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Analytic function

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Type of function in mathematics

Not to be confused withanalytic expression oranalytic signal.

This article is about both real and complex analytic functions. For analytic functions in complex analysis specifically, seeholomorphic function. For analytic functions in SQL, seeWindow function (SQL).

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Inmathematics, ananalytic function is afunction that is locally given by aconvergent power series. There exist bothreal analytic functions andcomplex analytic functions. Functions of each type areinfinitely 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 $x_{0}$ in itsdomain, itsTaylor series about $x_{0}$ converges to the function in someneighborhood of $x_{0}$ . This is stronger than merely beinginfinitely differentiable at $x_{0}$ , and therefore having a well-defined Taylor series; theFabius function provides an example of a function that is infinitely differentiable but not analytic.

Definitions

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Formally, a function $f {\displaystyle f}$ isreal analytic on anopen set $D {\displaystyle D}$ in thereal line if for any $x_{0}\in D$ one can write $f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-x_{0}\right)^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\cdots$

in which the coefficients $a_{0},a_{1},\dots$ are real numbers and theseries isconvergent to $f(x)$ for $x {\displaystyle x}$ in a neighborhood of $x_{0}$ .

Alternatively, a real analytic function is aninfinitely differentiable function such that theTaylor series at any point $x_{0}$ in its domain

$T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}$

converges to $f(x)$ for $x {\displaystyle x}$ in a neighborhood of $x_{0}$ pointwise.^[a] The set of all real analytic functions on a given set $D {\displaystyle D}$ is often denoted by ${\mathcal {C}}^{\,\omega }(D)$ , or just by ${\mathcal {C}}^{\,\omega }$ if the domain is understood.

A function $f {\displaystyle f}$ defined on some subset of the real line is said to be real analytic at a point $x {\displaystyle x}$ if there is a neighborhood $D {\displaystyle D}$ of $x {\displaystyle x}$ on which $f {\displaystyle f}$ is real analytic.

The definition of acomplex 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 isholomorphic i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.^[1]

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".^[2]

Examples

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Typical examples of analytic functions are

The followingelementary functions:
- Allpolynomials: if a polynomial has degreen, any terms of degree larger thann in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its ownMaclaurin series.
- Theexponential function is analytic. Any Taylor series for this function converges not only forx close enough tox₀ (as in the definition) but for all values ofx (real or complex).
- Thetrigonometric functions,logarithm, and thepower functions are analytic on any open set of their domain.
Mostspecial functions (at least in some range of the complex plane):

Typical examples of functions that are not analytic are

Theabsolute value function when defined on the set of real numbers orcomplex numbers is not everywhere analytic because it is not differentiable at 0.
Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
Thecomplex conjugate functionz →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 $\mathbb {R} ^{2}$ to $\mathbb {R} ^{2}$ .
Othernon-analytic smooth functions, and in particular any smooth function $f {\displaystyle f}$ with compact support, i.e. $f\in {\mathcal {C}}_{0}^{\infty }(\mathbb {R} ^{n})$ , cannot be analytic on $\mathbb {R} ^{n}$ .^[3]

Alternative characterizations

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The following conditions are equivalent:

$f {\displaystyle f}$ is real analytic on an open set $D {\displaystyle D}$ .
There is a complex analytic extension of $f {\displaystyle f}$ to an open set $G\subset \mathbb {C}$ which contains $D {\displaystyle D}$ .
$f {\displaystyle f}$ is smooth and for everycompact set $K\subset D$ there exists a constant $C {\displaystyle C}$ such that for every $x\in K$ and every non-negative integer $k {\displaystyle k}$ the following bound holds^[4] $\left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}k!$

Complex analytic functions are exactly equivalent toholomorphic 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 theFourier–Bros–Iagolnitzer transform.

In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization.^[5] Let $U\subset \mathbb {R} ^{n}$ be an open set, and let $f:U\to \mathbb {R}$ .Then $f {\displaystyle f}$ is real analytic on $U {\displaystyle U}$ if and only if $f\in C^{\infty }(U)$ and for every compact $K\subseteq U$ there exists a constant $C {\displaystyle C}$ such that for every multi-index $\alpha \in \mathbb {Z} _{\geq 0}^{n}$ the following bound holds^[6]

$\sup _{x\in K}\left|{\frac {\partial ^{\alpha }f}{\partial x^{\alpha }}}(x)\right|\leq C^{|\alpha |+1}\alpha !$

Properties of analytic functions

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The sums, products, andcompositions of analytic functions are analytic.
Thereciprocal of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whosederivative is nowhere zero. (See also theLagrange inversion theorem.)
Any analytic function issmooth, 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 differentiableonce on an open set is analytic on that set (see "analyticity and differentiability" below).
For anyopen set $\Omega \subseteq \mathbb {C}$ , the setA(Ω) of all analytic functions $u:\Omega \to \mathbb {C}$ is aFréchet 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 ofMorera's theorem. The set $A_{\infty }(\Omega )$ of allbounded analytic functions with thesupremum norm is aBanach 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 anaccumulation point inside itsdomain, then ƒ is zero everywhere on theconnected component containing the accumulation point. In other words, if (r_n) is asequence of distinct numbers such that ƒ(r_n) = 0 for alln and this sequenceconverges to a pointr in the domain ofD, then ƒ is identically zero on the connected component ofD containingr. This is known as theidentity 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 moredegrees of freedom than polynomials, they are still quite rigid.

Analyticity and differentiability

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As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or ${\mathcal {C}}^{\infty }$ ). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: seenon-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 thatany complex function differentiable (in the complex sense) in an open set is analytic. Consequently, incomplex analysis, the termanalytic function is synonymous withholomorphic function.

Real versus complex analytic functions

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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.^[7]

According toLiouville'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

$f(x)={\frac {1}{x^{2}+1}}.$

Also, if a complex analytic function is defined in an openball around a pointx₀, its power series expansion atx₀ is convergent in the whole open ball (holomorphic functions are analytic). This statement for real analytic functions (with open ball meaning an openinterval of the real line rather than an opendisk of the complex plane) is not true in general; the function of the example above gives an example forx₀ = 0 and a ball of radius exceeding 1, since the power series1 −x² +x⁴ −x⁶... diverges for |x| ≥ 1.

Any real analytic function on someopen 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 functionf(x) defined in the paragraph above is a counterexample, as it is not defined forx = ±i. This explains why the Taylor series off(x) diverges for |x| > 1, i.e., theradius of convergence is 1 because the complexified function has apole 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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One can define analytic functions in several variables by means of power series in those variables (seepower 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 neverdiscrete. This can be proved byHartogs's extension theorem.
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 ofpseudoconvexity.

Notes

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^This impliesuniform convergence as well in a (possibly smaller) neighborhood of $x_{0}$ .

^Churchill; Brown; Verhey (1948).Complex Variables and Applications. McGraw-Hill. p. 46.ISBN 0-07-010855-2.A functionf of the complex variablez isanalytic at pointz₀ if its derivative exists not only atz but at each pointz in some neighborhood ofz₀. It is analytic in a regionR if it is analytic at every point inR. The termholomorphic is also used in the literature to denote analyticity{{cite book}}:ISBN / Date incompatibility (help)
^Gamelin, Theodore W. (2004).Complex Analysis. Springer.ISBN 9788181281142.
^Strichartz, Robert S. (1994).A guide to distribution theory and Fourier transforms. Boca Raton: CRC Press.ISBN 0-8493-8273-4.OCLC 28890674.
^Krantz & Parks 2002, p. 15.
^Komatsu, Hikosaburo (1960)."A characterization of real analytic functions".Proceedings of the Japan Academy.36 (3):90–93.doi:10.3792/pja/1195524081.ISSN 0021-4280.
^"Gevrey class - Encyclopedia of Mathematics".encyclopediaofmath.org. Retrieved2020-08-30.
^Krantz & Parks 2002.

References

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Conway, John B. (1978).Functions of One Complex Variable I.Graduate Texts in Mathematics 11 (2nd ed.). Springer-Verlag.ISBN 978-0-387-90328-6.
Krantz, Steven;Parks, Harold R. (2002).A Primer of Real Analytic Functions (2nd ed.). Birkhäuser.ISBN 0-8176-4264-1.
Gamelin, Theodore W. (2004).Complex Analysis. Springer.ISBN 9788181281142.

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