The Inhomogeneity Expansion for Planar Q.E.D in a Magnetic Background

The effective action for Q.E.D in external magnetic field is constructed using the method of inhomogeneity expansion. We first treat the non-relativistic case where a Chern-Simons like term is generated. We then consider the full relativistic theory and derive the effective action for the $A_{\mu}$ fields. In the non-relativistic case we also add a 4-fermi type interaction and show that under certain circumstances, it corresponds to a Zeeman type term in the effective action.