Complex Analysis I offers a rigorous introduction to the theory of functions of a complex variable. The course begins with the geometry and algebra of the complex plane, leading to the study of analytic functions, the Cauchy-Riemann equations, and the mapping properties of elementary functions.
A cornerstone of the course is the theory of complex integration. We cover Cauchy’s Integral Theorem and Formula, which have profound consequences such as Liouville’s Theorem and the Fundamental Theorem of Algebra. The relationship between differentiation and integration in the complex domain is explored in depth.
The final part of the course focuses on series representations and singularities. Students learn about Taylor and Laurent series, the classification of singularities, and the Residue Theorem, applying these tools to evaluate definite real integrals and study the local behavior of analytic functions.
This document compiles selected notes and examples developed during the course. While not intended as a comprehensive textbook, these materials provide detailed explanations of key topics and are designed to serve as a robust supplementary resource for students.
As with the notes, this section contains a compilation of various homework assignments and problem sets. Although they do not follow a unified structure, they may serve as valuable practice material for anyone who wishes to reinforce their understanding of the topics covered in the course.
