Removable Singularity

(Theory)

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What is Removable singularity?

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point. For instance, the (unnormalized) sinc function has a singularity at z = 0. This singularity can be removed by defining , which is the limit of as z tends to 0. The resulting function is holomorphic. In this case the problem was caused by being given an indeterminate form. Taking a power series expansion for around the singular point shows that Formally, if is an open subset of the complex plane , a point of , and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on . We say is holomorphically extendable over if such a exists.

Technology Types

analytic functionbernhard riemanndifferential equationequationfunctionmathematical relationmathematical statementmathematical theoremmeromorphic functionordinary differential equationpropositionstatementtheoremtheory

Synonyms

Removable singular pointRiemann's theorem on removable singularities

Translations

Ophefbare singulariteit (nl)Riemannscher Hebbarkeitssatz (de)Singularidade removível (pt)Устранимая особая точка (ru)Усувна особлива точка (uk)可去奇点 (zh)可除特異点 (ja)

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