半无限希尔伯特变换的积分关系及其在奇异积分方程中的应用
Integral relations associated with the semi-infinite Hilbert transform and applications to singular integral equations
摘要：
导出了拉盖尔多项式、汇合型超几何函数和柱面函数在半轴上与柯西核的积分关系。这些公式的一部分是通过利用厄米多项式的一些性质得到的，包括它们的希尔伯特和傅里叶变换以及与拉盖尔多项式的联系。发现的关系产生新的正交函数的完整系统。在不含奇异积分的情况下，给出了半无限区间上特征完全奇异积分方程的精确解和近似解。通过对雅可比多项式和第二类雅可比函数在有限段上的Cauchy核的积分关系，使某些参数趋于无穷大，导出了另一组半轴上的希尔伯特变换.这些公式导致贝塞尔函数的积分关系。给出了它们在接触力学模型问题中的应用。基于Laguerre多项式与汇合型超几何函数的积分关系，导出了一个新的半轴上Cauchy积分求积公式，并进行了数值验证.找到了其余部分的边界。
英文摘要：
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.
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