Editing Analytic function (section)

== Examples ==
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''<sub>0</sub> (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

* 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.
* [[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>