  In the in-out formalism we advance a new method to represent the gamma
function for QED actions in supercritical fields, which is complementary to the
proper-time integral representation in Phys. Rev. D 78, 105013 (2008) and Phys.
Rev. D 84, 065004 (2011). The new method directly yields the QED action in
terms of the Hurwitz zeta function in a constant magnetic field and the complex
QED action in a constant electric field. The complex action exactly gives the
vacuum polarization and the vacuum persistence and thereby the pair-production
rate in the electric field. The QED actions exhibit the electromagnetic
duality.
