Editing Picard theorem (section)

==The theorems==
[[Image:Cplot of exp(1z).png|right|220px|thumb|[[Domain coloring]] plot of the function exp({{frac|1|''z''}}), centered on the essential singularity at ''z''&nbsp;=&nbsp;0. The hue of a point ''z'' represents the [[argument (complex analysis)|argument]] of exp({{frac|1|''z''}}), the luminance represents its absolute value. This plot shows that arbitrarily close to the singularity, all non-zero values are attained.]]

<blockquote>'''Little Picard Theorem:''' If a [[function (mathematics)|function]] <math display="inline">f: \mathbb{C} \to\mathbb{C}</math> is [[entire function|entire]] and non-constant, then the set of values that <math display="inline">f(z)</math>  assumes is either the whole complex plane or the plane minus a single point. </blockquote>

<blockquote>'''Sketch of Proof:''' Picard's original proof was based on properties of the [[modular lambda function]], usually denoted by <math display="inline">\lambda</math>, 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 <math display="inline">f</math> omits two values, then the composition of <math display="inline">f</math> with the inverse of the modular function maps the plane into
the unit disc which implies that <math display="inline">f</math> is constant by [[Liouville's theorem (complex analysis)|Liouville's theorem.]]</blockquote>

This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be [[unbounded function|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 <math display="inline">f</math> are missing a single point, this point is called a [[lacunary value]] of the function.

<blockquote>'''Great Picard's Theorem:''' If an analytic function <math display="inline">f</math> has an [[essential singularity]] at a point <math display="inline">w</math> , then on any [[punctured neighborhood]] of <math display="inline">w, f(z)</math>  takes on all possible complex values, with at most a single exception, infinitely often.</blockquote>

This is a substantial strengthening of the [[Casorati–Weierstrass theorem]], which only guarantees that the range of <math display="inline">f</math>  is [[dense set|dense]] in the complex plane. 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: 
*[[exponential function|e<sup>z</sup>]] is an entire non-constant function that is never 0,
*<math display="inline">e^{\frac{1}{z}}</math>  has an essential singularity at 0, but still never attains 0 as a value.