loo Art. 4— M. Kuniyeda 



where a > 0, q ami a denote certain condanti^, fi/ul p, r <irt)Ur<iry 

 real co7id(int>i, then the beh(wioiir of the coefficient a„, as 71-^00, is 

 deteinnined asi/mptotic<dly as follotrs. 



(i) If g = \, a = TT, or the si)ig%d'irity is of the type 



then 



a„ <-^ 2-^T-*a-2P+Te-è",i*p-f (log n^ sin {la^u^ — (iiJ — 1)^] • 



(ii) //■ <: g <: 1, a = (l + q)^^ or the singularity is of the type 



then 



a„ r^ -T l-y ( 1 + g)"^""*' (ga)' '■+^n^^~{\og n}' exp F {Ä; n ^'^- (i-^; - |)7r] i7\ , 



where Ä: = (1 +g)g"^«.^-^^. 



(iii) If ("> <q <^, « = ("^ — g)-o , or the singularity is of the type 



a. ^:^^) (1 +(?)''""^^ M'^+' H^W-(log ,,f exp [- [/; ,,ï^v_^i^_^);,]^J , 

 k being the same as in the case Çii). 



I\'^ Case in vjhich the Slngvlarity is of the Type 



58- More generally we obtain the following theoi'em, the 

 argument ])eing quite similar as in the i)rGceding case. 



CC 



Theorem X. let ^ «„ ,-/" he a ponder series, whose i^adius of 



n = 



convergence is unity, representing a function f{z) which has, on the 



