Analytic function: Difference between revisions

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{{shortShort description|FunctionType locallyof givenfunction byin a convergent power seriesmathematics}}

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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 its [[Taylor series]] about ''x''₀ converges to the function in some [[neighborhood (topology)|neighborhood]] for every ''x''₀ in its [[Domain of a function|___domain]].

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.

== Definitions ==

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 + a_3 (x-x_0)^3 + \cdots

:<math>

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 + a_3 (x-x_0)^3 + \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 ==

Typical examples of analytic functions are:

* AllThe following [[elementary function]]s:

** All [[polynomial]]s: if a polynomial has degree ''n'', any terms of degree larger than ''n'' in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own [[Maclaurin series]].

** The [[exponential function]] is analytic. Any Taylor series for this function converges not only for ''x'' close enough to ''x''₀ (as in the definition) but for all values of ''x'' (real or complex).

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** [[gamma function]]s

Typical examples of functions that are not analytic are:

* The [[absolute value]] function when defined on the set of real numbers or [[complex numbersnumber]]s is not everywhere analytic because it is not differentiable at 0.
* [[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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The following conditions are equivalent:

1. #<math>f</math> is real analytic on an open set <math>D</math>.

2. #There is a complex analytic extension of <math>f</math> to an open set <math>G \subset \mathbb{C}</math> which contains <math>D</math>.

3. #<math>f</math> is real 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.

2. There is a complex analytic extension of <math>f</math> to an open set <math>G \subset \mathbb{C}</math> which contains <math>D</math>.

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

3. <math>f</math> is real 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}}

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>.

::<math> \left | \frac{d^k f}{dx^k}(x) \right | \leq C^{k+1} k!</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{d\partial^k\alpha f}{dx\partial x^k\alpha}(x) \right | \leq C^{k|\alpha|+1} k\alpha!</math>

Complex analytic functions are exactly equivalent to [[Holomorphic function|holomorphic functions]], 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>.

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> \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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* The [[Multiplicative inverse|reciprocal]] of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose [[derivative]] is nowhere zero. (See also the [[Lagrange inversion theorem]].)

* Any analytic function is [[smooth function|smooth]], that is, infinitely differentiable. The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. For the complex numbers, the converse does hold, and in fact any function differentiable ''once'' on an open set is analytic on that set (see "analyticity and differentiability" below).

* For any [[open set]] Ω&nbsp;⊆&nbsp;'''<math>\Omega \subseteq \mathbb{C'''}</math>, the set ''A''(Ω) of all analytic functions ''<math>u''&nbsp;:&nbsp;Ω&nbsp;→&nbsp;'''\Omega \to \mathbb{C'''}</math> is a [[Fréchet space]] with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of [[Morera's theorem]]. The set <math>\scriptstyle A_\infty(\Omega)</math> of all [[bounded function|bounded]] analytic functions with the [[supremum norm]] is a [[Banach space]].

A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function ƒ has an [[accumulation point]] inside its [[___domain of a function|___domain]], then ƒ is zero everywhere on the [[connected space|connected component]] containing the accumulation point. In other words, if (''r_n'') is a [[sequence]] of distinct numbers such that ƒ(''r''_''n'')&nbsp;=&nbsp;0 for all ''n'' and this sequence [[limit of a sequence|converges]] to a point ''r'' in the ___domain of ''D'', then ƒ is identically zero on the connected component of ''D'' containing ''r''. This is known as the [[Identityidentity Theoremtheorem]].

Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component.

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==Analyticity and differentiability==

As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or ''C<supmath>∞\mathcal{C}^{\infty}</supmath>''). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see [[non-analytic smooth function]]. In fact there are many such functions.

The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that [[proof that holomorphic functions are analytic|any complex function differentiable (in the complex sense) in an open set is analytic]]. Consequently, in [[complex analysis]], the term ''analytic function'' is synonymous with ''[[holomorphic function]]''.

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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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*[[Quasi-analytic function]]

*[[Infinite compositions of analytic functions]]

*[[Non-analytic smooth function]]

==Notes==

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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==

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* [https://web.archive.org/web/20130615052245/http://ivisoft.org/index.php/software/8-soft/6-zersol Solver for all zeros of a complex analytic function that lie within a rectangular region by Ivan B. Ivanov]

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