Gauss-Kronrod integration rules for Cauchy principal value integrals

Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using n + 2 points which requires only values of the integrand. However, this new rule does not exist for all n. and when it does, several significant figures are lost in its use. 1. Introduction. In this paper we shall consider Kronrod extensions (KE) to integration rules based on Gauss and Lobatto points for the evaluation of Cauchy principal value (CPV) integrals of the form fix) vyx) 1 Since the existence theory of KE's for regular integrals insures such extensions for only certain classes of weight functions, w(x), we shall restrict our attention here to the most important class, namely w(x) = (1 — x2y~x/2, where 0 < p < 2 in the Gauss case and - \ < p < 1 in the Lobatto case (9). (The only other relevant weight functions are the Jacobi weight functions w(x) = (1 — x)a(l + x)& with a = \, - i < s < : or s = y - \ < a < f and w(x) = )/\ - x2 /(I - rx2), -oo < r < 1 (5).) For our case, w(x) = (1 — x2)M_1/2, the corresponding orthogonal polynomials are the Gegenbauer polynomials C (x) (usually written C£(x)) which have the following normalization (11, p. 174) (2) ( w(x)C(x)Cmil(x)dx = 8"mhnil, where (3) hn)i = fT(n + 2p)r(p + {)/ (n + u)n!r(p)r(2p),