Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity

Abstract: The standard classic integration rules give inaccurate results for where , are real numbers and f is any sufficiently smooth function on . These integrals have been investigated for the special case in Hascelik [A.I. Hascelik, On numerical computation of integrals with integrands of the form on (2007), in press] and for the case (, ) in Gautschi [W. Gautschi, Computing polynomials orthogonal with respect to densely oscillating and exponentially decaying weight functions and related integrals, J. Comput. Appl. Math. 184 (2005) 493–504]. In this work we construct suitable Gauss quadrature rules for approximating these integrals in high accuracy. The required three-term recurrence coefficients are computed by the Chebyshev algorithm using arbitrary precision arithmetic. We also give appropriate Filon-type methods for these integrals, with related error bounds. Some numerical examples are given to test the new methods. [Copyright &y& Elsevier]