This textbook explores a selection of topics in complex analysis. From core material in the mainstream of complex analysis itself, to tools that are widely used in other areas of mathematics, this versatile compilation offers a selection of many different paths. Readers interested in complex analysis will appreciate the unique combination of topics and connections collected in this book.

Beginning with a review of the main tools of complex analysis, harmonic analysis, and functional analysis, the authors go on to present multiple different, self-contained avenues to proceed. Chapters on linear fractional transformations, harmonic functions, and elliptic functions offer pathways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions lead into L -functions, while a chapter on entire functions opens pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give riseto Hilbert and Fourier transforms, with an emphasis on the connection to complex analysis. Valuable additional topics include Riemann surfaces, steepest descent, tauberian theorems, and the WienerHopf method.

Showcasing an array of accessible excursions, Explorations in Complex Functions is an ideal companion for graduate students and researchers in analysis and number theory. Instructors will appreciate the many options for constructing a second course in complex analysis that builds on a first course prerequisite; exercises complement the results throughout.

Illustrates a unique, accessible range of topics relevant across analysis and number theory Includes pathways toward applications of the Schwarzian, the Riemann hypothesis, and parametrization of Riemann surfaces Offers many self-contained options for exploring topics relevant to specific interests Enhances the theory with ample exercises and color illustrations throughout Includes supplementary material: sn.pub/extras

Autorentext
Richard Beals is Professor Emeritus of Mathematics at Yale University. His research interests include ordinary and partial differential equations, operator theory, integrable systems, and transport theory. He has authored many books, including Advanced Mathematical Analysis , published in 1973 as the twelfth volume in the series Graduate Texts in Mathematics. Roderick S. C. Wong is Professor Emeritus of Mathematics at the City University of Hong Kong. His research interests include asymptotic analysis, perturbation methods, and special functions. He has been president of the Canadian Applied Mathematics Society and the Hong Kong Mathematical Society, and received numerous professional honors, including election to the European Academy of Sciences in 2007. He has written and edited a wide variety of books, with several notable works in the area of special functions. This is the third book in the authors' collaboration, after two previous volumeson special functions.

Inhalt
Basics.- Linear Fractional Transformations.- Hyperbolic geometry.- Harmonic Functions.- Conformal maps and the Riemann mapping theorem.- The Schwarzian derivative.- Riemann surfaces and algebraic curves.- Entire functions.- Value distribution theory.- The gamma and beta functions.- The Riemann zeta function.- L-functions and primes.- The Riemann hypothesis.- Elliptic functions and theta functions.- Jacobi elliptic functions.- Weierstrass elliptic functions.- Automorphic functions and Picard's theorem.- Integral transforms.- Theorems of PhragménLindelöf and PaleyWiener.- Theorems of Wiener and Lévy; the WienerHopf method.- Tauberian theorems.- Asymptotics and the method of steepest descent.- Complex interpolation and the RieszThorin theorem.