A bstract We consider a set of two-loop sunrise master integrals with two different internal masses at pseudo-threshold kinematics and we solve it in terms elliptic polylogarithms to all orders the dimensional regulator.

A bstract We present a complete set of analytic helicity amplitudes for top quark pair production via gluon fusion at two-loops in QCD. For the first time, we include corrections due to massive fermion loops which give rise integrals over elliptic curves. results missing master needed compute amplitude and obtain an form finite remainders terms iterated using rationalised kinematics field sampl...

Hyperbolic beta integrals are analogues of Euler’s beta integral in which the role of Euler’s gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as (q, q̃ )-bibasic analogues of the beta integral in which the two bases q and q̃ are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under considerat...

Liouville proved that certain integrals, most famously ∫ e−x 2 dx, cannot be expressed in elementary terms. We explain how to give precise meaning to the notion of integration “in elementary terms”, and we formulate Liouville’s theorem that characterizes the possible form of elementary antiderivatives. Using this theorem, we deduce a practical criterion for proving such impossibility results in...

The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The ‘derivative’ NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen – Lee – Liu equation, are studied. More precisely, the complex envelope is separa...