Is there a power series representation for $\pi^{z}$, $z \in \mathbb{C}$?

Question

In the same way that we have a power series representation for $e^{z}$ as

$$e^{z}=\sum_{k=0}^{\infty}\frac{z^{k}}{k!}$$

does there exist a power series for $\pi^{z}$ as

$$\pi^{z}=\sum_{k=0}^{\infty}a_{k}z^{k}$$

Thanks.

Answer 1

We have $\pi^z=e^{(\log \pi) z}$. Substitute $(\log \pi)z$ for $w$ in the standard series for $e^w$.