ON THE IMAGINARY POWER OF A NUMBER 329 



In the same way from the expression for \l/{mx) there results 



^(x)=kx-^^^f + ^---- ■ (21) 



These series are one-valued absolutely convergent series for all assigned 

 real values of k and x. They define the functions (p and \p and construct 

 them as was required whenever the constant k is determined. 



10. The construction of e'" is therefore 



which can be written 



l + ikx+^+- 



l+ic + ik)x+^-^^^x'+---, (22) 



a series absolutely convergent for all real values of x, c and k. To deter- 

 mine the constants c and k we observe the defining law of e"'' requires 



and in particular (e''')' = e~'^ is a real function of the real variable x and 

 is the real exponential function when e = 2.71 . . . Furthermore (e'=')' 

 is nothing more or less than the value of e''' at the value ix of x, for this is 

 e'"'' or e~^. The number e^ is therefore constructed for all values of f 

 in the real and in the imaginary continuum. As x varies continuously 

 from X in the real continuum through zero to ix in the imaginary continuum 

 the function (22) varies continuously from its value in (22) to the value 



l + {ic-k}x+^^^^x^+--- 



which is identical with 



^-^•+2-.- 



for all values of x. Consequently 



(tc - k)- = (- 1)- 



