gQ Art. 4. —M. Knniyeda : 



Lemma 9. // 



(62) lmiJ(rO = 0, 



then «n = -o_y 1,^ ^rr ^z, 



^>^l ./ (C) ^ 



provided thd thh udegnd is convergent, the contour (C) of integration 

 being the circle of convergence. 



Proof. Siiice the function f{z) is regular at ever}- point on 

 tlie circle of convergence except only at the point A, the integral 



J BCD ^"+1 



dz. 



where the path of integration is the arc BCD, is convergent and 

 so also is the integral /(n) for every rj such that < ri < 1. Hence, 

 by means of Cauchy's Theorem, we have 



''" ~ 2-i Jbcdpb ^"+'^ ^' 



Now let vi tend to zero. Then the arc BCD tends to the whole 

 circle of convergence and we have 



a„ = 



since, by hypothesis, lini 7(/-i) = and the last mtegral is 

 convergent. 



45. Ijemm.a 10. f-<'t ^ be d small positive constant and 



(63) lin) = -j- ( /■ ' + /''^ ) fie'') e-""' dd (0 < c < ;r). 



Then, if tiie conditions of Lemma 9 are satisfied, the behaviour of the 

 coefficient a„ of the poiver scries (60), as n-^:D , is asymptotically 

 determined as follows : 



