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I Integration of Legendre Polynomials with different arguments

  1. Jun 17, 2017 #1
    Hi everybody,
    I'm trying to calculate this:

    $$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$

    where ##P_{l}## are the Legendre polynomials, ##\Omega## is the surface of a sphere of radius ##R##, and

    $$ \cos{\gamma} = \cos{\theta'} \cos{\theta} + \sin{\theta'}\sin{\theta}\,\cos({\phi' -\phi}) $$

    I am told that only the ##l=1## term survives due to orthogonality of Legendre polynomials (of course ##\cos{\theta'} = P_{1}(\cos{\theta'}) ##), but I'm don't see why, since the Legendre polynomials have different arguments.

    How can I show that this is true?
     
  2. jcsd
  3. Jun 20, 2017 #2

    Orodruin

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    Your angle ##\gamma## is an angle from the direction defined by ##\theta## and ##\phi##. The spherical harmonics of a fixed ##\ell## form an irrep of the rotation group and so your ##P_\ell## will be rotated into a linear combination of the spherical harmonics ##Y_\ell^m##. The Legendre polynomial ##P_1(\cos\theta') = \cos\theta'## is directly proportional to ##Y_1^0(\theta',\phi')## and so your integral will project out this component of ##P_\ell(\cos\gamma)##.
     
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