Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function

Robert Reynolds; Allan Stauffer; Robert Reynolds; Allan Stauffer

doi:10.3934/math.2021082

AIMS Mathematics

2021, Volume 6, Issue 2: 1324-1331. doi: 10.3934/math.2021082

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Research article

Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function

Robert Reynolds ^,,
Allan Stauffer

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

Received: 04 October 2020 Accepted: 12 November 2020 Published: 17 November 2020
MSC : 01A55, 11M06, 11M35, 30-02, 30D10, 30D30, 30E20

We evaluate definite integrals of the form given by $\int_{0}^{\infty}R(a, x)\log (\cos (\alpha) \text{sech}(x)+1)dx$. The function $R(a, x)$ is a rational function with general complex number parameters. Definite integrals of this form yield closed forms for famous integrals in the books of Bierens de Haan [4] and Gradshteyn and Ryzhik ^[5].
- entries of Gradshteyn and Ryzhik,
- hyperbolic integrals,
- Bierens de Haan,
- Hurwitz zeta function
Citation: Robert Reynolds, Allan Stauffer. Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function[J]. AIMS Mathematics, 2021, 6(2): 1324-1331. doi: 10.3934/math.2021082

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Abstract

We evaluate definite integrals of the form given by $\int_{0}^{\infty}R(a, x)\log (\cos (\alpha) \text{sech}(x)+1)dx$. The function $R(a, x)$ is a rational function with general complex number parameters. Definite integrals of this form yield closed forms for famous integrals in the books of Bierens de Haan [4] and Gradshteyn and Ryzhik ^[5].

References

[1]	F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions, 1st ed.; Springer-Verlag, Berlin Heidelberg, 1990.
[2]	M. Abramowitz, I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover, 1982.
[3]	D. H. Bailey, J. M. Borwein, N. J. Calkin, R. Girgensohn, D. R. Luke, V. H. Moll, Experimental Mathematics in Action, Wellesley, MA: A K Peters, 2007.
[4]	D. Bierens de Haan, Nouvelles Tables d'intégrales définies, Amsterdam, 1867
[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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