More Explorations in Complex Functions

More Explorations in Complex Functions is something of a sequel to GTM 287, Explorations in Complex Functions. Both texts introduce a variety of topics, from core material in the mainstream of complex analysis to tools that are widely used in other areas of mathematics and applications, but there is minimal overlap between the two books. The intended readership is the same, namely graduate students and researchers in complex analysis, independent readers, seminar attendees, or instructors for a second course in complex analysis. Instructors will appreciate the many options for constructing a second course that builds on a standard first course in complex analysis. Exercises complement the results throughout. There is more material in this present text than one could expect to cover in a year's course in complex analysis. A mapping of dependence relations among chapters enables instructors and independent readers a choice of pathway to reading the text. Chapters 2, 4, 5, 7, and 8 contain the function theory background for some stochastic equations of current interest, such as SLE.
The text begins with two introductory chapters to be used as a resource.  Chapters 3 and 4 are stand-alone introductions to complex dynamics and to univalent function theory, including deBrange's theorem, respectively.  Chapters 5-7 may be treated as a unit that leads from harmonic functions to covering surfaces to the uniformization theorem and Fuchsian groups.  Chapter 8 is a stand-alone treatment of quasiconformal mapping that paves the way for Chapter 9, an introduction to Teichmüller theory. The final chapters, 10-14, are largely stand-alone introductions to topics of both theoretical and applied interest: the Bergman kernel, theta functions and Jacobi inversion, Padé approximants and continued fractions, the Riemann-Hilbert problem and integral equations, and Darboux's method for computing asymptotics.

List of contents

1. Basics.- 2. Further preliminaries.- 3. Complex dynamics.- 4. Univalent functions and de Brange's theorem.- 5. Harmonic and subharmonic functions; the Dirichlet problem.- 6. General Riemann surfaces.- 7. The uniformization theorem.- 8. Quasiconformal mapping.- 9. Introduction to Teichmüller theory.- 10. The Bergman kernel.- 11. Theta functions.- 12. Padé approximants and continued fractions.- 13. Riemann-Hilbert problems.- 14. Asymptotic and Darboux's method.- References.- Index.

About the author

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.

Summary

More Explorations in Complex Functions is something of a sequel to GTM 287, Explorations in Complex Functions. Both texts introduce a variety of topics, from core material in the mainstream of complex analysis to tools that are widely used in other areas of mathematics and applications, but there is minimal overlap between the two books. The intended readership is the same, namely graduate students and researchers in complex analysis, independent readers, seminar attendees, or instructors for a second course in complex analysis. Instructors will appreciate the many options for constructing a second course that builds on a standard first course in complex analysis. Exercises complement the results throughout. There is more material in this present text than one could expect to cover in a year’s course in complex analysis. A mapping of dependence relations among chapters enables instructors and independent readers a choice of pathway to reading the text. Chapters 2, 4, 5, 7, and 8 contain the function theory background for some stochastic equations of current interest, such as SLE.
The text begins with two introductory chapters to be used as a resource.  Chapters 3 and 4 are stand-alone introductions to complex dynamics and to univalent function theory, including deBrange’s theorem, respectively.  Chapters 5—7 may be treated as a unit that leads from harmonic functions to covering surfaces to the uniformization theorem and Fuchsian groups.  Chapter 8 is a stand-alone treatment of quasiconformal mapping that paves the way for Chapter 9, an introduction to Teichmüller theory. The final chapters, 10–14, are largely stand-alone introductions to topics of both theoretical and applied interest: the Bergman kernel, theta functions and Jacobi inversion, Padé approximants and continued fractions, the Riemann—Hilbert problem and integral equations, and Darboux’s method for computing asymptotics.

Additional text

“This book covers a variety of topics in the area of complex analysis. … The text contains a nice diagram … . Every chapter includes a literature review … . the sheer scope of the text makes it an excellent source of advanced topics suitable for a graduate-level course.” (James Tipton, Mathematical Reviews, January, 2025) 
“The material is presented very well ... . The book is excellent as a quick introduction to an outsider to any of the topics included in the book, or as supplementary reading for a topics based second course in complex analysis.” (Amol Sasane, zbMATH 1529.30001, 2024) 

Report

This book covers a variety of topics in the area of complex analysis. The text contains a nice diagram . Every chapter includes a literature review . the sheer scope of the text makes it an excellent source of advanced topics suitable for a graduate-level course. (James Tipton, Mathematical Reviews, January, 2025) 
The material is presented very well ... . The book is excellent as a quick introduction to an outsider to any of the topics included in the book, or as supplementary reading for a topics based second course in complex analysis. (Amol Sasane, zbMATH 1529.30001, 2024) 

Product details

• Authors: Richard Beals, Roderick S C Wong, Roderick S. C. Wong, Roderick S.C. Wong
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 01.08.2023
• EAN: 9783031282874
• ISBN: 978-3-0-3128287-4
• No. of pages: 403
• Dimensions: 155 mm x 22 mm x 235 mm
• Illustrations: XII, 403 p. 50 illus., 43 illus. in color.
• Series: Graduate Texts in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Zahlentheorie, B, Funktionalanalysis und Abwandlungen, Mathematics and Statistics, Number Theory, Special Functions, Functions of a Complex Variable, Functional analysis and transforms