\(\DeclareMathOperator{\res}{Res}\)

The Cauchy Residue Theorem is a remarkable tool for evaluating contour integrals. Essentially, it says that, instead of computing an integral along a curve \(\gamma\), you can replace it with a sum of “residues” at some special points \(a_k\):

But what is a residue? What are the \(a_k\)? What’s really going on here?