OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
519 
Formula for Q,'" 1 (cosh xp) under special conditions. 
52. When /x is real and greater than unity, let g = cosh xp, we then have, provided 
the real parts of n + m + 1, | — m are positive, 
COS 7R7T . i ' s 
Q n m (cosh xp) = e mm . 2 m . IT (m — P ) IT (— i) --sinh® if; -- 
Jo V1 — 
h 
n-tm 
7 T 
2 ft//. + h 2 ) 
771 + 
- dh. 
Let h = e u , we have then, taking the path along the real axis, 
Q/(coshi/») = e mm . 2 m . Ii(m—^) n(— 
cos rmr 
7 r 
sinh m xp ( 
J d, 
g—(n+v)u 
(2 cosh u — 2 cosh xp)” 
- clu , (118). 
If we take the path to be from 0 to — co along the real axis, along an infinite 
semicircle from — co to + °o , along a straight path from co to —, avoiding the 
CKD f— 
\ 
■< - f 
-W— 7 
oo 
point z by describing a small semicircle ; the integrals along the semicircles vanish 
provided the real part of n — m is negative, we then have 
Q,® (cosh \p) = e mm .2 M .TI(m—^)Tl(—^) co s ~ l - sinh®i|»|| — 
-G 
(2ffl+1) " (2 cosh w —2 cosh f) r ‘ 
\ 0 e {m+iUn (2 
g—(n+ni+l) in- g(n+J)M 
cosh xp + 2 cosh u) m 
~(n+i)u 
du 
' (2 cosh xp —2 cosh u) 
m+i 
i du\. 
In a similar manner, by taking the semicircles above the real axis, we can show 
that 
f*OS?7?,7T f f 00 (?l+7/l +1) err (7t+^)w 
Q/(coshi//) = e m,r ‘.2 OT n(m—A-) n(—i) -sinh Wi iG - — - 
^ v v 27 v 2/ 7 r y lJ_» (2 cosh+ +2 cosh m) w+ * 
x du 
( 2771 + 1 ) ITT 
(2 cosh 2 cosh xp) 
m + 
t du — \ — 
1 Joe 
— (»+$) u 
(m+i) in ^ cos p ^ _ 2 cosh 
du L 
Multiplying the first expression by e (,l+w+1)l,r , and the second by e (!i+m+1) ‘’ r , and 
subtracting, we then have 
