olt> DR. E W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
P„~ m (cosh \p) 
= 2 «n (- i) n (m - i) ^ ( o 2 cosil ( w + i) <M 2 cosb ^ — 2 cosh <£)“"* d(j> . ( 112 ) 
where the real part of m -f- \ is positive, and in particular 
2 
P* (cosh xjj) = 7r J o 
cosh (% + 
cosh i|r — 2 cosh (£ 
dtp 
• (US). 
The path joining the points 2 , 1 /z can be placed as in the figure, along the real axis 
from 1/z to — 00 , except for a small semicircle round the point 0, then a semicircle 
of infinite radius, and lastly a straight path along the real axis from + oc to 2 . If 
the real part of m lies between \ and — ^, and if the real part of n — m + 1 is 
positive, and if n + m is negative, the straight portions of the path are the only ones 
which contribute anything to the value of the integral; in this way we find that 
under the conditions just specified. 
P„ “(costn//) — 
sinh m \}r 
9 m 
■2tsinh (A+-g<£ — lviv) (2 cosh cp — 2cosh 1//)"' * d(f> 
e («+i) <*> + (*■+£> ™ (2 cosh (f> + 2 cosh 
In a similar manner, we can prove, by taking the semicircles below the real axis 
that under the same conditions, P„ -m (cosh xp) is given by the formula 
Pr^cosh xjj) = 2 m U{-i!)U^n-h) {J, 2t siuh ( n + s </> + m7T ) ( 2 cosh </> ~ 2 cosh ^ 
_p | e (»+f)^“(«+i)«r ^2 cosh cf) + 2 cosh dtp j. 
