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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.

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1 Answer

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

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Imagine that you didn't know $\pi$ in advance, you wouldn't be able to use $\log(\pi)$. That's what I had in mind. –  Neves Jan 28 '13 at 9:14
    
I have no idea what it means, to imagine I didn't know $\pi$ in advance. But you can use the same idea to find a power series for $b^z$ where $b$ is an unknown-in-advance constant. –  Gerry Myerson Jan 28 '13 at 11:56
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