Analytic function: Difference between revisions

Revision as of 23:11, 19 April 2005

Revision as of 23:04, 1 July 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 in an open set is analytic]]. Consequently, in [[complex analysis]], the term ''analytic function'' is synonymous with ''[[holomorphic function]]''.

==Real versus complex analytic functions==