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

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