Liouville's theorem (complex analysis)

:For Liouville's theorem in Hamiltonian mechanics, see Liouville's theorem (Hamiltonian).

Related Topics:
Hamiltonian mechanics - Liouville's theorem (Hamiltonian)

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Liouville's theorem in complex analysis states that every bounded (i.e., there exists a real number M such that |f(z)| ≤ M for all z in C) entire function (a holomorphic function f(z) defined on the whole complex plane C) must be constant.

Related Topics:
Complex analysis - Real number - Entire function - Holomorphic function - Complex plane

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Liouville's theorem can be used to give an elegant short proof for the fundamental theorem of algebra.

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The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits at least two complex numbers must be constant.

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In the language of Riemann surfaces, the theorem can be generalized as follows: if M is a parabolic Riemann surface (such as the complex plane C) and N is a hyperbolic one (such as an open disk), then every holomorphic function f : M → N must be constant.

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