## You are here

Homeinfinitely-differentiable function that is not analytic

## Primary tabs

# infinitely-differentiable function that is not analytic

If $f\in\mathcal{C}^{{\infty}}$, then we can certainly *write* a Taylor series for $f$. However, analyticity requires that this Taylor series actually converge (at least across some radius of convergence) to $f$. It is not necessary that the power series for $f$ converge to $f$, as the following example shows.

Let

$f(x)=\begin{cases}e^{{-\frac{1}{x^{2}}}}&x\neq 0\\ 0&x=0\end{cases}.$ |

Then $f\in\mathcal{C}^{{\infty}}$, and for any $n\geq 0$, $f^{{(n)}}(0)=0$ (see below). So the Taylor series for $f$ around 0 is 0; since $f(x)>0$ for all $x\neq 0$, clearly it does not converge to $f$.

# Proof that $f^{{(n)}}(0)=0$

Let $p(x),q(x)\in\mathbb{R}[x]$ be polynomials, and define

$g(x)=\frac{p(x)}{q(x)}\cdot f(x).$ |

Then, for $x\neq 0$,

$g^{{\prime}}(x)=\frac{(p^{{\prime}}(x)+p(x)\frac{2}{x^{3}})q(x)-q^{{\prime}}(x% )p(x)}{q^{2}(x)}\cdot e^{{-\frac{1}{x^{2}}}}.$ |

Computing (e.g. by applying LβHΓ΄pitalβs rule), we see that $g^{{\prime}}(0)=\lim_{{x\to 0}}g^{{\prime}}(x)=0$.

Define $p_{0}(x)=q_{0}(x)=1$. Applying the above inductively, we see that we may write $f^{{(n)}}(x)=\frac{p_{n}(x)}{q_{n}(x)}f(x)$. So $f^{{(n)}}(0)=0$, as required.

## Mathematics Subject Classification

30B10*no label found*26A99

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier