The Rectangle Rule for Computing Cauchy Principal Value Integral on Circle

Abstract

The classical composite rectangle (constant) rule for the computation of Cauchy principle value integral with the singular kernel x s cot 2 − is discussed. We show that the superconvergence rate of the composite midpoint rule occurs at certain local coodinate of each subinterval and obtain the corresponding superconvergence error estimate. Then collation methods are presented to solve certain kind of Hilbert singular integral equation. At last, some numerical examples are provided to validate the theoretical analysis.