Real Analytic Functions

Definition: Real Analytic Function

A real analytic function is a real function $f:D\to \mathbb{R}$ on an open interval $D\subseteq \mathbb{R}$ for which there exists a real power series $\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$ on $D$ with

$$f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n\forall x\in D$$

Properties

Theorem: Differentiation of Real Analytic Functions

Every real analytic function $f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$ is differentiable on $D$ and its derivative $f^{\prime }(x)$ is also a real analytic function given by

$$f^{\prime }(x)=\sum _{n=1}^{\infty }n{c}_n(x-a{)}^{n-1}$$

PROOF

TODO

Theorem: Antiderivatives of Real Analytic Functions

The antiderivatives of any real analytic function $f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$ are also real analytic functions and $f$‘s indefinite integral is given by

$$\int f(x)\mathrm{dx}=C+\sum _{n=0}^{\infty }{c}_n\frac{(x-a{)}^{n+1}}{n+1}$$

PROOF

TODO