Uniform approximations to finite Hilbert transform and its derivative

Interpolatory integration rules of numerical stability are presented for approximating Cauchy principal value (p.v.) integrals ∫-11 f(t)/(t-c) dt and Hadamard finite part (f.p.) integrals ∫-11f(t)/(t - c)2 dt, -1 < c < 1, respectively, for a given smooth function f(t). Present quadrature rules consist of interpolating f(t) at abscissae in the interval of integration [-1, 1] except for the pole c, where neither the function value f(c) nor its derivative f'(c) is required, followed by subtracting out the singularities.We demonstrate that the use of both endpoints ±1 as abscissae in interpolating f(t) is essential for uniformly approximating the integrals, namely for bounding the approximation errors independently of the values of c. In fact, for the f.p. integrals the use of double abscissae at both endpoints ±1 as well as simple abscissae in (-1, 1) enables the uniform approximations, while the use of simple abscissae at both endpoints ±1 and those in (-1, 1) is sufficient for the p.v. integrals. These facts suggest that finite Hilbert transforms (p.v. integrals) and their derivatives (f.p. integrals) with varied values of c could be approximated efficiently with the same number of abscissae, respectively. Some numerical examples are given.