Complex Analysis

3 The gamma function

The gamma function \(\Gamma (s)\) is a analytic function that extends the factorial to a meromorphic function. It begins with integration by parts: \(\int _0^\infty t^se^{-t}dt= t^s(-e^{-t})|^\infty _0-\int _0^\infty st^{s-1}-e^{-t} = s\int _0^\infty t^{s-1}e^{-t}\). If we define \(\Gamma (s) = \int _0^\infty t^{s-1}e^{-t}dt\), then it will satisfy \(\Gamma (s) = (s-1)\Gamma (s-1)\). Moreover \(\Gamma (1) = 1\), and so by induction \(\Gamma (n) = (n-1)!\). Note that we can split the integral into the ones on \([0,1]\) and \([1,\infty ]\), and that each of these integrals converges uniformly when \(\real s\) is bounded above and below, so \(\Gamma (s)\) is a holomorphic function on \(\real (s)>0\). Moreover the equation \(\Gamma (s) = (s-1)\Gamma (s-1)\) shows that it extends to a meromorphic function with simple poles at the nonnegative integers.