In complex analysis, the residue is a complex number that reflects the behavior of a function near a singularity, specifically a pole. It plays a crucial role in residue theory, which helps evaluate contour integrals by connecting the integral of a function around a closed curve to the sum of its residues at singular points inside that curve. This concept is essential for series expansions, allowing for the simplification and evaluation of integrals and functions in mathematical physics.
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