imported>Sławomir Biały: Taylor series should be mentioned in the lead.

2026-05-30T08:35:31Z

Taylor series should be mentioned in the lead.

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{{short description|Type of function in mathematics}}
{{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 [[mathematical analysis]], an '''analytic function''' is a [[function (mathematics)|function]] that is locally represented by a convergent [[power series]]. More precisely, a real or complex function is analytic at a point if, in some neighborhood of that point, it is equal to a power series centered there. Analytic functions are therefore locally determined by their coefficients, or equivalently by their [[derivative]]s at the center of the expansion. In other words, an analytic function is a function that is locally represented by a convergent [[Taylor series]].

Analytic functions occur in both [[real analysis]] and [[complex analysis]], in slightly different ways. A real or complex analytic function is necessarily [[smooth function|smooth]], having derivatives of all orders. But a smooth real function need not be analytic. By contrast, a complex function on an open set is analytic if and only if it is [[holomorphic function|holomorphic]], that is, complex differentiable at every point of the set. For this reason, in complex analysis the terms ''analytic function'' and ''holomorphic function'' are often used interchangeably. The terms '''complex analytic''' and '''real analytic''' distinguish between these cases. In [[signal processing]], a complex analytic function is sometimes called an [[analytic signal]].

Analyticity is a strong regularity condition. Analytic functions have rigid local behavior: for example, on a connected domain, an analytic function whose zeros have an accumulation point must vanish identically. Standard examples include [[polynomial]]s, the [[exponential function]], and the [[trigonometric function]]s on their domains of analyticity.

== Definitions ==

Formally, a function <math>f</math> is ''real analytic'' on an [[open set]] <math>D</math> in the [[real line]] if for every <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>
in which the coefficients {{ tmath| a_0 }}, {{tmath| a_1 }}, ...are real numbers and this [[series (mathematics)|series]] (the right-hand side of this equation) is [[convergent series|convergent]] to <math>f(x)</math> for <math>x</math> in a [[Neighbourhood (mathematics)|neighborhood]] of <math>x_0</math> (that is a set containing an open set including {{tmath| x_0 }}).

Alternatively, a real analytic function is an [[smooth function|infinitely differentiable function]] such that the [[Taylor series]] at each 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 {{tmath| x_0 }}.}} The set of all real analytic functions on a given set <math>D</math> is often denoted by {{tmath| \mathcal{C}^{\omega}(D)}}, 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''<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 |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>

In complex analysis, a function is called analytic in an open set {{tmath| U }} if it is (complex) differentiable at each point in {{tmath| U }}.<ref>{{cite book |last= Gamelin |first= Theodore W. |author-link=Theodore Gamelin |title=Complex Analysis |publisher=Springer |year=2004 |isbn= 9788181281142 }}</ref>

== Examples ==
Typical examples of analytic functions are
* The following [[elementary function]]s:
** All [[polynomial]]s: if a polynomial has degree {{tmath| n }}, any terms of degree larger than {{tmath| 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 {{tmath| x }} close enough to {{tmath| x_0 }} (as in the definition) but for all values of {{tmath| x }} (real or complex).
** The [[trigonometric function]]s are analytic on any open set of their domain.
** The [[natural logarithm]] is analytic on any open set where its branch is single-valued, such as <math>(0,\infty)</math> or the complement in the [[Riemann sphere]] of any simple arc connecting <math>0</math> to <math>\infty</math>.
** [[Exponentiation|Power function]]s are analytic everywhere for non-negative integral powers, away from zero for negative integral powers, and for arbitrary complex powers on any open subset of the complex plane where the logarithm is analytic.
* Many [[special function]]s are analytic on a suitable domain:
** [[hypergeometric function]]s on suitable domains
** [[Bessel function]]s on suitable domains
** The [[gamma function]] away from its [[pole (complex analysis)|poles]] at zero and the negative integers
** The [[Riemann zeta function]] except for a simple pole at <math>1</math>
* [[Algebraic function]]s are analytic away from any poles and [[branch point]]s they may have. Near a branch point, an algebraic function can be represented by a convergent [[Puiseux series]]; equivalently, after a change of variable <math>z-a=t^e</math>, it becomes analytic as a function of <math>t</math>.

Typical examples of functions that are not analytic are
* The [[absolute value]] function <math>x\mapsto |x|</math> on the real numbers is not analytic at <math>0</math>. The corresponding function <math>z\mapsto |z|</math> on the complex numbers is not complex analytic on any nonempty open subset of <math>\mathbb C</math>.
* [[Piecewise|Piecewise defined]] functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
* The [[complex conjugate]] function {{tmath| z \to 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 {{tmath| \R^{2} }}.
* Other [[non-analytic smooth function]]s, and in particular any smooth function <math>f</math> with compact support that is not identically zero, i.e. <math>f \in C^\infty_0(\mathbb R^n)\setminus\{0\}</math>, cannot be analytic on all of <math>\mathbb R^n</math>.<ref>{{cite book |last=Strichartz |first=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 following conditions are equivalent:
# <math>f</math> is real analytic on an open set {{tmath| D }}.
# There is a complex analytic extension of <math>f</math> to an open set <math>G \subset \mathbb{C}</math> that contains {{tmath| D }}.
# <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.

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 {{tmath| f : U \to \R }}.
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 ==
* 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.
* In one variable, the inverse of a one-to-one analytic function whose derivative is nowhere zero is analytic. In several variables, the corresponding condition is that the derivative be an invertible linear map, or equivalently that the Jacobian determinant be nonzero.
* 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: any function with a single complex derivative on an open set is analytic on that set (see ''{{slink|#Analyticity and differentiability}}'').
* For any [[open set]] {{tmath| \Omega \subseteq \mathbb{C} }}, the set {{tmath| A(\Omega)}} 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]].
* In contrast, the real analytic functions on an open set are not complete under the topology of uniform convergence on compact subsets, since compact-open limits of real analytic functions need not be real analytic. Real analytic functions are dense in the space of continuous functions in the compact-open topology.

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 {{tmath| f }} has an [[accumulation point]] inside its [[domain of a function|domain]], then {{tmath| f }} is zero everywhere on the [[connected space|connected component]] containing the accumulation point. In other words, if {{tmath| (r_n) }} is a [[sequence]] of distinct numbers such that {{tmath|1= f(r_n) = 0 }} for all {{tmath| n }} and this sequence [[limit of a sequence|converges]] to a point {{tmath| r }} in the domain of {{tmath| D }}, then {{tmath| f }} is identically zero on the connected component of {{tmath| D }} containing {{tmath| r }}. This is known as the [[identity theorem]].

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

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 function (real or complex) is infinitely differentiable on a neighborhood where it is equal to a convergent power series. In a neighborhood of a point of analyticity, there is a convergent power series equal to the function in the neighborhood, so the function is infinitely differentiable there. 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. Any complex-differentiable function in an open disc centered at <math>z=z_0</math> ([[holomorphic function]]) is [[proof that holomorphic functions are analytic|analytic there]], and conversely any function given by a convergent power series in the complex variable <math>z-z_0</math> is complex differentiable (and infinitely differentiable) on the disk of convergence. 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 {{tmath| x_0 }}, its power series expansion at {{tmath| 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 {{tmath|1= x_0 = 0 }} and a ball of radius exceeding {{tmath| 1 }}, since the power series {{tmath|1 - x^2 + x^4 - x^6 + \ldots }} diverges for {{tmath| \vert x \vert \ge 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 {{tmath| f(x) }} defined in the paragraph above is a counterexample, as it is not defined for {{tmath|1= x = \pm i }}. This explains why the Taylor series of {{tmath| f(x) }} diverges for {{tmath| \vert x \vert > 1 }}, i.e., the [[radius of convergence]] is {{tmath| 1 }} because the complexified function has a [[Complex pole|pole]] at distance {{tmath| 1 }} from the evaluation point {{tmath| 0 }} and no further poles within the open disc of radius {{tmath| 1 }} around the evaluation point.

== Taylor series and radius of convergence ==
{{main|Taylor series|Radius of convergence}}

If a function is analytic at <math>a</math>, then its Taylor series converges ''to the function'' in some open neighborhood of <math>a</math>. More generally, for any power series
<math display="block">\sum_{n=0}^\infty c_n (x-a)^n,</math>
there is a number <math>R</math> called the [[radius of convergence]], which can be any non-negative number or <math>+\infty</math>, such that the power series converges absolutely for <math>|x-a|<R</math> and diverges for <math>|x-a|>R</math>.<ref>{{harvnb|Stein|Shakarchi|2003|p=15}}</ref><ref>{{harvnb|Freitag|Busam|2005|pp=111–112}}</ref> Thus, when a Taylor series converges, it does so in an open interval centered at <math>a</math> in the real case, or a disc centered at <math>a</math> in the complex case.<ref>{{harvnb|Stein|Shakarchi|2003|p=15}}</ref><ref>{{harvnb|Freitag|Busam|2005|pp=111–112}}</ref> A Taylor series may converge absolutely or conditionally at some, all, or none of the boundary points of the open interval or disc.<ref>{{harvnb|Freitag|Busam|2005|pp=112–113, 124}}</ref>

In complex analysis, the radius of convergence of a [[holomorphic function]] at a point is the radius of the largest open disc centered at that point on which the function remains holomorphic. In many common cases, this is the distance to the nearest [[singularity (mathematics)|singularity]] of the function in the complex plane.<ref>{{harvnb|Freitag|Busam|2005|pp=116–117}}</ref>

This explains the different radii of convergence for familiar Taylor series. The series for <math>e^z</math>, <math>\sin z</math>, and <math>\cos z</math> have infinite radius of convergence because these are [[entire function]]s, having no singularities in the complex plane.<ref>{{harvnb|Freitag|Busam|2005|p=53}}</ref><ref>{{harvnb|Freitag|Busam|2005|pp=113–115}}</ref> By contrast, the Taylor series for <math>\log(1+z)</math> around <math>z=0</math> has radius of convergence <math>1</math>, because the nearest singularity is at <math>z=-1</math>.<ref>{{harvnb|Stein|Shakarchi|2003|pp=98–100}}</ref>

Complex singularities can determine the radius of convergence even for functions that are smooth on the real line. For example, although
<math display="block">
f(x)=\frac{1}{1+x^2}
</math>
is smooth for all real <math>x</math>, the radius of convergence of its Taylor series around <math>x=0</math> is <math>1</math>, because the corresponding complex function has singularities at <math>x=i</math> and <math>x=-i</math>, which are points of the complex unit circle.<ref>{{harvnb|Freitag|Busam|2005|pp=116–117}}</ref> Thus, even for real-valued functions, the role of complex singularities is important: a function can be infinitely differentiable on the whole real line, and yet have a Taylor series with only a finite radius of convergence, because the limiting obstruction can come from singularities in the corresponding complex function rather than any failure of smoothness on the real axis.<ref>{{harvnb|Freitag|Busam|2005|pp=116–117}}</ref>

A power series may converge at every point of the boundary of its disc of convergence and still fail to extend holomorphically beyond that disc. For example, if <math>\alpha>0</math> is not an integer, then the [[binomial series]]
<math display="block">(1+z)^\alpha=\sum_{n=0}^\infty \binom{\alpha}{n}z^n</math>
has radius of convergence <math>R=1</math>.<ref>{{harvnb|Lang|1999|p=58}}</ref> The series converges everywhere on the closed unit disc, including every boundary point. However, for nonintegral <math>\alpha</math>, the function <math>(1+z)^\alpha</math> does not extend as a single-valued holomorphic function to any neighborhood of <math>z=-1</math>.<ref>{{harvnb|Freitag|Busam|2005|pp=29–30, 34}}</ref> Thus the obstruction to analytic continuation at the boundary point <math>z=-1</math> is not a failure of convergence of the power series, nor a [[Zeros and poles|pole]] or [[essential singularity]], but the branching of the analytic continuation. In effect, <math>z=-1</math> is a [[branch point]] of the function.<ref>{{harvnb|Freitag|Busam|2005|pp=29–30, 34}}</ref><ref>{{harvnb|Stein|Shakarchi|2003|pp=73, 98–100}}</ref> This illustrates that convergence on the closed disc is weaker than holomorphic extendibility beyond the boundary.

== Analytic continuation ==
{{main|analytic continuation}}
Because analytic functions are locally represented by power series, a value of the function on one small neighborhood can sometimes determine values on a larger region. This process is called [[analytic continuation]]. Starting with a power series representation in one neighborhood, one can then sometimes use this to define the function on overlapping neighborhoods, and continue this process along paths in the domain.

Analytic continuation is unique when it exists on a connected domain. More precisely, if two analytic functions on a connected open set agree on a nonempty open subset, or more generally on a set having an accumulation point in the domain, then they agree everywhere on that connected open set. This is a form of the [[identity theorem]].

However, analytic continuation need not be possible everywhere, and it need not be single-valued. For example, the [[natural logarithm]] is locally analytic on <math>\mathbb C\setminus\{0\}</math>, but continuation around a closed path encircling <math>0</math> changes its value by an integer multiple of <math>2\pi i</math>. For this reason, a single-valued branch of the logarithm can be defined on domains such as <math>\mathbb C\setminus(-\infty,0]</math>, but not on all of <math>\mathbb C\setminus\{0\}</math>. Taking a single analytic function in a disc and forming all possible analytic continuations of it leads in general to a [[Riemann surface]] that covers an open subset of the [[Riemann sphere]]. The [[global analytic function]] obtained in this way is naturally a [[sheaf (mathematics)|sheaf]] rather than a function: each [[germ (mathematics)|germ]] is a convergent power series in some disc, but multiple germs may be stacked on top of each other.<ref>{{citation
| last1 = Ahlfors
| first1 = Lars V. |author-link=Lars Ahlfors
| title = Complex analysis
| edition = 3rd
| publisher = McGraw-Hill
| location = New York
| year = 1979
| isbn = 978-0-07-000657-7
}}, Chapter 8.</ref>

[[Algebraic function]]s give another source of multi-valued analytic continuation. For example, the function <math>f(z)=\sqrt{1-z}</math> is analytic, where the branch is such that <math>f(0)=1</math>, has a convergent power series in the disc <math>|z|<1</math>. It can be continued around a loop enclosing <math>z=1</math>, but transforms to its negative after a single loop. The Riemann surface associated with an algebraic function is a finite [[ramified cover]] of the Riemann sphere.

Many special functions are first defined by a power series or an integral formula on a restricted domain and then extended by analytic continuation. For example, the [[Riemann zeta function]] is initially defined by the [[Dirichlet series]]
<math display="block">
\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s}
</math>
for <math>\operatorname{Re}(s)>1</math>, but it has a [[meromorphic]] continuation to the complex plane, with a single simple pole at <math>s=1</math>.

== 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 two or more complex dimensions:
* Zero sets of complex analytic functions in more than one variable are never [[discrete space|discrete]] if they are non-empty. 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]].

== Analytic functions over other valued fields ==
Analogous notions of analyticity can be formulated over other [[complete valued field]]s, the real and complex numbers being the two most prominent ones where the absolute value is [[archimedean field|archimedean]]. Analytic functions can also be defined over non-Archimedean [[local field]]s, such as the [[p-adic numbers]] <math>\mathbb Q_p</math> and its [[finite extension field]]s, and fields of formal [[Laurent series]] <math>\mathbb F_q((t))</math> over a [[finite field]].{{sfn|Serre|1979}}

If <math>K</math> is a complete valued field, a function on a neighborhood of a point <math>a\in K</math> is called analytic if it is locally represented by a convergent power series
<math display="block">
\sum_{n=0}^{\infty} c_n(x-a)^n
</math>
with coefficients in <math>K</math>. In the non-Archimedean case, convergence is governed by an [[ultrametric]] absolute value, and the resulting theory differs significantly from both real and complex analysis.{{sfn|Schikhof|1984}}

For example, a power series over <math>\mathbb Q_p</math>
<math display="block">
\sum_{n=0}^{\infty} a_nx^n
</math>
converges to an analytic function on the ''p''-adic integers <math>\mathbb Z_p=\{x\in\mathbb Q_p\mid |x|_p\le 1\}</math> if and only if
<math display="block">|a_n|_p \to 0.</math>
Likewise, the series on a finite extension field <math>K</math> of a ''p''-adic field converges on the ring of integers if and only if <math>|a_n|_K\to 0</math>. The reason is the ultrametric criterion: if <math>|x|_K\le 1</math>, then <math>|a_nx^n| \le |a_n|</math>, and ultrametricity implies that any middle segment of the series satisfies
<math display="block">\left|\sum_{n=N}^M a_nx^n\right|_K \le \max_{N\le n\le M} |a_n|.</math>

More generally, on the closed disc
<math display="block"> a+\pi^m\mathcal O_K=\{x:|x-a|\le |\pi|^m\}, </math>
a series
<math display="block"> \sum_{n=0}^{\infty} b_n(x-a)^n </math>
converges on that disc if and only if
<math display="block"> |b_n|\,|\pi|^{mn}\to 0. </math>
Here <math>\pi</math> denotes the uniformizer of <math>K|\mathbb Q_p</math>.

Over a non-Archimedean field the ring of analytic functions on a closed disc is thus related to the [[Tate algebra]], the algebra of power series whose coefficients tend to zero sufficiently fast. This point of view is fundamental in [[rigid analytic geometry]] and other forms of non-Archimedean analytic geometry.{{sfn|Bosch|Güntzer|Remmert|1984}}

== 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. |author-link=Theodore Gamelin |title=Complex Analysis |publisher=Springer |year=2004 |isbn=9788181281142 }}

== External links ==
* {{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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Analytic function - Revision history

imported>Sławomir Biały: Taylor series should be mentioned in the lead.