504 
DR, E. W. HOBSON ON A TYPE OP SPHERICAL HARMONICS 
liis formula he proves, for the case in which ra is a real integer, by a method of 
transformation which cannot be applied to obtain the more general result (91). 
42. In the formula 
Q» m (p) = 
TI (n + m) 
2^ H (n) 
(i — (/x — *)-»-*- 1 dt. . (ii) } 
which holds, provided the real part of n + 1 is positive ; let 
t = p — \/p 2 — 1. then 1 — £ 2 = \/p 3 — 1 . e“ {2p — 2\/p 2 — 1 cosh it], 
hence 
Q •- (p) 
6 ,,m TT (n + m) 
•log, \/e_ 1 
v m + i 
,»+l ' jr,, (p 2 1 ) 
11 O) ) (P 2 -1) 
Mo gev //i±i ' 
v m-i 
+ Ml + 
— 2 (»+m+l)it 
or 
IT / . \ i *‘°g« -\/t±i _ 
Q/' (p) = e” m . —^ ^ 1 {/-^ “ v /7 p 2 — 1 cosh w}" cosh mu du . (92). 
This formula holds for all values of n and m such that the real part of n + 1 is 
positive. 
The Evaluation of a certain Definite Integral. 
43. Suppose n and m are such that n — m is a real integer, and that they are 
otherwise unrestricted; in this case the integral 
IV, P 3 ~ ff C - !)“ (* - * 
taken round a closed path which includes the three singular points 1, — 1, p will 
satisfy the fundamental equation (2), since the integrand attains its original value 
after description of the closed path. We shall take the path to be a circle with centre 
at the point p; if we put £ = p + ^p 2 — 1. as in Art. 38, in this case we 
must have u > \ log mod 
P +1 
p - 1 
and the integral becomes 
f {p + Vf 2 ~ 1 cos (</> — 
J 0 
xjj ± iu)}"e~ m d<t>. 
