Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function
Abstract
:1. Introduction
2. Derivation of the Definite Integral of the Contour Integral
3. Derivation of the Contour Integral in Terms of the Lerch Function
3.1. The Lerch Function
3.2. Derivation of the Infinite Sum of the Contour Integral
3.2.1. Derivation of the First Contour Integral
3.2.2. Derivation of the Second Contour Integral
4. Definite Integral in Terms of the Lerch Function
5. Evaluation of Special Cases of Definite INTEGRALS
5.1. Special Case 1
5.2. Special Case 2
6. Derivation of Entry 3.514.4 in Gradshteyn, I.S.; et al.
7. Derivation of Entry (2.3.1.19) in Yu, A.; et al.
8. Derivation of a New Entry for Table 3.514 in Gradshteyn, I.S.; et al.
9. Definite Integral in Terms of the Hurwitz Zeta Function
10. Definite Integral in Terms of the Log-Gamma and Harmonic Number Functions
11. Derivation of Hyperbolic and Algebraic Forms
12. Discussion
13. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Gradshteyn, I.S.; Ryzhik, I.M. Tables of Integrals, Series and Products, 6th ed.; Academic Press: Cambridge, MA, USA, 2000. [Google Scholar]
- de Haan, D.B. Nouvelles Tables D’intégrales Définies; Engels, P., Ed.; Nabu Press: Amsterdam, The Netherlands, 1867. [Google Scholar]
- Reynolds, R.; Stauffer, A. A Method for Evaluating Definite Integrals inTerms of Special Functions with Examples. Int. Math. Forum 2020, 15, 235–244. [Google Scholar] [CrossRef]
- Reynolds, R.; Stauffer, A. A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function. Mathematics 2019, 7, 1148. [Google Scholar] [CrossRef] [Green Version]
- Reynolds, R.; Stauffer, A. Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions. Mathematics 2020, 8, 687. [Google Scholar] [CrossRef]
- Abramowitz, M.; Stegun, I.A. (Eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed.; Dover: New York, NY, USA, 1982. [Google Scholar]
- Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I. Integrals and Series, More Special Functions; USSR Academy of Sciences: Moscow, Russia, 1990; Volume 1. [Google Scholar]
- Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. (Eds.) NIST Digital Library of Mathematical Functions; U.S. Department of Commerce, National Institute of Standards and Technology: Washington, DC, USA; Cambridge University Press: Cambridge, UK, 2010; With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248 (2012a:33001). [Google Scholar]
- Myland, J.; Oldham, K.B.; Spanier, J. An Atlas of Functions: With Equator, the Atlas Function Calculator, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Yu, A.; Brychkov, O.I.; Marichev, N.V. Handbook of Mellin Transforms; CRC Press; Taylor & Francis Group: Boca Raton, FL, USA, 2019. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Reynolds, R.; Stauffer, A. Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function. Sci 2021, 3, 37. https://doi.org/10.3390/sci3040037
Reynolds R, Stauffer A. Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function. Sci. 2021; 3(4):37. https://doi.org/10.3390/sci3040037
Chicago/Turabian StyleReynolds, Robert, and Allan Stauffer. 2021. "Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function" Sci 3, no. 4: 37. https://doi.org/10.3390/sci3040037