Analytic Function

analysis

Definition

Analytic Function

A function $f:D\to \mathbb{R}$ is called analytic at a point $x_0\in D$ if it can be locally represented by a convergent power series. That is, there exists a neighbourhood $U$ of $x_0$ such that for all $x\in U$:

$$f(x)=\sum \_{n=0}^{\infty }{a}_n(x-x_0{)}^n$$

A function is called analytic on $D$ if it is analytic at every point in $D$. This implies the function is infinitely differentiable ($C^{\infty }$) 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 $C^{\infty }$, but the converse is false (e.g., the function $f(x)=e^{-1/x^2}$ for $x\ne 0$ and $f(0)=0$ is $C^{\infty }$ but not analytic at $x=0$).
• Local Power Series: The coefficients $a_n$ are necessarily given by the derivatives at the point:$a_n=\frac{f^{(n)}(x_0)}{n!}$.