Definition:Isolated Singularity

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Definition

Complex Function

Let $U \subseteq \C$ be an open set.

Let $f : U \to \C$ be a holomorphic function.

An isolated singularity of $f$ is a point $z_0 \in \C$ for which $U$ is a punctured neighborhood.

Riemann Surface

Let $U$ be an open set of a Riemann surface.

Let $z_0 \in U$.

Let $f: U \setminus \set {z_0} \to \C$ be a holomorphic function.

Then $f$ has an isolated singularity at $z_0$.

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(Equivalently, an isolated singularity is an isolated point of the complement of the domain of definition of $f$.)

Types of isolated singularities

Removable Singularity

The isolated singularity $z_0$ is called removable if and only if $f$ can be extended to a holomorphic function $f: U \to \C$.

Pole

Let $U$ be an open subset of a Riemann surface.

Let $z_0 \in U$.

Let $f: U \setminus \set {z_0} \to \C$ be a holomorphic function.

Definition 1

Let $z_0$ be an isolated singularity of $f$.

Then $z_0$ is a pole if and only if:

$\ds \lim_{z \mathop \to z_0} \cmod {\map f z} \to \infty$

Definition 2

Let $z_0$ be an isolated singularity of $f$.

Then $z_0$ is a pole if and only if $f$ can be written in the form:

$\map f z = \dfrac {\map \phi z} {\paren {z - z_0}^k}$

where:

$\phi$ is analytic at $z_0$

$\map \phi {z_0} \ne 0$

$k \in \Z$ such that $k \ge 1$.

Essential Singularity

An isolated singularity $z_0$ which is neither a removable singularity nor a pole is called an essential singularity.

Note that the first two cases can be combined by saying that $f$ extends to a meromorphic function on $U$.

Characterization using Laurent series

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If $U \subset \C$, let:

$\ds \map f z = \sum_{j \mathop = -\infty}^\infty a_j \paren {z - z_0}^j$

be the Laurent series expansion of $f$ near $z_0$.

Then:

$z_0$ is an isolated singularity if and only if $a_j = 0$ for $j < 0$

$z_0$ is a pole if and only if there are at least one but at most finitely many nonzero coefficients $a_j$ with $j < 0$

$z_0$ is an essential singulary if and only if there are infinitely many nonzero coefficients $a_j$ with $j < 0$.

In particular, if $f: \C \to \C$ is an entire function, then:

$\infty$ is a removable singularity if and only if $f$ is constant

$\infty$ is a pole if and only if $f$ is a (complex) polynomial function

$\infty$ is an essential singularity if and only if $f$ is a transcendental entire function.

Equivalent characterizations

By the Riemann Removable Singularities Theorem and the Big Picard Theorem, we can say the following:

$z_0$ is a removable singularity if and only if $f$ is bounded near $z_0$

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$z_0$ is an essential singularity if and only if, for every value $a \in \C$ with at most one exception, every neighborhood of $z_0$ contains a preimage of $a$ under $f$.

Isolated singularities of meromorphic functions

We can analogously define and classify isolated singularities of meromorphic functions.

However, note that in general a meromorphic function does not have a Laurent series expansion near an essential singularity.

Also see

• Results about isolated singularities can be found here.