Analytic function: Difference between revisions

Revision as of 22:15, 12 February 2005

Revision as of 22:20, 12 February 2005

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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.

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

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

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

==Analytic functions of several variables==