The Residue Theorem

Complex Analysis · Axiom Academy

One equation turns a contour integral into a sum of residues — and turns hard real integrals into algebra. Let f be analytic on and inside a simple closed contour C (traversed counterclockwise), except at finitely many isolated singularities. The integral around C depends only on the singularities trapped inside — each contributes its residue: Sum over the singularities inside C only 2. Close the Contour to Crack a Real Integral The theorem's most famous payoff: evaluating a real improper integral that has no elementary antiderivative. Take . View the integrand as a complex function and close the path with a big upper-half-plane arc of radius R . The bottom of the contour runs along the real axis from -R to R — that piece becomes the integral we actually want. On |z|=R the integrand decays like 1/R^2 , so the arc's contribution is bounded by . Only z=i lies in the upper half-plane. Its residue is , since x^2+1=(z-i)(z+i) . The closed loop equals , and the arc vanished — so the real integral is . The whole trick is choosing a closure whose contribution vanishes. Here the half-circle's length grows like R but the integrand shrinks like 1/R^2 — the product goes to zero, so the loop integral collapses onto the real axis. (For integrands with an e^ ix factor, Jordan's lemma guarantees the same vanishing.) 3. Counting with Residues: The Argument Principle

This is the written version of the interactive lesson above. See the full Complex Analysis course.