Picard theorem

is entire and non-constant, then the set of values that

Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by

, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc.

with the inverse of the modular function maps the plane into the unit disc which implies that

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.

Great Picard's Theorem: If an analytic function

takes on all possible complex values, with at most a single exception, infinitely often.This is a substantial strengthening of the Casorati–Weierstrass theorem, which only guarantees that the range of

A result of the Great Picard Theorem is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception.

The "single exception" is needed in both theorems, as demonstrated here: Suppose

is an entire function that omits two values

By a similar argument using the quadratic formula, there is an entire function

omits all complex numbers of the form

But from above, any sufficiently large disk contains at least one number that the range of h omits.

Suppose f is an analytic function on the punctured disk of radius r around the point w, and that f omits two values z0 and z1.

By considering (f(p + rz) − z0)/(z1 − z0) we may assume without loss of generality that z0 = 0, z1 = 1, w = 0, and r = 1.

Because the right half-plane is simply connected, similar to the proof of the Little Picard Theorem, there are analytic functions G and H defined on the right half-plane such that F(z) = e2πiG(z) and G(z) = cos(H(z)).

For any w in the right half-plane, the open disk with radius Re(w) around w is contained in the domain of H. By Landau's theorem and the observation about the range of H in the proof of the Little Picard Theorem, there is a constant C > 0 such that |H′(w)| ≤ C / Re(w).

Because G is continuous and its domain is connected, the difference G(z + 2πi) − G(z) = k is a constant.

Thus, there is an analytic function g defined in the punctured disk with radius e−2 around 0 such that G(z) − kz / (2πi) = g(e−z).

Using the bound on G above, for all real numbers x ≥ 2 and 0 ≤ y ≤ 2π, holds, where A′ > A and C′ > 0 are constants.

By Riemann's theorem on removable singularities, g extends to an analytic function in the open disk of radius e−2 around 0.

By Riemann's theorem on removable singularities, f(z)zk extends to an analytic function in the open disk of radius e−3 around 0.

Therefore, if the function f has an essential singularity at 0, the range of f in any open disk around 0 omits at most one value.

So f(z) takes all possible complex values, except at most one, infinitely often.

Great Picard's theorem is true in a slightly more general form that also applies to meromorphic functions: Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P1(C) = C ∪ {∞} denotes the Riemann sphere and f : M\{w} → P1(C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f(z) attains all but at most two points of P1(C) infinitely often.Example: The function f(z) = 1/(1 − e1/z) is meromorphic on C* = C - {0}, the complex plane with the origin deleted.

With this generalization, Little Picard Theorem follows from Great Picard Theorem because an entire function is either a polynomial or it has an essential singularity at infinity.

As with the little theorem, the (at most two) points that are not attained are lacunary values of the function.

The following conjecture is related to "Great Picard's Theorem":[1] Conjecture: Let {U1, ..., Un} be a collection of open connected subsets of C that cover the punctured unit disk D \ {0}.

Suppose that on each Uj there is an injective holomorphic function fj, such that dfj = dfk on each intersection Uj ∩ Uk.

In the special case where the residue of g at 0 is zero the conjecture follows from the "Great Picard's Theorem".