Fundamental Theorem for Contours

Complex Analysis · Axiom Academy

LESSON The Fundamental Theorem for Contours When an analytic function has an antiderivative, a contour integral depends only on its endpoints — the path can do whatever it likes. 1. Endpoints Are All That Matter In real calculus, whenever F'=f . The complex version reads identically : if F'(z)=f(z) on a region D , then for any contour inside D running from z_0 to z_1 , only the endpoints survive. The Fundamental Theorem for contour integrals The animation drives this home: three completely different paths from z_0=0 to z_1=1+i — a straight line, a parabola, a wide bulge — are each integrated against f(z)=z . Watch the running value. Every path lands on the same answer, F(z_1)-F(z_0) . 2. Why It Works: the Chain Rule The proof is one line of chain rule. Parametrize the contour as for . Then is just a function of the real variable t , and its t -derivative is exactly the integrand: So the contour integral is an ordinary integral of a perfect derivative — and the real FTC finishes the job. The animation slides a point along ; the moving arrow is , which stays locked to the entire way. Telescoping that ordinary integral over [a,b] leaves only the boundary terms — the path's interior contributes nothing. No parametrization needed. Since , the antiderivative is . Evaluate at the two endpoints and subtract. Expand (1+i)^3 = 1 + 3i + 3i^2 + i^3 = 1 + 3i - 3 - i = -2 + 2i . Then divide by 3: 4. The Payoff: Closed Loops Give Zero

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