Analytic Continuation

Complex Analysis · Axiom Academy

A power series only lives inside one disk — yet the function it names can stretch across the whole plane. Here is how, and where it stops. 1. The Series Sees Only Its Disk The geometric series converges only when . Its sum is — a function that is perfectly well-behaved everywhere except the single point z = 1 . The series cannot see past the circle |z| = 1 because it runs straight into that lone singularity sitting on the boundary. Converges only inside the disk …but the function lives on all of 2. Re-Expanding Reaches New Territory Pick a new center z_1 inside the first disk and re-expand the very same function as a power series about z_1 . Because the nearest singularity is still only z = 1 , the new series converges on a disk of radius |z_1 - 1| — which can poke outside the original disk, defining the function at points the first series never reached. Chaining such disks is analytic continuation . centered at 0 , radius 1 . Stops at the circle |z|=1 . New radius , reaching out to z=-2 . Both series equal wherever the two disks meet. By the Identity Theorem, this is the only analytic extension. Why it is unique — the Identity Theorem If two analytic functions agree on a set with an accumulation point inside their common domain, they agree everywhere. So once the extension exists on the overlap, it is forced — there is nothing to choose. Methods differ (power-series chains, functional equations like , integral formulas), but they all land on the same function.

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