Integration involving spherical harmonics

[lordkelvin]

Homework Statement

Evaluate the integral ∫∫dΩ V(Ω)Y^m_l(Ω) for V(Ω) = +V₀ for 0<θ<π/2 ; -V₀ for π/2<θ<π

Homework Equations

I was hoping to apply the orthonormality properties of the spherical harmonics but this is a little more difficult since the integral breaks into two integrals over half spheres instead of one integral over a full sphere.

I guess it may be useful to rewrite the V(Ω) as ±√(4π)Y₀⁰(Ω)V₀ in order to invoke the orthogonality properties of the spherical harmonics when integrated on a sphere.

The Attempt at a Solution

Basically I've split the integral into two integrals over the upper and lower halves of the sphere. Now what? Surely I'm not to use the definition of the spherical harmonics; there must be a cleaner way to do this by invoking the properties of Ylm(omega). There is nothing I can say about ∫∫_{half of sphere}dΩY_l^m(Y₀⁰)* as far as I know. Is there some way to combine the two integrals again so that I can use the orthonormality property?

 

[vela]

Orthonormality won't really help here because you're not integrating the product of two spherical harmonics. The integral should be pretty straightforward though. Start by integrating φ out first. Then you'll be left with an integral involving a Legendre polynomial.

 

[lordkelvin]

I get zero. I think this is so since Y_l^m(θ,$\phi$) = const* e^(i*$\phi$)*P_l^m(cos($\theta$) and so the integral over phi yields zero.

 

[vela]

It's $e^{im\phi}$, not $e^{i\phi}$. It makes a difference.