Difference between revisions of "Uniformly bounded derivatives implies globally analytic"
From Calculus
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==Statement== | ==Statement== | ||
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| + | ===Global statement=== | ||
Suppose <math>f</math> is an infinitely differentiable function on <math>\R</math> such that, for any fixed <math>a,b \in \R</math>, there is a constant <math>C</math> (possibly dependent on <math>a,b</math>) such that for all nonnegative integers <math>n</math>, we have: | Suppose <math>f</math> is an infinitely differentiable function on <math>\R</math> such that, for any fixed <math>a,b \in \R</math>, there is a constant <math>C</math> (possibly dependent on <math>a,b</math>) such that for all nonnegative integers <math>n</math>, we have: | ||
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Then, <math>f</math> is a [[globally analytic function]]: the [[Taylor series]] of <math>f</math> about any point in <math>\R</math> converges to <math>f</math>. In particular, the Taylor series of <math>f</math> about 0 converges to <math>f</math>. | Then, <math>f</math> is a [[globally analytic function]]: the [[Taylor series]] of <math>f</math> about any point in <math>\R</math> converges to <math>f</math>. In particular, the Taylor series of <math>f</math> about 0 converges to <math>f</math>. | ||
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| + | ==Examples== | ||
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| + | The functions <math>\exp, \sin, \cos</math> all fit this description. | ||
Revision as of 14:51, 7 July 2012
Statement
Global statement
Suppose 
is an infinitely differentiable function on 
such that, for any fixed 
, there is a constant 
(possibly dependent on 
) such that for all nonnegative integers 
, we have:
![|f^{(n)}(x)| \le C \ \forall x \in [a,b]](/w/images/math/f/9/2/f92c06eca67577b309548232ef380cb0.png)
Then, is a globally analytic function: the Taylor series of about any point in converges to . In particular, the Taylor series of about 0 converges to .
Examples
The functions 
all fit this description.
