TheoremProved

Liouville's theorem (complex analysis)

Statement

Every bounded entire function is constant: if $f : \mathbb{C} \to \mathbb{C}$ is holomorphic on the entire complex plane $\mathbb{C}$ and there exists a real constant $M \ge 0$ such that $|f(z)| \le M$ for all $z \in \mathbb{C}$ , then $f$ is constant.

Why is it true?

On the real line $\mathbb{R}$ , smooth functions like $\sin x$ can oscillate forever while staying bounded between $-1$ and $1$ . In the complex plane $\mathbb{C}$ , holomorphic functions cannot do this: the mean value property forces the value at any point to be the exact average over circles of arbitrarily large radius $R$ centered at that point. If the function is trapped inside a disk of radius $M$ everywhere on $\mathbb{C}$ , then taking averages over larger and larger circles leaves no room for the derivative to be nonzero — to vary at all, a non-constant entire function must grow without bound in some direction (for example, $|\sin(iy)| = \sinh|y| \to \infty$ ).

Proof sketch

Fix any $z_0 \in \mathbb{C}$ and apply Cauchy's integral formula for the first derivative on the circle $C_R = \{|z - z_0| = R\}$ of radius $R > 0$ : $f'(z_0) = \frac{1}{2\pi i} \oint_{C_R} \frac{f(z)}{(z - z_0)^2}\,dz$ . Estimating the integral with $|f(z)| \le M$ and $|z - z_0| = R$ along the circumference $2\pi R$ gives Cauchy's estimate $|f'(z_0)| \le \frac{1}{2\pi} \cdot \frac{M}{R^2} \cdot 2\pi R = \frac{M}{R}$ . Letting $R \to \infty$ forces $f'(z_0) = 0$ for every $z_0 \in \mathbb{C}$ ; since $\mathbb{C}$ is connected, $f$ is constant. (Historically, Augustin-Louis Cauchy published this proof in 1844; Joseph Liouville presented it in his 1847 lectures, from which the theorem acquired its name.)

Stated by

stated 1844Augustin-Louis Cauchy

Proved by

proved 1844Augustin-Louis Cauchy

Topics that use this theorem

Functions of a complex variable

Related theorems

Cauchy's integral formula Fundamental theorem of algebra Maximum principle for harmonic functions

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Lars V. Ahlfors (1979). Complex Analysis
Reinhold Remmert (1991). Theory of Complex Functions