Cauchy's Theorem Introduction

Complex Analysis · Axiom Academy

Integrate an analytic function once around any closed loop and the answer is always the same: exactly zero. One trip around, and everything cancels In ordinary calculus, integrating around a closed path almost never gives a clean answer. In the complex plane, a single condition changes that completely. If a function is analytic — complex-differentiable — everywhere inside a closed loop, then integrating it once around the loop yields precisely zero , no matter the function or the shape of the loop. This one fact is the foundation the whole subject is built on. Watch why the answer is zero. A marker makes one full trip around a closed loop C ; at every step it adds a tiny contribution to a running total — the partial sum of the contour integral, drawn as a growing vector on the right. Because f is analytic everywhere inside C , those contributions perfectly cancel: the vector wanders out, curls around, and lands right back on the origin. Analytic inside the loop → the partial sums close up perfectly: . Why zero? Because an antiderivative is waiting Inside a simply connected region, every analytic f has an antiderivative F , so a path integral only depends on its endpoints: . Drag the moving end of the path along the real axis. The running integral tracks the height difference of F between the two ends — and as you slide the end back toward the fixed start, that difference shrinks. Bring them together (a closed loop) and the integral is forced to .

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