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Spherical Harmonic Integration

  1. May 24, 2008 #1
    Spherical Harmonic Integration plz help!!!

    Hi guys,

    I have a spherical harmonic integration problem that I would like to solve

    [tex]\int_S Y_{nm}^*(\omega)Y_{nm}^*(\omega) d\omega[/tex]
    which I re-write as follows:

    [tex]= \int_S \left|Y_{nm}^*(\omega)\right|^2 d\omega [/tex]

    Am I right to say that

    [tex]\int_S \left|Y_{nm}^*(\omega)\right|^2 d\omega = \frac{2n+1}{4\pi}[/tex] ???

    since we know that

    [tex] \left|Y_{nm}^*(\omega)\right|^2 = \frac{2n+1}{4\pi}[/tex]

    Thanks in advance

    Kostas
     
  2. jcsd
  3. May 24, 2008 #2
    Spherical Harmonic Integration plz help!!!

    Hi,

    Please accept my apologies but I would like to re-phrase the problem as it might make things clearer for those of you who are experts in this area of mathematics. I have also made some corrections to my last post.

    I have a spherical harmonic integration problem that I would like to solve. The integral follows:

    [tex]\int_S Y_{nm}(\omega)Y_{nm}^*(\omega) d\omega[/tex]

    My understanding is that the above can be re-written as follows:

    [tex]\int_S Y_{nm}(\omega)Y_{nm}^*(\omega) d\omega= \int_S \left|Y_{nm}(\omega)\right|^2 d\omega [/tex]

    and since we know that

    [tex]\int_S \left|Y_{nm}(\omega)\right|^2 d\omega = \frac{2n+1}{4\pi}[/tex]

    I conclude that

    [tex]\int_S Y_{nm}(\omega)Y_{nm}^*(\omega) d\omega = \frac{2n+1}{4\pi}[/tex]

    Would you agree with the above approach???

    My problem is that using the orthogonality principle of Spherical Harmonics the same problem can be handled (or can it be handled?).

    [tex]\int_S Y_{nm}(\omega) Y_{n'm'}^*(\omega) d\omega = \delta_{nn'} \delta_{mm'}[/tex]

    so that if [tex] n = n' [/tex] the result will be equal to 1 and 0 otherwise. Am I wrong?

    I would highly appreciate your advice

    Kostas
     
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