Taylor Series in ℂ

Complex Analysis · Axiom Academy

A power series doesn't just sum — it fills a disk that reaches exactly to the nearest place the function breaks. In the complex plane, a series doesn't converge on a line — it fills a disk On the real line you met power series that converge on an interval . Step into ℂ and that interval inflates into a disk : a single circular region around the center where the series locks onto the function. And the disk's reach isn't arbitrary — it stops exactly where the function first misbehaves. Watch it happen before you drive it yourself. Here is the geometric series 1 + z + z 2 + z 3 + ··· trying to reproduce f(z) = 1/(1−z) , centered at z 0 = 0. Press play: as more terms are added, the matched region spreads outward from the center — but it slams to a halt at the circle that just kisses the pole at z = 1. That circle is the boundary of convergence. More terms fill more of the disk — never more than the disk. The radius is locked at R = 1, the distance from the center to the pole. The radius IS the distance to the nearest singularity Take the same function f(z) = 1/(1−z), with its single pole nailed at z = 1. Now drag the center z 0 anywhere you like. The Taylor series about that new center has its own disk — and the disk always swells until its edge just grazes the pole. Its radius is never a free choice: it is exactly R = |z 0 − 1| , the straight-line distance from the center to the singularity.

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