Analytic function: Difference between revisions

Revision as of 00:14, 24 May 2021

Revision as of 10:59, 30 May 2021

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The following conditions are equivalent:

1. #<math>f</math> is real analytic on an open set <math>D</math>.

2. #There is a complex analytic extension of <math>f</math> to an open set <math>G \subset \mathbb{C}</math> which contains <math>D</math>.

3. #<math>f</math> is real smooth and for every [[compact set]] <math>K \subset D</math> there exists a constant <math>C</math> such that for every <math>x \in K</math> and every non-negative integer <math>k</math> the following bound holds{{sfn |Krantz |Parks |2002|p=15}} <math display="block"> \left| \frac{d^k f}{dx^k}(x) \right| \leq C^{k+1} k!</math>

2. There is a complex analytic extension of <math>f</math> to an open set <math>G \subset \mathbb{C}</math> which contains <math>D</math>.

3. <math>f</math> is real smooth and for every [[compact set]] <math>K \subset D</math> there exists a constant <math>C</math> such that for every <math>x \in K</math> and every non-negative integer <math>k</math> the following bound holds{{sfn |Krantz |Parks |2002|p=15}}

::<math> \left | \frac{d^k f}{dx^k}(x) \right | \leq C^{k+1} k!</math>

Complex analytic functions are exactly equivalent to [[Holomorphic function|holomorphic functions]], and are thus much more easily characterized.