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

Robert Reynolds and Allan Stauffer
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Robert Reynolds: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada
Allan Stauffer: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada

Mathematics, 2021, vol. 9, issue 6, 1-12

Abstract: 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 ? i e ? i m x ( log ( a ) ? i x ) k ? e i m x ( log ( a ) + i x ) k 2 e ? x ? 1 d x and ? 0 ? i e ? i m x ( log ( a ) ? i x ) k ? e i m x ( log ( a ) + i x ) k 2 e ? x + 1 d x 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.

Keywords: entries in Erdélyi; Lerch function; hypergeometric function; incomplete beta function; definite integral; Mellin transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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