Although the generalized complex planes have the same form as the complex plane, theorems and concepts that have been proven true for complex numbers cannot necessarily be extended to dual and double numbers. In this paper, we explore analytic functions of a dual and double variable and disprove two Liouville theorems in these cases. We also modify a domain coloring scheme in order to visualize analytic functions of a generalized complex variable.

Author Bio

Kyle DenHartigh is majoring in mathematics and minoring in German at Calvin College. He plans to pursue a postgraduate degree in mathematics after completing his undergraduate studies. This research was done the summer between his junior and senior years.

Rachel Flim is an undergraduate at Calvin College double majoring in mathematics and sociology honors with a minor in business. This research was done the summer between her junior and senior years.