Definition:Isolated Singularity

Complex Functions
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 Surfaces
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 $z_0$.

In most applications, the Riemann surface in question is the complex plane or the Riemann sphere.

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

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

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
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 $a_j = 0$ for $j < 0$
 * $z_0$ is a pole there are at least one but at most finitely many nonzero coefficients $a_j$ with $j < 0$
 * $z_0$ is an essential singulary 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 $f$ is constant
 * $\infty$ is a pole $f$ is a (complex) polynomial function
 * $\infty$ is an essential singularity $f$ 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 $f$ is bounded near $z_0$


 * $z_0$ is an essential singularity, 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.