Spherical Harmonic Integration

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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
 

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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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