Definition:Isolated Singularity

Let $$U$$ be an open subset of a Riemann surface, and let $$z_0\in U$$. A holomorphic function $$f:U\setminus\{z_0\}\to \C$$ is said to have 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 $$f$$.)

Types of isolated singularities

 * The isolated singularity $$z_0$$ is called removable if $$f$$ can be extended to a holomorphic function $$f:U\to\C$$.
 * The isolated singularity $$z_0$$ is called a pole if $$\lim_{z\to z_0} |f(z)| = \infty$$.
 * Otherwise, $$z_0$$ 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
 * $$f(z)=\sum_{j=-\infty}^{\infty} a_j (z-z_0)^j$$

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 polynomial.
 * $$\infty$$ 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 $$z_0$$.
 * $$z_0$$ 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.