A \(q\)-logarithmic analogue of Euler's sine integral
Euler calculated the integral \[ \int_0^{\frac\pi2} \log(\sin x)\,dx=-\frac\pi2\log2. \] Let \(q>1\) and let \[ \ell_q(x)=(q-1)\sum_{m=1}^\infty\frac{x-1}{x-1+q^{m}} \] be a \(q\)-analogue of the usual logarithm. For \(0<a\leq1\) the authors consider generalizations \[ \int_0^{\frac\pi2} \log(1-ae^{2ix})\,dx\quad \text{and}\quad\int_0^{\frac\pi2} \ell_q(1-ae^{2ix})\,dx. \] The first integral is evaluated by Euler's dilogarithm function, the second is evaluated by the quantum dilogarithm function. In the limit case, when \(q\) tends to 1, both integrals coincide.
scientific article; zbMATH DE number 5147109