A Property of the Gamma Function at its Singularities

Abstract

The singularities of the $\Gamma$ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers $n$ and $k$, $$ \lim_{z\rightarrow 0} \frac{\Gamma(nz)}{\Gamma(z)} = \frac 1 n; \hspace{0.4in} \lim_{z\rightarrow -k} \frac{\Gamma(nz)}{\Gamma(z)} = \f{(-1)^{k}\ \Gamma(k)}{n^2\ \Gamma(nk)}.$$ The above relations add to the list of the known fundamental Gamma function identities.