Pole (of a function)

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[edit]

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.

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