588 



where k is a^aiii a positive nuinhei- iiol depending- on n and /. 

 Hence, cori-esponding lo any fixed |){)sitive (piantily f chosen arhi- 

 traiilv small, thei-e is an inlegra! nnndtei- N, sncli that tl)e left-hand 

 mend)er of the laüer inecpialKj is less in valne than t, for every 

 value of t in the interval {vA), if only /i > xV. For these values of 



n we have therefore 



1 1 



1 ^- ^ ^-^ dt i <Ckt (1 -O'*^^' '/« 



\J rof-fn) n J' ' 



if /j*U') ]> — i. For any such value of /■, i.e. a fortiori for 

 /?(,?•) ^ >/, since ).' was supposed greatei- than — 1, the part of the 

 integral (11) taken over the interval (r,l) has zero for its limit 

 foi" n = X) . 



For the integration over the remaining interval (ü,i') we apply 

 the mode of treatnient of § 3 and the inetpiality (17). According to 

 the latter there is, corresponding to any lixed <) and f, chosen as 

 small as we please, an integei' A such that we have aniforiidy 

 ill the interval (0,1), and hence in (0,c), 



I r(/' + ff+n + l) 

 For the interval (0,r) it follows from this that, for n > N 



r(.^-fn) 



l<p,if7i>iV. 



i 

 this that. 



thus 



ƒ 









If now lii.r) ^ f-' , we can have chosen the numbers d" and ffi 

 so small that /?(.?■) is also greater than )' -\- d -f o^, and in this case 

 we infer from the latter inequality that the integral over the interval 

 (0,r), too, has zero as a limit for ii = x , if R{.v) ^ A'. Thus the 

 theorem of Niklskn has heen proved, in case t ^= is an ordinary 

 point of the function (f{t). 



If a function (p{t) has the point ^=1 for its only singular point 

 on the circumference of the circle of convergence (0,1), and if, 

 moreover, it satisties the conditions of Hadamakd, i.e. if it is con- 

 tinuous and "a écarl fini" on that circumference, or if a certain 

 derivative of negative oi-der — vj has this property, then we always have 



a> = ;. =r ;' -f 1, 

 and the theorem of Nielsen has ceased having anything particular. 



