##  [Liouville's Theorem](/liouvilles-theorem-0) 

 Definition

A result in complex analysis stating that any entire (holomorphic on the whole complex plane) function that is bounded must be constant.

 

 

 

 

 

 





## Principle

Principle

Global boundedness plus holomorphicity on C forces all complex derivatives to vanish, so the function cannot vary.

 

 

 

 

 





## Demonstration

Demonstration

If f is entire and |f(z)| ≤ M for all z, Cauchy estimates give |f^{(n)}(0)| ≤ n! M / R^n for any R; letting R → ∞ yields all derivatives beyond order zero equal zero, so f is constant.

 

 

 

 

## Misapplication

Misapplication

Applying the theorem to functions that are only bounded on a proper subset of C, to meromorphic functions, or to real-differentiable functions; or concluding boundedness on a domain implies constancy without global holomorphicity.

 

 

 

 

 





## Consequence

Consequence

Provides a sharp restriction on the growth of entire functions, underpins the Fundamental Theorem of Algebra (by applying to 1/p(z) for a nonconstant polynomial p), and rules out nonconstant bounded entire behaviors.

 

 

 

 

## Reversal

Reversal

The contrapositive: any nonconstant entire function must be unbounded on C; equivalently, existence of a nonconstant entire function implies it attains arbitrarily large values.

 

 

 

 

 





## Boundary

Boundary

Requires holomorphicity on the entire complex plane and boundedness on C. It does not apply to functions with poles, to functions only holomorphic on a proper domain, or to functions bounded only on subsets or along sequences.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises with real-analysis intuition where bounded differentiable functions can be nonconstant; in complex analysis, holomorphicity is much stronger and interacts with global topology.

 

 

 

 

 





## Synthesis

Synthesis

Liouville's Theorem ties global holomorphic regularity and boundedness: in the full complex plane these two conditions force triviality, making constant functions the only bounded entire maps.