Difference between revisions of "Pole (of a function)"
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− | + | The ''pole of a function'' is | |
− | + | 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} | \begin{equation} | ||
\label{eq1} | \label{eq1} | ||
− | f(z) = \sum_{k=-m}^\infty c_k (z-a)^k,\ | + | f(z) = \sum_{k=-m}^\infty c_k (z-a)^k,\qquad a \neq \infty, c_{-m} \neq 0, z \in V, |
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\end{equation} | \end{equation} | ||
or, respectively, | or, respectively, | ||
\begin{equation} | \begin{equation} | ||
\label{eq2} | \label{eq2} | ||
− | f(z) = \sum_{k=-m}^\infty \frac{c_k}{z^k},\ | + | f(z) = \sum_{k=-m}^\infty \frac{c_k}{z^k},\qquad a = \infty, c_{-m} \neq 0, z \in V', |
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\end{equation} | \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 $a$. | 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 $a$. | ||
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however, for $n \geq 2$, poles, as with singular points in general, cannot be isolated. | however, for $n \geq 2$, poles, as with singular points in general, cannot be isolated. | ||
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====Comments==== | ====Comments==== | ||
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====References==== | ====References==== | ||
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− | |valign="top"|{{Ref|Ah}}||valign="top"| L.V. Ahlfors, "Complex analysis", McGraw-Hill (1979) pp. Chapt. 8 | + | |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}} | ||
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− | |valign="top"|{{Ref| | + | |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| | + | |valign="top"|{{Ref|Sh}}||valign="top"| B.V. Shabat, "Introduction of complex analysis", '''2''', Moscow (1976) (In Russian) {{ZBL|0799.32001}} |
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|} | |} |
Latest revision as of 10:10, 18 January 2014
2010 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 |
Pole (of a function). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pole_(of_a_function)&oldid=25728