A Comparison of Different Line, Geometric Representations for Functions of a Complex Variable a Dissertation Submitted to the Faculty of the Ogden of Philosophy, Department or Mathematics

Explore how line-geometric methods reveal the shape of complex functions.

This dissertation compares different ways to represent functions of a complex variable using line and plane geometry, showing how the same function can look different when you view it through parallel or non-parallel reference planes. It also extends Wilczynski’s ideas by generalizing the Riemann sphere approach to visualize relationships between the two complex variables.

The work frames the study around two main representations. One uses parallel planes to build a congruence from analytic functions, while another allows non-parallel planes, yielding a broader view of the geometric properties. Throughout, the author analyzes how focal points, developable surfaces, and ruled surfaces relate to the underlying functions. The discussion moves toward a general method that connects these geometric pictures to the classic Riemann-sphere idea.

Key ideas include how changing the reference geometry affects the associated coordinates, the conditions under which certain surfaces become developable, and how these viewpoints illuminate common properties of analytic functions. The text also considers simple function examples to illustrate how focal points can be real or imaginary depending on the representation.

• Learn how line and plane configurations encode complex functions and their congruences
• See how parallel versus non-parallel plane setups influence focal structures and developables
• Explore a generalized Riemann-sphere approach to visualize complex-variable relations
• Gain intuition from concrete examples that bridge geometry and complex analysis

Ideal for readers interested in the geometric methods of complex analysis and how different representations shape our understanding of analytic functions.