IJAM: Volume 37, No. 3 (2024)

DOI: 10.12732/ijam.v37i3.2

HJORTNAES INTEGRALS INVOLVING

THE DI- AND TRILOGARITHMS

Lubomir Markov

Department of Mathematics and CS

Barry University

11300 N.E. Second Avenue

Miami Shores, FL 33161, USA

Abstract. At the 1953 Scandinavian Mathematical Congress, M. Hjortnaes presented the following transformation of $\sum_{k=1}^\infty 1/k^3=\zeta(3)$ to a definite integral:
$\zeta(3) = 10\dint_0^{\log[(1+\sqrt{5})/2]} t^2 \coth t\, dt$. We show that a similar formula holds for $\zeta(2)$, namely $\zeta(2) = \frac{10}{3}\int_0^{\log[(1+\sqrt{5})/2]} t \coth t\, dt$, and refer to the above and related integrals as Hjortnaes integrals. Based on the known exact values of the first two polylogarithms, we derive corresponding Hjortnaes integrals and also some series representations.

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How to cite this paper?
DOI: 10.12732/ijam.v37i3.2
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 3

References

[1] M. Munthe Hjortnaes, Overfшring av rekken … til et bestemt integral, Tolfte Skandinaviska Matematikerkongressen, Lund, 1953 (1954), 211–213.

[2] L. Lewin, Polylogarithms and Associated Functions, Elsevier, North-Holland, New York/London/Amsterdam, 1981.

[3] L. Markov, A functional expansion and a new set of rapidly convergent series involving Zeta values, Stud. Comput. Intelligence 793 (2019), 267–276.

[4] L. Markov, Two short proofs of the formula … Gazette 106 (2022), 28–31.

[5] H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht/Boston/London, 2001.

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