Explorations in Complex Functions

1. Front Matter

   Pages i-xvi

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2. Basics

   • Richard Beals, Roderick S. C. Wong

   Pages 1-19

3. Linear Fractional Transformations

   • Richard Beals, Roderick S. C. Wong

   Pages 21-31

4. Hyperbolic geometry

   • Richard Beals, Roderick S. C. Wong

   Pages 33-40

5. Harmonic functions

   • Richard Beals, Roderick S. C. Wong

   Pages 41-49

6. Conformal maps and the Riemann mapping theorem

   • Richard Beals, Roderick S. C. Wong

   Pages 51-66

7. The Schwarzian derivative

   • Richard Beals, Roderick S. C. Wong

   Pages 67-82

8. Riemann surfaces and algebraic curves

   • Richard Beals, Roderick S. C. Wong

   Pages 83-104

9. Entire functions

   • Richard Beals, Roderick S. C. Wong

   Pages 105-119

10. Value distribution theory

    • Richard Beals, Roderick S. C. Wong

    Pages 121-140

11. The gamma and beta functions

    • Richard Beals, Roderick S. C. Wong

    Pages 141-153

12. The Riemann zeta function

    • Richard Beals, Roderick S. C. Wong

    Pages 155-165

13. L-functions and primes

    • Richard Beals, Roderick S. C. Wong

    Pages 167-183

14. The Riemann hypothesis

    • Richard Beals, Roderick S. C. Wong

    Pages 185-204

15. Elliptic functions and theta functions

    • Richard Beals, Roderick S. C. Wong

    Pages 205-218

16. Jacobi elliptic functions

    • Richard Beals, Roderick S. C. Wong

    Pages 219-227

17. Weierstrass elliptic functions

    • Richard Beals, Roderick S. C. Wong

    Pages 229-238

18. Automorphic functions and Picard’s theorem

    • Richard Beals, Roderick S. C. Wong

    Pages 239-254

19. Integral transforms

    • Richard Beals, Roderick S. C. Wong

    Pages 255-268

20. Theorems of Phragmén–Lindelöf and Paley–Wiener

    • Richard Beals, Roderick S. C. Wong

    Pages 269-281

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.

• 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

“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)

• Department of Mathematics, Yale University, New Haven, USA

  Richard Beals

• Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

  Roderick S. C. Wong

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.

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