510 
DR. E. W. HOBSON ON A TYPE OE SPHERICAL HARMONICS 
Generalization o/Dieichlet’s and Mehlee’s Expressions for P„ (cos 6) as a 
Definite Integral. 
46. It has been shown in Art. 33, that provided the real part of m + ^ is positive, 
1 p— L7T (?/l— j) f 2 
P.- (/*) = 5; ■ n(-t)no„-i) ( E “ ]! 7t ”'“ (i - 2 A + ft*)"'* dh. 
let /x = cos 6 0 . i, the line joining the points 2 , — on tlie A-plane is perpendicular 
to the real axis, and the path of integration may be taken to be a circular arc with 
centre at the origin; let A = e^, then remembering that the phase of 1 — 2/xA -j- Jr 
increases from 27r — 6, at the lower limit to 27 t -fi- 9, at the upper one, we have 
(1 — 2/jlJi -{- A 2 )™ - - = e 2,r(m_i)t (2 cos <f> — 2 cos 9) m ~-, 
hence 
P„ m (cos 6) = e imm P„ m (cos 6 + 0. 1 ) 
g — hum _ 
p — L~ (wi—J) 
2 “ n (- 1 ) n (m-j) 
g-Jmm TO p ^2*- (m-i)i 
I e (B - wW . e (m ~ iH (2 cos <£ — 2 cos . ie‘* cA^>, 
J -e 
or 
P ;; w (cos 0) = 
2 “ n ( — J) n (m — 4 ) 
sm' 
gf „ “AAA #■ (102). 
Jo (2 cos $ — 2 cos 0p -Mi r v 1 
From Art. 11 we find 
P ;i “ (cos 6) = jj ^ + | cos W17r • ( cos ~ sin rrnr . Q,” 1 (cos A) [■> 
IT (n — m) f 2 . „ , 
—-- 1 cos nin . V, n (cos 6) --sm mv . Q,/“ (cos 6) 
n (n + m) [ v ’ 7T v v ' 
* rin-g.r . 9 co i ( "t a V .# 
J o (2 cos 0 — 2 cos Ay m r 
2 m IT (— 4) II (?» — 
(103). 
hence 
