From Surf Wiki (app.surf) — the open knowledge base
Isolated singularity
Has no other singularities close to it
Has no other singularities close to it
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 is an isolated singularity of a function if there exists an open disk centered at such that f is holomorphic on , that is, on the set obtained from by removing .
Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function is any isolated point of the boundary \partial \Omega of the domain . In other words, if U is an open subset of , and is a holomorphic function, then a is an isolated singularity of .
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 has as an isolated singularity.
- The cosecant function 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
- The function \tan\left(\frac{1}{z}\right) is meromorphic on , with simple poles at , for every . Since , every punctured disk centered at 0 has an infinite number of singularities within it, so no Laurent expansion is available for around , which is in fact a cluster point of its poles.
- The function has a singularity at that is not isolated, since there are additional singularities at the reciprocal of every integer, which are located arbitrarily close to (though the singularities at these reciprocals are themselves isolated).
- The function defined via the Maclaurin series converges inside the open unit disk centred at and has the unit circle as its natural boundary.
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about Isolated singularity — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report