View source for Pole (of a function)

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An isolated [[Singular point|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,\quad
\text{$a \neq \infty$, $c_{-m} \neq 0$, $z \in V$},
\end{equation}
or, respectively,
\begin{equation}
\label{eq2}
f(z) = \sum_{k=-m}^\infty \frac{c_k}{z^k},\quad
\text{$a = \infty$, $c_{-m} \neq 0$, $z \in V'$},
\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 $a$.

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.

====References==== 

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|valign="top"|{{Ref|Sh}}||valign="top"| B.V. Shabat, "Introduction of complex analysis", '''2''', Moscow (1976) (In Russian)
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====Comments==== 

For $n=1$ see {{Cite|Ah}}. For $n \geq 2$ see {{Cite|GrFr}}, {{Cite|Ra}}.

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

====References==== 

{|
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|valign="top"|{{Ref|Ah}}||valign="top"| L.V. Ahlfors, "Complex analysis", McGraw-Hill (1979) pp. Chapt.  8
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|valign="top"|{{Ref|GrFr}}||valign="top"| H.  Grauert, K. Fritzsche, "Several complex variables", Springer (1976) (Translated from German)
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|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
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