The Gamma function, defined by \(\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}\dt\) for \(\Re s>0\), extends to a meromorphic function on \(\CC\) whose only singularities are simple poles at \(s=0,-1,-2,\ldots\).
For \(\Re s>0\), integration by parts gives \(\Gamma(s+1)=s\Gamma(s)\), and \(\Gamma\) is holomorphic there by PR-NLV6Q. For \(n\ge1\) define, on \(\ts{s\st\Re s>-n}\), \[ \Gamma_n(s)\coloneqq\frac{\Gamma(s+n)}{s(s+1)\cdots(s+n-1)}. \] Since \(\Re(s+n)>0\), the numerator is holomorphic on this half-plane, so \(\Gamma_n\) is meromorphic there with at most simple poles at \(0,-1,\ldots,-(n-1)\). Iterating \(\Gamma(s+1)=s\Gamma(s)\) shows \(\Gamma_n=\Gamma\) on \(\Re s>0\), so by the identity theorem \(\Gamma_n\) and \(\Gamma_{n+1}\) agree on \(\ts{\Re s>-n}\) minus the poles. The functions \(\Gamma_n\) therefore define one meromorphic function on \(\CC\). At \(s=-k\) with \(0\le k\le n-1\), the numerator is \(\Gamma(n-k)=(n-k-1)!\neq0\) and exactly one factor of the denominator vanishes, simply, so \(-k\) is a simple pole.