Does De Moivre's Theorem hold for all real n?

Question

I have seen the proof by induction for all integers, and I have also seen in a textbook that we can use Euler's formula to prove it true for all rational n, but nowhere in the book does it say its true for irrational n.

I have also looked over the internet and there seems to be some problem with non-integer values for n (as I understand, a problem of uniqueness, but I'm not sure).

I would appreciate it if someone could just clarify this for me.

Thanks in advance!

Answer 1

The formula is actually true in a more general setting: if $z$ and $w$ are complex numbers, then $\left(\cos z + i\sin z\right)^w$ is a multi-valued function while $\cos (wz) + i \sin (wz)$ is not. However, it still holds that $\cos (wz) + i \sin (wz)$ is one value of $\left(\cos z + i\sin z\right)^w$.