A Note on Some Definite Integrals of Arthur Erdélyi and George Watson

This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and the Fermi-Dirac functions in the integrands, separately, with numerators slightly modified with a difference in two expressions that contain the Fourier kernel multiplied by a polynomial and its complex conjugate. In this work, we use our contour integral method to derive these definite integrals, which are given by ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx−1dx and ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx+1dx in terms of the Lerch function. We use these two definite integrals to derive formulae by Erdéyli and Watson. We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.

On Computation of Highly Oscillatory Integrals with Bessel Kernel

In this paper, we introduce a new numerical scheme for approximation of highly oscillatory integrals having Bessel kernel. We transform the given integral to a special form having improper nonoscillatory Laguerre type and proper oscillatory integrals with Fourier kernels. Integrals with Laguerre weights over [0, ∞) will be solved by Gauss-Laguerre quadrature and oscillatory integrals with Fourier kernel can be evaluated by meshless-Levin method. Some numerical examples are also discussed to check the efficiency of proposed method.