View source for Casorati-Sokhotskii-Weierstrass theorem

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''Casorati-Weierstrass theorem, Sokhotskii theorem''

A theorem which characterizes isolated [[Essential singular point|essential singularities]] of [[Holomorphic function|holomorphic functions]] of one complex variable

'''Theorem'''
Let $f: U\to \mathbb C$ be an holomorphic function and $z_0$ a point for which $U$ is a punctured neighborhood. Then either the limit
\[
\lim_{z\to z_0} f(x)
\]
exists in the [[Riemann sphere|extended complex plane]] $\bar{\mathbb C}$, or otherwise the [[Cluster set|cluster set]] $C(z_0, f)$ (namely the set of points $w\in \bar{\mathbb C}$ for which there is a sequence $z_n \to z_0$ with $f(z_n)\to w$) is the entire $\bar{\mathbb C}$.

In the latter case, the singularity is called [[Essential singular point|essential]]. When the limit exists, then $z_0$ is either a [[Removable singular point|removable singularity]], in which case the limit belongs to $\mathbb C$, or a [[Pole (of a function)|pole]]. Removable singularities, poles and essential singularities can also be characterized using the [[Laurent series]]. The assumption that $f$ is defined on a punctured neighborhood of $z_0$ can be weakend (see [[Essential singular point]]). Instead there is no direct generalization to the case of holomorphic functions of several complex variables (see {{Cite|Sh}}).

The Casorati-Sokhotskii-Weierstrass theorem was the first result characterizing the [[Cluster set|cluster set]] of an analytic function $f$ at an essential singularity. A stronger theorem from which the Casorati-Sokhotskii-Weierstrass theorem can be inferred is the [[Picard theorem]]. 

The theorem was proved by Sokhotskii {{Cite|So}} and Casorati {{Cite|Ca}} in 1868 [[#References|[1]]]. Weierstrass stated it in 1876 (see {{Cite|We}}). It was, however, proved earlier by C. Briot and C. Bouquet and  appears in the first edition {{Cite|BB}} of their book on  elliptic functions (1859), though it is missing from the second edition  of this work; cf. the discussion in {{Cite|CL}}, pp. 4–5.

====References====
{|
|-
|valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}}
|-
|valign="top"|{{Ref|BB}}||  C. Briot,   C. Bouquet,   "Théorie des fonctions doublement périodiques  et, en particulier, des fonctions elliptiques" , Mallet–Bachelier   (1859)
|-
|valign="top"|{{Ref|Ca}}|| F. Casorati,   "Teoria delle funzioni di variabili complesse" , Pavia  (1868)
|-
|valign="top"|{{Ref|CL}}|| E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)
|-
|valign="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of   a complex variable" ,  '''1–3''' , Chelsea (1977) (Translated from   Russian) {{MR|0444912}}  {{ZBL|0357.30002}}
|-
|valign="top"|{{Ref|Sh}}|| B.V. Shabat, "Introduction of complex  analysis" , '''1–2''' , Moscow  (1976) (In Russian) {{MR|}}  {{ZBL|0799.32001}} {{ZBL|0732.32001}}  {{ZBL|0732.30001}}  {{ZBL|0578.32001}} {{ZBL|0574.30001}}  
|-
|valign="top"|{{Ref|So}}|| Yu.V. Sokhotskii,   "Theory of integral residues with some applications" , St. Petersburg  (1868)  (In Russian)
|-
|valign="top"|{{Ref|We}}|| K. Weierstrass,   "Zur Theorie der eindeutigen analytischen Funktionen" , ''Math. Werke'' , '''2''' , Mayer &amp; Müller  (1895)  pp. 77–124
|-
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