Quadrature rules for singular integrals with application to Schwarz-Christoffel mappings

Abstract

Numerical quadrature rules for singular integrals are presented and error bounds are derived. The rules are simple modifications of composite Newton-Cotes formulas. For singularities of type x α , α > −1, the lowest order rule (modified midpoint rule) has error terms of order Δ 2, Δ 2+ α , and Δ 2 log ( 1 Δ ) , where Δ is the subinterval length. The rule proposed by Davis for integration of the Schwarz-Christoffel equation for conformal mapping of polygons is shown to have error terms of the same order. For polygons with sharp corners, i.e., α close to −1, the number of integration subintervals required for the Schwarz-Christoffel equation can be reduced by several orders of magnitude by use of higher order rules given here. Explicit formulas are given for four rules of most likely utility; they are extensions of the midpoint, trapezoidal, Simpson's, and 4-point rules.