Liouville's Theorem

Complex Analysis · Axiom Academy

A function that is analytic everywhere and never gets large has nowhere to go — it must be constant. 1. Bounded Everywhere Leaves Nowhere to Go Suppose f is entire — complex-differentiable at every point of — and bounded , meaning some fixed M satisfies for all z . Liouville's theorem says these two demands cannot coexist with any variation at all: f is forced to be a single constant value. entire on all of , and never exceeds M the only escape is to stop moving — f is constant Liouville needs both words. Drop "entire": 1/z is bounded for |z| > 1 but has a pole at 0 , so it isn't constant. Drop "bounded": is entire yet un bounded — on the imaginary axis . And "bounded on a disk" is not enough either: f(z)=z is bounded on |z| < 1 without being constant. 2. Why It's True: a Derivative Bound That Vanishes The whole proof is one estimate. Fix any point z_0 and apply Cauchy's formula for the derivative on a circle |z - z_0| = R . Because f is entire we may take that circle as large as we please — and the bigger it gets, the smaller the bound on f'(z_0) becomes. On the circle |z - z_0| = R , so |z-z_0|^2 = R^2 ; with and contour length : Since z_0 was arbitrary, f'(z_0) = 0 at every point. A function with zero derivative throughout a connected domain is constant — so f is constant.

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