Taylor series operator commutes with composition: Difference between revisions
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Suppose <math>x_0</math> is a real number. Suppose <math>g</math> is a function defined on a subset of the reals that is infinitely differentiable at <math>x_0</math>. Suppose <math>f</math> is a function defined on a subset of the reals that is infinitely differentiable at <math>g(x_0)</math>. Then, the [[composite of two functions]] <math>f \circ g</math> is infinitely differentiable at <math>x_0</math>, and its Taylor series can be computed formally by ''composing'' the Taylor series for <math>f</math> at <math>g(x_0)</math> with the Taylor series for <math>g</math> at <math>x_0</math>. Formally, what this means is that we write down the Taylor series for <math>f</math> at <math>g(x_0)</math>, then plug in for <math>x</math> the entire expression for the Taylor series of <math>g</math>, then simplify. | Suppose <math>x_0</math> is a real number. Suppose <math>g</math> is a function defined on a subset of the reals that is infinitely differentiable at <math>x_0</math>. Suppose <math>f</math> is a function defined on a subset of the reals that is infinitely differentiable at <math>g(x_0)</math>. Then, the [[composite of two functions]] <math>f \circ g</math> is infinitely differentiable at <math>x_0</math>, and its Taylor series can be computed formally by ''composing'' the Taylor series for <math>f</math> at <math>g(x_0)</math> with the Taylor series for <math>g</math> at <math>x_0</math>. Formally, what this means is that we write down the Taylor series for <math>f</math> at <math>g(x_0)</math>, then plug in for <math>x</math> the entire expression for the Taylor series of <math>g</math>, then simplify. | ||
==Related facts== | |||
===Similar facts=== | |||
* [[Taylor series operator is linear]] | |||
* [[Taylor series operator is multiplicative]] | |||
* [[Taylor series operator commutes with differentiation]] | |||
Latest revision as of 16:15, 30 June 2012
Statement
Suppose is a real number. Suppose is a function defined on a subset of the reals that is infinitely differentiable at . Suppose is a function defined on a subset of the reals that is infinitely differentiable at . Then, the composite of two functions is infinitely differentiable at , and its Taylor series can be computed formally by composing the Taylor series for at with the Taylor series for at . Formally, what this means is that we write down the Taylor series for at , then plug in for the entire expression for the Taylor series of , then simplify.