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Journal of Undergraduate Research

Keywords

complex analysis, computer aid, theoretical analysis, complex numbers

College

Physical and Mathematical Sciences

Department

Mathematics

Abstract

Complex analysis is a fundamental course in mathematics required by some engineering and physical science disciplines. Since complex numbers are two dimensional – they have a real and imaginary part – visualizing their graphs, a very important part of practical and theoretical analysis, can be difficult. Complex valued functions of complex numbers can only be graphed in a space of four or more dimensions. To see the entire graph at once is not a feasible option for the human mind. Instead, analysts concern themselves with small sets of the plane and the effect that complex functions have on them. For example, the action of a certain function on a circle or line may be considered by viewing what the image of this set is under the action of the function. This cross-sectional type analysis requires several iterations to gain a true visual understanding of what a complex function actually does. Computers lend themselves naturally to the task. To begin their foray into the world of computer aided graphing of complex functions, the authors chose a relatively simple, yet extremely prevalent, class of functions called fractional linear, or Moebius, transformations.

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