  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.
