OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
505 
This integral has been evaluated in Art. 38, when u < | log mod '-- ; we pro 
ceed to evaluate it in the present case u > \ log mod 
definite integral by -— I (n, m) 
fi + 1 
A* ~ 1 ’ 
p 4" 1 
P 
We shall denote the 
7rt 
Denote by L, M, N the integrals of — (£ 2 — 1)" (tf — p) n m 1 taken along loops 
from the point A, round the points — 1, 1, p respectively, then 
r(jx+, l+, m—> i—) 1 
llT) 1 
- ( t 3 — l)« (t — dt = ~N + — Ne 2!Wt — M 
= (1 — e 2nm ) {N + Me- 4mr ‘(l + e 2nm )} 
(/-if* f 
Also 
( t 2 — 1) ,! (t — p) -m m 1 dt = (L — M)e 2n,r ‘. 
I (w, m) = N -fi Le + Me 2n7U , 
hence 
, r('n+, 1+,/a-, X—) 1 
(1 — e 2 ™ 1 ) I (??, w) = (p 2 — l) im — (£ 2 — I)" (£ — [x )~ n ~ m ~ 1 c/£ 
- (1 - e~ 2 ™) (/x 2 - l)* m f ( ~ 1+ ’ 1 > i(« 3 - 1 ) n (t - ix)- n ~ m - 1 dt, 
or 
n (n) 
— e nm . 2i sin mr. I (n, m) — --:. 47t sin Mr. e nm ~P n m (p) 
v ’ ' rr (ft + m) vrv 
n (n) 
IT (?i + m) 
8 sin 2 mr. Q„ m (p), 
we thus have 
— f {p + ^/p, 2 — 1 COS ((f) — xp i iu)} n e~ mi(<t> 2 ' )±w “ d(f> 
ATT Jo 
= n o, + } ^ ) { F>r M ~ l e ~ nm sin niT Q/ ' (/*)} 5 
L 
3 T 
MDCCCXCVI.—A. 
