592 



if f = 1 is not a singular point of 7 (0, but if it is, the proposition 

 is not a matter of course. 



Since, if the coefficients of the power-series in question are complex, 

 the two series formed separately by means of the real and of the 

 imaginary parts of those coefficients must both corjverge, we may 

 without loss of generality suppose the coefficients to be real 

 quantities. We then consider, together with the function 



the function 

 where 



(33) 





Since .?„, as n becomes indefinitely large, approaches to a definite 

 limit s, the series (33) t)ehaves, so far as regards its terms for large 

 values of a, as the power-series of the function 



s 



and according to the reasoning of Cksako we have not only 



1 



lim f {I) : 



but also 



lim ƒ(" 



t=\ L 



\t) 



(34) 



(1 -tY+^ 



KiH'ther, from ?i-fold differentiation of the identity 



rfit =(l-«)/u) 

 we obtain the new one 



(1 — 0"</^'"'(t) _ (1— 0""t-\/''"^(0 (1 - 0"/^""^^(0 



^7 "" ^ (n-l).' * 



The limit of the right-hand side of the latter equation for t=\, 

 is, by (34), equal to zero for all positive integral ?i-values, and the 

 required formula (32) has thus been proved. 



By substituting t = t' e^f we obtain : If the expansion in a power- 

 series of a function 7 {t) convertfes at the point t = e^'r of its circle 

 of convergence (0,1), then, for all positive integral vahies of n and 

 for real values of t' , we have 



lim{\ -t'Y ffi"\t'e''f) = 



