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