Spherical Harmonic Integration

[onlyk]

Spherical Harmonic Integration please help!

Hi guys,

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

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

$$= \int_S \left|Y_{nm}^*(\omega)\right|^2 d\omega$$

Am I right to say that

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

since we know that

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

Thanks in advance

Kostas

 

[onlyk]

Spherical Harmonic Integration please 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:

$$\int_S Y_{nm}(\omega)Y_{nm}^*(\omega) d\omega$$

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

$$\int_S Y_{nm}(\omega)Y_{nm}^*(\omega) d\omega= \int_S \left|Y_{nm}(\omega)\right|^2 d\omega$$

and since we know that

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

I conclude that

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

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?).

$$\int_S Y_{nm}(\omega) Y_{n'm'}^*(\omega) d\omega = \delta_{nn'} \delta_{mm'}$$

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

I would highly appreciate your advice

Kostas