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Is it possible to integrate this analytically:

$$ \int_{0}^{2\pi} P_l(\cos(\theta-\theta')) P_l(\cos(\theta)) \sin(\theta) d\theta$$

I mean the integral would be pretty easy by using the orthogonality definition of the $l$-th Legendre polynomial $P_l$ if we only had $\cos(\theta)$ in there, but this difference of angles, makes it somewhat hard for me.

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  • $\begingroup$ Do you need the explicit form of $G(\theta')$ (the integral you introduce) or is it sufficient for your applications to study some properties of it like symmetry, value at $z=0$, (some) derivatives etc...? $\endgroup$ – Avitus Aug 13 '13 at 15:15
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If you expand the $\cos(\theta - \theta')$, I think you'll find the antiderivative is a linear combination of terms of the form $\cos(m \theta + n \theta')$ for integers $m$, $n$, and thus is periodic in $\theta$ with period $2 \pi$.

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There is always the Legendre addition theorem

$$P_l(\cos\gamma) = P_l(\cos\theta_1)P_(\cos\theta_2) + 2\sum_{m=1}^l \frac{(l-m)!}{(l+m)!} P_l^m(\cos\theta_1)P_l^m(\cos\theta_2)\cos[m(\phi_1-\phi_2)]$$

where $\cos\gamma = \cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2\cos(\phi_1-\phi_2)$ and $P_l^m(x)$ are the associated Legendre polynomials. I have no idea of how the integral actually get evaluated to but you can probably get the answer by using the Gaunt's formula on the wiki page.

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