Definition
Analytic Function
A function
is called analytic at a point if it can be locally represented by a convergent power series. That is, there exists a neighbourhood of such that for all : A function is called analytic on
if it is analytic at every point in . This implies the function is infinitely differentiable ( ) and its Taylor series converges to the function itself.
Properties
- Holomorphicity: In complex analysis, a function is analytic if and only if it is holomorphic (differentiable in the complex sense).
- Identity Theorem: If two analytic functions on a connected domain agree on a set with an accumulation point, they are identical everywhere on that domain.
- Smoothness: Every analytic function is
, but the converse is false (e.g., the function for and is but not analytic at ). - Local Power Series: The coefficients
are necessarily given by the derivatives at the point: .