cinfinity - A C ∞ function which is not everywhere...

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Unformatted text preview: A C ∞ function which is not everywhere analytic Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 13 December, 2008 (at 21:07 ) The goal of this note is to produce a C ∞-function L : R whose Taylor series ( centered at zero ) converges to a different fnc —namely, to the zero-function. On U := Rr { } , the following fnc L () is strictly positive; thus it differs from the zero-fnc on all of U . M ( x ) := e- 1 x 2 ; L ( x ) := ( , if x = 0 M ( x ) , if x 6 = 0 ) . 1 : Generalizing, a degree-D poly P ( z ) := ∑ D n =0 C n z n , with each C n real, defines a function L P by L P ( x ) := ( , if x = 0 M ( x ) · P ( 1 x ) , if x 6 = 0 ) . 2 : It may not be evident that L P is differentiable at 0. But at a non-zero x , we can use the Product Rule to compute as follows. [ L P ] ( x ) = M ( x ) P ( 1 x ) + M ( x ) · P ( 1 x ) ·- 1 x 2 = M ( x ) 2 x 3 P ( 1 x )- M ( x ) · 1 x 2 P ( 1 x ) = M ( x ) · h 2[ 1 x ] 3 P ( 1 x )- [ 1 x ] 2 P ( 1 x ) i ....
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

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cinfinity - A C ∞ function which is not everywhere...

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