INFINITE SKKI&S OF ANALYTIC FUNCTIONS. i- -. 



niul the getienil solution of LAMP'S equation for large values of n is approximately 



]. ..... (7), 



where C, and C, are arbitrary constants. 

 In the notation of BYBRLY* 



and 



F/(snz) = 



approximately, for largo values of n, where 

 Ilenco the series 



|.F/(snO 



corresponding to LAMB'S equation converges if 



| dna; + kcnx \ < | dnt + kcnt \ . 

 Write 



The condition reduces to 



is, in the notation of HEINE,! satisfied by the functions 



E.M and F.(*). 

 When n is large, 



and 



F n (<f>) = C, N/SCC^. e - ( "* )l11 *, approximately. 



Hence the series 



corresponding to the equation of the elliptic cylinder converges if 



* 'FouniKR's Series anil Spherical Harmonics,' pi 255. 

 t ' Kugelfimctionon,' 2nd c-lition, p. 404. 



4. The equation of the elliptic cylinder 



It . (9) 



