Editing Casorati–Weierstrass theorem (section)

==Formal statement of the theorem==
Start with some [[open set|open subset]] <math>U</math> in the [[complex number|complex plane]] containing the number <math>z_0</math>, and a function <math>f</math> that is [[holomorphic function|holomorphic]] on <math>U \setminus \{z_0\}</math>, but has an [[essential singularity]] at <math>z_0</math>&nbsp;. The ''Casorati–Weierstrass theorem'' then states that 
{{block indent | em = 1.5 | text = if <math>V</math> is any [[Neighborhood (mathematics)|neighborhood]] of <math>z_0</math> contained in <math>U</math>, then <math>f(V \setminus \{z_0\})</math> is [[dense set|dense]] in <math>\Complex</math>.}}

This can also be stated as follows: 
{{block indent | em = 1.5 | text = for any <math>\varepsilon > 0, \delta > 0 </math>, and a complex number <math>w</math>, there exists a complex number <math>z</math> in <math>U</math> with <math>0<|z-z_0|<\delta</math> and <math>|f(z)-w| < \varepsilon</math>.}}

Or in still more descriptive terms:
{{block indent | em = 1.5 | text = <math>f</math> comes arbitrarily close to ''any'' complex value in every neighborhood of <math>z_0</math>.}}

The theorem is considerably strengthened by [[Picard's great theorem]], which states, in the notation above, that <math>f</math> assumes ''every'' complex value, with one possible exception, infinitely often on <math>V</math>.

In the case that <math>f</math> is an [[entire function]] and <math>a = \infty</math>, the theorem says that the values <math>f(z)</math> approach every complex number and <math>\infty</math>, as <math>z</math> tends to infinity.
It is remarkable that this does not hold for [[holomorphic map]]s in higher dimensions, as the famous example of [[Pierre Fatou]] shows.<ref>{{cite journal |last1=Fatou |first1=P |title=Sur les fonctions méromorphes de deux variables |journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris |date=1922|volume=175 |pages=862–865|jfm=48.0391.02|url=http://gallica.bnf.fr/ark:/12148/bpt6k3128v.f862}}
, {{cite journal |last1=Fatou |first1=P |title=Sur certaines fonctions uniformes de deux variables |journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris |date=1922|volume=175 |pages=1030–1033|jfm=48.0391.03|url=http://gallica.bnf.fr/ark:/12148/bpt6k3128v.f1030}}</ref>

[[Image:Essential singularity.png|right|220px|thumb|Plot of the function exp(1/''z''), centered on the essential singularity at ''z''&nbsp;=&nbsp;0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which would be uniformly white).]]