Definition:Isolated Singularity

Riemann Surface
(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.