  This paper provides an analytical solution for the deformation of an elastic
half-space caused by a cylindrical roller. The roller is considered rigid, and
is forced into the half space and rolls across its surface, with contact
modelled by Coulomb friction. In general, portions of the surface of the roller
in contact with the half space may slip across the surface of the half space,
or may stick to it. In this paper, we consider only the regime where all of the
rollers contact surface is slipping. This results in a mixed boundary value
problem, which is formulated as a $2\times 2$ matrix Wiener-Hopf problem. The
exponential factors in the Wiener-Hopf matrix allows a solution by following
the iterative method of Priddin, Kisil, and Ayton (Phil. Trans. Roy. Soc. A
378, p. 20190241, 2020) which is implemented numerically by computing Cauchy
transforms using a spectral method following Slevinsky and Olver (J. Comput.
Phys. 332, pp. 290-315, 2017). The limits of the contact region are located a
posteriori by applying an optimisation method. The solution is illustrated with
several examples, and numerical code to compute the solutions in general is
included in the supplementary material.
