Contour Integral Definition

Complex Analysis · Axiom Academy

One definition turns an integral along a curve in the complex plane into an ordinary integral in t . 1. A Path in the Complex Plane A contour is just a curve you can walk along . Describe it with a parametrization for t running over [a,b] : as t increases, the point sweeps out the curve. Each tiny step you take is the vector — its direction is the tangent , and its length is how fast you're moving. The point on the curve at parameter t The infinitesimal step along the path To integrate f along , substitute the parametrization everywhere: replace z by and dz by . The complex integral becomes an ordinary integral in the real variable t — one you already know how to compute. Write the curve as , , and differentiate to get . Form , then multiply by the step . Integrate the product over [a,b] — a real-variable integral. A single complex number: the integral of f along that path. Worked example — along the upper unit semicircle Let for , the half-circle from z=1 to z=-1 . Then and , so the integrand is Integrating from 0 to gives a clean complex number: (Since z^2 has the antiderivative , this also equals — a useful check.) 3. How Big Can It Be? The ML Estimate Often you don't need the exact value — just a guarantee that the integral is small. If f never exceeds M in size anywhere on , and the path has length L , the integral can't beat the simple product .

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