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.

Author Bio

Conrad Blom grew up in Cutlerville, Michigan near Grand Rapids where he attended South Christian high school. He has attended Calvin College the past three years, and he is currently in his senior year there. He is majoring in mathematics with a minor in physics, and he is planning on going to graduate school in mathematics. His ultimate goal is to become a professor at a 4 year college or university.

Timothy Devries is from Grand Rapids, Michigan. He is currently a sophomore at Calvin College and is studying Electrical Engineering and Mathematics. After completing his double major, Timothy expects to attend graduate school for electrical engineering.

Andrew Hayes is from Pittsburgh, Pennsylvania. He is a junior at Calvin college studying mathematics and computer science