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.
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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 $(z^{1/a})^b = (z^b)^{1/a}$?
Best Answer
There really is not a coherent notion of "principal" nth root of a complex number, because of the inherent and inescapable ambiguities.
For example, we could declare that the principal nth root of a positive real is the positive real root (this part is fine), but then the hitch comes in extending this definition to include all or nearly all complex numbers. For example, we could try to require continuity, but if we go around 0 clockwise, versus counter-clockwise, we'd obtain two different nth roots for number we've "analytically continued" to. A/the traditional "solution" (which is not a real solution) is to "slit" the complex plane along the negative real axis to "prevent" such comparisons. And some random choice about whether the negative real axis is lumped together with one side or the other.
But even avoiding that ambiguity leaves us with a root-taking function that substantially fails to be a group homomorphism, that is, fails to have the property that the nth root of a product is the product of the nth roots.
The expression in terms of radius and argument "solves" the problem by not really giving a well-defined function on complex numbers, but only well-defined on an infinite-fold (ramified) covering of the complex plane... basically giving oneself a logarithm from which to make nth roots. But logarithms cannot be defined as single-valued functions on the complex plane, either, for similar reasons. Partially defined in artificial ways, yes, but then losing the fundamental property that log of a product is sum of the logs.