Computing ∫z dz Along a Line

Complex Analysis · Axiom Academy

EXAMPLE Computing Along a Line A first contour integral, worked from parametrization to final value Compute the contour integral of f(z) = z along the straight line segment from z_1 = 0 to z_2 = 1 + i . Nicely done — you turned a contour integral into an ordinary real integral and got . The recipe: parametrize , then evaluate — a single-variable integral in t . Lines are easy to parametrize: on , so is constant. Path didn't matter: the L-shaped detour gives too. Since f(z)=z is entire, Cauchy's theorem makes the integral depend only on the endpoints. Antiderivative shortcut: because everywhere, — same answer, far faster. Parametrize-and-integrate always works; when an antiderivative exists, F(z_2) - F(z_1) is the express lane.

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