For \(s\in\CC\) with \(\Re(s)>0\), the Gamma function satisfies \[ \Gamma(s+1) = s\,\Gamma(s). \] Consequently \(\Gamma(n+1)=n!\) for every integer \(n\ge0\).
For \(0<\varepsilon<1\), integration by parts on \([\varepsilon,1/\varepsilon]\) gives \[ \int_\varepsilon^{1/\varepsilon}\frac{d}{dt}\qty{e^{-t}t^s}\dt=-\int_\varepsilon^{1/\varepsilon}e^{-t}t^s\dt+s\int_\varepsilon^{1/\varepsilon}e^{-t}t^{s-1}\dt. \] The left side equals \(e^{-1/\varepsilon}\varepsilon^{-s}-e^{-\varepsilon}\varepsilon^s\), which tends to \(0\) as \(\varepsilon\to0\) because \(\abs{\varepsilon^s}=\varepsilon^{\Re s}\to0\) and \(e^{-1/\varepsilon}\) decays faster than any power of \(\varepsilon\). The right side tends to \(-\Gamma(s+1)+s\Gamma(s)\), so \(\Gamma(s+1)=s\Gamma(s)\). Since \(\Gamma(1)=\int_0^\infty e^{-t}\dt=1\), induction on \(n\) gives \(\Gamma(n+1)=n\Gamma(n)=n!\).