Principal $n$th root of a complex number This is really two questions.


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*Is there a definition of the principal $n$th root of a complex number? I can't find it anywhere.

*Presumably, the usual definition is $[r\exp(i\theta)]^{1/n} = r^{1/n}\exp(i\theta/n)$ for $\theta \in [0,2\pi)$, but I have yet to see this anywhere. Is this because it has bad properties? For instance, according to this definition is it true that for all complex $z$ it holds that $(z^{1/a})^b = (z^b)^{1/a}$?

If $z=|z|e^{i\theta}\in\mathbb{C}^*$, then there's exactly $n$ numbers $w_k,k\in\{0,\ldots,n-1\}$ such that $w_k^n=z$:
$$w_k=\sqrt[n]{|z|}e^{i(2k\pi+\theta)/n},\quad k=0,\ldots,n-1.$$
The principle root is when $k=0$.
