Title: Cosine Integrals for the Clausen Function and Its Fourier Series Expansion
Author(s): F. M. S. Lima
Pages: 102-107
Cite as:
F. M. S. Lima, Cosine Integrals for the Clausen Function and Its Fourier Series Expansion, Int. J. Anal. Appl., 15 (1) (2017), 102-107.


In a recent work, on taking into account certain finite sums of trigonometric functions I have derived exact closed-form results for some non-trivial integrals, including $\int_0^\pi{\sin(k\,\theta) \, \mathrm{Cl}_2(\theta) \, d \theta}$, where $k$ is a positive integer and $\,\mathrm{Cl}_2(\theta)\,$ is the Clausen function. There in that paper, I pointed out that this integral has the form of a Fourier coefficient, which suggest that its cosine version $\int_0^\pi{\cos(k\,\theta) \, \mathrm{Cl}_2(\theta) \, d \theta}$, $k \ge 0$, is worthy of consideration, but I could only present a few conjectures at that time. Here in this note, I derive exact closed-form expressions for this integral and then I show that they can be taken as Fourier coefficients for the series expansion of a periodic extension of $\,\mathrm{Cl}_2(\theta)$. This yields new closed-form results for a series involving harmonic numbers and a partial derivative of a generalized hypergeometric function.

Full Text: PDF



  1. T. Clausen, ¨ Uber die Function sinφ+(1/2 2 )sin2φ+(1/3 2 )sin3φ+etc., J. Reine Angew. Math. (Crelle) 8, 298–300 (1832). Google Scholar

  2. D. F. Connon, Fourier series and periodicity. arXiv:1501.03037 [math.GM]. Google Scholar

  3. L. Lewin, Polylogarithms and associated functions, North Holland, New York, 1981. Google Scholar

  4. L. Lewin, Structural properties of polylogarithms, American Mathematical Society, Providence, 1991. Google Scholar

  5. F. M. S. Lima, Evaluation of some non-trivial integrals from finite products and sums, Turkish J. Anal. Number Theory 4, 172–176 (2016). Google Scholar

  6. K. F. Riley and M. P. Hobson, Essential Mathematical Methods for the Physical Sciences, Cambridge University Press, New York, 2011. Google Scholar

  7. C. Zhang, Further Discussion on the Calculation of Fourier Series, Appl. Math. 6, 594–598 (2015). Google Scholar

  8. I. S. Gradshteyn and I. M. Rhyzik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007. Google Scholar


Copyright © 2020 IJAA, unless otherwise stated.