Overview
- 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
Part of the book series: Graduate Texts in Mathematics (GTM, volume 287)
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Table of contents (23 chapters)
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Front Matter
Keywords
- Complex analysis textbook
- Complex analysis for number theory
- Linear fractional transformations
- Harmonic functions complex analysis
- Elliptic functions complex analysis
- Hyperbolic geometry
- Conformal mappings
- Automorphic functions
- Schwarzian derivative
- Applications of the Schwarzian
- Gamma function
- Beta function
- Zeta function
- L-functions
- Entire functions complex analysis
- Riemann hypothesis complex analysis
- Nevanlinna theory
- Cauchy transform
- Hilbert transform
- Fourier transform
About this book
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.
Reviews
“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)
Authors and Affiliations
About the authors
Bibliographic Information
Book Title: Explorations in Complex Functions
Authors: Richard Beals, Roderick S. C. Wong
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-030-54533-8
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-54532-1Published: 20 October 2020
Softcover ISBN: 978-3-030-54535-2Published: 21 October 2021
eBook ISBN: 978-3-030-54533-8Published: 19 October 2020
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XVI, 353
Number of Illustrations: 1 b/w illustrations, 29 illustrations in colour
Topics: Functions of a Complex Variable, Special Functions, Number Theory