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

The standard classic integration rules give inaccurate results for∫01tαf(t)sin(ω/tr)dtand∫01f(t)tαcos(ω/tr)dt where ω,r>0ω,r>0, α+r>−1α+r>−1 are real numbers and f   is any sufficiently smooth function on [0,1][0,1]. These integrals have been investigated for the special case α=0α=0 in Hascelik [A.I. Hascelik, On numerical computation of integrals with integrands of the form f(x)sin(1/xr)f(x)sin(1/xr) on [0,1][0,1] (2007), in press] and for the case (r=1r=1, α=0α=0) 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.