  In this article we obtain some exact results for the 2D Ising model with a
general boundary magnetic field and for a finite size system, by an alternative
method to that developed by B. McCoy and T.T. Wu. This method is a
generalization of ideas from V.N. Plechko presented for the 2D Ising model in
zero field, based on the representation of the Ising model using a Grassmann
algebra. In this way, a Gaussian 1D action is obtained for a general
configuration of the boundary magnetic field. In the special case where the
magnetic field is homogeneous, we check that our results are in agreement with
McCoy and Wu's previous work, and we also compute the two point correlation
functions on the boundary. We use this correlation function to obtain the exact
partition function and the free energy in the special case of an inhomogeneous
boundary magnetic field.
