591 



I If' ('0 "''(<- ") ""da <:^~ ((<— '0 ' dn K{\—t)-^, 



SO that this integral, too, is for t^l at most of order (1-0 "• 

 The same holds therefore for (he fund ion 7 [t), and since a may 

 be supposed arbitrarily little greater than A' -|- 8^, we have certainly 

 for every d ^ 



Urn (1 - <)>'-Hi+'' <ƒ(«) = 



and thus, as we proposed to show 



Secondly, let X -\- 0^*[\e between the integers /> — 1 and p\ we 

 may choose rt so small that the same holds foi' X' -\- O^ -\- (\ =z n. 

 We write 



a = p—l + ^/ (30) 



so that 



<r/ < 1 (31) 



In this case we have the following reduction 



' (« +/>) V' ¥^77rr—^ — 



^ r{a + m) 



= />'-! 



Owing to (31) we may, as in the former case, using here the 

 inequality (29) for n = in, prove that the expression 



is at most equivalent to (1 — ~ *""*"'"" ^' ö,nd thus ff> it), as in is 

 no greater than ^j, is of an order no higher than that of (1 — "'"' + /' ~*', 

 that is, according to (30), of the order (1 — t)'~"-. Thus the required 

 result is obtained completely. 



9. We now give a proof of the proposition used in the preceding- 

 paragraph. It may be stated as follows: 



/ƒ the expansion in a poiuer-series of a function (p (t) convercjes 

 at the point t=^i of the circle of convergence (0,1), ire have for all 

 poaitive integral value.^ of n 



Lim {\ -t)" <f " ' (t) = (32) 



1=1 



This proposition, of course, ceases to have a particular meaning, 



') Properly speaking it should be D/'D=<' !■ but this comes to the same thing 

 as D^'-^ Di'. l)ecniisp ;f - 0' i)-limos the subjert of operation is 7,!M0 for ^ = 0. 



