I have the complex function $f(z) = \sqrt[n]{r^n - z^n}$, where $z$ is a complex variable, $r$ is a positive real number, and $n$ is any non-zero real number. For any value of $n$ besides 1, this function does not fill the entire complex plane, instead occupying only a portion of it. I know that analytic continuation can be used to extend analytic functions to the full complex plane, but I can't figure out how to
- Determine if $f(z) = \sqrt[n]{r^n - z^n}$ is analytic, and, if it is,
- Carry out the analytic continuation of $f(z)$.
The highest level of math class I have taken is Calc 1, so while I know the basics of complex numbers, I don't have any knowledge of Complex Analysis besides what Google, Wikipedia, and WolframAlpha can tell me.
Can someone explain how to go about figuring this out?
UPDATE: I figured out that the function $f(z) = \sqrt[n]{r^n - z^n}$ can be represented by the Taylor Series
$$ f(z) = r × \sum_{k = 0}^∞ \left({\left(\frac{z}{r}\right)^{k × n} × \frac{Γ\left(k - \frac{1} {n}\right)} {Γ(k + 1) × Γ\left(\frac{-1} {n}\right)}}\right) $$
which I think means that it is indeed analytic. However, I'm still confused as to how to carry out analytic continuation on the function.