683 



l-t 



l- 



< 



a — kn 



so that the left-hand member of (21) for these values of aipproaches 

 uniformly to zero for n =z qc (the factor //. r(n): r{<f f n) is only 

 equivalent to ii^-^ and does therefore not aifect this statement). 

 The integral 



1 



r nF (n) / 1 - < Y' ( 1 - «)°-i 



consequently has zero for its limit if ^iincreases indelinitely, however 

 small the value of ö may be fixed. 



For the interval (0, r) we have, independently of t and n, 



nr{n) f \ — tY (i—ty^-^ 



r(d + n) \ei^—t 



old 



— t 



< 



hi r{n) 



r(rf+n) 



where k is again a positive number not depending on // and t. ^) 

 Thus, considering (23), it follows 



I r nr{7i) r 



1 ~. --^- dt\ <yfcn'-« X V 



We therefore need only choose ^f, less than (f, to see that also 

 the integral over the interval (0,v) is zero for n = cc. Thus the 

 whole remainder (11) is zero for 71 = 00, if only /? (.j;) > 0, i.e., 

 since X = — cc and a' = 0, if R (./■) > ;.' and R {x) > X. For these 

 values of .v the integral 



1 



ƒ 



(1-0 



a — 1 



e^^—t 



dt 



can therefore be expanded into a series of factorials; and the theorem 

 of NiELSfEN holds in this case. 

 Again we take the example 



<p{t) = 



1 



1 



0<^<2jt 



eiB-t {\-ty VO<fi< 1 

 Here X' = 0, on account of the first term, and A = n, on account 



^) We shall always, in future, denote by k a finite pusitive number, without 

 always meaning the same number by this letter. This will not cause any am- 

 biguity, because the exact value of k is of no importance in our reasonings. For 

 the sake of clearness, however, we shall often mention the quantities on which 

 k does not depend. 



