Theorem

Picard Theorem

Analysis

An entire function that is not a polynomial takes every complex value, with at most one exception, infinitely often. Proved by Emile Picard, the 'great' Picard theorem is one of the deepest classical results on the behavior of holomorphic functions near an essential singularity.

Facts

Statement

Picard's little theorem states that a non-constant entire function on the complex plane takes every complex value, with at most one exception. Picard's great theorem strengthens this: near an essential singularity, a holomorphic function takes every value, with at most one exception, infinitely often. 1

Classification

Statement Form

Uniqueness Theorem 1

Statement Form

Inequality 1

Connections

Source Picard Theorem (Wikipedia)

Sources

1. Picard Theorem (Wikipedia)

Wikimedia Foundation

Lead section, opening sentence
Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function.
Proved By: Émile Picard, Lead
They are named after Émile Picard.

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Picard Theorem

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