# Thread: taylor polynomial for composition

1. ## taylor polynomial for composition

hello all

i have a bit problem with finding the taylor polynomial for a composed function.

if anyone can help me find the taylor polynomial for the function f(x) = e^sin(x) and how i would appriciate it alot

thanks!

2. ## Re: taylor polynomial for composition

Which degree does the Taylor polynomial needs to have?

3. ## Re: taylor polynomial for composition

doesnt matter, i want to know how is it done. lets say up to 5'th degree around 0

4. ## Re: taylor polynomial for composition

It's in general not easy to find the Taylor expansion of the composition of two function. There exists a formula, due to Faà di Bruno.

5. ## Re: taylor polynomial for composition

Originally Posted by dorubin
hello all

i have a bit problem with finding the taylor polynomial for a composed function.

if anyone can help me find the taylor polynomial for the function f(x) = e^sin(x) and how i would appriciate it alot

thanks!
You can always do it the long way. Let \displaystyle \displaystyle \begin{align*} e^{\sin{x}} = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots \end{align*}.

We know that \displaystyle \displaystyle \begin{align*} e^{\sin{0}} = 1 \end{align*}, so let \displaystyle \displaystyle \begin{align*} x = 0 \end{align*} and we find \displaystyle \displaystyle \begin{align*} c_0 = 1 \end{align*}.

Differentiate both sides and we find

\displaystyle \displaystyle \begin{align*} e^{\sin{x}}\cos{x} &= c_1 + 2c_2x + 3c_3x^2 + 4c_4x^3 + \dots \end{align*}

We know \displaystyle \displaystyle \begin{align*} e^{\sin{0}}\cos{0} = 1 \end{align*}, so let \displaystyle \displaystyle \begin{align*} x = 0 \end{align*} and we find \displaystyle \displaystyle \begin{align*} c_1 = 1 \end{align*}.

Differentiate both sides and we have

\displaystyle \displaystyle \begin{align*} e^{\sin{x}}\cos^2{x} - e^{\sin{x}}\sin{x} = 2c_2 + 2\cdot 3c_3x + 3\cdot 4c_4x^2 + 4\cdot 5c_5x^3 + \dots \end{align*}

We know that \displaystyle \displaystyle \begin{align*} e^{\sin{0}}\cos^2{0} - e^{\sin{0}}\sin{0} = 1 \end{align*} so let \displaystyle \displaystyle \begin{align*} x = 0 \end{align*} and we find \displaystyle \displaystyle \begin{align*} c_2 = \frac{1}{2} \end{align*}.

Follow this process to find as many terms as you need. I doubt that there will be a pattern.

thanks guys