Coulomb-nuclear interference in elastic scattering: eikonal calculation to all orders of $\alpha$

The Coulomb-nuclear interference (CNI) has recently been used by the TOTEM Collaboration to analyse proton-proton elastic-scattering data from the LHC and to draw physics conclusions. This paper will present an eikonal calculation of the CNI effects performed to all orders of the fine structure constant, $\alpha$. This calculation will be used as a reference to benchmark several widely-used CNI formulae and to verify several recent claims by other authors.