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I came across the following integral

$$\int_{-1}^{+1} (1-x^{2}) \frac{\partial P_{lm}(x)}{\partial x} \frac{\partial P_{km}(x)}{\partial x} dx$$ where $P_{lm}(x)$ is an associated Legendre polynomial, while trying to derive an expression in a physics paper. I am not entirely sure if this is solvable, though. I have tried to integrate the expression using the recurrence and orthogonality relations listed here. However, no matter which recurrence relation I use, I am unable to exactly evaluate this integral. Is there an way to find this integral exactly?

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  • $\begingroup$ Do you mean $P_l^m$ and $P_k^m$? $\endgroup$
    – Gary
    Commented Apr 6, 2022 at 8:02
  • $\begingroup$ yes, but the paper I am studying uses subscripts on the $Y_{lm}$ (spherical harmonics) $\endgroup$
    – saad
    Commented Apr 6, 2022 at 8:05

1 Answer 1

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According to the definitions of the associated Legendre polynomials and assuming $m+1\leq l,k$ we have that

$$\int_{-1}^1 (1-x^2)\partial_x P_l^m \partial_x P_k^m \:dx = \int_{-1}^1 P_l^{m+1}P_k^{m+1}\:dx = \frac{2(l+m+1)!}{(2l+1)(l-m-1)!}\delta_{kl}$$

the orthogonality condition for fixed $m$.

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  • $\begingroup$ Short, simple and nice (as usual !). Cheers :-) and (+1) for sure $\endgroup$
    – Claude Leibovici
    Commented Apr 6, 2022 at 10:08
  • $\begingroup$ @ClaudeLeibovici thank you :) $\endgroup$
    – Ninad Munshi
    Commented Apr 7, 2022 at 7:25

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