Cosine Integrals for the Clausen Function and Its Fourier Series Expansion

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.