Abstract
In this paper we study the representation of integrals whose integrand involves the product of a polylogarithm and an inverse or inverse hyperbolic trigonometric function. We further demonstrate many connections between these integrals and Euler sums. We develop recurrence relations and give some examples of these integrals in terms of Riemann zeta values, Dirichlet values and other special functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ablinger, J., Blümlein, J.: Harmonic sums, polylogarithms, special numbers, and their generalizations. In: Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation, pp. 1–32. Springer, Vienna (2013)
Bailey, D.H., Borwein, J.M.: Computation and structure of character polylogarithms with applications to character Mordell–Tornheim–Witten sums. Math. Comput. 85(297), 295–324 (2016)
Choi, Junesang: Log-sine and log-cosine integrals. Honam Math. J. 35(2), 137–146 (2013)
Devoto, A., Duke, D.W.: Table of integrals and formulae for Feynman diagram calculations. Riv. Nuovo Cimento (3) 7(6), 1–39 (1984)
Espinosa, O., Moll, V.: On some integrals involving the Hurwitz zeta function: part 1. Ramanujan J. 6, 159–188 (2002)
Espinosa, O., Moll, V.: On some integrals involving the Hurwitz zeta function: part 2. Ramanujan J. 6, 440–468 (2002)
Flajolet, P., Salvy, B.: Euler sums and contour integral representations. Exp. Math. 7(1), 15–35 (1998)
Freitas, P.: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums. Math. Comput. 74(251), 1425–1440 (2005)
Kölbig, K.S.: Nielsen’s generalized polylogarithms. SIAM J. Math. Anal. 17(5), 1232–1258 (1986)
Lewin, R.: Polylogarithms and Associated Functions. North Holland, New York (1981)
Mezo, I.: A family of polylog-trigonometric integrals. Ramanujan J. 46, 161–171 (2018)
NIST: Digital Library of Mathematical Functions. https://www.dlmf.nist.gov
Sofo, A.: Integral identities for sums. Math. Commun. 13(2), 303–309 (2008)
Sofo, A.: Quadratic alternating harmonic number sums. J. Number Theory 154, 144–159 (2015)
Sofo, A.: Summation formula involving harmonic numbers. Anal. Math. 37(1), 51–64 (2011)
Sofo, A.: New classes of harmonic number identities. J. Integer Seq. 15(7), Article 12.7.4 (2012)
Sofo, A.: Integrals of polylogarithmic functions with negative argument. Acta Univ. Sapientiae Math. 10(2), 347–367 (2018)
Sofo, A.: Families of integrals of polylogarithmic functions, special functions and applications. In Choi J, Shilin I (eds) Mathematics, vol. 7, p. 143. MDPI AG, Basel (2019). https://doi.org/10.3390/math7020143
Sofo, A., Cvijović, D.: Extensions of Euler harmonic sums. Appl. Anal. Discrete Math. 6(2), 317–328 (2012)
Sofo, A., Srivastava, H.M.: A family of shifted harmonic sums. Ramanujan J. 37(1), 89–108 (2015)
Xu, C., Yan, Y., Shi, Z.: Euler sums and integrals of polylogarithm functions. J. Number Theory 165, 84–108 (2016)
Acknowledgements
The author is grateful to a referee for carefully considered suggestions and meticulous reading of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sofo, A. Integrals of inverse trigonometric and polylogarithmic functions. Ramanujan J 54, 291–307 (2021). https://doi.org/10.1007/s11139-020-00319-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-020-00319-1