Applications of Hilbert transform theory to numerical quadrature

Abstract:

Some finite integrals are difficult to evaluate numerically because the integrand has a high peak or contains a rapidly oscillating function as a factor. If the integrand is an analytic function Cauchy’s theorem may be applied to replace the integral by a contour integral, the path being chosen to avoid singularities of the integrand, together with a possible residue contribution. If the integrand has branch singularities in the complex plane close to the interval of integration, the direct application of Cauchy’s theorem is not practical. In this paper we show how the theory of Hilbert transforms may be applied to replace the integrand by a different complex valued function whose real part coincides with the integrand on the real line, but which has no singularities in the upper half plane. Using these transformations, integrands whose difficult behavior arises from a factor whose Hilbert transform is known analytically may be treated by carrying out a contour integral of a different function and taking the real part of the result. It is shown by means of examples that such a procedure may result in significant savings in terms of computational effort.