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''Weierstrass theorem, Weierstrass–Sokhotskii–Casorati theorem''
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{{MSC|30}}
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{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860301.png" /> be an [[Essential singular point|essential singular point]] of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860302.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860303.png" />. Given any complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860304.png" /> (including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860305.png" />), there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860306.png" /> converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860307.png" /> such that
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''Casorati-Weierstrass theorem, Sokhotskii theorem''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860308.png" /></td> </tr></table>
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A theorem which characterizes isolated [[Essential singular point|essential singularities]] of [[Holomorphic function|holomorphic functions]] of one complex variable
  
This theorem was the first result characterizing the [[Cluster set|cluster set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s0860309.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s08603010.png" /> at an essential singularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s08603011.png" />. According to the theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s08603012.png" /> is total, that is, it coincides with the extended plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s08603013.png" /> of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s08603014.png" />. The theorem was proved by Yu.V. Sokhotskii [[#References|[1]]] (see also [[#References|[2]]]). K. Weierstrass stated this theorem in 1876 (see [[#References|[3]]]). Further information about the behaviour of an analytic function in a neighbourhood of an essential singularity is contained in the [[Picard theorem|Picard theorem]].
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'''Theorem'''
 
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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
This theorem does not admit a direct generalization to the case of analytic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s08603015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086030/s08603016.png" /> (see [[#References|[5]]]).
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\[
 
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\lim_{z\to z_0} f(x)
====References====
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\]
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Sokhotskii,  "Theory of integral residues with some applications" , St. Petersburg  (1868)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Casorati,   "Teoria delle funzioni di variabili complesse" , Pavia  (1868)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Weierstrass,  "Zur Theorie der eindeutigen analytischen Funktionen" , ''Math. Werke'' , '''2''' , Mayer &amp; Müller  (1895)  pp. 77–124</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  pp. Chapt. 2  (In Russian)</TD></TR></table>
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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 $\bar{\mathbb C}$.
  
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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}}).
  
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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]].
  
====Comments====
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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.
In the West, this theorem is known universally as the Casorati–Weierstrass theorem. It was, however, proved earlier by C. Briot and C. Bouquet and appears in the first edition [[#References|[a1]]] of their book on elliptic functions (1859), though it is missing from the second edition of this work; cf. the discussion in [[#References|[a2]]], pp. 4–5.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Briot,  C. Bouquet,  "Théorie des fonctions doublement périodiques et, en particulier, des fonctions elliptiques" , Mallet–Bachelier (1859)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}}
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|-
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|valign="top"|{{Ref|BB}}|| C. Briot,  C. Bouquet,  "Théorie des fonctions doublement périodiques et, en particulier, des fonctions elliptiques" , Mallet–Bachelier   (1859)
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|-
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|valign="top"|{{Ref|Ca}}|| F. Casorati,  "Teoria delle funzioni di variabili complesse" , Pavia  (1868)
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|-
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|valign="top"|{{Ref|CL}}|| E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)
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|-
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|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}}
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|-
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|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}} 
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|-
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|valign="top"|{{Ref|So}}|| Yu.V. Sokhotskii,  "Theory of integral residues with some applications" , St. Petersburg  (1868)  (In Russian)
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|-
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|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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|-
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|}

Latest revision as of 20:00, 14 December 2014

2020 Mathematics Subject Classification: Primary: 30-XX [MSN][ZBL]

Casorati-Weierstrass theorem, Sokhotskii theorem

A theorem which characterizes isolated essential singularities of 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 extended complex plane $\bar{\mathbb C}$, or otherwise the 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. When the limit exists, then $z_0$ is either a removable singularity, in which case the limit belongs to $\mathbb C$, or a 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 [Sh]).

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

References

[Al] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904
[BB] C. Briot, C. Bouquet, "Théorie des fonctions doublement périodiques et, en particulier, des fonctions elliptiques" , Mallet–Bachelier (1859)
[Ca] F. Casorati, "Teoria delle funzioni di variabili complesse" , Pavia (1868)
[CL] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966)
[Ma] A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[Sh] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[So] Yu.V. Sokhotskii, "Theory of integral residues with some applications" , St. Petersburg (1868) (In Russian)
[We] K. Weierstrass, "Zur Theorie der eindeutigen analytischen Funktionen" , Math. Werke , 2 , Mayer & Müller (1895) pp. 77–124
How to Cite This Entry:
Casorati-Sokhotskii-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Casorati-Sokhotskii-Weierstrass_theorem&oldid=11218
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article