Liouville's Theorem

Liouville's theorem states that every bounded entire function is constant. This simple but profound result has many consequences, including an elegant proof of the fundamental theorem of algebra.

Statement

Theorem2.4Liouville's theorem

If $f: \mathbb{C} \to \mathbb{C}$ is entire (holomorphic on all of $\mathbb{C}$ ) and bounded, then $f$ is constant.

RemarkInterpretation

An entire function that is not constant must be unbounded: as $|z| \to \infty$ , we must have $|f(z)| \to \infty$ (or at least $|f(z)|$ is unbounded along some sequence).

Proof via Cauchy's Estimates

Proof

Suppose $|f(z)| \leq M$ for all $z \in \mathbb{C}$ . By Cauchy's integral formula for derivatives,

$f'(z_0) = \frac{1}{2\pi i} \oint_{|z-z_0|=R} \frac{f(z)}{(z-z_0)^2} dz$

for any $R > 0$ . Taking absolute values and using the ML inequality:

$|f'(z_0)| \leq \frac{1}{2\pi} \cdot \frac{M}{R^2} \cdot 2\pi R = \frac{M}{R}.$

Since this holds for all $R > 0$ , letting $R \to \infty$ gives $|f'(z_0)| = 0$ . Thus $f'(z_0) = 0$ for all $z_0 \in \mathbb{C}$ , so $f$ is constant.

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Examples

ExamplePolynomials are unbounded

Every non-constant polynomial $p(z) = a_n z^n + \cdots + a_0$ with $n \geq 1$ satisfies $|p(z)| \to \infty$ as $|z| \to \infty$ . By Liouville's theorem, no non-constant polynomial is bounded.

For instance, $p(z) = z$ satisfies $|p(z)| = |z| \to \infty$ .

ExampleExponential is unbounded

$e^z$ is entire but unbounded: $|e^{x+iy}| = e^x \to \infty$ as $x \to \infty$ . By Liouville's theorem, $e^z$ cannot be constant.

ExampleSine is unbounded

$\sin z = \frac{e^{iz} - e^{-iz}}{2i}$ is entire. On the imaginary axis, $\sin(iy) = i\sinh y \to \infty$ as $y \to \infty$ . Thus $\sin z$ is unbounded, consistent with Liouville's theorem.

Application: Fundamental Theorem of Algebra

Theorem2.5Fundamental theorem of algebra (via Liouville)

Every non-constant polynomial $p(z)$ with complex coefficients has at least one root in $\mathbb{C}$ .

Proof

Suppose $p(z)$ has no roots. Then $f(z) = 1/p(z)$ is entire (holomorphic on all of $\mathbb{C}$ ).

Since $|p(z)| \to \infty$ as $|z| \to \infty$ (for a non-constant polynomial), we have $|f(z)| = 1/|p(z)| \to 0$ as $|z| \to \infty$ . Thus $f$ is bounded on $\mathbb{C}$ .

By Liouville's theorem, $f$ is constant. Therefore $p$ is constant, contradicting the assumption that $p$ is non-constant. Hence $p$ must have a root.

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RemarkElegance

This is one of the most elegant proofs in mathematics: a deep algebraic result (existence of roots) follows from a simple analytic theorem (Liouville).

Generalizations

RemarkFunctions of polynomial growth

Liouville's theorem generalizes: if $f$ is entire and $|f(z)| \leq C|z|^n$ for some $n \in \mathbb{N}$ and all large $|z|$ , then $f$ is a polynomial of degree at most $n$ .

Proof idea: The $k$ -th derivative $f^{(k)}$ satisfies $|f^{(k)}(z)| \leq C_k |z|^{n-k}$ for large $|z|$ . For $k > n$ , this gives $|f^{(k)}(z)| \to 0$ as $|z| \to \infty$ , so by Cauchy's estimates, $f^{(k)} \equiv 0$ . Thus $f$ is a polynomial of degree $\leq n$ .

RemarkPicard's theorem

A stronger result is Picard's great theorem: an entire function omits at most one value (unless it is constant). For instance:

$e^z$ omits $0$ (and only $0$ ).
A non-constant polynomial omits no values (by the fundamental theorem).
Constant functions omit all values except one.

Picard's theorem is much deeper than Liouville's and requires advanced techniques (normal families, modular functions).

Application: Harmonic Functions

RemarkLiouville for harmonic functions

A bounded harmonic function on $\mathbb{C}$ is constant. This follows from Liouville's theorem applied to a holomorphic function $f$ with $u = \text{Re}(f)$ (every harmonic function is locally the real part of a holomorphic function).

Summary

RemarkKey points

Liouville's theorem: A bounded entire function is constant.
Proof: Uses Cauchy's integral formula and estimates.
Applications:
- Fundamental theorem of algebra.
- Classification of entire functions of polynomial growth.
- Harmonic function theory.
Intuition: Entire functions have "no escape": if bounded, they cannot grow, so they must be constant.
Liouville's theorem is a cornerstone of complex analysis, with applications to PDE, number theory, and algebraic geometry.

See also: Entire functions and Picard's theorem.