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Why books say that is necessary in Legendre equation to have $l$ integer if you want regular solutions in $\vert x\vert \leq 1$. It seems not necessary.

Thanks in advance.

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"It seems not necessary." - what gave you the idea? Both kinds of Legendre functions will necessarily display singularities at $\pm 1$, with the notable exception of the Legendre polynomials (and that bit has a whole lot to do with the integer restriction)... –  J. M. Nov 8 '12 at 6:27
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