I Integration of Legendre Polynomials with different arguments

[Alan Sammarone]

Hi everybody,
I'm trying to calculate this:

$$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$

where $P_{l}$ are the Legendre polynomials, $\Omega$ is the surface of a sphere of radius $R$, and

$$ \cos{\gamma} = \cos{\theta'} \cos{\theta} + \sin{\theta'}\sin{\theta}\,\cos({\phi' -\phi}) $$

I am told that only the $l=1$ term survives due to orthogonality of Legendre polynomials (of course $\cos{\theta'} = P_{1}(\cos{\theta'})$), but I'm don't see why, since the Legendre polynomials have different arguments.

How can I show that this is true?

 

[Orodruin]

Your angle $\gamma$ is an angle from the direction defined by $\theta$ and $\phi$. The spherical harmonics of a fixed $\ell$ form an irrep of the rotation group and so your $P_\ell$ will be rotated into a linear combination of the spherical harmonics $Y_\ell^m$. The Legendre polynomial $P_1(\cos\theta') = \cos\theta'$ is directly proportional to $Y_1^0(\theta',\phi')$ and so your integral will project out this component of $P_\ell(\cos\gamma)$.