A Property of the Gamma Function at its Singularities

The singularities of the $\mathrm{\Gamma }$ 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 $k$, $$\underset{z\to 0}{\lim }\frac{\mathrm{\Gamma }(nz)}{\mathrm{\Gamma }(z)}=\frac{1}{n};\underset{z\to -k}{\lim }\frac{\mathrm{\Gamma }(nz)}{\mathrm{\Gamma }(z)}=\text{f}(-1{)}^k\mathrm{\Gamma }(k)n^2\mathrm{\Gamma }(nk)\mathrm{.}$$ The above relations add to the list of the known fundamental Gamma function identities.