The Operator – ^ to the power of .. – is not only applicable to real numbers ( 2^3 =8) . You can use this operator even with complex numbers z=a + b*i . The formula from Moivre allows this operation in an easy way using polar coordinates even if x equals a complex number. – z^x = r^x (cos(x*phi + i*sin(x*phi) -. The math computer languages Octave(R) and matlab(R) are able to do these calculations.
The very simple function y = x^i has its definition and equals in y = e^(i*log(x). For the graphical representation we use negative x and positive x separately. Here we see unexpected results at least to the autor. First of all if we take x > 0 and look along the x-axis to the projection of the yz-plane all results y lay on a unit cicle with r = 1 . . at x = 23.1407 , log(23.1407) =pi we get the famous Euler-formula
e ^(i * pi ) = 1 ,
other prominent results were given with x = 111.32 , 335.4917 and 2567.8 (y = i ) etc. . When we consider negative values x < 0 with again the same projection, we see that all results lay on a cicle with r = 0.0432 = 1/e^pi. The prominent values were valid too and the results for x = -23.1407 is 0.0432 .