Isolated singularity

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 Template:Tmath is an isolated singularity of a function Template:Tmath if there exists an open disk Template:Tmath centered at Template:Tmath such that f is holomorphic on Template:Tmath, that is, on the set obtained from Template:Tmath by removing Template:Tmath .

Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function Template:Tmath is any isolated point of the boundary $\partial \mathrm{\Omega }$ of the domain Template:Tmath. In other words, if $U$ is an open subset of Template:Tmath, Template:Tmath and Template:Tmath is a holomorphic function, then $a$ is an isolated singularity of Template:Tmath.

Every singularity of a meromorphic function on an open subset $U\subset \mathbb{ℂ}$ 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. Isolated singularities may be classified into three distinct types: removable singularities, poles and essential singularities.

Examples

• The function Template:Tmath has Template:Tmath as an isolated singularity.
• The cosecant function Template:Tmath has every integer as an isolated singularity.

Nonisolated singularities

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).

Examples

File:Natural boundary example.gif

The natural boundary of this power series is the unit circle (read examples).

• The function $\tan \left(\frac{1}{z}\right)$ is meromorphic on Template:Tmath, with simple poles at Template:Tmath, for every Template:Tmath. Since Template:Tmath, every punctured disk centered at $0$ has an infinite number of singularities within it, so no Laurent expansion is available for Template:Tmath around Template:Tmath, which is in fact a cluster point of its poles.
• The function Template:Tmath has a singularity at Template:Tmath that is not isolated, since there are additional singularities at the reciprocal of every integer, which are located arbitrarily close to Template:Tmath (though the singularities at these reciprocals are themselves isolated).
• The function defined via the Maclaurin series Template:Tmath converges inside the open unit disk centred at Template:Tmath and has the unit circle as its natural boundary.