Lesson plan
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 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.
Article 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 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.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.