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Unformatted text preview: Smooth versus Analytic functions Henry Jacobs December 6, 2009 Functions of the form f ( x ) = X i ≥ a i x i that converge everywhere are called analytic. We see that analytic functions are equal to there Taylor expansions. Obviously all analytic functions are smooth or C ∞ but not all smooth functions are analytic. For example g ( x ) = e 1 /x 2 Has derivatives of all orders, so g ∈ C ∞ . This function also has a Taylor series expansion about any point. In particular the Taylor expansion about 0 is g ( x ) ≈ 0 + 0 x + 0 x 2 + . . . So that the Taylor series expansion does in fact converge to the function ˜ g ( x ) = 0 We see that g and ˜ g are competely different and only equal each other at a single point. So we’ve shown that g is not analytic. This is relevent in this class when finding approximations of invariant man ifolds. Generally when we ask you to find a 2nd order approximation of the center manifold we just want you to express it as the graph of some function...
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 Fall '09
 Marsden
 Derivative, Taylor Series, Analytic function, Holomorphic function

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