Evaluation of some integrals involving associated Legendre polynomials and the arctanh function

Nyengeri and al

Résumé

A method for computing the definite integrals is developped. This method consists of representing the product of two associated
Legendre polynomials (ALPs) by a finite series in ALPs with constant coefficients. Each of the two integrals in question is
then transformed in a finite series in simple integrals, each involving a single ALP. Recursive formulas for computing the
latter integrals are proposed. Our method is tested through the computation of the two integrals in question for some values of
degrees and orders of the ALPs. We compare the method with the numerical computation by means of the Simpson method.