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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.

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

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

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