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I found a definite integral form of Legendre polynomial of second kind, $$ Q_{n}(z)=\frac{1}{2}\int^{+1}_{-1}\frac{P_{n}(t)}{z-t}dt $$ when n is an integer. I wonder how to evaluate this integral. I tried to use contour integral by changing $t$ to $e^{i\theta}$, but it failed.

I'll appreciate your help.

tard
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  • This is pretty much a definition of the Legendre function of the second kind. So you're asking for an expression for $Q_n(z)$. Which one?.. – metamorphy Apr 01 '22 at 09:15

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