Liouville's theorem

Liouville's theorem states that every bounded entire function on the complex plane is constant. It sharply constrains holomorphic functions defined everywhere.

Entire function: A complex-valued function holomorphic at every point of the complex plane. The theorem applies precisely to functions holomorphic throughout the plane.
Fundamental theorem of algebra: The theorem that every nonconstant polynomial with complex coefficients has a complex root. A standard proof assumes a rootless polynomial's reciprocal is bounded and entire, contradicting Liouville's theorem.
Joseph Liouville: A French mathematician whose work spanned number theory, analysis, and mathematical physics. The theorem bears his name, though its standard proof uses later-developed complex analysis.
Liouville's theorem (Hamiltonian mechanics): A theorem stating that Hamiltonian flow preserves volume in phase space. This unrelated theorem has the same name but concerns classical mechanics, not entire functions.
Bounded function: A function whose values remain within a fixed finite bound on its domain. A global bound on the entire function is the theorem's decisive hypothesis.
Liouville's theorem (number theory): A result in Diophantine approximation bounding how closely algebraic numbers can be approximated by rationals. It shares Liouville's name but is a distinct theorem from this complex-analysis result.
Augustin-Louis Cauchy: A French mathematician who established foundational results in complex analysis. Cauchy's integral methods provide the standard proof mechanism.
Liouville's theorem (differential equations): A formula expressing the determinant of a fundamental matrix for a linear differential system. This is another distinct result called Liouville's theorem.
Cauchy's integral formula: A formula expressing holomorphic-function values and derivatives through contour integrals around them. Its derivative estimates force every positive-order derivative to vanish.
Picard theorem: A complex-analysis theorem stating that a nonconstant entire function takes every complex value, with at most one exception. Its stronger range restriction on entire functions builds on Liouville's rigidity.

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