ON THE IMAGINARY POWER OF A NUMBER 327 



is a regular sequence assigning a finite positive number k as a. superior 

 limit. By repeated application of (20) 



^(g) ^ /x\ (A . . . (A. ^ ^2;/2") 



v2/^\2V ^\2°/ x/2^ 



The product of the ^'s in this expression can be written 



[i-2,.(|)]...[:-2,.(^.)]. 



This product is absolutely convergent and different from zero when n = <» 

 since the series 



2V + ^H2^^+" 



(having for its quotient of convergence J in virtue of the limit k) is abso- 

 lutely convergent. Hence when n = oo 



*M = ..(|).(J)... (20a) 



in which for x ^ the product of the <p's is a positive number greater than 

 and less than 1, since each ^ is less than 1. This product continually 

 increases as x converges to since each factor increases, hence it has a 

 superior limit equal to or less than 1, and the product is convergent for 

 a; = 0. But in any convergent infinite product the limit of the product 

 of all terms after the nth is unity when n = co . Whatever e be chosen 

 there can be always assigned an n such that 



^-'^{^^^{-^^■■- <^' 

 for any x for which the product is convergent. All the more so is 



and since for any assigned n 





