De Moivre's theorem, which typically is stated as
[r(cos(theta)+isin(theta))]^n=r^n(cos(ntheta)+isin(ntheta))
does not apply to non integer values for n. This is because non integer powers of complex numbers are multiple-valued, whereas the trigonometric representation is single-valued.
For example, letting z=i and n=1/2, we have
i = cos(pi/2)+isin(pi/2) = cos((5pi)/2)+isin((5pi)/2)
Using Euler's formula e^(itheta)=cos(theta)+isin(theta), we can see that these representations, despite both being equal to i, produce different results when taken to the power of 1/2:
(e^(ipi/2))^(1/2) = e^(ipi/4) = cos(pi/4)+isin(pi/4) = sqrt(2)/2+sqrt(2)/2i
and
(e^(i(5pi)/2))^(1/2) = e^(i(5pi)/4)=cos((5pi)/4)+isin((5pi)/4)=-sqrt(2)/2-sqrt(2)/2i
Thus (cos(pi/2)+isin(pi/2))^(1/2) has two values, whereas (cos(1/2*pi/2)+isin(1/2*pi/2)) has only one.
As De Moivre's theorem requires the left hand side to be single valued to match with the right hand side, n is restricted to integers.
