# How do I integrate a product of Legendre polynomials over a volume?

If I have a legendre polynomial $P_n(\cos \theta)$ where $\cos \theta = z/r$, how to I perform the integral of $$\int d^3 \mathbf x P_n(\cos \theta) P_m(\cos \theta) f(r)$$ I know I can rewrite the volume of integrate $d^3 \mathbf x$ as $r dr d(\cos \theta)$ but then I need to evaluate the integral of the product of $P_n P_m$ and I can't find a simple expression for it.

• there are orthogonality relations for the Legendre polynomials which i suppose will be helpful here – tired Mar 19 '16 at 9:36

$$\int_{-1}^{1} P_{m}(\cos \theta) P_{n}(\cos \theta) d(\cos \theta)= \frac{2\delta_{mn}}{2n+1}$$