Analytic Continuation

Complex Analysis · Axiom Academy

One formula runs out of room — yet the function it describes keeps going. The art of following it past its own edge launches the final chapter of complex analysis. A formula that quits — and a function that doesn't Write the simplest infinite sum there is, 1 + z + z^2 + z^3 + . It only adds up to something when |z| 1 — step outside that disk and the series flies off to infinity. But you already know its value: it equals 1-z , and that expression makes perfect sense almost everywhere. The function was never trapped in the disk; only its first formula was. Following the function out past where its formula breaks is analytic continuation — and it is the engine behind the deepest results in all of mathematics. Watch it happen on the complex plane. The series lives inside the unit disk where it converges. When the animation plays, the function spills past the circle and floods the whole plane — every point except the single pole at z=1 , where 1-z truly blows up. The disk is where the series works. The whole plane is where the function works. Continuation carries you from one to the other. Drag past the edge and see which one survives Drag the point z anywhere on the plane. Inside the disk, the partial sum _ n=0 ^ N z^n and the closed form 1-z march together — the same value. Cross the circle and the partial sum runs away to infinity, yet 1-z keeps handing you a clean answer. That answer is the continued function.

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