Definition:Isolated Singularity

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Complex Functions

Let $U \subset \C$ 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 $f: U \setminus \left\{{z_0}\right\} \to \C$ 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 $f$.)


Types of isolated singularities

Removable Singularity

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


Pole

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

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

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


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

$\displaystyle f \left({z}\right) = \sum_{j = -\infty}^{\infty} a_j \left({z - z_0}\right)^j$

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 $z_0$.
$z_0$ 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.