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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Formally, a function <math>f</math> is ''real analytic'' on an [[open set]] <math>D</math> in the [[real line]] if for any <math>x_0\in D</math> one can write

:<math display="block">

f(x) = \sum_{n=0}^\infty a_{n} \left( x-x_0 \right)^{n} = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots

</math>

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Alternatively, a real analytic function is an [[smooth function|infinitely differentiable function]] such that the [[Taylor series]] at any point <math>x_0</math> in its ___domain

:<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]].

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>

:<math display="block"> \sup_{x \in K} \left | \frac{\partial^\alpha f}{\partial x^\alpha}(x) \right | \leq C^{|\alpha|+1}\alpha!</math>

==Properties of analytic functions==

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According to [[Liouville's theorem (complex analysis)|Liouville's theorem]], any bounded complex analytic function defined on the whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by

:<math display="block">f(x)=\frac{1}{x^2+1}.</math>

Also, if a complex analytic function is defined in an open [[Ball (mathematics)|ball]] around a point ''x''₀, its power series expansion at ''x''₀ is convergent in the whole open ball ([[analyticity of holomorphic functions|holomorphic functions are analytic]]). This statement for real analytic functions (with open ball meaning an open [[interval (mathematics)|interval]] of the real line rather than an open [[disk (mathematics)|disk]] of the complex plane) is not true in general; the function of the example above gives an example for ''x''₀&nbsp;=&nbsp;0 and a ball of radius exceeding&nbsp;1, since the power series {{nowrap|1 − ''x''² + ''x''⁴ − ''x''⁶...}} diverges for |''x''|&nbsp;≥&nbsp;1.

Any real analytic function on some [[open set]] on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function ƒ''f''(''x'') defined in the paragraph above is a counterexample, as it is not defined for ''x''&nbsp;=&nbsp;±''i''. This explains why the Taylor series of ƒ''f''(''x'') diverges for |''x''|&nbsp;>&nbsp;1, i.e., the [[radius of convergence]] is 1 because the complexified function has a [[Complex pole|pole]] at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.

==Analytic functions of several variables==

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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==