  It is widely accepted that the free energy F(H) of the two-dimensional Ising
model in the ferromagnetic phase T<T_c has an essential branch cut singularity
at the origin H=0. The phenomenological droplet theory predicts that near the
cut drawn along the negative real axis H<0, the imaginary part of the free
energy per lattice site has the form Im F[exp (\pm i\pi |H|)] = \pm B |H| exp
(-A/|H|) for small |H|. We verify this prediction in analytical perturbative
transfer matrix calculations for the square lattice Ising model for all
temperatures 0<T<T_c and arbitrary anisotropy ratio J_1/J_2. We obtain an
expression for the constant A which coincides exactly with the prediction of
the droplet theory. For the amplitude B we obtain B =\pi M/18, where M is the
equilibrium spontaneous magnetization. In addition we find discrete-lattice
corrections to the above mentioned phenomenological formula for ImF, which
oscillate in H^{-1}.
