Articles tagged with "analysis"

Cauchy Residue Theorem


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 \(a_k\):

$$ \oint_\gamma f(z)~dz = 2 \pi i \sum_k \res(f, a_k) $$

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