Properties of Contour Integrals

Complex Analysis · Axiom Academy

LESSON Properties of Contour Integrals Linearity, additivity over paths, and what reversing the direction does — the algebra that makes contour integrals tractable. 1. Linearity: Constants Pull Out, Sums Split Underneath, is an ordinary integral of — and ordinary integration is linear . So the integral of a scaled sum is the scaled sum of the integrals: each function f and g contributes its own total, and the constants simply rescale those totals. α scales the contribution of f, β scales the contribution of g Take C : the straight segment from 0 to 1+i , with f(z)=z and g(z)=1 . Then and , so — exactly what you get integrating 2z+3 in one shot. 2. Additivity: Joined Paths, Added Integrals If a contour C runs from A to B by first traversing C_1 and then C_2 (joined end-to-end), the underlying integral splits at the junction. Walk the first leg, walk the second leg, and the totals simply add. This is what makes piecewise-smooth contours computable: integrate each smooth piece, then sum. From A to the junction. Parametrize it on its own and integrate; that gives . From the junction to B . Parametrize separately; that gives . C_2 has to start where C_1 ends — the path is continuous, no jumps. The rule chains: splits into n separate integrals you add up. Go from 0 to 1+i in two legs: C_1 straight along the real axis , then C_2 straight up , with f(z)=z . Then and , and they sum to i — the same value as the direct segment, since z is analytic.

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