  We study both analytically and numerically the role of orbital effects caused
by a magnetic field applied along the axis of a semiconducting Rashba nanowire
in the topological regime hosting Majorana fermions. We demonstrate that the
orbital effects can be effectively taken into account in a one-dimensional
model by shifting the chemical potential, and, thus modifying the topological
criterion. We focus on the energy splitting between two Majorana fermions in a
finite nanowire and find a striking interplay between orbital and Zeeman
effects on this splitting. In the limit of strong spin-orbit interaction, we
find regimes where the amplitude of the oscillating splitting stays constant or
even decays with increasing magnetic field, in stark contrast to the commonly
studied case where orbital effects of the magnetic field are neglected. The
period of these oscillations is found to be almost constant in many parameter
regimes.
