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Integral involving product of derivatives of Legendre polynomials

  1. Jan 2, 2013 #1
    Anyone how to evaluate this integral?

    [itex]\int_{-1}^{1} (1-x^2) P_{n}^{'} P_m^{'} dx [/itex], where the primes represent derivative with respect to [itex]x [/itex]?

    I tried using different recurrence relations for derivatives of the Legendre polynomial, but it didn't get me anywhere...
  2. jcsd
  3. Jan 2, 2013 #2


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    Homework Helper

    Use the facts

    [tex]\left((1-x^2)P_n^\prime \right)^\prime=-n(n+1)P_n[/tex]


    [tex]\int_{-1}^1 P_m P_n \text{ dx}=\dfrac{2}{2n+1} \delta_{mn}[/tex]

    to integrate by parts

    or just use

    [tex]P_n=\frac{1}{(2n)!!} \dfrac{d^n}{dx^n} (x^2-1)^n[/tex]
    Last edited: Jan 2, 2013
  4. Feb 1, 2013 #3
    Thank you very much!
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