analytic

Let $U$ be a domain in the complex numbers (resp., real numbers). A function $f:U⟶ℂ$ (resp., $f:U⟶ℝ$) is analytic (resp., real analytic) if $f$ has a Taylor series about each point $x\in U$ 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 equivalent, 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