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Is there a closed form solution for the indefinite integral of an associated Legendre polynomial: $$\int dx\, P_l^m(x)=\,\,?$$

Where the associated polynomial is defined as:

$$P_l^m(x)\equiv (-1)^m\left(1-x^2\right)^{\frac {m}{2}}\frac{d^mP_l(x)}{dx^m},m\ge0$$ $$P_l^{-m}=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(x)$$

And $P_l(x)$ is a Legendre polynomial of the first kind:

$$P_l(x)\equiv \frac{1}{2^ll!}\frac{d^l}{dx^l}\left(x^2-1\right)^2$$

For "closed form" I'm including as a function of other well-known special functions. In particular, if possible I would like to have a solution in terms of functions implemented in Boost 1.41.0.

Thanks in advance!

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  • $\begingroup$ you can integrate the recursion formula $\endgroup$ – tired Aug 27 '16 at 14:55
  • $\begingroup$ @tired Well, of course you can if I leave the $(1-x^2)^{m/2}$ term out. D'oh. Fixed now. $\endgroup$ – Chris Aug 27 '16 at 21:09

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