1921-22.] 
A Fourier-Bessel Exp 
ansion. 
95 
of the first term in the asymptotic expansion of (p{C) taken round the 
rectangle bounded by $=S{>0), >; = 0, ?;=N, and then, when the 
integrals have been added to give the integral of O(^), make 
It is thus found that 
where is the first term in the asymptotic expansion of . 
Accordingly 
m'{- 
^ n/r 
Now the integrand of each integral on the right is of the form 
±M + ^’ 
where x(v) remains finite throughout the path of integration. Thus, by 
increasing M, each of these integrals can be made arbitrarily small. 
Hence 
fmdi+., (10) 
where e-^0 as M-»oo . 
Lastly, to evaluate j 6{C)dl, deform C into the ^-axis indented at the 
zeros of sin {^(6 — a)}. As before, the integrals along the axis cancel, and 
the integrals round the semicircles give 
ZTTt- 
n (STT(x-a)\ . iS7r(r-a)'i 
sin - — ^ 1 sill ' 
Z sin - — 7 r sill — j r 
^ [ h - a ) [ 0 - a j 
b - a 
where jul denotes the number of positive zeros of sin {^(b — a)} between 
0 and M. 
This is equal to 
7n 
J(xr) 
— — y ( 
■) b-a-^\ 
COS 
f stt{x - r) 
\ b - a 
cos 
[sir{x + r - 2a\ ) 
'] 
h - a 
f f (x-?-)] f 
I sin ( ^(/. + J) ) ‘^1“ (’^(/^ + i) 
(x + r - 2a) ^ 
{b-a) / 
I ^ h_-a 1 -^{x-j) ( c,b-a ) ir(x + r- 2a) \ 
1 2(6 - a)l 
sm 
2(6 - a) I 
^ This method could also be employed in the discussion of the integral of 0(0, and then 
it would be unnecessary to introduce the restriction 
