431 



F'or this reason Nielsen lias to add the new condition under 3" 

 expressing that tiie inequality (5) is also valid for an interval 

 0<.t<(f for values of ./■ lying in the half-plane on the right of 

 R{.v) = X' . Then it follows from the reasoning just mentioned, in 

 connection with the condition 3° that the remainder given l)y (11) 

 certainly approaches to zero for ?i = oo, if we have at the same time 

 A^(.r)>/' + l, and /?(.«)> ^. + 1 .... (12) 

 For we may write 



/t„ = / (.r) I at. 







and since /^(,r — 1) is both greater than I' and greater than P., the 

 function under the sign of integratioji unifonnlii approaciies to zero 

 in the interval ()</<], as n becomes indetinitely large. But noliiing 

 more can, in oui- opinion, be derived from the reasoning of Nielsen. 

 It is remarkable that in his book the author gives the equality (10) in 

 an erroneous form, in such a manner that undei- the sign of integration 

 a derivative of ff{t) occurs of an order too higii by one unity ; and 

 it might therefore be supposed that this circumstance has led Nielsen 

 to his erroneous conclusion. But in a memoir published by him in 

 Annal. de I'Ecole Normale (1902) the formula (10) occurs ina right 

 manner, and still Nielsen draws his conclusion without any further 

 explanation. It would have been of interest, if the author had 

 explained a little more amply how he arriv^ed at it. It seems not 

 impossible that the example given by him, and perhaps others, have 

 led him to the erroneous opinion that his theorem should have 

 been proved by the reasoning he has given. 



3. Nevertheless this theorem might be right and then, of course, 

 there is every reason to introduce the characteristic number P. together 

 with y . But still it seems not unuseful to show that we may develop 

 the integral (1) in an absolutely converging series of factorials under 

 the only conditions 1° and 3°, and then for /?(.i') > ;.' + 1 . We 

 shall even resti'ict the latter condition a little: not assume that the 

 inequalit}^ occurring in it is satistied for a certain small interval 

 (0,<f) of t, but only for the endpoint / := 0. Since the values of the 

 different derivatives of </(0 in ^ = depend on the coefficients of 

 the power-series of that function by means of tjie formula 



"" = — 1- • ' (13) 



n! 



this supposition amounts to the following as to those coefficients 

 themselves : 



28* 



