588 



known trutli that the (.'onvei'geiice of (25) is a coiisequence of these 

 two facts. The same thing would hold with regard to the series 



if 



Urn <f{n) -^ 71''+^, and Um A </(?<) -- ?i''+^-i. 



n= CO 



Therefore, in the statement of the above theorem snch a series 

 may be chosen as well. We fni-ther remark that A', which was 

 hitherto supposed to be greater than — 1, may also be less than the 

 latter number: the theorem of Nielsen, in the particular case demon- 

 strated in § 6, keeps its validity for those values of A', though we 

 should have to apply our reasonings to an integral of the form (8) 

 (in a footnote of ^ 1) in the latter case. 



By substituting t ■= i' e^'i' in the [)0wer-series for (f {(.) we obtain 

 the more general theorem : 



//' the coefjicients a-n of a pofner-series in the letter t are equi- 

 valent to )//' for n = 00, the function <p (t) represented by that series 

 has, on the circumference of its circle of convergence {being the circle 

 (0,1), singularities at all points inhere the series 



ant" 



diverges. We may add that this theorem already holds, if only the 

 upper limit of the coefticients a„ is, in the sense of equation (14'), 

 equivalent to n'' for 7i = oo. 



Finitely we observe that the reverse of the proposition does not 

 hold: if tfie series (25) converges, the point t=l need not be an 

 ordinary [)oint. To make this clear we need only think of the case 

 that the coefficients a„ ditfer from zero only foi' values of n lying 

 at a certain distance from each other; it may happen then that the 

 series (25) convei-ges absolutely, but the function </ {t) has its whole 

 circle of convergence as a singular line. 



8. As already remarked, we doubt of the general validity of 

 Nielsen's theorem, though we are not in a position to furnish a case 

 of the non-validity. It is our opinion that, if A' <:^ P. <^ a' -|- 1, there 

 will be cases in which the integral (1) cannot, for all values of 

 R (x) ^ )., be expanded into a series of factorials. On the other hand 

 we can prove that such an expansion is not possible for any value 

 of R {x) <^ /, which is a thing not immediately evident if X lies 

 between X' and A' -|- 1. ^) 



1) If R (.T) < A', the impossibility is at once evident, since the series-terms have 

 not zero for their limits then. 



