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

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...

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Bibliographic Details
Published inAIMS mathematics Vol. 6; no. 2; pp. 1324 - 1331
Main Authors Reynolds, Robert, Stauffer, Allan
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2021
Subjects
bierens de haan
entries of gradshteyn and ryzhik
hurwitz zeta function
hyperbolic integrals
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ISSN2473-6988
2473-6988
DOI10.3934/math.2021082

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Summary: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 ].
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2021082