429 



3. There exists a real number ;/ with the property that, corre- 

 sponding to any assigned number a, however small, an integer N 

 may be chosen, such that we have, tmi/ormlij in a.u \utevvsi\0<t<d, 

 which, for the rest, we may think as small as we please 



< ^1 

 ^^—^^ ^— 1 for « > iV . . . . (5) 



s ' 



according as R{.ii) '^ X' or /^(,r) < A'. (R(.t') means the real part of .r). 

 The series of factorials corresponding to an integral of the form 

 (J) is 



a. a, 2/ a. nf a„ 



where a^, a^, . . . are the coefficients of the power-series (2). 



This series converges, according to Niei.sp:n, for the values of u: 

 satisfying the two conditions 



R{x) > ;., and R{.v) > ;.' (7) 



and if at least one of the two chai-acteristics X and X' is not negative, 

 the series (6) will represent the integral (1) for the values of x 

 mentioned ^). 



From the first of the inequalities (5), applied for t = 0, and from 

 the consideration that 



a,i = ~ — 7-, • • (9) 



n/ 



it may at once be derived that the series (6) converges absolutely 

 for R^A-) <^ X' -\- 1 . In connection with (7) Nielskn therefore infers 

 that the nuuiber X is at most equal to X' -\- \ . 



The thing to be remarked in the statement of Nielsen is however 



1) If A and // are both negative, so that f(t) is finite for t= I, the integral (1) 

 has in general only a meaning for Rix) > 0, and the development in question is 

 valuable for these values of x only. But tlien we may consider the integral 



1 



"''^'""^ J 4.r+l)...(.t' + p-l) 







which has a meaning for Rix) > A. To this integral a certain remainder of the 

 series (6), viz. 



H — ; — r~r^ 7 — ; ttt + • • • » • • \^) 



corresponds and the integral would then be equal to this remainder for the values 

 of X determined by (7). 



28 

 Proceedings Royal Acad. Amsterdam Vol. XXI. 



