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An isolated [[Singular point|singular point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734901.png" /> of single-valued character of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734902.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734903.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734904.png" /> increases without bound when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734905.png" /> approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734906.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734907.png" />. In a sufficiently small punctured neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734908.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p0734909.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349010.png" /> in the case of the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349011.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349012.png" /> can be written as a [[Laurent series|Laurent series]] of special form:
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{{MSC|30}}
 
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{{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/p/p073/p073490/p07349013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$
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\newcommand{\abs}[1]{\left| #1 \right|}
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\newcommand{\set}[1]{\left\{ #1 \right\}}
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$
  
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The ''pole of a function'' is
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an isolated [[Singular point|singular point]] $a$ of single-valued character of an [[Analytic function|analytic function]] $f(z)$ of the complex variable $z$ for which $\abs{f(z)}$ increases without bound when $z$ approaches $a$: $\lim_{z\rightarrow a} f(z) = \infty$. In a sufficiently small punctured neighbourhood $V=\set{z\in\C : 0 < \abs{z-a} < r}$ of the point $a \neq \infty$, or $V'=\set{z\in\C : r < \abs{z} < \infty}$ in the case of the point at infinity $a=\infty$, the function $f(z)$ can be written as a [[Laurent series]] of special form:
 +
\begin{equation}
 +
\label{eq1}
 +
f(z) = \sum_{k=-m}^\infty c_k (z-a)^k,\qquad a \neq \infty, c_{-m} \neq 0, z \in V,
 +
\end{equation}
 
or, respectively,
 
or, respectively,
 +
\begin{equation}
 +
\label{eq2}
 +
f(z) = \sum_{k=-m}^\infty \frac{c_k}{z^k},\qquad a = \infty, c_{-m} \neq 0, z \in V',
 +
\end{equation}
 +
with finitely many negative exponents if $a\neq\infty$, or, respectively, finitely many positive exponents if $a=\infty$. The natural number $m$ in these expressions is called the order, or multiplicity, of the pole $a$; when $m=1$ the pole is called simple. The expressions \ref{eq1} and \ref{eq2} show that the function $p(z)=(z-a)^mf (z)$ if $a\neq\infty$, or $p(z)=z^{-m}f(z)$ if $a=\infty$, can be [[Analytic continuation|analytically continued]] to a full neighbourhood of the pole $a$, and, moreover, $p(a) \neq 0$. Alternatively, a pole $a$ of order $m$ can also be characterized by the fact that the function $1/f(z)$ has a zero of multiplicity $m$ at $a$.
  
<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/p/p073/p073490/p07349014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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A point $a=(a_1,\ldots,a_n)$ of the complex space $\C^n$, $n\geq2$, is called a pole of the analytic function $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$ if the following conditions are satisfied: 1) $f(z)$ is holomorphic everywhere in some neighbourhood $U$ of $a$ except at a set $P \subset U$, $a \in P$; 2) $f(z)$ cannot be analytically continued to any point of $P$; and 3) there exists a function $q(z) \not\equiv 0$, holomorphic in $U$, such that the function $p(z) = q(z)f(z)$, which is holomorphic in $U \setminus P$, can be holomorphically continued to the full neighbourhood $U$, and, moreover, $p(a) \neq 0$. Here also
 
+
$$
with finitely many negative exponents if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349015.png" />, or, respectively, finitely many positive exponents if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349016.png" />. The natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349017.png" /> in these expressions is called the order, or multiplicity, of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349018.png" />; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349019.png" /> the pole is called simple. The expressions (1) and (2) show that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349021.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349022.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349023.png" />, can be analytically continued (cf. [[Analytic continuation|Analytic continuation]]) to a full neighbourhood of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349024.png" />, and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349025.png" />. Alternatively, a pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349026.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349027.png" /> can also be characterized by the fact that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349028.png" /> has a zero of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349030.png" />.
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\lim_{z\rightarrow a}f(z) =  
 
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\lim_{z\rightarrow a}\frac{p(z)}{q(z)} = \infty;
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349031.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349033.png" />, is called a pole of the analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349034.png" /> of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349035.png" /> if the following conditions are satisfied: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349036.png" /> is holomorphic everywhere in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349038.png" /> except at a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349040.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349041.png" /> cannot be analytically continued to any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349042.png" />; and 3) there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349043.png" />, holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349044.png" />, such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349045.png" />, which is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349046.png" />, can be holomorphically continued to the full neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349047.png" />, and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349048.png" />. Here also
+
$$
 
+
however, for $n \geq 2$, poles, as with singular points in general, cannot be isolated.
<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/p/p073/p073490/p07349049.png" /></td> </tr></table>
 
 
 
however, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349050.png" />, poles, as with singular points in general, cannot be isolated.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
 
  
 +
====Comments====
  
====Comments====
+
For $n=1$ see {{Cite|Ah}}. For $n \geq 2$ see {{Cite|GrFr}}, {{Cite|Ra}}.
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349051.png" /> see [[#References|[a1]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073490/p07349052.png" /> see [[#References|[a2]]]–[[#References|[a3]]].
 
  
For the use of poles in the representation of analytic functions see [[Integral representation of an analytic function|Integral representation of an analytic function]]; [[Cauchy integral|Cauchy integral]].
+
For the use of poles in the representation of analytic functions see [[Integral representation of an analytic function]]; [[Cauchy integral]].
  
====References====
+
====References====  
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"L.V. Ahlfors,   "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"H. Grauert,   K. Fritzsche,   "Several complex variables" , Springer (1976) (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 1, Sect. 3</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ah}}||valign="top"| L.V. Ahlfors, "Complex analysis", McGraw-Hill (1979) pp. Chapt. 8 {{MR|0510197}} {{ZBL|0395.30001}}
 +
|-
 +
|valign="top"|{{Ref|GrFr}}||valign="top"| H. Grauert, K. Fritzsche, "Several complex variables", Springer (1976) (Translated from German) {{MR|0414912}}  {{ZBL|0381.32001}}
 +
|-
 +
|valign="top"|{{Ref|Ra}}||valign="top"| R.M. Range, "Holomorphic functions and integral representation in several complex variables", Springer (1986) pp. Chapt. 1, Sect. 3 {{MR|0847923}} 
 +
|-
 +
|valign="top"|{{Ref|Sh}}||valign="top"| B.V. Shabat, "Introduction of complex analysis", '''2''', Moscow (1976) (In Russian)  {{ZBL|0799.32001}}
 +
|-
 +
|}

Latest revision as of 09:10, 18 January 2014

2020 Mathematics Subject Classification: Primary: 30-XX [MSN][ZBL] $ \newcommand{\abs}[1]{\left| #1 \right|} \newcommand{\set}[1]{\left\{ #1 \right\}} $

The pole of a function is an isolated singular point $a$ of single-valued character of an analytic function $f(z)$ of the complex variable $z$ for which $\abs{f(z)}$ increases without bound when $z$ approaches $a$: $\lim_{z\rightarrow a} f(z) = \infty$. In a sufficiently small punctured neighbourhood $V=\set{z\in\C : 0 < \abs{z-a} < r}$ of the point $a \neq \infty$, or $V'=\set{z\in\C : r < \abs{z} < \infty}$ in the case of the point at infinity $a=\infty$, the function $f(z)$ can be written as a Laurent series of special form: \begin{equation} \label{eq1} f(z) = \sum_{k=-m}^\infty c_k (z-a)^k,\qquad a \neq \infty, c_{-m} \neq 0, z \in V, \end{equation} or, respectively, \begin{equation} \label{eq2} f(z) = \sum_{k=-m}^\infty \frac{c_k}{z^k},\qquad a = \infty, c_{-m} \neq 0, z \in V', \end{equation} with finitely many negative exponents if $a\neq\infty$, or, respectively, finitely many positive exponents if $a=\infty$. The natural number $m$ in these expressions is called the order, or multiplicity, of the pole $a$; when $m=1$ the pole is called simple. The expressions \ref{eq1} and \ref{eq2} show that the function $p(z)=(z-a)^mf (z)$ if $a\neq\infty$, or $p(z)=z^{-m}f(z)$ if $a=\infty$, can be analytically continued to a full neighbourhood of the pole $a$, and, moreover, $p(a) \neq 0$. Alternatively, a pole $a$ of order $m$ can also be characterized by the fact that the function $1/f(z)$ has a zero of multiplicity $m$ at $a$.

A point $a=(a_1,\ldots,a_n)$ of the complex space $\C^n$, $n\geq2$, is called a pole of the analytic function $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$ if the following conditions are satisfied: 1) $f(z)$ is holomorphic everywhere in some neighbourhood $U$ of $a$ except at a set $P \subset U$, $a \in P$; 2) $f(z)$ cannot be analytically continued to any point of $P$; and 3) there exists a function $q(z) \not\equiv 0$, holomorphic in $U$, such that the function $p(z) = q(z)f(z)$, which is holomorphic in $U \setminus P$, can be holomorphically continued to the full neighbourhood $U$, and, moreover, $p(a) \neq 0$. Here also $$ \lim_{z\rightarrow a}f(z) = \lim_{z\rightarrow a}\frac{p(z)}{q(z)} = \infty; $$ however, for $n \geq 2$, poles, as with singular points in general, cannot be isolated.

Comments

For $n=1$ see [Ah]. For $n \geq 2$ see [GrFr], [Ra].

For the use of poles in the representation of analytic functions see Integral representation of an analytic function; Cauchy integral.

References

[Ah] L.V. Ahlfors, "Complex analysis", McGraw-Hill (1979) pp. Chapt. 8 MR0510197 Zbl 0395.30001
[GrFr] H. Grauert, K. Fritzsche, "Several complex variables", Springer (1976) (Translated from German) MR0414912 Zbl 0381.32001
[Ra] R.M. Range, "Holomorphic functions and integral representation in several complex variables", Springer (1986) pp. Chapt. 1, Sect. 3 MR0847923
[Sh] B.V. Shabat, "Introduction of complex analysis", 2, Moscow (1976) (In Russian) Zbl 0799.32001
How to Cite This Entry:
Pole (of a function). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pole_(of_a_function)&oldid=15756
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article