Evaluation of the real parts of polylogarithm expressions containing complex arguments via certain logarithmic integrals

Narendra Bhandari
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 3, Pages 613–633
DOI: 10.7546/nntdm.2024.30.3.613-633
Full paper (PDF, 314 Kb)

Details

Authors and affiliations

Narendra Bhandari
Department of Mathematics, Rajdhani College, University of Delhi
New Delhi, India
Budiganga-01, Bajura, Sudurpaschim Province, Nepal

Abstract

We consider a polylogarithm expression containing complex arguments, namely

    \[\mathcal{P}_{{\pm}}(n)=\Re\left(\mathrm{Li}_n\left(\frac{1\pm i}{2}\right)\right).\]

The central notion of the present paper is to evaluate the real parts of \mathcal{P}_{\pm}(n) for first four orders, specifically n=1, 2, 3, and 4, by constructing certain logarithmic integrals. To extract the real parts, we demonstrate an organized approach, and the proofs solely rely on the calculation of the logarithmic integrals. Additionally, we present a potential closed form of \mathcal{P}_{{\pm}}(5).

Keywords

  • Polylogarithm function
  • Dilogarithm function
  • Logarithmic integral
  • Real part
  • Harmonic number
  • Gamma function
  • Beta function

2020 Mathematics Subject Classification

  • 33B30
  • 40C10
  • 33B15

References

  1. Adamchik, V. S. (2002). A certain series associated with Catalan’s constant. Journal of Analysis and its Application, 21(3), 817–826.
  2. Au, K. C. (2022). Mathematica package multiple zeta values. Preprint. Available online at: https://www.researchgate.net/publication/357601353.
  3. Bastien, G. Elementary methods for evaluating Jordan’s sums
    \sum_{n\geq 1}\left(1+\frac{1}{3}+\cdots +\frac{1}{2n-1}\right)\frac{1}{n^{2a}} and \sum_{n\geq 1}\left(1+\frac{1}{3}+\cdots+\frac{1}{2n-1}\right)\frac{1}{(2n-1)^{2a}} and analogous Euler’s type sums and for setting a \sigma–sum theorem. Preprint. Available online at: https://arxiv.org/abs/1301.7662.
  4. Bradley, D. V. (2001). Representations of Catalan’s constant. Preprint. Available online at: https://www.researchgate.net/publication/2325473.
  5. Campbell, J. M., Levrie, P., & Nimbran, A. S. (2021). A natural companion to Catalan’s constant. Journal of Classical Analysis, 18(2), 117–135.
  6. Chen, K. (2020). On the relationship between \Re\mathrm{Li}_n(1+i) and \mathrm{Li}_n\left(1/2\right), when n\geq 5. Available online at: https://math.stackexchange.com/questions/3447529.
  7. Gradshteyn, I. S., & Ryzhik, I. M. (2007). Table of Series, Integrals, and Products (7th Ed.). Academic Press, Elsevier.
  8. Lewin, L. (1981). Polylogarithms and Associated Functions. North Holland, New York.
  9. Olaikhan, A. S. (2023). An Introduction to the Harmonic Series and Logarithmic Integrals. Phoenix, AZ.
  10. Shadhar, A. (2020). How to find \sum_{n=1}^{\infty}\frac{(-1)H_{2n}}{n^3} and \sum_{n=1}^{\infty}\frac{(-1)H^{(2)}_{2n}}{n^2} using real methods?. Available online at: https://math.stackexchange.com/questions/3803424.
  11. Sofo, A., & Nimbran, A. S. (2020). Euler-like sums via powers of log, arctan, and arctanh functions. Integral Transforms and Special Functions, 31, 966–981.
  12. Stewart, S. M. (2020). Explicit Evaluation of some quadratic Euler-type sums containing double-index harmonic numbers. Tatra Mountains Mathematical Publications, 77, 73–98.
  13. Vălean, C. I. (2019). (Almost) Impossible Integrals, Sums, and Series. Springer, New York.
  14. Vălean, C. I. (2023). More (Almost) Impossible Integrals, Sums, and Series. Springer, New York.
  15. Vălean, C. I. (2019). A special way of extracting the real part of the Trilogarithm, \mathrm{Li}_3\left(\frac{1 \pm i}{2}\right). Preprint. Available online at: https://www.researchgate.net/publication/337868999.
  16. Vălean, C. I. (2019). A simple strategy of calculating two alternating harmonic series generalizations. Preprint. Available online at: https://www.researchgate.net/publication/333339272.
  17. Zhao, M. H. (2020). On logarithmic integrals, harmonic sums and variations. Preprint. Available online at: https://arxiv.org/abs/1911.12155.

Manuscript history

  • Received: 6 August 2023
  • Revised: 25 June 2024
  • Accepted: 22 October 2024
  • Online First: 24 October 2024

Copyright information

Ⓒ 2024 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Related papers

Cite this paper

Bhandari, N. (2024). Evaluation of the real parts of polylogarithm expressions containing complex arguments via certain logarithmic integrals. Notes on Number Theory and Discrete Mathematics, 30(3), 613-633, DOI: 10.7546/nntdm.2024.30.3.613-633

Comments are closed.