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INFINITE SERIES OF ANALYTIC FUNCTIONS. 497 



Then <f>(z) can be expanded* in the infinite series of LEOENDRK'S associated 

 functions 



(+...} .... (38), 



\ ~r */ 



where 



The series is convergent if z is inside C, divergent if z is outside C, and, in general, 

 oscillatory if z is on C, in which case the expansion fails. 



* The position of this result in the historical development of the subject is noteworthy. If m is a real 

 positive integer as well as n, P n m () vanishes so long as n<m. 

 With the further restriction on <(*) that (1 + ) l ~*~^() vanishes when t- - 1, we find that 



whero 



This result is given by HEINE (' Kugelfunctionen,' 2nd edition, p. 252). 

 In his notation 



/*) denotes 



If in the last equation m~0, we have the well-known expansion, valid within the ellipse C, in terms of 

 the simple Legendre functions, 



*) + ... +a n l' n (*)+.... 



where 



rt,, = |(2n + l)f 1 P,(0 



(WaiTTAKER, ' Modern Analysis,' p. 230). 



VOL. (XIV. A. 



