Date of Graduation

5-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

Loredana Lanzani

Committee Member

Mark E. Arnold

Second Committee Member

Phillip S. Harrington

Third Committee Member

Andrew Raich

Keywords

Pure sciences; Applied sciences; Complex plane; Nature of real coefficients; Riemann map; Schwarz-christoffel maps; Symmetrical domains; Taylor series

Abstract

The Riemann mapping theorem guarantees the existence of a conformal mapping or Riemann map in the complex plane from the open unit disk onto an open simply-connected domain, which is not all of the complex plane. Although its existence is guaranteed, the Riemann map is rarely known except for special domains like half-planes, strips, etc. Therefore, any information we can determine about the Riemann map for any class of domains is interesting and useful.

This research investigates how symmetry affects the Riemann map. In particular, we define domains with symmetries called Rectangular Domains or RDs. The Riemann map of an RD has real-valued coefficients, as opposed to complex-valued, and therefore we can determine the sign of the coefficients of the Taylor series about the origin of the Riemann map, f(z), from the unit disk onto RDs determined by f(0)=0 and f '(0)>0. We focus on the form of the Riemann map for specific RD polygons. These include rhombi, rectangles and non-equilateral octagons which have 2-fold symmetries. We also investigate equilateral polygons with more than 2-fold symmetries such as squares (rotated diamond), regular polygons, and equilateral octagons.

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