DE-TYPE QUADRATURE FORMULAE FOR CAUCHY PRINCIPAL-VALUE INTEGRALS AND FOR HADAMARD FINITE-PART INTEGRALS

摘要

The double exponential formula (abbreviated to the DE formula) for numerical integration, which was first proposed by Takahasi and Mori in 1974, is recognized as one of the most efficient quadrature formulae and has been widely used in a variety of area, physics, engineering, and so on. In this paper, DE-type quadrature formulae are proposed for evaluating the Cauchy principal-value integral p.v. ∫_(-1)~1 f(x)/(x ―λ)dx and for evaluating the Hadamard finite-part integral f .p. ∫_(-1)~1 f(x)/(x ―λ)~2dx, where f(x) is a function that is analytic on the interval (―1,1), and where λ is a constant such that ―1 < λ< 1. For the evaluation of Cauchy principal-value integrals and for the evaluation of Hadamard finite-part integrals, there have already been proposed general variable transformation-type formulae in Bialecki's papers, which are based on the so-called "Sinc Numerical Methods". It has been also shown there, that when employing the transformation x = tanh(t/2), which we call the single exponential transformation, the quadrature error is of the order O(exp(―cN~(1/2)), where N is the number of nodes used, and where c is a positive constant independent of N. DE-type formulae are obtained simply by replacing the single exponential transformation with the double exponential transformation x = tanh(( π/2) sinh t). Yet they enjoy a good convergence property. In fact, it is demonstrated that the quadrature error is of the order O(exp(―c_λN/logN)), where c is a constant dependent only on λ. The paper is organized as follows: first, we derive DE-type formulae in 2; the quadrature error of the DE-type formulae is estimated theoretically in 3; a numerical example is shown in 4; we conclude the present paper in 5.