Linear chemical potential leading to a closed form of the magnetization of a 2DEG in a perpendicular magnetic field

A two-dimensional electron gas (2DEG) under a perpendicular magnetic field is considered. For a given Landau level, the chemical potential can be approximated to have a linear dependence on the magnetic field. This approximation is valid except when complete filling is reached (integer filling factor). The chemical potential oscillation yields a recursion relation for every Landau level. This gives way to an exact form of the magnetization. The derivations herein hold at low temperature.