Casorati-Sokhotskii-Weierstrass theorem

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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.


[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:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article