Namespaces
Variants
Actions

Difference between revisions of "Essential singular point"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
An [[Isolated singular point|isolated singular point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362601.png" /> of single-valued character of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362602.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362603.png" /> at which the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362604.png" />, whether finite or infinite, does not exist. In a sufficiently small punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362605.png" /> of an essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362606.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362607.png" /> in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362608.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e0362609.png" /> can be expanded into a Laurent series:
+
{{MSC|30}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626010.png" /></td> </tr></table>
+
====Definition====
 +
An [[Isolated singular point|isolated singular point]] $z_0$ of an [[Analytic function|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.  
  
or, correspondingly,
+
====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 a function)|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 singular point|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 $U$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626011.png" /></td> </tr></table>
+
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 $\infty$.  
  
where in the principal part of these series there is an infinite number of non-zero coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626012.png" /> with negative indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626013.png" />.
+
If we introduce the [[Cluster set|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.
  
The [[Sokhotskii theorem|Sokhotskii theorem]] asserts that every complex value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626014.png" /> in the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626015.png" /> is a limit value for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626016.png" /> in any neighbourhood, however small, of an essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626017.png" />. According to the [[Picard theorem|Picard theorem]], every finite complex value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626018.png" />, with one possible exception, is a value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626019.png" /> taken infinitely often in any neighbourhood of an essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626020.png" />. The Sokhotskii theorem can also be expressed in another way, by stating that the [[Cluster set|cluster set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626021.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626022.png" /> at an essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626023.png" /> coincides with the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626024.png" />. For regular points and poles, this set, on the other hand, is degenerate, i.e. it reduces to a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626025.png" />. Therefore, in a more general sense, the name essential singular point of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626026.png" /> is applied to every singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626027.png" /> (not necessarily isolated) at which no finite or infinite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626028.png" /> exists, or, in other words, at which the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626029.png" /> is non-degenerate. The theorems of Sokhotskii and Picard for such essential singular points, not being isolated points of the set of all singular points, have only been proved under certain additional assumptions. For example, these theorems still hold for an isolated point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626030.png" /> of the set of essential singular points, in particular for a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626031.png" /> of the poles of a [[Meromorphic function|meromorphic function]].
+
====Casorati-Weierstrass and Picard theorems====
 +
The following assertion is called [[Sokhotskii theorem]] or Casorati-Weierstrass theorem.
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626032.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626034.png" />, is called a point of meromorphy of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626035.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626036.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626037.png" /> is a meromorphic function in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626039.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626040.png" /> can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626041.png" /> as a quotient of two holomorphic functions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626043.png" />. Singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626045.png" /> that are not points of meromorphy are called essential singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626046.png" />. In these cases the non-degeneracy of the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036260/e03626047.png" /> ceases to be a characteristic property of essential singular points.
+
'''Theorem 1'''
 +
If $z_0$ is an essential singularity of the holomorphic function $f$ defined in a punctured neighborhood of $z_0$, then $C (z_0, f) = \bar{\mathbb C}$.
  
====References====
+
The theorem is in fact a corollary of the following stronger statement, called [[Picard theorem]]:
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Fuks,  "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR></table>
+
 
 +
'''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|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 $n\geq 2$ [[Pole (of a function)|poles]] cannot be isolated.
  
====Comments====
+
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.
In Western literature the Sokhotskii theorem is known as the Casorati–Weierstrass theorem.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Saks,  A. Zygmund,  "Analytic functions" , Elsevier  (1971)  (Translated from Polish)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}}
 +
|-
 +
|valign="top"|{{Ref|Fu}}|| B.A. Fuks, "Theory of analytic functions of several complex variables" ,  '''1–2''' , Amer. Math. Soc.  (1963–1965) (Translated from Russian)  {{MR|0188477}} {{MR|0174786}}  {{MR|0168793}} {{MR|0155003}}  {{MR|0037915}} {{MR|0027069}}  {{ZBL|0146.30802}} {{ZBL|0138.30902}}  {{ZBL|0040.19002}}
 +
|-
 +
|valign="top"|{{Ref|GF}}|| H. Grauert, K. Fritzsche, "Several  complex variables" , Springer (1976)  (Translated from German)  {{MR|0414912}} {{ZBL|0381.32001}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of  a complex variable" ,  '''1–3''' , Chelsea (1977) (Translated from  Russian) {{MR|0444912}}  {{ZBL|0357.30002}}
 +
|-
 +
|valign="top"|{{Ref|SZ}}|| S. Saks,  A. Zygmund,  "Analytic functions" , Elsevier  (1971)  (Translated from Polish)
 +
|-
 +
|}

Latest revision as of 10:11, 18 January 2014

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

[Al] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904
[Fu] B.A. Fuks, "Theory of analytic functions of several complex variables" , 1–2 , Amer. Math. Soc. (1963–1965) (Translated from Russian) MR0188477 MR0174786 MR0168793 MR0155003 MR0037915 MR0027069 Zbl 0146.30802 Zbl 0138.30902 Zbl 0040.19002
[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
[SZ] S. Saks, A. Zygmund, "Analytic functions" , Elsevier (1971) (Translated from Polish)
How to Cite This Entry:
Essential singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Essential_singular_point&oldid=14859
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article