Cauchy Residue Theorem

The Cauchy Residue Theorem is a fundamental result in complex analysis that helps evaluate certain types of integrals. It states that if you integrate a complex function around a closed loop, the integral's value depends only on the singularities (points where the function behaves badly) inside that loop. Specifically, the integral equals 2πi times the sum of the residues (special numbers capturing the behavior) at those singularities. This theorem simplifies complex integral calculations by focusing on these key singular points, turning a potentially difficult integral into straightforward algebraic computations.