On three classes of logarithmic integrals
Necdet Batır, Nandan Sai Dasireddy
TL;DR
This work addresses the evaluation of three families of logarithmic integrals by extending the Herglotz-Zagier-Novikov framework to odd positive integers $m$. It develops closed-form expressions for $J(m)$ and for $\int_{0}^{1} \frac{\log(x^m+1)}{x^2+1}\,dx$ when $m$ is odd, expressing results in terms of the dilogarithm $\operatorname{Li}_2$, the inverse tangent integral $\operatorname{Ti}_2$, Catalan's constant $G$, and elementary functions; a key step uses a factorization of $z^m+1$ and a substitution $x=(1-u)/(1+u)$. In addition, the paper provides a general functional-equation framework for HZN-type sums via $\mathcal{F}$ and $\mathcal{G}$, demonstrating symmetry under $x\to 1/x$ and giving applications to explicit identities and examples, including connections to Zagier's Kronecker limit formula. The results yield concrete closed forms for odd $m$, extend known even-$m$ results, and offer new identities such as relations among $\operatorname{Ti}_2$, $G$, and logarithmic terms, thereby enabling precise evaluations of related logarithmic integrals and enriching the theory surrounding modular-type and Kronecker-limit phenomena.
Abstract
In this paper, we evaluate the following families of definite integrals in closed form and we show that they are expressible only in terms of the dilogarithm function and the inverse tangent integral, and elementary functions. \begin{equation*} \int_{0}^{1}\frac{\log\big(x^m+1\big)}{x+1}\thinspace{\rm d}x \quad \mbox{and}\quad \int_{0}^{1}\frac{\log\big(x^m+1\big)}{x^2+1}\thinspace{\rm d}x, \end{equation*} where $m$ is a positive odd integer. When $m$ is a positive even integer, these integrals have been evaluated previously by Sofo and Batır, and the case where $m$ is an odd integer has been left as open problems. The integrals of the first kind arise in Zagier's work on the Kronecker limit formula. In addition, we demonstrate that a functional equation satisfied by the Herglotz-Zagier-Novikov function is a very specific case of of a more general formula, and give numerous illustrative examples.