Is there any series for $x^y$?

Question

I wonder if there is any kind of generic series expansion for:

$$x^y$$

Where $x,y \in \mathbb{C}$. And the serie uses integer powers and other functions. Something like:

$$x^y = \sum\limits_{k=0}^{\infty}{y^k\cdot f(k,x)}$$

where $f(k,x)$ is function of complex value.

Answer 1

Specifying a branch for the complex logarithm, $\log(x)$, and using $x^y=e^{\log(x)\,y}$, we have

$$x^y=\sum_{n=0}^\infty \frac{(\log(x))^n\,y^n}{n!}$$

And we are.done.