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In complex analysis, not every singularity is equally bad. A function such as $1/z$ fails to be [holomorphic](/page/Holomorphic%20Function) at $0$, but its failure is completely organized: multiplying by $z$ repairs it. By contrast, $e^{1/z}$ has infinitely many negative powers near $0$, and no finite correction captures its behaviour. A pole isolates the first kind of singularity: a point where a holomorphic function blows up like a finite negative power.

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Poles are the singularities that make [Meromorphic Function](/page/Meromorphic%20Function) theory possible. They are also the local source of residues, the obstruction measured by contour integrals, and the singularities controlled by a [Laurent Series](/page/Laurent%20Series). The concept is useful because it turns local blow-up into finite algebraic data: an integer order and finitely many principal coefficients.

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## Definition

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Throughout this page, $B(a,r)=\{z\in\mathbb{C}: |z-a|<r\}$ denotes the open disk of radius $r$ centered at $a$. A punctured neighbourhood of $a$ means a set of the form $0<|z-a|<r$, obtained from such a disk by deleting its center. This is the natural local domain for studying an isolated singularity: the function is defined near $a$, but not necessarily at $a$ itself.

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The defining feature of a pole is not merely that a function is undefined at one point, but that the failure is repaired by multiplying by a finite power of the distance from that point. This distinguishes a pole from both a removable singularity, where no negative power is needed, and an essential singularity, where no finite power is enough.

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[definition: Pole] Let $U \subset \mathbb{C}$ be open, let $a \in U$, and let $f: U \setminus \{a\} \to \mathbb{C}$ be holomorphic. The point $a$ is a pole of $f$ if there exists a positive integer $m$, a radius $r > 0$ with $B(a,r) \subset U$, and a holomorphic function $g: B(a,r) \to \mathbb{C}$ such that $g(a) \ne 0$ and \begin{align*} f(z) &= \frac{g(z)}{(z-a)^m} \end{align*} for all $z \in B(a,r) \setminus \{a\}$. [/definition]

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This definition already contains the main local data: the singularity is finite, and the exponent $m$ is a positive integer measuring the amount of finite blow-up. The supporting notions used to compute that exponent are collected next, so the primary definition remains separate from the surrounding bookkeeping.

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[example: Simple Pole at the Origin] The function $f(z)=1/z$ on $\mathbb{C}\setminus\{0\}$ has a pole at $0$. Indeed, take $m=1$ and $g(z)=1$ in the definition, so \begin{align*} f(z) &= \frac{g(z)}{z} \end{align*} for every $z\ne 0$. The principal part is the single term $z^{-1}$, so this is a simple pole with residue $1$. [/example]

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Here the residue means the coefficient of $(z-a)^{-1}$ in the Laurent expansion about the pole $a$. This page uses that coefficient-level convention when discussing contour integrals; no separate global residue theory is needed for the local computations below.

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## Supporting Local Notions

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Laurent expansions and local contour integrals require a punctured disc around the point with no other singularities interfering. When singularities accumulate, there is no annulus on which the local Laurent machinery can be applied to one point at a time, so isolation must be stated relative to the whole singular set.

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[definition: Isolated Singularity] Let $U \subset \mathbb{C}$ be open, let $S \subset U$, let $a \in S$, and let $f: U \setminus S \to \mathbb{C}$ be holomorphic. The point $a$ is an isolated singularity of $f$ if there exists $r > 0$ such that $B(a,r) \subset U$ and \begin{align*} B(a,r) \cap S &= \{a\}. \end{align*} [/definition]

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The word isolated matters because if singular points accumulate at $a$, the local annulus on which Laurent theory works may not exist. For a pole, the punctured disc is the stage on which the finite blow-up can be factored cleanly.

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To compare two poles, or to compute residues and contour integrals, it is not enough to know that some exponent works. The exact exponent is the local invariant that measures the strength of the singularity. The function $g$ is the holomorphic part left after the finite blow-up has been factored out, and the condition $g(a) \ne 0$ prevents overestimating the exponent.

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[definition: Order of a Pole] Let $U \subset \mathbb{C}$ be open, let $a \in U$, and let $f: U \setminus \{a\} \to \mathbb{C}$ be holomorphic. Suppose $a$ is a pole of $f$. The order of the pole at $a$ is the unique positive integer $m$ for which there exists a radius $r > 0$ with $B(a,r) \subset U$ and a holomorphic function $g: B(a,r) \to \mathbb{C}$ with $g(a) \ne 0$ such that \begin{align*} f(z) &= \frac{g(z)}{(z-a)^m} \end{align*} for all $z \in B(a,r) \setminus \{a\}$. [/definition]

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To use a pole in calculations, the order alone is not always enough; contour integrals and local subtraction formulas require the actual negative-power terms. Laurent expansion gives a compact way to collect exactly the terms responsible for the singularity and separate them from the holomorphic remainder. A pole of order $1$ is called a simple pole, while higher order poles have a longer finite singular part.

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[definition: Principal Part at a Pole] Let $U \subset \mathbb{C}$ be open, let $a \in U$, and let $f: U \setminus \{a\} \to \mathbb{C}$ be holomorphic. Suppose $a$ is a pole of order $m$ and suppose the Laurent expansion of $f$ about $a$ is \begin{align*} f(z) &= \sum_{k=-m}^{\infty} c_k (z-a)^k \end{align*} for $0 < |z-a| < r$, where $r > 0$ and $B(a,r) \subset U$. The principal part of $f$ at $a$ is \begin{align*} \sum_{k=-m}^{-1} c_k (z-a)^k. \end{align*} [/definition]

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Quotients force a second local measurement: not only how a function blows up, but how another holomorphic function vanishes. The cancellation between numerator and denominator is controlled by the order of a zero, so pole computations need a precise definition of that order.

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[definition: Zero of Order] Let $U \subset \mathbb{C}$ be open, let $a \in U$, and let $h: U \to \mathbb{C}$ be holomorphic. The function $h$ has a zero of order $m$ at $a$ if $m$ is a positive integer and there exists a radius $r > 0$ with $B(a,r) \subset U$ and a holomorphic function $q: B(a,r) \to \mathbb{C}$ such that $q(a) \ne 0$ and \begin{align*} h(z) &= (z-a)^m q(z) \end{align*} for all $z \in B(a,r)$. [/definition]

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For global complex analysis, singularities usually occur at many points rather than just one, so the next useful class consists of functions whose singularities are all controlled in the finite Laurent sense. This class should allow algebraic operations while excluding essential singularities. The definition of a meromorphic function supplies that global framework.

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