Conformal Mapping and Physical Modelling

Abstract

This project presents a study of conformal mappings and some of their applications. We emphasize Möbius transformations and their algebraic and geometric properties. These transformations, which preserve angles and the structure of circles, serve as foundational tools in the broader theory of conformal maps. The Schwarz-Christoffel formula is introduced to construct conformal mappings from the upper half-plane to polygonal regions. Applications are explored in two main areas: fluid dynamics, where harmonic and conformal functions model ideal flows, and cartography, where classical projections such as the Mercator and stereographic projections are interpreted through conformal methods. This study demonstrates the powerful interplay between abstract complex analysis and its concrete applications in physics and geometry.