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A function <math>f</math> defined on some subset of the real line is said to be real analytic at a point <math>x</math> if there is a neighborhood <math>D</math> of <math>x</math> on which <math>f</math> is real analytic.
The definition of a ''complex analytic function'' is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is [[Holomorphic function|holomorphic]] i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.<ref>{{cite book |quote=A function ''f'' of the complex variable ''z'' is ''analytic'' at point ''z''<sub>0</sub> if its derivative exists not only at ''z'' but at each point ''z'' in some neighborhood of ''z''<sub>0</sub>. It is analytic in a region ''R'' if it is analytic at every point in ''R''. The term ''holomorphic'' is also used in the literature to denote analyticity |
In complex analysis, a function is called analytic in an open set "U" if it is (complex) differentiable at each point in "U" and its complex derivative is continuous on "U".<ref>{{Cite book |last= Gamelin |first= Theodore W. |title=Complex Analysis |publisher=Springer |year=2004|isbn= 9788181281142}}</ref>
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* [[Piecewise|Piecewise defined]] functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
* The [[complex conjugate]] function ''z'' → ''z''* is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from <math>\mathbb{R}^{2}</math> to <math>\mathbb{R}^{2}</math>.
* Other [[non-analytic smooth function]]s, and in particular any smooth function <math>f</math> with compact support, i.e. <math>f \in \mathcal{C}^\infty_0(\R^n)</math>, cannot be analytic on <math>\R^n</math>.<ref>{{Cite book|last=Strichartz, Robert S.
==Alternative characterizations==
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In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization.<ref>{{Cite journal|last=Komatsu|first=Hikosaburo|date=1960|title=A characterization of real analytic functions|url=https://projecteuclid.org/euclid.pja/1195524081|journal=Proceedings of the Japan Academy|language=EN|volume=36|issue=3|pages=90–93|doi=10.3792/pja/1195524081|issn=0021-4280|doi-access=free}}</ref> Let <math>U \subset \R^n</math> be an open set, and let <math>f: U \to \R</math>.
Then <math>f</math> is real analytic on <math>U</math> if and only if <math>f \in C^\infty(U)</math> and for every compact <math>K \subseteq U</math> there exists a constant <math>C</math> such that for every multi-index <math>\alpha \in \Z_{\geq 0}^n</math> the following bound holds<ref>{{Cite web|title=Gevrey class - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Gevrey_class#References|access-date=2020-08-30|website=encyclopediaofmath.org}}</ref>
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