Falling Factorial of Complex Number as Summation of Unsigned Stirling Numbers of First Kind

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

Then:


 * $z^{\underline r} = \ds \sum_{k \mathop = 0}^m{r \brack r - k} \paren {-1}^k z^{r - k} + \map \OO {z^{r - m - 1} }$

where:
 * $\ds {r \brack r - k}$ denotes the extension of the unsigned Stirling numbers of the first kind to the complex plane
 * $z^{\underline r}$ denotes $z$ to the $r$ falling
 * $\map \OO {z^{r - m - 1} }$ denotes big-$\OO$ notation.