About: Analyticity of holomorphic functions

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In complex analysis 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

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rdfs:label	Analyticity of holomorphic functions (en)
rdfs:comment	In complex analysis 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 (en)
sameAs	Analyticity of holomorphic functions Analyticity of holomorphic functions
dbp:wikiPageUsesTemplate	dbt:Short_description dbt:Planetmath_reference dbt:Complex_analysis_sidebar
Subject	Analytic functions Theorems in complex analysis Article proofs
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions?oldid=916962772&ns=0
Wikipage page ID	481100 (xsd:integer)
page length (characters) of wiki page	5387 (xsd:nonNegativeInteger)
Wikipage revision ID	916962772 (xsd:integer)
Link from a Wikipage to another Wikipage	Analytic function Radius of convergence Taylor series Bump function Analytic functions Theorems in complex analysis Cauchy's integral formula Differentiable function Disk (mathematics) Power series Complex analysis Complex number Connected space Convergent series Holomorph Partition of unity Article proofs Entire function Function (mathematics) Identity theorem Weierstrass M-test Limit point Singularity (mathematics)
has abstract	In complex analysis 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 S with an accumulation point inside the intersection of their domains also agree everywhere in every connected open subset of their domains that contains the set S, 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. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions
is Wikipage redirect of	All holomorphic functions are analytic Analytic nature of holomorphic functions Holomorphic functions are analytic Proof of Taylor Series for complex and real variables Proof that holomorphic functions are analytic
is Link from a Wikipage to another Wikipage of	Non-analytic smooth function Analytic function Morera's theorem Cauchy's integral formula Cauchy–Riemann equations Lee–Yang theory Liouville's theorem (complex analysis) List of mathematical proofs Complex plane Holomorph List of complex analysis topics Identity theorem All holomorphic functions are analytic Analytic nature of holomorphic functions Holomorphic functions are analytic
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions

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