At a simple pole, the residue is the surviving coefficient after multiplying by $z-z_0$ and taking the limit.Suggest a slogan
Let \(f\) be holomorphic on a punctured disc about \(z_0\) with a simple pole at \(z_0\). Then its residue at \(z_0\) is \[ \Res_{z=z_0}f = \lim_{z\to z_0} (z-z_0) f(z). \]
Near \(z_0\), \(f(z)=\frac{a_{-1}}{z-z_0}+G(z)\) with \(G\) holomorphic at \(z_0\), so \((z-z_0)f(z)=a_{-1}+(z-z_0)G(z)\to a_{-1}=\Res_{z=z_0}f\).