Residue at a pole
Theorem 12.2 (Residue at a pole). If \(f\) has a pole of order \(m\) at \(z_0\), then
\[ \Res (f; z_0) = \frac {1}{(m-1)!} \lim _{z \to z_0} \frac {\dd ^{m-1}}{\dd z^{m-1}} \bigl [ (z - z_0)^m f(z) \bigr ]. \]
For a simple pole, \(\Res (f; z_0) = \lim _{z \to z_0} (z - z_0) f(z)\).
Source: [1, ch. 6]
Proof. Near \(z_0\),
\[ (z - z_0)^m f(z) = a_{-m} + a_{-m+1}(z - z_0) + \dots + a_{-1}(z - z_0)^{m-1} + \dotsb , \]
a power series once its removable singularity at \(z_0\) is filled in. By Theorem 7.1, its \((m-1)\)st derivative at \(z_0\) is \((m-1)! \, a_{-1}\), and this derivative is continuous, so it equals the limit.