infinitely-differentiable function that is not analytic
If , then we can certainly write a Taylor series for . However, analyticity requires that this Taylor series actually converge (at least across some radius of convergence) to . It is not necessary that the power series for converge to , as the following example shows.
Then , and for any , (see below). So the Taylor series for around 0 is 0; since for all , clearly it does not converge to .
Let be polynomials, and define
Then, for ,
Computing (e.g. by applying L’Hôpital’s rule (http://planetmath.org/LHpitalsRule)), we see that .
Define . Applying the above inductively, we see that we may write . So , as required.
|Title||infinitely-differentiable function that is not analytic|
|Date of creation||2013-03-22 12:46:15|
|Last modified on||2013-03-22 12:46:15|
|Last modified by||ariels (338)|