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Integrating legendre polynomials with weighting function

  1. Jan 3, 2010 #1
    1. The problem statement, all variables and given/known data
    I recently came across this integral while doing a problem in electromagnetism (I'm not sure if there exists a nice analytic answer):

    [tex]\int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) = \int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2},[/tex]

    2. Relevant equations

    P_m(x) is the m^th Legendre Polynomial.

    3. The attempt at a solution

    There are lots of close integrals in Gradshteyn and Ryzhik 7.1-7.2 but nothing close enough for me to use.

    One way to evaluate it would be to expand the square root as a Taylor series, and then change basis to re-expand it as a series of Legendre polynomials, then use tricks involving triple integrals of Legendre polynomials (such as those in Arfken and Weber 12.9). However this is incredibly messy and I can't see how I could get an analytic expression from it.

    Can anyone think of a nice way of approaching this integral? I can't use for example differentiation under the integral because I don't know how to integrate the product of Legendre functions on intervals other than [-1,1] (and [0,1]).
  2. jcsd
  3. Jan 3, 2010 #2


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    Never mind...
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