Residues When to Use Use this skill when working on residues problems in complex analysis. Decision Tree 1. Computing Residues - Simple pole at z0: Res(f, z0) = lim {z- z0} (z - z0)f(z) - Pole of order n: Res(f, z0) = (1/(n-1)!) lim d^{n-1}/dz^{n-1}[(z-z0)^n f(z)] - L'Hopital shortcut for f = g/h with simple pole: Res(f, z0) = g(z0)/h'(z0) 2. Identify Pole Order - Simple pole: (z - z0)f(z) has finite limit - Order n: (z - z0)^n f(z) has finite limit, but (z - z0)^{n-1} f(z) doesn't - 3. Essential Singularities - Neither pole nor removable (e.g., e^{1/z} at z=0) - Compute residue via Laurent s…