  In Ref. 3, we presented an asymptotic formula for the fermion-induced
effective energy in 3+1 dimensions in the presence of a cylindrically symmetric
inhomogeneous strong magnetic field. However, there are some points which were
not clearly explained. In fact, the arguments, which led us to the asymptotic
formula, are based on a numerical study of the integral of Eq. (10), as we will
see in the main part of this paper. The aim of this work is to present this
study in detail.
