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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AIMS Press
2021-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | http://www.aimspress.com/article/doi/10.3934/math.2021082?viewType=HTML |
| _version_ | 1818322645978972160 |
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| author | Robert Reynolds Allan Stauffer |
| author_facet | Robert Reynolds Allan Stauffer |
| author_sort | Robert Reynolds |
| collection | DOAJ |
| description | 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 <sup>[<span class="xref"><a href="#b5">5</a></span>]</sup>. |
| first_indexed | 2024-12-13T11:00:06Z |
| format | Article |
| id | doaj.art-d351760757ca4aae8a5c424d9d03bb21 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-13T11:00:06Z |
| publishDate | 2021-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-d351760757ca4aae8a5c424d9d03bb212022-12-21T23:49:18ZengAIMS PressAIMS Mathematics2473-69882021-12-01621324133110.3934/math.2021082Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta functionRobert Reynolds0Allan Stauffer1Department of Mathematics, York University, 4700 Keele Street, Toronto, M3J1P3, CanadaDepartment of Mathematics, York University, 4700 Keele Street, Toronto, M3J1P3, CanadaWe 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 <sup>[<span class="xref"><a href="#b5">5</a></span>]</sup>.http://www.aimspress.com/article/doi/10.3934/math.2021082?viewType=HTMLentries of gradshteyn and ryzhikhyperbolic integralsbierens de haanhurwitz zeta function |
| spellingShingle | Robert Reynolds Allan Stauffer Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function AIMS Mathematics entries of gradshteyn and ryzhik hyperbolic integrals bierens de haan hurwitz zeta function |
| title | Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function |
| title_full | Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function |
| title_fullStr | Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function |
| title_full_unstemmed | Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function |
| title_short | Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function |
| title_sort | definite integral of the logarithm hyperbolic secant function in terms of the hurwitz zeta function |
| topic | entries of gradshteyn and ryzhik hyperbolic integrals bierens de haan hurwitz zeta function |
| url | http://www.aimspress.com/article/doi/10.3934/math.2021082?viewType=HTML |
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