Liouville's theorem in complex analysis

Statement

If $f:\mathbb C\to\mathbb C$ is an entire function and bounded, then $f$ is constant.

Cauchy-estimate proof

If $|f|\le M$ , the derivative form of the Cauchy integral formula on the circle $|z-a|=R$ gives

|f'(a)|\le\frac{M}{R}.

Letting $R\to\infty$ yields $f'(a)=0$ for every $a$ , hence $f$ is constant.

Consequences

The theorem gives a complex-analytic proof of the fundamental theorem of algebra. More generally, an entire function satisfying $|f(z)|\le C(1+|z|^m)$ is a polynomial of degree at most $m$ , by the higher Cauchy estimates.

Disambiguation

This theorem is unrelated to Liouville's theorem in Hamiltonian mechanics or the Liouville–Arnold theorem.

References

Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 4, §3.