Casorati-Sokhotskii-Weierstrass theorem

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