Explorations in Complex Functions

This textbook offers a diverse range of topics in complex analysis, covering essential tools and concepts while providing various self-contained pathways for readers, making it suitable for graduate students and researchers.

Explorations in Complex Functions is a comprehensive textbook that delves into the intricate world of complex analysis, making it an invaluable resource for graduate students and researchers in the fields of analysis and number theory. The book begins with an overview of essential tools from complex analysis, harmonic analysis, and functional analysis, setting a solid foundation for the topics that follow.

The authors present a diverse range of self-contained chapters, each exploring different aspects of complex functions. Notable topics include linear fractional transformations, harmonic functions, and elliptic functions, which serve as gateways to more advanced concepts such as hyperbolic geometry and automorphic functions. The text also introduces the Schwarzian derivative and discusses important functions like the gamma, beta, and zeta functions, leading into the realm of L-functions. Additionally, a chapter dedicated to entire functions paves the way for discussions on the Riemann hypothesis and Nevanlinna theory.

In addition to its thorough exploration of core material, Explorations in Complex Functions highlights the connections between complex analysis and other mathematical disciplines. The inclusion of topics such as Riemann surfaces, steepest descent, tauberian theorems, and the Wiener–Hopf method enriches the reader's understanding. With exercises accompanying each chapter, instructors will find ample opportunities to design a second course in complex analysis that builds on prior knowledge, making this book an essential companion for anyone looking to deepen their understanding of complex functions.