Difference between revisions of "Casorati-Sokhotskii-Weierstrass theorem"
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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 $\bar{\mathbb C}$. | 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 | + | 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 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]]. | ||
Revision as of 10:33, 18 January 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 1968 [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 |
Casorati-Sokhotskii-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Casorati-Sokhotskii-Weierstrass_theorem&oldid=31264