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Integral relations with the Cauchy kernel on a semi-axis for the Laguerrepolynomials, the confluent hypergeometric function, and the cylindricalfunctions are derived. A part of these formulas is obtained by exploiting someproperties of the Hermite polynomials, including their Hilbert and Fouriertransforms and connections to the Laguerre polynomials. The relationsdiscovered give rise to complete systems of new orthogonal functions. Free ofsingular integrals, exact and approximate solutions to the characteristic andcomplete singular integral equations in a semi-infinite interval are proposed.Another set of the Hilbert transforms in a semi-axis are deduced from integralrelations with the Cauchy kernel in a finite segment for the Jacobi polynomialsand the Jacobi functions of the second kind by letting some parameters involvedgo to infinity. These formulas lead to integral relations for the Besselfunctions. 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 anintegral relation for the Laguerre polynomials and the confluent hypergeometricfunction is derived and tested numerically. Bounds for the remainder are found.

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