Evaluating log-tangent integrals via Euler sums
Abstract
An investigation into the representation of integrals involving the product of the logarithm and the arctan functions, reducing to log-tangent integrals, will be undertaken in this paper. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.
Keyword : Dirichlet beta functions, log-tangent integral, Euler sums, Dirichlet lambda function, zeta functions
How to Cite
Sofo, A. (2022). Evaluating log-tangent integrals via Euler sums. Mathematical Modelling and Analysis, 27(1), 1–18. https://doi.org/10.3846/mma.2022.13100
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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H. Alzer and J. Choi. Four parametric linear Euler sums. Journal of Mathematical Analysis and Applications, 484(1):123661, 2020. https://doi.org/10.1016/j.jmaa.2019.123661
N. Batir. On some combinatorial identities and harmonic sums. International Journal of Number Theory, 13(7):1695–1709, 2017. https://doi.org/10.1142/S179304211750097X
H.S. Bear and D. Myers. A simpler approach to integration and the Fubini theorem. The American Mathematical Monthly, 112(1):51–60, 2005. https://doi.org/10.1080/00029890.2005.11920167
D. Borwein, J.M. Borwein and D.M. Bradley. Parametric Euler sum identities. Journal of Mathematical Analysis and Applications, 316(1):328–338, 2006. https://doi.org/10.1016/j.jmaa.2005.04.040
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L. Elaissaoui and Z. El Abidine Guennoun. Log-tangent integrals and the Riemann zeta function. Mathematical Modelling and Analysis, 24(3):404–421, 2019. https://doi.org/10.3846/mma.2019.025
L. Elaissaoui and Guennoun Zine El Abidine. Evaluation of log-tangent integrals by series involving ζ (2n + 1). Integral Transforms and Special Functions, 28(6):460–475, 2017. https://doi.org/10.1080/10652469.2017.1312366
P. Flajolet and B. Salvy. Euler sums and contour integral representations. Experimental Math, 7:15–35, 1996.
G. Huvent. Formules BBP. online, 2001. Available from Internet: http://gery.huvent.pagesperso-orange.fr/index_explorer_net.htm
K. Knopp. Theory and Application of Infinite Series. Blackie, London, 1951. reprint, Dover 1981.
K. S. Kölbig. The polygamma function ψ(k) (x) for x = 1/4 and x = 3/4. Journal of Computational and Applied Mathematics, 75(1):43–46, 1996. https://doi.org/10.1016/S0377-0427(96)00055-6
R. Lewin. Polylogarithms and Associated Functions. North Holland, New York, 1981.
D. Orr. Generalized log-sine integrals and Bell polynomials. Journal of Computational and Applied Mathematics, 347:330–342, 2019. https://doi.org/10.1016/j.cam.2018.08.026
N.J.A. Sloane. The On-Line Encyclopedia of Integer Sequences. https://oeis.org/A000364, 1973.
A. Sofo. Integral identities for sums. Mathematical Communications, 13(2):303– 309, 2008.
A. Sofo. New classes of harmonic number identities. Journal of Integer Sequences, 15(7):12, 2012.
A. Sofo. Shifted harmonic sums of order two. Communications of the Korean Mathematical Society, 29(2):239–255, 2014. https://doi.org/10.4134/CKMS.2014.29.2.239
A. Sofo. General order Euler sums with multiple argument. Journal of Number Theory, 189:255–271, 2018. https://doi.org/10.1016/j.jnt.2017.12.006
A. Sofo. General order Euler sums with rational argument. Integral Transforms and Special Functions, 30(12):978–991, 2019. https://doi.org/10.1080/10652469.2019.1643851
A. Sofo and D. Cvijović. Extensions of Euler harmonic sums. Applicable Analysis and Discrete Mathematics, 6(2):317–328, 2012. https://doi.org/10.2298/AADM120628016S
A. Sofo and A.S. Nimbran. Euler sums and integral connections. Mathematics, 7(9):833, 2019. https://doi.org/10.3390/math7090833
A. Sofo and H.M. Srivastava. A family of shifted harmonic sums. The Ramanujan Journal, 37(1):89–108, 2015. https://doi.org/10.1007/s11139-014-9600-9
J. Spanier and K.B. Oldham. An atlas of functions. Taylor & Francis/Hemispher, New York, USA, 1987.
H.M. Srivastava and J. Choi. Series associated with the zeta and related functions. Springer Netherlands, Dordrecht, 2001.
C.I. Vălean. (Almost) impossible integrals, sums, and series. Problem Booksin Mathematics. Springer, 2019.
E. Alkan. Approximation by special values of harmonic zeta function and logsine integrals. Communications in Number Theory and Physics, 7(3):515–550, 2013. https://doi.org/10.4310/CNTP.2013.v7.n3.a5
H. Alzer and J. Choi. Four parametric linear Euler sums. Journal of Mathematical Analysis and Applications, 484(1):123661, 2020. https://doi.org/10.1016/j.jmaa.2019.123661
N. Batir. On some combinatorial identities and harmonic sums. International Journal of Number Theory, 13(7):1695–1709, 2017. https://doi.org/10.1142/S179304211750097X
H.S. Bear and D. Myers. A simpler approach to integration and the Fubini theorem. The American Mathematical Monthly, 112(1):51–60, 2005. https://doi.org/10.1080/00029890.2005.11920167
D. Borwein, J.M. Borwein and D.M. Bradley. Parametric Euler sum identities. Journal of Mathematical Analysis and Applications, 316(1):328–338, 2006. https://doi.org/10.1016/j.jmaa.2005.04.040
T.J. Bromwich and G.N. Watson. An introduction to the theory of infinite series. Merchant Books, New York, 2008.
L. Elaissaoui and Z. El Abidine Guennoun. Log-tangent integrals and the Riemann zeta function. Mathematical Modelling and Analysis, 24(3):404–421, 2019. https://doi.org/10.3846/mma.2019.025
L. Elaissaoui and Guennoun Zine El Abidine. Evaluation of log-tangent integrals by series involving ζ (2n + 1). Integral Transforms and Special Functions, 28(6):460–475, 2017. https://doi.org/10.1080/10652469.2017.1312366
P. Flajolet and B. Salvy. Euler sums and contour integral representations. Experimental Math, 7:15–35, 1996.
G. Huvent. Formules BBP. online, 2001. Available from Internet: http://gery.huvent.pagesperso-orange.fr/index_explorer_net.htm
K. Knopp. Theory and Application of Infinite Series. Blackie, London, 1951. reprint, Dover 1981.
K. S. Kölbig. The polygamma function ψ(k) (x) for x = 1/4 and x = 3/4. Journal of Computational and Applied Mathematics, 75(1):43–46, 1996. https://doi.org/10.1016/S0377-0427(96)00055-6
R. Lewin. Polylogarithms and Associated Functions. North Holland, New York, 1981.
D. Orr. Generalized log-sine integrals and Bell polynomials. Journal of Computational and Applied Mathematics, 347:330–342, 2019. https://doi.org/10.1016/j.cam.2018.08.026
N.J.A. Sloane. The On-Line Encyclopedia of Integer Sequences. https://oeis.org/A000364, 1973.
A. Sofo. Integral identities for sums. Mathematical Communications, 13(2):303– 309, 2008.
A. Sofo. New classes of harmonic number identities. Journal of Integer Sequences, 15(7):12, 2012.
A. Sofo. Shifted harmonic sums of order two. Communications of the Korean Mathematical Society, 29(2):239–255, 2014. https://doi.org/10.4134/CKMS.2014.29.2.239
A. Sofo. General order Euler sums with multiple argument. Journal of Number Theory, 189:255–271, 2018. https://doi.org/10.1016/j.jnt.2017.12.006
A. Sofo. General order Euler sums with rational argument. Integral Transforms and Special Functions, 30(12):978–991, 2019. https://doi.org/10.1080/10652469.2019.1643851
A. Sofo and D. Cvijović. Extensions of Euler harmonic sums. Applicable Analysis and Discrete Mathematics, 6(2):317–328, 2012. https://doi.org/10.2298/AADM120628016S
A. Sofo and A.S. Nimbran. Euler sums and integral connections. Mathematics, 7(9):833, 2019. https://doi.org/10.3390/math7090833
A. Sofo and H.M. Srivastava. A family of shifted harmonic sums. The Ramanujan Journal, 37(1):89–108, 2015. https://doi.org/10.1007/s11139-014-9600-9
J. Spanier and K.B. Oldham. An atlas of functions. Taylor & Francis/Hemispher, New York, USA, 1987.
H.M. Srivastava and J. Choi. Series associated with the zeta and related functions. Springer Netherlands, Dordrecht, 2001.
C.I. Vălean. (Almost) impossible integrals, sums, and series. Problem Booksin Mathematics. Springer, 2019.