Explorations in Complex Functions

This textbook covers various topics in complex analysis, including harmonic analysis, functional analysis, linear fractional transformations, elliptic functions, hyperbolic geometry, automorphic functions, the Schwarzian derivative, L-functions, Riemann surfaces, steepest descent, tauberian theorems, and the Wiener-Hopf method. It is suitable for graduate students and researchers in analysis and number theory, and instructors can use it to construct a second course in complex analysis.

\n Format: Hardback
\n Length: 353 pages
\n Publication date: 20 October 2020
\n Publisher: Springer Nature Switzerland AG
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This comprehensive textbook delves into a diverse range of topics within the realm of complex analysis. Spanning from fundamental concepts in the mainstream of complex analysis to widely employed tools in other branches of mathematics, this versatile compilation presents a multitude of diverse pathways. Those with an interest in complex analysis will find great value in the unique blend of topics and interconnectedness curated within this book.

The journey begins with a thorough review of the essential tools of complex analysis, including harmonic analysis, functional analysis, and complex integration. The authors then proceed to offer a diverse array of self-contained approaches, each designed to facilitate a deeper understanding of the subject matter. Chapters on linear fractional transformations, harmonic functions, and elliptic functions serve as gateways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions pave the way for L-functions, while a chapter on entire functions opens up pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give rise to Hilbert and Fourier transforms, with a particular emphasis on their connection to complex analysis.

Additional valuable topics covered include Riemann surfaces, steepest descent, tauberian theorems, and the Wiener–Hopf method. These subjects provide valuable insights and practical applications, making Explorations in Complex Functions an indispensable companion for graduate students and researchers in analysis and number theory. Instructors will also appreciate the flexibility of this textbook, as it offers numerous options for constructing a second course in complex analysis that builds upon a foundational prerequisite. The comprehensive exercises accompanying the results further enhance the learning experience.

\n Weight: 758g\n
Dimension: 164 x 243 x 27 (mm)\n
ISBN-13: 9783030545321\n
Edition number: 1st ed. 2020\n