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{{Short description|Type of function in mathematics}}

In [[mathematics]], an '''analytic function''' is a [[function (mathematics)|function]] that is locally given by a convergent [[power series]]. Analytic functions can be thought of as a bridge between [[polynomial]]s and general functions. There exist '''real analytic functions''' and '''complex analytic functions''', which have similarities as well as differences.

{{Distinguish|analytic expression|analytic signal}}

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

{{Complex analysis sidebar}}

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.

== Definitions ==

== Definitions ==

Formally, function ''f'' defined on an [[open set]] ''D'' in the [[real number]]s is called '''analytic''', if for any ''x''<sub>0<sub> in ''D'' one can write ''f''&nbsp;(''x'') as

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>

<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

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>

in which the coefficients ''a''<sub>0</submath>a_0, ''a''<sub>1a_1, \dots</submath>, ... are real numbers and the [[series (mathematics)|series]] is [[convergent series|convergent]] to <math>f(x)</math> for <math>x</math> in a neighborhood of <math>x_0</math>.

[[neighborhood_(topology)|neighborhood]] of ''x''<sub>0<sub>.

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

The definition of a '''complex analytic function''' is obtained by replacing everywhere above "real" with "complex". Note that a complex analytic function depends only on the complex coordinate, and not on the [[complex conjugate]]. Thus, if ''z'' is the coordinate on the [[complex plane]], then the series expansion does not contain any terms with the complex conjugate ''z''^*. For an in-depth article about complex analytic functions see [[holomorphic function]].

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

== Properties of analytic functions==

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.

* The sum, product, and composition of analytic functions are analytic.

* The [[reciprocal]] of a non-zero analytic function is analytic, and the inverse of an invertible analytic function with non-zero [[derivative]] is analytic.

* Any polynomial is an analytic function. For a polynomial, the power series expansion contains only a finite number of non-zero terms.

* Any analytic function is [[smooth function|smooth]].

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.

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 ''f'' has an [[accumulation point]] inside its [[___domain (mathematics)|___domain]], then ''f'' is zero everywhere on the [[connected space|connected]] component containing the accumulation point.

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 to denote analyticity |last1=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>

More formally this can be stated as follows. If (''r''_''n'') is a [[sequence]] of distinct numbers such that ''f''(''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 ''f'' is identically zero on the connected component of ''D'' containing ''r''.

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>

Also, if all the derivatives of an analytic function at a point are zero, the same conclusion as above holds.

== Examples ==

These statements imply that while analytic functions do have more [[degrees of freedom]] than polynomials, they are still quite inflexible.

Typical examples of analytic functions are

* The 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).

** The [[trigonometric function]]s, [[logarithm]], and the [[Exponentiation|power functions]] are analytic on any open set of their ___domain.

* Most [[special function]]s (at least in some range of the complex plane):

** [[hypergeometric function]]s

** [[Bessel function]]s

** [[gamma function]]s

Typical examples of functions that are not analytic are

== Analyticity and differentiability==

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

Any analytic function (real or complex) is differentiable, actually infinitely differentiable (that is, smooth). There exist smooth real functions which are not analytic, see the following [[an infinitely differentiable function that is not analytic|example]]. The real analytic functions are much "fewer" than the real (infinitely) differentiable functions.

* [[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.|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==

The situation is quite different for complex analytic functions. It can be proved that [[proof that holomorphic functions are analytic|any complex function differentiable in an open set is analytic]]. Consequently, in [[complex analysis]], the term ''analytic function'' is synonymous with ''[[holomorphic function]]''.

The following conditions are equivalent:

==Real versus complex analytic functions==

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

Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Complex analytic functions are more rigid in many ways.

#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>.

#<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 [[holomorphic function]]s, and are thus much more easily characterized.

According to [[Liouville's theorem (complex analysis)|Liouville's theorem]], any bounded complex analytic function defined on the whole complex plane is constant. This statement is clearly false for real analytic functions, as illustrated by

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

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

Also, if a complex analytic function is defined in an [[open ball]] around a point ''x''₀, its power series expansion at ''x''₀ is convergent in the whole ball. This is not true for in general for real analytic functions. (Note that an open ball in the complex plane would be a [[disk (mathematics)|disk]], while on the real line it would be an [[interval (mathematics)|interval]].)

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>.

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 any real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function ''f''&nbsp;(''x'') defined in the paragraph above is a counterexample.

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>

In particular, note that for a complex coordinate ''z'', the complex function that gives the real part of ''z'', namely Re ''z'' = (''z'' + ''z''^*)/2, is '''not''' analytic. This follows, as a complex analytic function cannot depend on the [[complex conjugate]]. This is true as well for the imaginary part Im ''z'' = -i (''z'' - ''z''^*)/2. In particular, one does not get a real analytic function by taking the real part of a complex analytic function.

==Properties of analytic functions==

It also follows that complex analytic functions are [[Laplace's equation|harmonic]], whereas real analytic functions in general are not. This is most easily seen by writing the [[Laplacian]] in complex coordinates:

* The sums, products, and [[function composition|compositions]] of analytic functions are analytic.

* 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]] <math>\Omega \subseteq \mathbb{C}</math>, the set ''A''(Ω) of all analytic functions <math>u:\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>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 [[identity theorem]].

:<math>\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} =

\left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right)

\left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right)

= 4 \frac{\partial}{\partial z} \frac{\partial}{\partial z^*}

</math>.

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

Since a complex analytic function does not depend on ''z''^*, the Laplacian vanishes trivially for complex analytic functions. Equivalently, this is a consequence of the [[Cauchy-Riemann equations]].

These statements imply that while analytic functions do have more [[degrees of freedom (physics and chemistry)|degrees of freedom]] than polynomials, they are still quite rigid.

==Analyticity and differentiability==

As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or <math>\mathcal{C}^{\infty}</math>). (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]]''.

==Real versus complex analytic functions==

Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.{{sfn |Krantz |Parks |2002}}

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 ''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 ''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==

One can define analytic functions in several variables by means of power series in those variables (see [[power series]]). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in 2 or more complex dimensions:

* Zero sets of complex analytic functions in more than one variable are never [[discrete space|discrete]]. This can be proved by [[Hartogs's extension theorem]].

* [[Domain of holomorphy|Domains of holomorphy]] for single-valued functions consist of arbitrary (connected) open sets. In several complex variables, however, only some connected open sets are domains of holomorphy. The characterization of domains of holomorphy leads to the notion of [[pseudoconvexity]].

==See also==

*[[Cauchy–Riemann equations]]

*[[Holomorphic function]]

*[[Paley–Wiener theorem]]

*[[Quasi-analytic function]]

*[[Infinite compositions of analytic functions]]

*[[Non-analytic smooth function]]

==Notes==

{{Notelist}}

{{Reflist}}

==References==

*{{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==

One can define analytic functions in several variables by means of power series in those variables (see [[power series]]). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in many dimensions.

* {{springer|title=Analytic function|id=p/a012240}}

* {{MathWorld | urlname= AnalyticFunction | title= Analytic Function }}

* [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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