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.
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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 $f$ 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.
