Symmetry Rule for Binomial Coefficients/Complex Numbers

Theorem
For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer:
 * $\dbinom z w = \dbinom z {z - w}$

where $\dbinom z w$ is a binomial coefficient.

Proof
From the definition of the binomial coefficient:


 * $\dbinom z w := \displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}$

where $\Gamma$ denotes the Gamma function.