Proof of Residue Theorem

Complex Analysis · Axiom Academy

EXAMPLE Proof of the Residue Theorem Work the derivation step by step — from contour deformation to the formula that only the residue survives. Suppose f is analytic inside and on a simple closed contour (traced counterclockwise) except at finitely many isolated singularities lying inside . Prove that Each step below asks for the next move; answer it to reveal that line of the proof. Each singularity z_k is ringed by its own small counterclockwise circle C_k . The shaded region between and the circles is where f is analytic. Proof complete. The Residue Theorem falls out of three ideas working together. Deformation: in the region where f is analytic the outer integral equals the sum of the small-circle integrals — Cauchy's theorem for a multiply connected region. Laurent series: near each singularity , and integrating term by term, every power vanishes except m = -1 , which gives . Only the residue counts: , so . Equivalently, split off the simple-pole part with g analytic: the first piece integrates to and the analytic piece g integrates to 0 . The coefficient a_ -1 is the one number that records how the function winds around the singularity — that is exactly the residue.

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