Cauchy residues theorem

Cauchy's Residue Theorem is a fundamental concept in complex analysis that helps evaluate complex integrals. It states that if you have a function with isolated singularities (points where it behaves badly) inside a closed curve, the integral of the function around that curve equals 2πi times the sum of the residues at those singularities. Residues are values capturing the function’s behavior near these points. Essentially, the theorem simplifies complex integrals by focusing on these key singularities, transforming difficult calculations into manageable algebraic sums.