OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
499 
1 [ 2lT 
9 ~ & + 
■j ‘r J o 
cos 
v//x 2 — 1 cos (</> — ifj ± ' nnf) d(f) 
sin 
t-. , . n (%) cos ., __ , 
P '”' (/ X ) rTTUTUU olv, m “F t? 0 • • • 
II (% + m sin 
If we change n into — (n -f 1), we have 
(81). 
-if 
2tt ] 
J o 
cos 
■ m<b 
sin r 
{fj, + \//x a — 1 COS (0 — yfr + iv)} 
n+l 
d(f) 
-r, , . IT (n — m) . „. „ cos , , \ 
P«* (/a) ^ ( v ■ (— 1) • m (i// =F ^) 
v 7 II (w) v ' sm v r ' 
(82). 
In these formulae n is unrestricted, m is a real integer, and u is any real positive 
quantity less than log mod ^ + , and the real part of /x is positive. 
fM l 
Formulae corresponding to (81), (82) have been given by Heine in the case in 
which n is a positive integer.* 
If in (81), (82) we put u = 0, x(j = 0, we have 
1 ( 2k , , /—2 - T J\n C0S X 7 X n ( W ) 
— (/x + v//a 2 — 1 cos 6y l . md) dd> = 
2 tt J o v 1 r 7 sm T T 
IT (n + m) 
COS , 
. mxb 
sin r 
2ir J 0 (/a + \/fi~ — 1 COS </>) 
n +1 
n (it) 
P/‘ (g) ■ • 
. (83). 
’P/AO • 
GO 
^4- 
Definite Integral Expressions for Q,/" (/x). 
39. When the real parts of m -J - b, n — m + 1 are both positive, the formula 
(54), for Q,“ (/x) reduces to 
Q/ (jx) = . 2 ? 
n (n + m) II(-£) (jf - l) h 
IT (n — m ) II (m — J) 
hm fl 
M+OT+1 
f (i n)"-® ( I-^ 
Jo v ’ \ # 
—n—m—l 
du 
V — 1 
on changing the independent variable to v, where u = --- , we then have 
Q„ m (/x) = e ffl7ri . 2“ . P ^ + - m ^ . —% . (/x 2 — 1 ) im . 2"~ ; " +1 
IT (7i — m) ’ II (m — 4) 
| (F 2 — 1 ) m ~' 12 + + 
v z 
1 \ 1 — i 
} 
cZy, 
* See ‘ Kugelfunctionen,’ vol. 1, p. 211. 
3 s 2 
