Picard Theorem - The Theorems

The Theorems

Little Picard Theorem: If a function 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.

This theorem was proved by Picard in 1879. It is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded. Later many different proofs of Picard's theorem were later found and Schottky's theorem is a quantitative version of it.

Big Picard 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.

This is a substantial strengthening of the Weierstrass–Casorati theorem, which only guarantees that the range of f is dense in the complex plane.

The "single exception" is needed in both theorems:

• ez is an entire non-constant function that is never 0,
• e1/z has an essential singularity at 0, but still never attains 0 as a value.