  We present an identity relating the partition function of N=4 supersymmetric
QED to that of its dual under mirror symmetry. The identity is a generalized
Fourier transform. Many known properties of abelian theories can be derived
from this formula, including the mirror transforms for more general gauge and
matter content. We show that N=3 Chern-Simons QED and N=4 QED with BF-type
couplings are conformal field theories with exactly marginal couplings. Mirror
symmetry acts on these theories as strong-weak coupling duality. After
identifying the mirror of the gauge coupling (sometimes called the ``magnetic
coupling'') we construct a theory which is exactly mirror -- at all scales --
to N=4 SQED. We also study vortex-creation operators in the large $N_f$ limit.
