Essential singular point

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

Definition

An isolated singular point $z_0$ of an holomorphic function $f: U\setminus \{z_0\}\to \mathbb C$ (where $U$ is an open set) at which the limit \begin{equation}\label{e:limit} \lim_{z\to z_0} f(z)\, , \end{equation} whether finite or infinite, does not exist.

Laurent series

In a sufficiently small punctured neighbourhood $V\setminus \{z_0\}$ of an isolated singular point $z_0$, any holomorphic function $f$ can be expanded into a Laurent series: \[ \sum_{n=-\infty}^\infty a_n (z-z_0)^n\, . \] Then $z_0$ is an essential singularity of $f$ if and only if there are infinitely many negative indices $n$ for which $a_n\neq 0$. If there are finitely many and at least one nonzero $a_n$ with $n< 0$, then $z_0$ is a pole of $f$ (and the limit in \eqref{e:limit} is then $\infty$). If $a_n=0$ for every $n<0$ then $z_0$ is a removable singularity: in this case the limit in \eqref{e:limit} is $a_0$ and $f$ can be extended to an holomorphic function on the whole domain $U$.

These notions are usually extended to the case $z_0=\infty$. In this case the domain of $f$ is assumed to contain a set of the form $\{|z|> R\}$ for some $R$ and the Laurent series expansion is given \[ \sum_{n=-\infty}^\infty a_n z^{-n}\, \] and the paragraph above remains literally valid. The two cases $z_0\in \mathbb C$ and $z_0 = \infty$ can be unified introducing the Riemann sphere $\bar{\mathbb C} = \mathbb C \cup \{\infty\}$ (in this case a set of the form $\{z\in \mathbb C: |z|>R\}$ is simply a punctured neighborhood of $\infty$.

If we introduce the cluster set $C(z_0, f)$, namely the subset of $w\in \bar{\mathbb C}$ for which there is a sequence $\{z_n\}$ in the domain where $f$ is defined, such that $z_n \to z_0$ and $f(z_n) \to w$, then poles and removable singularities are characterized by the property that $C (z_0, f)$ consists of a single point: in this case one says that $C(z_0, f)$ is degenerate. Therefore, in a more general sense, the name essential singular point of an analytic function $f$ is applied to every accumulation point of the domain of definition of $f$ where the cluster set $C (z_0, f)$ is not degenerate.

Casorati-Weierstrass and Picard theorems

The following assertion is called Sokhotskii theorem or Casorati-Weierstrass theorem.

Theorem 1 If $z_0$ is an essential singularity of the holomorphic function $f$ defined in a punctured neighborhood of $z_0$, then $C (z_0, f) = \bar{\mathbb C}$.

The theorem is in fact a corollary of the following stronger statement, called Picard theorem:

Theorem 2 Under the assumption of Theorem 1, any finite complex value $w$, with at most one possible exception, is taken by $f$ infinitely often in any neighbourhood of $z_0$.

When the domain of definition of $f$ does not contain a punctured neighborhood of $z_0$, the theorems of Sokhotskii and Picard for essential singular points have only been proved under certain additional assumptions. For example, these theorems still hold for an isolated point of the set of essential singular points, in particular for a limit point of the poles of a meromorphic function.

Several complex variables

Assume $f$ is a holomorphic function defined on an open set $V$ and $z_0\in \mathbb C^n$ is an accumulation point of $V$. Then $z_0$ is called a point of meromorphy of $f$ if there is a neighborhood $U$ of $z_0$ and two holomorphic functions $p,q: U \to \mathbb C$ such that \[ f(z) = \frac{p(z)}{q(z)} \qquad \mbox{for every}\qquad z\in U\cap C\, . \] Observe that, by the Hartogs theorem, when $n\geq 2$, if the domain of definition of $f$ contains a punctured neighborhood of $z_0$, then $f$ can in fact be holomorphically extended to $z_0$: so in the case $n\geq 2$ poles cannot be isolated.

Accumulation points of $V$ which are not points of meromorphy are called essential singular points of $f$. In this cases the non-degeneracy of the cluster set $C (z_0, f)$ ceases to be a characteristic property of essential singular points.

References

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[GF]	H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) (Translated from German) MR0414912 Zbl 0381.32001
[Ma]	A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
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