  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.
