Picard's theorem

In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

Little Picard Theorem: If a function f : C → C is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point.

Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by λ, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If f omits two values, then the composition of f with the inverse of the modular function maps the plane into the unit disc which implies that f is constant by Liouville's theorem.

This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded. Many different proofs of Picard's theorem were later found and Schottky's theorem is a quantitative version of it. In the case where the values of f are missing a single point, this point is called a lacunary value of the function.

Great Picard's Theorem: If an analytic function f has an essential singularity at a point w, then on any punctured neighborhood of w, f(z) takes on all possible complex values, with at most a single exception, infinitely often.

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