Expanding function with spherical harmonics

[Integratethis]

Homework Statement

The function cos(theta)*cos(phi) in spherical coordinates cannot be expanded to a series of spherical harmonics. Explain why.

Homework Equations

As far as I can recall, the spherical harmonics are a complete set over a sphere, meaning every function which is SI over a sphere can be expanded to such a series...including this one.

The Attempt at a Solution

Tried everything I can think of - got to the point in which I need to prove that the coefficient for each Y(l,m=+-1) = 0, but couldn't find a way around it.

 

[dextercioby]

You have to check if the function is a member of L^2 (S^2). Then apply the theory from http://en.wikipedia.org/wiki/Spherical_harmonics - real form and expansion of real functions in terms of real forms of spherical harmonics.

And pay attention to integrations, that's all.

 

[vela]

Tried everything I can think of - got to the point in which I need to prove that the coefficient for each Y(l,m=+-1) = 0, but couldn't find a way around it.

You can't prove that because it's not true. The coefficient of Y₂₁, for instance, is not 0. I'm not sure what the problem is getting at because, as you said, the spherical harmonics are a complete set and the function is square-integrable on the sphere.