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

Theorem: Automatic Differentiation via Dual Numbers

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real analytic function and let $a+b\epsilon$ be a dual number.

If $a\in \mathcal{D}$, then

$$f(a+b\epsilon )=f(a)+b{f}^{\prime }(a)\epsilon$$

Tip: Automatic Differentiation

This theorem allows us to find the derivative of $f$ at any $a\in \mathcal{D}$ without the need to find an expression for $f^{\prime }$, so long as we have a closed-form expression for $f$. All we have to do is set $b=1$ and then use this expression to evaluate $f(a+b\epsilon )$. In the end, the value of the derivative $f^{\prime }(a)$ will be the coefficient before $\epsilon$.

PROOF

TODO