ON THE IMAGINARY POWER OF A NUMBER 323 



. , C{x) ,, , Six) 



VC\x)-\-S\x) V C-{x)->rS\x) 



wherein the absolute value of the radical is taken. Clearly 



^\x) + iKx) = 1. (12) 



Divide equations (8) and also (9) by the square roots of the corresponding 

 members of (11). Whence 



<p{x ± y) = 'p{x)ip{y) =F ^Pix) yp{y), (13) 



^{x ± y) = ^{x) <p(y) ± ^(x) xpiy). 



Subtract the second from the first in (13), and then add the last two. 



There result 



vix + y) - <p{x - y) = - 2 4>{x) i{y), (14) 



^p{x + y) - i'ix- y) = +2 4^{x) ^(y). 



Since ,S(0) = it follows that r/'(0) = 0, thence <p{0) = 1. The first of 

 (14), when x = 0, shows that ip{y) = ip{— y), the second that ^{y) = — 

 yp{— y). Hence ^ is an even and ^p an odd function of x. 



It has been demonstrated for real functions that any function F{x) 

 defined by the relation 



F{x + 2/) = F{x) . F{y) 



must be A^ where A is some positive constant. Equation (11) shows that 

 C-ix) + *S^(a;) is such a function. 



.-. C\x) + S\x) = A^ = e^^^ 



where 2c = loge A. Hence 



eix = e'^^lvix) + i^'ix)]. (15) 



6. The relations (12) and (13) together with the continuity of <p and i^ 

 will serve to completely construct the numbers we seek. In (13) y = x 

 gives 



<p{2x) = <p\x) - i^^x), i/'(2.t) = 2<p{x) xp(x). (16) 



These show that if <p and \p are continuous in any assigned interval {— h, 

 + h) then they must be continuous in the interval ( — 2°/i, + 2°/j) for any 

 assigned integer n. Therefore it was only necessary to assume continuity 

 in the neighborhood of zero. 



