Functions of a complex variable Complex integration Theorem

Liouville's theorem

Builds on Entire function Cauchy's estimates

Statement

Let $f : \mathbb{C} \to \mathbb{C}$ be an entire function. Assume that $f$ is bounded: there is a real number $M$ with $|f(z)| \le M$ for each $z \in \mathbb{C}$ . Then $f$ is constant.

Remarks

Proof: let $c \in \mathbb{C}$ . By Cauchy's estimates, $|f'(c)| \le M / r$ for each $r > 0$ . So $f'(c) = 0$ for each $c$ , and $f$ is constant. For example, $\sin$ is entire and not constant, so it is not bounded on $\mathbb{C}$ . On $\mathbb{R}$ it is bounded, which shows that the theorem has no real analogue. The theorem gives a short proof of the fundamental theorem of algebra.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Complex numbers object
Real numbers object
Set bounded above definition
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Supremum definition
Circle integral definition
Disc of convergence definition
Completeness axiom statement
Estimation lemma statement
Existence of square roots of nonnegative reals statement
Square root definition
Modulus of a complex number definition
Complex differentiability definition
Holomorphic function definition
Cauchy's integral formula statement
Entire function definition
Power series are holomorphic in the disc of convergence statement
Holomorphic functions are analytic statement
Holomorphic functions are infinitely differentiable statement
Cauchy's integral formula for derivatives statement
Cauchy's estimates statement
Liouville's theorem statement

Where this leads

9 pages build on this one.

Tag	`0EHKVRF4J6`
URI	`m2://0EHKVRF4J6`
Aliases	`liouvilles-theorem-complex-analysis`
Kind	statement, theorem
Writing	W1 stated
MSC	30D20
Status	This copy of the site has no build data. Run `m2 status 0EHKVRF4J6`.
Uses	Entire function, Cauchy's estimates
Same as	Wikidata Q576478 Wikipedia Liouville's theorem (complex analysis)

Markdown · JSON · Edit on GitHub

What this page needs for W2

edge about out: the concepts that the statement is about (0 of 1)
edge proves in: at least one proof node (0 of 1)
edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
explanation L2 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.