Integração Complexa: aplicações do Teorema dos Resíduos

Abstract

This undergraduate thesis presents an in-depth study of Complex Analysis, emphasizing integration theory and the Residue Theorem. Initially, the necessary theoretical foundations are established, addressing the properties of complex numbers, holomorphic functions, Cauchy-Riemann conditions, and function representation via Power and Laurent series. The study proceeds to integration in the complex plane, detailing core results such as the Cauchy-Goursat Theorem and the Cauchy Integral Formula. The primary focus of the work lies in the formulation, proof, and application of the Residue Theorem, exploring techniques for calculating residues at isolated singularities. Finally, the effectiveness of this mathematical tool is demonstrated in solving improper real integrals and in interdisciplinary applications within Physics and Engineering, highlighting how complex analysis simplifies problems that are difficult to resolve using traditional real calculus methods.