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Concept
Isolated singularity
Summary
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z0 is an isolated singularity of a function f if there exists an open disk D centered at z0 such that f is holomorphic on D \ {z0}, that is, on the set obtained from D by taking z0 out.
Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function is any isolated point of the boundary of the domain . In other words, if is an open subset of , and is a holomorphic function, then is an isolated singularity of .
Every singularity of a meromorphic function on an open subset is isolated, but isolation of singularities alone is not sufficient to guarantee a function is meromorphic. Many important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the function be isolated.
There are three types of isolated singularities: removable singularities, poles and essential singularities.
The function has 0 as an isolated singularity.
The cosecant function has every integer as an isolated singularity.
Other than isolated singularities, complex functions of one variable may exhibit other singular behavior. Namely, two kinds of nonisolated singularities exist:
Cluster points, i.e. limit points of isolated singularities: if they are all poles, despite admitting Laurent series expansions on each of them, no such expansion is possible at its limit.
Natural boundaries, i.e. any non-isolated set (e.g. a curve) around which functions cannot be analytically continued (or outside them if they are closed curves in the Riemann sphere).
The function is meromorphic on , with simple poles at , for every . Since , every punctured disk centered at has an infinite number of singularities within, so no Laurent expansion is available for around , which is in fact a cluster point of its poles.
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