ON THE IMAGINARY POWER OF A NUMBER 325 



8. In (13) put |.i; for x and y, 



= 2<pKhx) - 1, (18) 



= 1-2 ^2(|x). 

 Hx) = 2^(|.r) ^Klx). (19) 



^(Itt) = gives ^(iTr) = >A(i7r) = 1/ l/T 

 also shows 



(p(|7r — a;) = i/'(a;), i^dx — a;) = (p{x). 



X = ^TT, y = Itt in (13) gives ^(ir) = — 1, ■/'(tt) = 0. It can be shown 

 in like manner for n any integer 



^(n,^) = (-l)^ ^(^^'^^^=0, 



The same equations show by an easy induction 



ipix + 2n7r) = <^(a;), \l/{x + 2nx) = iA(a:), 



consequently (p and i/' are periodic functions having the common period 

 2ir. 



9. Theorem. When x converges to zero in any manner the quotient 

 4'{x)/x converges to a determinate superior limit k. 



The number \l/{x) increases continuously from to 1 and (p(x) decreases 

 continuously from 1 to as x increases from to |x. In particular \p{x) 

 increases continuously from to | l/ 2 and (p{x) decreases from 1 to this 

 same value as x increases from to Iw. Let ?i be any arbitrarily great 

 assigned integer and h = lir/n. Then in 



xPix + h) - xPix) = 2<p{x + i/i) ./.(i/i), 



as X varies from to Itt the left side is positive and continually diminishes. 

 Let Xo = and x^ = Itt, then 



[xkixr,) - vKa;„_i)] + . . . + [xPix,) - X^iXo)] = il/^ 



Dividing by h there results 



^ (xn) - 4' (a^n-i) I . . . _^ 'P (xi) - xp (xo) ^ 2V2 -• 

 h h T 



