1921-22.] 
A Fourier-Bessel Expansion. 
93 
Then 
(8) 
f %v 
The first term in the asymptotic expansion of d> \ M + 
I A — y 
IS 
cot 
M-f- 
tv 
(A- a) I 
•2nm‘+iM(A.-y -v 
SO that the integrals on the right of (8) are convergent even when X — y is 
negative, provided that the asymptotic expansions of 0{M + iy/(X — y)} and 
• <p{ivl(X — y)} are still valid. 
Now consider the arguments of the factors of ^{M + i'y/(X — y)} as given 
by (7), and let \ — y = de^^, where d<y. 
{ / %V V 
\[M -1- j j ~ ^ d ^ 
increases from 0 to tt, this remains positive or zero for M^O: thus the 
amplitude of the argument varies between — 7t/2 and 7 t/ 2. 
II. 
The real 'part of | /x(^M + \'^y ) | 
d 
This remains 
positive if is positive, or negative if /x is negative, for M>0: thus the 
amplitude varies between — tt/ 2 and tt/ 2, or tt/ 2 and 37 t/2. 
Accordingly, the asymptotic expansions of the integrands on the right 
of (8) remain valid as 0 increases from 0 to tt, except for the initial point 
of the integral along the ly-axis. At this point, however, the integral is 
convergent, so that (8) is still valid. * 
Similarly the equation 
Jc" 
(9) 
where C" is a path from E to O above the real axis, can be shown to hold 
even when X — y is negative. In this case the amplitudes of /x, p, and cr 
are assumed to be 0 or — tt, and amp (X — y) is made to decrease from 
0 to — TT. 
Now 
[ F(OdC= f F(Od^+ f F(0d^. 
Jc Jc Jc" 
Along C' and C'" 
replace Jn(t^) by 
X7T 
and 
* Note . — The factor the asymptotic expansion of (^^M +- j is equal to 
giM(« cos 0-Md Sind when A — 7 = SO that the integral remains convergent when 9 varies 
from 0 to 7T. 
