Derivation of some integrals in Gradshteyn and Ryzhik

Robert Reynolds; Allan Stauffer; Robert Reynolds; Allan Stauffer

doi:10.3934/math.2021109

AIMS Mathematics

2021, Volume 6, Issue 2: 1816-1821. doi: 10.3934/math.2021109

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Research article

Derivation of some integrals in Gradshteyn and Ryzhik

Robert Reynolds ^,,
Allan Stauffer

Department of Mathematics, York University, 4700 Keele Street, Toronto, M3J1P3, Canada

Received: 04 October 2020 Accepted: 29 November 2020 Published: 30 November 2020
MSC : 01A55, 11M06, 11M35, 30-02, 30D10, 30D30, 30E20

In this work we present derivations of the formula listed in entry 4.113 in the sixth edition of Gradshteyn and Rhyzik's table of integrals. We evaluate two definite integrals of the form $ \begin{equation*} \int_{0}^{\infty}\frac{e^{-iay}(-iy+\log(z))^k+e^{iay}(iy+\log(z))^k}{\cosh(by)}dy \end{equation*} $ and $ \begin{equation*} \int_{0}^{\infty}\frac{e^{iay}(iy+\log(z))^k-e^{-iay}(-iy+\log(z))^k}{\sinh(b y)}dy \end{equation*} $ in terms of the Lerch function where $ k $, $ a $, $ z $ and $ b $ are arbitrary complex numbers. The entries in the table(s) are obtained as special cases in the paper below.
- entries of Gradshteyn and Ryzhik,
- hyperbolic integrals,
- hypergeometric function,
- Lerch function
Citation: Robert Reynolds, Allan Stauffer. Derivation of some integrals in Gradshteyn and Ryzhik[J]. AIMS Mathematics, 2021, 6(2): 1816-1821. doi: 10.3934/math.2021109

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Abstract

In this work we present derivations of the formula listed in entry 4.113 in the sixth edition of Gradshteyn and Rhyzik's table of integrals. We evaluate two definite integrals of the form $ \begin{equation*} \int_{0}^{\infty}\frac{e^{-iay}(-iy+\log(z))^k+e^{iay}(iy+\log(z))^k}{\cosh(by)}dy \end{equation*} $ and $ \begin{equation*} \int_{0}^{\infty}\frac{e^{iay}(iy+\log(z))^k-e^{-iay}(-iy+\log(z))^k}{\sinh(b y)}dy \end{equation*} $ in terms of the Lerch function where $ k $, $ a $, $ z $ and $ b $ are arbitrary complex numbers. The entries in the table(s) are obtained as special cases in the paper below.

References

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[5]	I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 6 Ed, Academic Press, USA, 2000.
[6]	R. Reynolds, A. Stauffer, A Method for Evaluating Definite Integrals in Terms of Special Functions with Examples, International Mathematical Forum, 15, 2020, 235–244. doi: 10.12988/imf.2020.91272
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