z) =r (v+y) r1 (1-t)veizt dt,
= (. + 1) p (. o
1 v rr
(6) J(z) (( ) j(Z () eiZcosOsin2Y OdO.
( (v + )P ( ) J0o
The formula obtained by a partial integration of (5), namely
(7) J( ) (V + -1) (0 1 sin (z cos 0) sinv-2 0 cos Od,
is soetimes useful; it is valid only when ( >
is sometimes useful; it is valid only when R (y)> >.