520 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
Q„ ?n (cosh xjj) sin (n + m) 7 t 
cos Tfiir r r 
= e mm . 2 m .II(m— i-)n( — i) —-—- sinh m xjj -I sin (n — m)n — 
TT I 
N 
— sin (n + |) 77 
J 0 
(2 cosh u—2 cosh -\|r) m+4 
du 
-(■A. + i)v. 
(2 cosh yjr — 2 cosh u) m+ 
du > ; 
where the real part of m is less than and the real parts of n -f- m + 1, wi — n 
are positive. 
Put m — 0, we have then 
Q„ (cosh xjj) = 
g(n+})« 
(2 cosh u — 2 cosh xjrf 
du — cot mr 
» —(»+i)« 
(2 cosh \Jr — 2 cosh uf 
du, (119) 
where the real part of n lies between 0 and — 1. 
Expressions for Q/' (cosh xjj), when n — \ is a reed integer. 
53. When n — \ is a real integer, the formula 
Qp fa) = t e < m - n)nL . 2 ” 
may be replaced by 
47r sin (n + m) t r 
0*"f 
0 1'-, 0, 1/e—, 0 —) 
h n+n 
(1 — 2 fill + h 2 ) m+i 
dh 
„ n (m - 4) n ( - 4) 
p2mm nm - v _2_/_\_2/ 
2ir 
( 1./2 + , 0 +) 
h n+n 
(1 -2 fih + Id) 
dh. 
i+i 
the path may, as in the figure, consist of a circle of infinite radius, straight paths 
along the real axis from cc to z, and a small circle round the point 2 . 
J. / A.v z 
*1 
oo 
oo 
If the real parts of m—n, \ — m are positive, the only effective parts of the integral 
are those along the real axis. The phase of 1 — 2 gh + Id, at A is 7r ; we thus find, 
Q" (cosh xp) 
n(m-|)II(- i) . 
— e 2mni . 2 m . ---—- sinh" 
27r 
du 
2 cosh u — 2 cosh \jr) m+ * 
g(n—m)2jri g(n+J)u 
_ g(»l + «) 7T1 f) 
l sin (m — n) n . 
n(m-i)n(-i) . 
'-IT 
sinh m xjj ( — 
J \!/ 
(2 cosh u—2 cosh \p) m+ i 
e (n + i)u 
(2 cosh u — 2 cosh \fr) m+ i 
du >• ’ 
du, 
