PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (192)
Orphanage
Unclass'd
Unproven (496)
Corrections (20)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
[parent] infinitely-differentiable function that is not analytic (Example)

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

$\displaystyle f(x)=\begin{cases} e^{-\frac{1}{x^2}}& x \ne 0 \ 0 & x = 0 \end{cases}. $
Then $ f\in \mathcal{C}^{\infty}$, and for any $ n\ge 0$, $ f^{(n)}(0)=0$ (see below). So the Taylor series for $ f$ around 0 is 0; since $ f(x)>0$ for all $ x\ne 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

$\displaystyle g(x)=\frac{p(x)}{q(x)} \cdot f(x). $
Then, for $ x\ne 0$,
$\displaystyle g'(x) = \frac{(p'(x) + p(x)\frac{2}{x^3})q(x) - q'(x)p(x)}{q^2(x)} \cdot e^{-\frac{1}{x^2}}. $
Computing (e.g. by applying L'Hôpital's rule), we see that $ g'(0)=\lim_{x\to 0}g'(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.



"infinitely-differentiable function that is not analytic" is owned by ariels.
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: polynomials, power series, necessary, radius of convergence, converge, Taylor series
There are 2 references to this entry.

This is version 2 of infinitely-differentiable function that is not analytic, born on 2002-06-09, modified 2002-06-09.
Object id is 3081, canonical name is InfinitelyDifferentiableFunctionThatIsNotAnalytic.
Accessed 5824 times total.

Classification:
AMS MSC: 30B10 (Functions of a complex variable :: Series expansions :: Power series )
 26A99 (Real functions :: Functions of one variable :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)