This graduate-level mathematics textbook offers a comprehensive and accessible exploration of selected topics in complex analysis, suitable for first-graduate classes and advanced studies. It emphasizes the elegant applications of complex analysis to geometry and number theory, featuring illustrations that clarify many concepts and proofs. Key topics include asymptotic analysis, conformal mapping, Riemann mapping theory, the Euler gamma function, the Riemann zeta function, and a proof of the prime number theorem. Additionally, it covers elliptic functions and modular forms. Notably, the final chapter presents the first detailed account of the recent solution to the sphere packing problem in dimension 8, achieved by Maryna Viazovska in 2016, for which she received the Fields Medal in 2022. This text is ideal for self-study by graduate students or advanced undergraduates interested in complex analysis and its applications, as well as for use in graduate mathematics courses, offering enough content for 2-3 semester-long classes. Researchers in complex analysis, analytic number theory, modular forms, and sphere packing theory will also find valuable insights and new material not typically included in standard textbooks.
