Analytic Function - Alternate Characterizations

Alternate Characterizations

If ƒ is an infinitely differentiable function defined on an open set D ⊂ R, then the following conditions are equivalent.

1) ƒ is real analytic.

2) There is a complex analytic extension of ƒ to an open set G ⊂ C which contains D.

3) For every compact set K ⊂ D there exists a constant C such that for every x ∈ K and every non-negative integer k the following bound holds:

The real analyticity of a function ƒ at a given point x can be characterized using the FBI transform.

Complex analytic functions are exactly equivalent to holomorphic functions, and are thus much more easily characterized.