Formulae concerning the computation of the Clausen integral Cl2(Θ)

摘要

The main problem under study concerns the expression of the Clausen integral Cl2(Θ) in closed form in terms of known constants and special functions when Θ is equal to a rational multiple of π belonging to [0, 2π]. A general formula giving Cl2(pπq) in terms of the derivative of the di-gamma function and the sine function is deduced from an appropriate Fourier series expansion. Some variants of this formula are obtained. In further sections, the formulae expressing Cl2(2Θ) and, more generally, Cl2(mΘ)(m=2,3,4,…) as linear combinations of terms of the form Cl2(Θ+α) (α: const.) are established. The various results are illustrated by means of typical examples of practical application. The last section contains two simple approximations enabling the computation of Cl(Θ) for any Θ in [0,π] with a relative error smaller than 0.63% and 0.003%, resp.. The paper ends with an appendix in which, among other things, a peculiar trigonometric identity is established as a by-product.