Derivative Expansion of the Effective Action for QED in 2+1 and 3+1 dimensions

The derivative expansion of the one-loop effective action in QED$_3$ and QED$_4$ is considered. The first term in such an expansion is the effective action for a constant electromagnetic field. An explicit expression for the next term containing two derivatives of the field strength $F_{\mu\nu}$, but exact in the magnitude of the field strength, is obtained. The general results for fermion and scalar electrodynamics are presented. The cases of pure electric and pure magnetic external fields are considered in details. The Feynman rules for the perturbative expansion of the one-loop effective action in the number of derivatives is developed.