Power of Complex Number as Summation of Stirling Numbers of Second Kind

Theorem
Let $z \in \C$ be a complex number whose real part is positive.

Then:


 * $z^r = \displaystyle \sum_{k \mathop \in \Z} \left\{ {r \atop r - k}\right\} z^{\underline {r - k} }$

where:
 * $\displaystyle \left\{ {r \atop r - k}\right\}$ denotes the extension of the Stirling numbers of the second kind to the complex plane
 * $z^{\underline {r - k} }$ denotes $z$ to the $r - k$ falling.