  We review equivalent formulations of nonlinear and higher derivatives
theories of electromagnetism exhibiting electric-magnetic duality rotations
symmetry. We study in particular on shell and off shell formulations of this
symmetry, at the level of action functionals as well as of equations of motion.
We prove the conjecture that the action functional leading to Born-Infeld
nonlinear electromagnetism, that is duality rotation invariant off shell and
that is known to be a root of an algebraic equation of fourth order, is a
hypergeometric function.
