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.

Abstract


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.

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References


  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).

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

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

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

  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).

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

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

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