Theorem of residues

The Residue Theorem is a powerful tool in complex analysis that helps evaluate integrals of functions around closed curves. It states that the integral of a complex function around a loop equals \( 2\pi i \) times the sum of the residues (specific values related to the function's behavior) at the function's singularities inside that loop. Essentially, it links the behavior of a function near certain points to the overall integral, allowing us to compute complex integrals efficiently by analyzing these specific points rather than the entire path.