# Casorati-Sokhotskii-Weierstrass theorem

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2010 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 . 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.

How to Cite This Entry:
Sokhotskii theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sokhotskii_theorem&oldid=31261