Explorations in Complex Functions
Springer International Publishing (Verlag)
978-3-030-54532-1 (ISBN)
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 Wiener-Hopf 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.
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.
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 Phragmén-Lindelöf and Paley-Wiener.- Theorems of Wiener and Lévy; the Wiener-Hopf method.- Tauberian theorems.- Asymptotics and the method of steepest descent.- Complex interpolation and the Riesz-Thorin theorem.
"This is a suitable book with a proper concept at the right time. It is suitable because it shows the beauty, power and profundity of complex analysis, enlightens how many sided it is with all its inspirations and cross-connections to other branches of mathematics." (Heinrich Begehr, zbMATH 1460.30001, 2021)
| Erscheinungsdatum | 21.10.2020 |
|---|---|
| Reihe/Serie | Graduate Texts in Mathematics |
| Zusatzinfo | XVI, 353 p. 30 illus., 29 illus. in color. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 744 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| Schlagworte | Applications of the Schwarzian • automorphic functions • Beta function • Cauchy transform • Complex analysis for number theory • Complex analysis textbook • Conformal Mappings • Elliptic functions complex analysis • Entire functions complex analysis • Fourier transform • gamma function • Harmonic functions complex analysis • Hilbert transform • Hyperbolic Geometry • L-functions • Linear Fractional Transformations • Nevanlinna theory • Riemann hypothesis complex analysis • Schwarzian derivative • zeta function |
| ISBN-10 | 3-030-54532-6 / 3030545326 |
| ISBN-13 | 978-3-030-54532-1 / 9783030545321 |
| Zustand | Neuware |
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