In 1944, J. Ferrand introduced the concept of a discrete analytic function. Many properties of discrete analytic functions were developed by R. J. Duffin in 1956. In the first part of this dissertation convergence properties of pseudo-power series are discussed and a proof is offered for an important inequality involving pseudo-powers. The next part of the paper discusses a group of transformations on the discrete analytic functions. Finally, the theory of discrete analytic functions on a radial lattice is developed.