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Spherical Harmonics Normalization

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data
    I'm trying to solve

    [tex] I_l = \int^{\pi}_{0} d \theta \sin (\theta) (\sin (\theta))^{2l} [/tex]

    2. Relevant equations

    the book suggest:

    [tex] I_l = \int^{+1}_{-1} du (1 - u^2)^l [/tex]

    3. The attempt at a solution

    I think it's something related to Legendre polynomials

    [tex] P_l (u) = \frac{(-1)^l}{2^l l!} \frac{d^l}{d u^l} (1- u^2)^l [/tex]


    but i don't know how to manage it... how it works?

    thank u,
    mahblah
     
  2. jcsd
  3. Nov 29, 2011 #2

    dextercioby

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    Science Advisor
    Homework Helper

    This is amenable in terms of the Euler Beta function. Look up the definiton of Beta in terms of the integral of polynomials or sine/cosine and use it to express your integral in terms of Beta.
     
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