计算Cauchy和Mangler积分主值的新方法

对于Cauchy积分主值曾提出过一些数值积分法,主要是抽去奇异性方法和内插型或Gauss型方法。对于前者,要求积分核在积分区间上有连续的一阶导数;对于后者,实际上只能求出节点(某类函数零点)处的积分主值,且当积分核不是多项式时

A NEW ALGORITHM FOR CAUCHY AND MANGLAR PRINCIPAL VALUE INTEGRALS

The kernel of Cauchy integrals is expanded in an asymptotic expansion of Chebyshef polynomials, and then the principal value of Cauchy integrals is calculated by integrating it into a close form term by term and summing the finite terms. Based on this, distinguishing the different types, by means of analytic differentiation term by term or cubic spline derivative with respect to the principal value of Cauchy integrals obtained, the principal value of Mangier is calculated. It will not restrict the positions of calculated points and the integral kernel, except the entirely natural condition that the kernel is " quadratical integrable". With present method the principal values of Cauchy and Mangier integrals of highly oscillatory kind can be treated. Eight numerical examples are given. The comparison between the calculated results of some exact solutions and the author's method shows clearly, that the present algorithm is of fine accuracy.