Cauchy's residue theorem

Cauchy's residue theorem is a powerful idea in complex analysis that simplifies calculating certain integrals. It states that for a function with isolated singularities (points where it behaves badly) inside a closed loop, the integral of the function around that loop depends only on those singularities and their residues—specific values capturing the function's behavior near each singularity. Instead of complex calculations, the integral can be found by summing these residues multiplied by \( 2\pi i \). This theorem allows for efficient evaluation of integrals in complex analysis by focusing on local information around singularities.