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

352

THEORY OF BESSEL FUNCTIONS

[CHAP. X

The integral is convergent when arg z < rr, and it may be seen without
difficulty that it is 0 (z-2p).
It may be shewn from the asymptotic expansion of the Gamma function
that the same integrand, when integrated round a semicircle, of radius R with
centre at -p - 2, on the right of the contour, tends to zero as R -. oo, provided
that R tends to infinity in such a manner that the semicircle never passes
through any of the poles of the integrand.
It follows that the expression given above is equal to the sum of the
residues of
r(2-2,_+,P-s)r(~-t- I-s) 7r (Z)2
z r- F ( - I ) +lr ) (- sin s7r
at the points
0,-1, -2,...,-(p-1),
1,2,3,...,
1 11/3- 1 l1__ 1,",
~-_2+1. 3-_~+kv, 2_L+ ^V,....
2 2 2 + 2 V) 32 —U 2 2 V 2-2 2 U T 2 V
When we calculate these residues we find that
p- =o( (i) - ( + 1 +v m) r (1 - 1- - + n)
M=0 12 ZY"" ( 21 - 1 2 + 1 2 V) Pr (2 2 -2 2U-2V
_+ ~ (_r + 1 2 +r2 2 V+I1) r ( 2 +
mi r(+l -l V+m) r( + 1 + 2V+m)
2_ 1 7F (2 + 1 r ( + 2 + ) () (_ 1 Z)-v+2m
r ( - 1/ + lr) sin v7r mofo! F (1 - + m)
21-1, 17 r( + -,- ) 'P (_)M (1,Z)+2m
- 1 - ) sin rr =o r (z-( + +),
so that
p- (- 1r(-i +i1 + m) r (1-1 - +,)
Pn =o:;Q,)2m r (1 1 +1. r 1(1_
2-1 r (1 +1 -- ) r (2 + 2+ v)
sin vrr
X [COS 1 (( - V) w7. J_ (Z) Lo + V) 7. J (z)] = 0 (Z-2p),
and so, by ~ 1071 (2), we have the formula
SA" (Z) = ~ -, -2 2_2 I2 9 + (Ztk-2
V=O (.)Q zr(u - + )r-]4 1 +( - I)
and this is equivalent to the asymptotic expansion stated in (1).
and this is equivalent to the asymptotic expansion stated in (1).