DE-Type Quadrature Formulae for Cauchy Principal-Value Integrals and for Hadamard Finite-Part Integrals

Abstract

The double exponential formula (abbreviated to the DE formula) for numerical integration, which was first proposed by Takahasi and Mori[3] 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. $\int_{ - 1}^1 {f(x)} /(x - \lambda )dx$ and for evaluating the Hadamard finite-part integral f.p. ${\int_{ - 1}^1 {f(x)/(x - \lambda )} ^2}dx$, 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.