Definition:Isolated Singularity
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Definition
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 $z_0$.
It is not clear under what conditions $z_0$ is an isolated singularity. The suggestion is that it just needs to be a point which, for whatever arbitrary reason, happens not to be included in the domain of an arbitrary holomorphic function.
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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
The isolated singularity $z_0$ is called a pole if $\displaystyle \lim_{z\to z_0} |f(z)| = \infty$.
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 iff $a_j = 0$ for $j<0$.
- $z_0$ is a pole iff there are at least one but at most finitely many nonzero coefficients $a_j$ with $j<0$.
- $z_0$ is an essential singulary iff 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 iff $f$ is constant.
- $\infty$ is a pole iff $f$ is a polynomial function.
- $\infty$ is an essential singularity iff $f$ 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 iff $f$ is bounded near $z_0$.
- $z_0$ is an essential singularity iff, 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.
