The residue at a pole is a complex number that describes the behavior of a complex function near its singularities, specifically poles. It is defined as the coefficient of the $(z-a)^{-1}$ term in the Laurent series expansion of the function around the pole 'a'. Residues play a crucial role in evaluating complex integrals through the residue theorem, which relates the sum of residues to the integral of a function around a closed contour.
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