Analytic Extension and Conformal Mapping in the Dual and the Double Planes

Many theorems in the complex plane have analogues in the dual (x+jy, j2=0) and the double (x+ky, k2=1) planes. In this paper, we prove that Schwarz reflection principle holds in the dual and the double planes. We also show that in these two planes the domain of an analytic function can usually be extended analytically to a larger region. In addition, we find that a certain class of regions can be mapped conformally to the upper half plane, which is analogous to the Riemann mapping theorem