Analytic Function

Definition

Analytic at a point

$\forall \delta >0,\exists f(x)=\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$ on a $(c-\delta ,c+\delta )$, then $f$ is analytic at a point $c$

If $f$ can be expressed as a power series on a $(c-\delta ,c+\delta )$ for all $\delta >0$, then $f$ is analytic at a point $c$

Analytic function

If $f$ is analytic at a point $x$ in any open interval $I$, then $f$ is called an analytic function

Properties

Operations

The sums, subtractions, products, and compositions of analytic functions are analytic

Suppose $f$ is analytic on $I$, and $g$ is analytic on $J$ $f\pm g$, $f\cdot g$ is analytic on $I\cap J$ $\forall x_0\in I\cap J,g(x_0)\ne 0$, $\frac{f}{g}$ is analytic on $\forall ϵ>0,(x_0-ϵ,x_0+ϵ)$

Suppose $f$ is analytic on $I$, $g$ is analytic on $J$, and $f(I)\subset J$ Then, $g\circ f$ is analytic on the $I$

Facts

A real analytic function is infinitely differentiable

Analytic function can be expressed as Taylor Series