Contour Integration Summary

Complex Analysis · Axiom Academy

SUMMARY Contour Integration Summary Section 6 — integrating complex functions along paths, and why analyticity makes the path stop mattering. A contour integral turns into an ordinary real integral the moment you parametrize the path: . For an analytic function with an antiderivative, the integral depends only on the endpoints — the path you take doesn't matter. The single integral around an enclosing circle is the seed of all of residue theory. Topology decides the answer: whether a singularity sits inside or outside a closed loop determines whether the integral is or 0 . The ML inequality bounds an integral by without ever computing it — the workhorse of every estimate to come. Core Concept Contour Integrals Integration moves off the real line and onto a path in the complex plane. Plug the parametrization in, and the contour integral collapses to a single-variable integral in t . The result is a complex number. When to use: any integral written — your first move is always to parametrize. Watch out for: don't forget the factor; it carries the path's direction and speed. Two parametrizations cover most contours: a line from z_1 to z_2 (so is constant), and a circle of radius R about z_0 (so ). Reparametrizing the same oriented curve never changes the integral. When to use: lines for segments, Re^ it for arcs and loops, for a full circle. Watch out for: reversing orientation flips the sign of the integral. Core Concept Path Independence

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