Representation of a polyparametric family of improper integrals by the method of hybrid integral transform of the type (Lebedev) 1st kind -- Legendre (Q2753523)

A system of differential equations \((B_\alpha-q_1^2)u_1(r) = -g_1(r)\), \(0\leq r\leq R_1\), \((\Lambda_\mu-q_2^2)u_2(r) = -g_2(r)\), \(R_1\leq r\leq\infty\) is considered. Here \(B_\alpha =r^2\,d^2/dr^2 + r(2\alpha+1)d/dr +\alpha^2 - \lambda^2 r^2\), \(0\leq\lambda < \infty\), \(\alpha\geq -1/2\), is the Bessel differential operator, \(\Lambda_\mu=d^2/dr^2 + \coth r\,d/dr + 1/4 - \mu^2/\sinh^2r\), \(\mu \geq -1/2\), is the Legendre differential operator. By comparison of solutions, constructed using (i) the Cauchy function technique and (ii) the hybrid integral transform of the type (Lebedev) 1st kind -- Legendre, a representation of a family of improper integrals containing Legendre and Bessel functions with imaginary index is obtained.