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Math Help - taylor polynomial for composition

  1. #1
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    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!
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  2. #2
    Super Member girdav's Avatar
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    Re: taylor polynomial for composition

    Which degree does the Taylor polynomial needs to have?
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  3. #3
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    Re: taylor polynomial for composition

    doesnt matter, i want to know how is it done. lets say up to 5'th degree around 0
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  4. #4
    Super Member girdav's Avatar
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    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 Fa di Bruno.
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    Re: taylor polynomial for composition

    Quote Originally Posted by dorubin View Post
    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 \begin{align*} e^{\sin{x}} = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots \end{align*}.

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

    Differentiate both sides and we find

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

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

    Differentiate both sides and we have

    \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 \begin{align*} e^{\sin{0}}\cos^2{0} - e^{\sin{0}}\sin{0} = 1 \end{align*} so let \displaystyle \begin{align*} x = 0  \end{align*} and we find \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.
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    Re: taylor polynomial for composition

    thanks guys
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