Applications of Complex Analysis to Selected Problems in Engineering.

Abstract

This master thesis aims to explore the profound applications of complex analysis in solving diverse engineering problems. With this goal in mind, the document is divided into two distinct but interlocked parts: a brief but comprehensive examination of the general theory of complex analysis, and the application of these principles to selected engineering problems. Beginning with a discussion of general results in complex analysis, we progress to examining specific applications in engineering, such as the Laplace equation and its connection to holomorphic mappings. Through the lens of complex analysis, topics such as the Laplace and Fourier transforms are investigated, by leveraging the power of the residue theorem. Furthermore, the unique solutions offered by complex analysis are highlighted with illustrative examples. This study encompasses various domains in physics and engineering, including electrostatics, gravitation, fluid dynamics, and steady-state heat conduction, providing a holistic understanding of the impact of complex analysis in these areas.

Keywords: Complex numbers, complex analysis, residue theorem, Laplace equation, harmonic functions, conformal mappings, integration methods, Laplace transform, Fourier transform.