A simple derivation of the integrals of products of Legendre polynomials with logarithmic weight

Sebastian Schmutzhard-H\"ofler

A simple derivation of the integrals of products of Legendre polynomials with logarithmic weight

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arxiv 2404.11173 v1 pith:BMTAOEYX submitted 2024-04-17 math.CA

Sebastian Schmutzhard-Höfler This is my paper · ORCID

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keywords integralslegendrepolynomialslogarithmicproductssimpleweightderivation

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abstract

We explore integrals of products of Legendre polynomials with a logarithmic weight function. More precisely, for Legendre polynomials $P_m$ and $P_n$ of orders $m$ and $n$, respectively, we provide simple derivations of the integrals $$\int \limits_0^1 P_n(2x-1) P_m(2x-1)\log (x)dx.$$

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