Complex Analysis II focuses on the geometric aspects of the subject and bridges the gap between analytic and geometric perspectives. The course starts with advanced applications of the Residue Theorem, including the evaluation of improper integrals via Fourier and Mellin transforms, and the summation of infinite series.
We then move to the study of analytic continuation, the Schwarz reflection principle, and an introduction to Riemann surfaces for multivalued functions like logarithms and roots. The local behavior of analytic functions is analyzed using the Argument Principle and Rouché’s Theorem, providing essential tools for zero counting and stability analysis.
The final part of the course covers advanced topics such as Elliptic Functions and the Schwarz-Christoffel formula. A major highlight is the rigorous proof of the Riemann Mapping Theorem, developed through the theory of normal families and Montel’s Theorem.
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.
