# To the Power of Root(-1) , -i-

I – 5    Potenzieren mit Wurzel -1 ( i )

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  is  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 author. 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 cycle 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 cycle with r = 0.0432 = 1/e^pi. The prominent values were valid too and the result for x = -23.1407 is 0.0432 .

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