Analytic form for the effective Lagrangian of QED and its application to pair production and photon splitting

We derive an analytic form for the Heisenberg-Euler Lagrangian in the limit where the component of the electric field parallel to the magnetic field is small. We expand these analytic functions to all orders in the field strength ${(F}_{\mu \nu }{F}^{\mu \nu })$ in the limits of weak and strong fields, and use these functions to estimate the pair-production rate in arbitrarily strong electric fields and the photon-splitting rate in arbitrarily strong magnetic fields.