Let U be a domain in the complex numbersMathworldPlanetmathPlanetmath (resp., real numbers). A function f:U (resp., f:U) is analyticPlanetmathPlanetmath (resp., real analytic) if f has a Taylor seriesMathworldPlanetmath about each point xU that converges to the function f in an open neighborhood of x.

1 On Analyticity and Holomorphicity

A complex function is analytic if and only if it is holomorphic. Because of this equivalence, an analytic function in the complex case is often defined to be one that is holomorphic, instead of one having a Taylor series as above. Although the two definitions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, it is not an easy matter to prove their equivalence, and a reader who does not yet have this result available will have to pay attention as to which definition of analytic is being used.

Title analytic
Canonical name Analytic
Date of creation 2013-03-22 12:04:36
Last modified on 2013-03-22 12:04:36
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 30B10
Classification msc 26A99
Synonym real analytic
Synonym real analytic function
Synonym complex analytic
Synonym complex analytic function
Synonym analytic function
Related topic Holomorphic
Related topic TaylorSeries
Related topic CauchyKowalewskiTheorem