Analytic function: Difference between revisions

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{{about|both real and complex analytic functions|analytic functions in complex analysis specifically|holomorphic function|analytic functions in SQL|Window function (SQL)}}

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In [[mathematics]], an '''analytic function''' is a [[function (mathematics)|function]] that is locally given by a [[convergent series|convergent]] [[power series]]. There exist both '''real analytic functions''' and '''complex analytic functions'''. Functions of each type are [[smooth function|infinitely differentiable]], but complex analytic functions exhibit properties that do not generally hold for real analytic functions.

A function is analytic if and only if for every <math> x_0 </math> in its [[Domain of a function|___domain]], its [[Taylor series]] about <math> x_0 </math> converges to the function in some [[neighborhood (topology)|neighborhood]] of <math> x_0 </math>. This is stronger than merely being [[smoothness|infinitely differentiable]] at <math> x_0 </math>, and therefore having a well-defined Taylor series; the [[Fabius function]] provides an example of a function that is infinitely differentiable but not analytic.

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<math display="block"> T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n}</math>

converges to <math>f(x)</math> for <math>x</math> in a neighborhood of <math>x_0</math> [[pointwise convergence| pointwise]].{{efn|This implies [[uniform convergence]] as well in a (possibly smaller) neighborhood of <math>x_0</math>.}} The set of all real analytic functions on a given set <math>D</math> is often denoted by <math>\mathcal{C}^{\,\omega}(D)</math>, or just by <math>\mathcal{C}^{\,\omega}</math> if the ___domain is understood.

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''₀ if its derivative exists not only at ''z'' but at each point ''z'' in some neighborhood of ''z''₀. 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 doto denote analyticity |lastlast1=Churchill |last2=Brown |last3=Verhey |title=Complex Variables and Applications |publisher=McGraw-Hill |year=1948 |isbn=0-07-010855-2 |page=[https://archive.org/details/complexvariable00chur/page/46 46] |url-access=registration |url=https://archive.org/details/complexvariable00chur/page/46 }}</ref>

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>

== Examples ==

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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''&nbsp;&rarr; ''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.|url=https://www.worldcat.org/oclc/28890674|title=A guide to distribution theory and Fourier transforms|date=1994|publisher=CRC Press|isbn=0-8493-8273-4|___location=Boca Raton|oclc=28890674}}</ref>

==Alternative characterizations==

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#<math>f</math> is smooth and for every [[compact set]] <math>K \subset D</math> there exists a constant <math>C</math> such that for every <math>x \in K</math> and every non-negative integer <math>k</math> the following bound holds{{sfn |Krantz |Parks |2002|p=15}} <math display="block"> \left| \frac{d^k f}{dx^k}(x) \right| \leq C^{k+1} k!</math>

Complex analytic functions are exactly equivalent to [[Holomorphicholomorphic function|holomorphic functions]]s, and are thus much more easily characterized.

For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the [[Fourier–Bros–Iagolnitzer transform]].

For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the [[Fourier–Bros–Iagolnitzer transform]].

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>.

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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*{{cite book |last=Conway |first=John B. |author-link=John B. Conway |title=Functions of One Complex Variable I |series=[[Graduate Texts in Mathematics]] 11 |publisher=Springer-Verlag |year=1978 |isbn=978-0-387-90328-6 |edition=2nd }}

*{{cite book |last1=Krantz |first1=Steven |author-link1=Steven G. Krantz |last2=Parks |first2=Harold R.|author2-link=Harold R. Parks |title=A Primer of Real Analytic Functions |edition=2nd |year=2002 |publisher=Birkhäuser |isbn=0-8176-4264-1 }}

*{{Cite book |last= Gamelin |first= Theodore W. |title=Complex Analysis |publisher=Springer |year=2004|isbn= 9788181281142}}

==External links==