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Homework Help: Messy integral

  1. Oct 1, 2006 #1
    Problem:

    Show that

    [tex]\int_{-1}^{1} x P_n(x) P_m(x) dx = \frac{2(n+1)}{(2n+1)(2n+3)}\delta_{m,n+1} + \frac{2n}{(2n+1)(2n-1)}\delta_{m,n-1}[/tex]

    I guess I should use orthogonality with the Legendre polynomials, but if I integrate by parts to get rid of the x my integral equals zero.
    Any tip on how to start working with this?
     
  2. jcsd
  3. Oct 1, 2006 #2

    Astronuc

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    Staff Emeritus
    Science Advisor

    First thought would be to use one of the recursion relationships on xPn(x).

    For example -

    [tex](l+1)P_{l+1}(x)\,-\,(2l+1)xP_l(x)\,+\,lP_{l-1}(x)\,=\,0[/tex]

    BTW, has one shown -

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

    That was demonstrated here on PF recently.
     
    Last edited: Oct 1, 2006
  4. Oct 1, 2006 #3
    Yes, I've got the last equation and I'll try with the recursion, thank you. =)
     
  5. Oct 1, 2006 #4
    Another thing I would recommend is to try using the Rodriguez formula for the Legendre polynomials, then play games with integration by parts.
     
  6. Oct 1, 2006 #5
    And why is that? I solved the problem by the way. Pretty simple when you know about the recursion relationships.
     
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