Conformal Mapping Methods for Solving Boundary Value Problems in Complex Domains

Abstract

This study focuses on the fundamental role that conformal transformations play in complex analysis due to their angle-preserving properties, making them particularly important in both pure mathematics and applied fields such as physics and engineering. This paper investigates the mathematical principles of conformal mappings, starting with a review of harmonic and analytic functions, and their relationship through the Cauchy-Riemann equations. A conformal mapping is formally defined as a transformation w=f(z) that preserves both the magnitude and orientation of angles between curves in the complex plane, provided that f(z) is analytic and its derivative does not vanish. Representative examples of conformal mappings, including linear transformations, rotations, translations, and power mappings are presented. In addition, it demonstrates how these mappings simplify complex geometric problems by preserving essential geometric properties and enabling solutions across different coordinate systems. The paper briefly contrasts conformal mappings with isogonal mappings, preserving angles but not orientations.