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- Analyticity_of_holomorphic_functions abstract "In complex analysis, a branch of mathematics, a complex-valued function ƒ of a complex variable z:is said to be holomorphic at a point a if it is differentiable at every point within some open disk centered at a, and is said to be analytic at a if in some open disk centered at a it can be expanded as a convergent power series(this implies that the radius of convergence is positive).One of the most important theorems of complex analysis is that holomorphic functions are analytic. Among the corollaries of this theorem are the identity theorem that two holomorphic functions that agree at every point of an infinite set with an accumulation point inside the intersection of their domains also agree everywhere in some open set, and the fact that, since power series are infinitely differentiable, so are holomorphic functions (this is in contrast to the case of real differentiable functions), and the fact that the radius of convergence is always the distance from the center a to the nearest singularity; if there are no singularities (i.e., if ƒ is an entire function), then the radius of convergence is infinite. Strictly speaking, this is not a corollary of the theorem but rather a by-product of the proof. no bump function on the complex plane can be entire. In particular, on any connected open subset of the complex plane, there can be no bump function defined on that set which is holomorphic on the set. This has important ramifications for the study of complex manifolds, as it precludes the use of partitions of unity. In contrast the partition of unity is a tool which can be used on any real manifold.".
- Analyticity_of_holomorphic_functions wikiPageID "481100".
- Analyticity_of_holomorphic_functions wikiPageRevisionID "577877992".
- Analyticity_of_holomorphic_functions hasPhotoCollection Analyticity_of_holomorphic_functions.
- Analyticity_of_holomorphic_functions id "3298".
- Analyticity_of_holomorphic_functions title "Existence of power series".
- Analyticity_of_holomorphic_functions subject Category:Analytic_functions.
- Analyticity_of_holomorphic_functions subject Category:Article_proofs.
- Analyticity_of_holomorphic_functions subject Category:Theorems_in_complex_analysis.
- Analyticity_of_holomorphic_functions type Abstraction100002137.
- Analyticity_of_holomorphic_functions type AnalyticFunctions.
- Analyticity_of_holomorphic_functions type ArticleProofs.
- Analyticity_of_holomorphic_functions type Cognition100023271.
- Analyticity_of_holomorphic_functions type Communication100033020.
- Analyticity_of_holomorphic_functions type Evidence105823932.
- Analyticity_of_holomorphic_functions type Function113783816.
- Analyticity_of_holomorphic_functions type Information105816287.
- Analyticity_of_holomorphic_functions type MathematicalRelation113783581.
- Analyticity_of_holomorphic_functions type Message106598915.
- Analyticity_of_holomorphic_functions type Proof105824739.
- Analyticity_of_holomorphic_functions type Proposition106750804.
- Analyticity_of_holomorphic_functions type PsychologicalFeature100023100.
- Analyticity_of_holomorphic_functions type Relation100031921.
- Analyticity_of_holomorphic_functions type Statement106722453.
- Analyticity_of_holomorphic_functions type Theorem106752293.
- Analyticity_of_holomorphic_functions type TheoremsInComplexAnalysis.
- Analyticity_of_holomorphic_functions comment "In complex analysis, a branch of mathematics, a complex-valued function ƒ of a complex variable z:is said to be holomorphic at a point a if it is differentiable at every point within some open disk centered at a, and is said to be analytic at a if in some open disk centered at a it can be expanded as a convergent power series(this implies that the radius of convergence is positive).One of the most important theorems of complex analysis is that holomorphic functions are analytic.".
- Analyticity_of_holomorphic_functions label "Analyticity of holomorphic functions".
- Analyticity_of_holomorphic_functions sameAs m.02fnyv.
- Analyticity_of_holomorphic_functions sameAs Q4751162.
- Analyticity_of_holomorphic_functions sameAs Q4751162.
- Analyticity_of_holomorphic_functions sameAs Analyticity_of_holomorphic_functions.
- Analyticity_of_holomorphic_functions wasDerivedFrom Analyticity_of_holomorphic_functions?oldid=577877992.
- Analyticity_of_holomorphic_functions isPrimaryTopicOf Analyticity_of_holomorphic_functions.