At a pole of order at most $n$, the residue is the $(n-1)$st derivative of $(z-z_0)^nf$ divided by $(n-1)!$.Suggest a slogan
Let \(n\ge1\), and let \(f\) be holomorphic on a punctured disc about \(z_0\) with a pole of order at most \(n\) at \(z_0\). Then the residue of \(f\) at \(z_0\) is \[ \Res_{z=z_0} f = \lim_{z\to z_0} {1 \over (n-1)!} \frac{d^{n-1}}{dz^{n-1}}\qty{(z-z_0)^n f(z)}. \]
State the residue formula at a pole of order $n$.