Integral relations associated with the semi-infinite Hilbert transform and applications to singular integral equations

TL;DR

This paper derives integral relations involving the semi-infinite Hilbert transform for special functions, leading to new orthogonal systems, solutions to singular integral equations, and applications in contact mechanics.

Contribution

It introduces novel integral relations for special functions and develops methods for solving singular integral equations on semi-infinite intervals.

Findings

Derived new integral relations for special functions

Proposed exact and approximate solutions to singular integral equations

Developed a new quadrature formula for the Cauchy integral

Abstract

Integral relations with the Cauchy kernel on a semi-axis for the Laguerre polynomials, the confluent hypergeometric function, and the cylindrical functions are derived. A part of these formulas is obtained by exploiting some properties of the Hermite polynomials, including their Hilbert and Fourier transforms and connections to the Laguerre polynomials. The relations discovered give rise to complete systems of new orthogonal functions. Free of singular integrals, exact and approximate solutions to the characteristic and complete singular integral equations in a semi-infinite interval are proposed. Another set of the Hilbert transforms in a semi-axis are deduced from integral relations with the Cauchy kernel in a finite segment for the Jacobi polynomials and the Jacobi functions of the second kind by letting some parameters involved go to infinity. These formulas lead to integral relations for the Bessel functions. Their application to a model problem of contact mechanics is given. A new quadrature formula for the Cauchy integral in a semi-axis based on an integral relation for the Laguerre polynomials and the confluent hypergeometric function is derived and tested numerically. Bounds for the remainder are found.