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Suppose we have a function $f(x):mathbbRrightarrow mathbbR$ which is analytic. We can write its Taylor expansion around $x=0$ as: $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + dots$.

The function $f^(k)(x) = f(x)circdotscirc f(x)$, that is the composition of $f(x)$ with itself $k$ times, is also analytic as a composition of analytic functions. So we can also write its Taylor expansion around $x=0$ as $f^(k)(x)=b_0 + b_1x + b_2x^2 + b_3x^3 + dots$.

My question: is there a way to get a close form of the coefficients $b_0,b_1,dots$ in terms of $a_0,a_1,dots$?

In particular, suppose the coefficients of $f(x)$ goes to zero as $Oleft(frac1n!right)$ (e.g. $f(x)=exp(x)$), how fast the coefficients of $f^(k)(x)$ will go to zero?

calculus taylor-expansion analyticity analytic-functions

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asked Mar 29 at 21:49

giladudegiladude

39229

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1

$begingroup$

Suppose we have a function $f(x):mathbbRrightarrow mathbbR$ which is analytic. We can write its Taylor expansion around $x=0$ as: $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + dots$.

The function $f^(k)(x) = f(x)circdotscirc f(x)$, that is the composition of $f(x)$ with itself $k$ times, is also analytic as a composition of analytic functions. So we can also write its Taylor expansion around $x=0$ as $f^(k)(x)=b_0 + b_1x + b_2x^2 + b_3x^3 + dots$.

My question: is there a way to get a close form of the coefficients $b_0,b_1,dots$ in terms of $a_0,a_1,dots$?

In particular, suppose the coefficients of $f(x)$ goes to zero as $Oleft(frac1n!right)$ (e.g. $f(x)=exp(x)$), how fast the coefficients of $f^(k)(x)$ will go to zero?

calculus taylor-expansion analyticity analytic-functions

share|cite|improve this question

asked Mar 29 at 21:49

giladudegiladude

39229

$endgroup$

add a comment | 

$begingroup$

Suppose we have a function $f(x):mathbbRrightarrow mathbbR$ which is analytic. We can write its Taylor expansion around $x=0$ as: $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + dots$.

The function $f^(k)(x) = f(x)circdotscirc f(x)$, that is the composition of $f(x)$ with itself $k$ times, is also analytic as a composition of analytic functions. So we can also write its Taylor expansion around $x=0$ as $f^(k)(x)=b_0 + b_1x + b_2x^2 + b_3x^3 + dots$.

My question: is there a way to get a close form of the coefficients $b_0,b_1,dots$ in terms of $a_0,a_1,dots$?

In particular, suppose the coefficients of $f(x)$ goes to zero as $Oleft(frac1n!right)$ (e.g. $f(x)=exp(x)$), how fast the coefficients of $f^(k)(x)$ will go to zero?

calculus taylor-expansion analyticity analytic-functions

share|cite|improve this question

asked Mar 29 at 21:49

giladudegiladude

39229

$endgroup$

share|cite|improve this question

asked Mar 29 at 21:49

giladudegiladude

39229

share|cite|improve this question

asked Mar 29 at 21:49

giladudegiladude

39229

asked Mar 29 at 21:49

giladudegiladude

39229

asked Mar 29 at 21:49

giladudegiladude

39229