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Math Help - Legendre-Fourier series question

  1. #1
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    Legendre-Fourier series question

    Consider the equation:

    \sum_{n=0}^\infty d_{n}P_{n}(x) \hspace{1mm} = \hspace{1mm} sin(x)

    where P_{n}(x) is the Legende polynomial of degree n.

    How can I find out the value of each  d_{1}, d_{2}, d_{3},...?
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  2. #2
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    Quote Originally Posted by garunas View Post
    Consider the equation:

    \sum_{n=0}^\infty d_{n}P_{n}(x) \hspace{1mm} = \hspace{1mm} sin(x)

    where P_{n}(x) is the Legende polynomial of degree n.

    How can I find out the value of each  d_{1}, d_{2}, d_{3},...?
    Note that

    \int_{-1}^{1}P_n(x)P_m(x)dx=\frac{2}{2n+1}\delta_{n,m}

    Where the right hand side is the Kronecker delta.

    So multiply both sides by the nth Legendre polynomial and integrate

    \frac{2 d_n}{2n+1}=\int_{-1}^{1}P_n(x)\sin(x)dx

    Can you finish from here?
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  3. #3
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    Sure can! Thanks mate
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