  We show that the two-loop Euler-Heisenberg effective Lagrangian for scalar
QED in a constant Euclidean self-dual background has a simple explicit closed
form expression in terms of the digamma function. This result leads to a simple
analysis of the weak- and strong-field expansions, the two-loop scalar QED beta
function, and the analytic continuation properties of the effective Lagrangian
and its imaginary part.
