Computation of Fourier and Laplace transforms of singular functions using modified moments

AbstractIn this paper a recurrence formula for the computation of Mk = ∫−1+1 (1 − x)α(1 + x)β exp[−a/(1+x)]Tk(x)dx is presented. The numerical stability is discussed. The starting values are confluent hypergeometric functions which can be evaluated using Luke's results on Chebyshev series expansions and Padé approximations of hypergeometric functions. Applications of this recurrence relation are the evaluation of the Fourier transform of singular functions by modified Clenshaw-Curtis integration, the construction of Gaussian quadrature formulae for Fourier integrals and the numerical inversion of the Laplace transform