### AIMS Mathematics

2021, Issue 2: 1324-1331. doi: 10.3934/math.2021082
Research article

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

• 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].

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

### Related Papers:

• 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].

 [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.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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