  The QED effective action encodes nonlinear interactions due to quantum vacuum
polarization effects. While much is known for the special case of electrons in
a constant electromagnetic field (the Euler-Heisenberg case), much less is
known for inhomogeneous backgrounds. Such backgrounds are more relevant to
experimental situations. One way to treat inhomogeneous backgrounds is the
"derivative expansion", in which one formally expands around the soluble
constant-field case. In this talk I use some recent exactly soluble
inhomogeneous backgrounds to perform precision tests on the derivative
expansion, to learn in what sense it converges or diverges. A closely related
question is to find the exponential correction to Schwinger's pair-production
formula for a constant electric field, when the electric background is
inhomogeneous.
