Liouville's Theorem

A bounded entire function is constant — via the Cauchy estimate.

The idea

Theorem (Liouville's theorem).

Let $f$ be entire, meaning holomorphic at every point of $\mathbb{C}$, and bounded, meaning some constant $M$ satisfies $|f(z)| \le M$ for every $z$. Then $f$ is constant.

No real analogue holds: $\sin x$ is bounded and infinitely differentiable on the whole real line, and it is not constant. Differentiability in the complex sense on the whole plane is a far stronger condition, and this theorem is one measure of how much stronger. The hypothesis is used on circles of every radius, as the figure shows: an entire function is available on the whole plane, not just near a point.

Ways to work on it

• Walkthrough. Build the Cauchy estimate and let the radius go to infinity.
• Practice. Decide when Liouville's hypotheses actually force a function constant.
• Hardest. Derive the fundamental theorem of algebra from Liouville.

Not sure where to start? Take the ten-question placement test.