Senior Project Advisor

Smit Vega Garcia, Mariana

Document Type

Project

Publication Date

Winter 2020

Keywords

Polygonal domains

Abstract

Conformal maps are functions from subsets of the complex plane to the complex plane that locally preserve angles. Our goal is to understand conformal maps that pass to and from polygonal domains. In order to do so, we derive some of the basic theory of harmonic functions on simply connected domains. In particular, our goal with the first few sections is to prove the Schwarz Reflection principle. Using this, as well as other tools from complex analysis, we give an in-depth explanation of Tao’s proof of the Schwarz-Christoffel formula. This is a differential equation that allows one to compute a conformal map from either a half plane or a disk into the interior of a polygonal domain. We apply the result to some basic examples in the analysis of fluid flow.

Department

Mathematics

Subjects - Topical (LCSH)

Conformal mapping; Schwarz function; Christoffel-Darboux formula; Polygons

Genre/Form

student projects; term papers

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Rights Statement

http://rightsstatements.org/vocab/InC/1.0/

Language

English

Format

application/pdf

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