Analytic function: Difference between revisions

Revision as of 03:42, 2 February 2022

Revision as of 03:54, 2 February 2022

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:<math> 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>.

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.

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* The [[absolute value]] function when defined on the set of real numbers or complex numbers 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 [[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 [[identity theorem]].

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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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==External links==

* {{Eomspringer| title = Analytic function | author-last1 id= Gonchar| author-first1 = A.A.| author-last2 = Shabat| author-first2 =B.V.| oldid = 51340p/a012240}}

* {{Eom| title = Real analytic function | oldid = 51339}}

* {{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]