OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
517 
TXT 
—*- 
*7 
oo 
Multiply the two formulae by e e (fl+i)l77 , and subtract them, we then have the 
formula 
sink m \[r 00 _ 
p,; m (cosh fi) = 2 ^n (- 1) n (rn - I) C0SeC ' f i) 77 ^ sinh (n + + ^tt) 
_|_ g fr+W" g ' n p ( n _j_ _ 7 m7r )j (2 cosh <j> — 2 cosh d<f> (114), 
where the real part of n is between 0 and —1 ; and the real part of m is between dz 
If m = 0, we have 
P„ (cosh \p) = — cot (n + |-) 7r| 
sinh (n + J) <£ 
^x/2 cosh <fi — 2 cosh t/t 
d(p 
(115). 
Definite Integral Formulae for Q fJ ®* (cos 0), under Special Conditions. 
51. When the real parts of ?& + m + Pi — m are positive, we have 
Q/'(cos 6+ 0. i) =e ,,l,r ‘. 2™n (m—£) II (— £) 
cos mir — 
vim 
e/* 
sm' 1 
6 
IT 
rl/z 
Jo 0^2 
Jjll + M 
(1 — 2/u.h + Id) 
m+Jr dh . (43), 
take the path of integration to be from 0 to 1, along the real axis, then from 1 to - 
along an arc of a circle of unit radius with its centre at the origin ; along the straight 
path, 1 — 2 gh + A 3 has the phase zero, and along the circular arc it has the same 
phase as h, hence, writing in the first part of the integral A = e~“ and in the second 
part h = e~ 
Q** (cos 6 + 0 . i) 
, „ / ,. i \ cos rmr . n 
— e 2 . 2" z . IT [m — i) n ( — i) - • sin® 6 
7r 
g-(»+i) u 
o (2 cosh u — 2 cos 6 ) m+ * 
jg— (»+j) i<t> 
du 
o(2 cos — 2 cos 6) 
m+h 
d(f> 
