infinitely-differentiable function that is not analytic

If fβˆˆπ’žβˆž, then we can certainly write a Taylor seriesMathworldPlanetmath for f. However, analyticity requires that this Taylor series actually converge (at least across some radius of convergenceMathworldPlanetmath) to f. It is not necessary that the power seriesMathworldPlanetmath for f converge to f, as the following example shows.


f⁒(x)={e-1x2xβ‰ 00x=0.

Then fβˆˆπ’žβˆž, and for any nβ‰₯0, f(n)⁒(0)=0 (see below). So the Taylor series for f around 0 is 0; since f⁒(x)>0 for all xβ‰ 0, clearly it does not converge to f.

Proof that f(n)⁒(0)=0

Let p⁒(x),q⁒(x)βˆˆβ„β’[x] be polynomials, and define


Then, for x≠0,


Computing (e.g. by applying L’HΓ΄pital’s rule (, we see that g′⁒(0)=limxβ†’0⁑g′⁒(x)=0.

Define p0⁒(x)=q0⁒(x)=1. Applying the above inductively, we see that we may write f(n)⁒(x)=pn⁒(x)qn⁒(x)⁒f⁒(x). So f(n)⁒(0)=0, as required.

Title infinitely-differentiable function that is not analytic
Canonical name InfinitelydifferentiableFunctionThatIsNotAnalytic
Date of creation 2013-03-22 12:46:15
Last modified on 2013-03-22 12:46:15
Owner ariels (338)
Last modified by ariels (338)
Numerical id 5
Author ariels (338)
Entry type Example
Classification msc 30B10
Classification msc 26A99