The residue theorem

The residue theorem is a powerful tool in complex analysis used to evaluate integrals of complex functions around closed loops. It states that the integral of a function around a loop can be found by summing the residues (special values) of the function's singularities (points where it behaves badly) inside that loop, multiplied by 2πi. Essentially, it transforms a complex integral into a straightforward sum of these residues, making calculations easier, especially for functions with multiple singularities, and is fundamental in fields like engineering and physics for analyzing waves, signals, and other complex phenomena.