View source for Casorati-Weierstrass theorem

The Casorati-Weierstrass theorem is a statement about the behavior of [[Holomorphic function|Holomorphic function]] in the vicinity of [[Complex Analysis/Isolated singularity|Isolated singularity]]. It essentially states that in every neighborhood of an essential singularity, every complex number can be arbitrarily closely approximated by the values of the function. It is a significantly easier-to-prove weakening of the great Picard theorem, which states that in every neighborhood of an essential singularity, every complex number (except possibly one) occurs infinitely often as a value.
== Statement ==
Let  <math>G \subseteq \mathbb C</math> be open, and <math>z_0 \in G</math>. Let <math>f\colon G \setminus \{z_0\} \to \mathbb C</math> be a [[Holomorphic function|Holomorphic function]]. Then, <math>f</math> has an [[Complex Analysis/Isolated singularity|Isolated singularity]] at <math>z_0</math> if and only if for every neighborhood <math>U \subseteq G</math> of <math>z_0</math>:
<math>\overline{f(U \setminus {z_0})} = \mathbb{C}</math>.
=== Proof === 
First, assume that <math>z_0</math> is an essential singularity of <math>f</math>, and suppose there exists an <math>r > 0</math> such that<math>f(B_r(z_0)\setminus\{z_0\})</math> is not dense in <math>\mathbb{C}</math>. Then there exists an <math>\epsilon > 0</math> and a <math>w_0 \in \mathbb{C}</math> such that <math>B_\epsilon(w_0)</math> and <math>f(B_r(z_0)\setminus\ {z_0})</math> are disjoint. Consider <math>B_r(z_0)\setminus\{z_0\}</math> the function. <math>g(z) := \frac{1}{f(z) - w_0}</math>.Let <math>r</math> be chosen so that <math>z_o</math> is the only <math>w_0</math>-pole in <math>f(B_r(z_0))</math>. This is possible by the [[Complex Analysis/Identity Theorem|Identity Theorem]] for non-constant holomorphic functions. Since <math>f</math> is not constant (as it has an essential singularity), it is holomorphic and bounded by <math>\frac{1}{\epsilon}</math>. By the [[Riemann Removability Theorem]], <math>g</math> is therefore holomorphically extendable to all of <math>B_r(z_0)</math>. Since <math>g \ne 0</math>
there exists an <math>m \ge 0</math> and a holomorphic function <math>g_0 \colon B_r(z_0) \to \mathbb C</math> with <math>g_0(z_0) \ne 0</math>, such that
<center><math> g(z) = (z-z_0)^m g_0(z), \qquad |z-z_0| < r. </math></center>
It follows that
<center><math> f(z) = w_0 + \frac{1}{(z-z_0)^m g_0(z)} </math></center>
and thus
<center><math> f(z)(z-z_0)^m = w_0(z-z_0)^m + \frac1{g_0(z)} </math></center>
Since <math>g_0(z_0) \ne 0</math>,is <math>\frac1g</math> is holomorphic in a neighborhood of <math>z_0</math>. Therefore, <math>f \cdot (\cdot -z_0)^m</math> is holomorphic in a neighborhood of <math>z_0</math>, meaning that <math>f</math> has at most a pole of order <math>m</math> at <math>z_0</math>, which leads to a contradiction.Conversely. 
let <math>z_0</math> be a removable singularity or a pole of <math>f</math>. Is <math>z_0</math> is a removable singularity, there exists a neighborhood <math>U</math> of <math>z_0</math>,where <math>f</math> is bounded, say <math>|f(z)| \le M</math> for  <math>z \in U\setminus \{z_0\}</math>.Then it follows that
<center><math> \overline{f\bigl(U\setminus \{z_0\})} \subseteq \bar B_M(0) \ne \mathbb C. </math></center>
If <math>z_0</math> is a pole of order <math>m</math> for <math>f</math>, there exists a neighborhood <math>U</math> of <math>z_0</math> and a holomorphic function <math>g \colon U \to \mathbb C</math> with <math>g(z_0) \ne 0</math> and <math>f(z) = g(z)(z-z_0)^{-m}</math>. Choose a neighborhood <math>\epsilon > 0</math> such that <math>|g(z)| \ge \frac 12|g(z_0)|</math> for <math>|z-z_0| < \epsilon</math>. Then it follows that
<center><math>|f(z)| = |g(z)| |z-z_0|^{-m} \ge \frac 12|g(z_0)|\cdot \epsilon^{-m}, \quad 0 < |z-z_0| <\epsilon </math></center>
Thus, <math> 0 \not\in \overline{f(B_\epsilon(z_0)\setminus\{z_0\})}</math> and this proves the claim.
== see also==
*[[w:en:Complex analysis|Complex Analysis]]

*[[Complex Analysis/Identity Theorem|Identity Theorem]]
==Translation and Version Control==

This page was translated based on the following [https://de.wikiversity.org/wiki/Satz_von_Casorati-Weierstraß Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Satz von Casorati-Weierstraß|Satz von Casorati-Weierstraß]] - URL:
https://de.wikiversity.org/wiki/Satz_von_Casorati-Weierstraß
* Date: 1/2/2025
<span type="translate" src="Satz von Casorati-Weierstraß" srclang="de" date="1/2/2025" time="12:55" status="inprogress"></span>

<noinclude>
[[de:Satz von Casorati-Weierstraß]]
</noinclude>

[[Category:Wiki2Reveal]]
[[Category:Complex Analysis]]