HILBERT AND HADAMARD TRANSFORMS BY GENERALIZED CHEBYSHEV EXPANSION

An automatic quadrature is presented for approximating Hadamard finite-part (fp) integrals of a smooth function, with a double pole singularity within the range of integration. The quadrature rule is derived from the differentiation of an approximation to a Cauchy principal value integral or the Hilbert transform. The approximation to the fp integral is represented as a function of the value of pole by using Chebyshev polynomials of the second kind. Since the error can be estimated independently of the value of pole, a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance, with the same number of function evaluations. Numerical examples are also included to illustrate the performance of the methods.