Linear relations between logarithmic integrals of high weight and some closed-form evaluations

Abstract

The focus of our investigation will be integrals of form ${\int }_0^1{\log }^a(1-x){\log }^{b}x{\log }^c(1+x)/f(x)dx$, where $f$ can be either $x,1-x$ or $1+x$. We show that these integrals possess a plethora of linear relations, and give systematic methods of finding them. In lower weight cases, these linear relations yields remarkable closed-forms of individual integrals; in higher weight, we discuss the $\mathbb{Q}$-dimension that such integrals span. Our approach will not feature Euler sums.

References in corpus (1)

• Logarithmic Integrals, Polylogarithmic Integrals and Euler sums