  We compute helicity amplitudes for the one-loop QCD corrections to top-quark
pair production analytically in terms of a set of uniformly transcendental
master integrals. We provide corrections up to $O(\epsilon^2)$ in the
dimensional regulator for the first time which are relevant at NNLO. Four
independent pentagon integral topologies appear in the complete description of
the colour structure for which we provide numerical solutions using canonical
form differential equations and the method of generalised power series
expansions. Analytic forms of the boundary values are obtained in all cases
except one where we find a one-dimensional integral representation.
