A Remarkable Formula

Complex Analysis · Axiom Academy

Know an analytic function only on the edge of a region, and you already know it everywhere inside. The whole interior, hiding on the boundary Pick an analytic function and a point a sitting somewhere inside a closed loop. You'd expect that to find f(a) you'd have to actually go to a . Cauchy's Integral Formula says something stranger: the values of f all the way around the loop already pin down f(a) exactly. Watch it happen first, then drive it yourself. A sweep travels once around the loop C , sampling f on the boundary. As it goes, those boundary values are gathered into the integral, and the running total settles on a single number — the value of f at the interior point a . The boundary held all the information — the formula just collected it. Here the analytic function is . Drag the point a along the axis. While a stays inside the loop, the formula returns the true value f(a) — compare the two stats and watch them agree. Drag a outside the loop and the whole integral collapses to zero : with no singularity enclosed, is analytic inside, so Cauchy's theorem sends the loop integral to 0. Inside, the formula reconstructs f(a) ; outside, it returns nothing — the loop only "sees" what it encloses.

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