| Hardcover ISBN: | 978-0-8218-4479-3 |
| Product Code: | GSM/97 |
| List Price: | $135.00 |
| MAA Member Price: | $121.50 |
| AMS Member Price: | $108.00 |
| eBook ISBN: | 978-1-4704-1163-3 |
| Product Code: | GSM/97.E |
| List Price: | $85.00 |
| MAA Member Price: | $76.50 |
| AMS Member Price: | $68.00 |
| Hardcover ISBN: | 978-0-8218-4479-3 |
| eBook: ISBN: | 978-1-4704-1163-3 |
| Product Code: | GSM/97.B |
| List Price: | $220.00 $177.50 |
| MAA Member Price: | $198.00 $159.75 |
| AMS Member Price: | $176.00 $142.00 |
| Hardcover ISBN: | 978-0-8218-4479-3 |
| Product Code: | GSM/97 |
| List Price: | $135.00 |
| MAA Member Price: | $121.50 |
| AMS Member Price: | $108.00 |
| eBook ISBN: | 978-1-4704-1163-3 |
| Product Code: | GSM/97.E |
| List Price: | $85.00 |
| MAA Member Price: | $76.50 |
| AMS Member Price: | $68.00 |
| Hardcover ISBN: | 978-0-8218-4479-3 |
| eBook ISBN: | 978-1-4704-1163-3 |
| Product Code: | GSM/97.B |
| List Price: | $220.00 $177.50 |
| MAA Member Price: | $198.00 $159.75 |
| AMS Member Price: | $176.00 $142.00 |
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Book DetailsGraduate Studies in MathematicsVolume: 97; 2008; 489 ppMSC: Primary 30
Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from?
In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work.
Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains. Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem.
This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis.
ReadershipGraduate students interested in complex analysis.
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Table of Contents
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Part 1. Complex made simple
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Chapter 0. Differentiability and the Cauchy-Riemann equations
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Chapter 1. Power series
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Chapter 2. Preliminary results on holomorphic functions
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Chapter 3. Elementary results on holomorphic functions
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Chapter 4. Logarithms, winding numbers and Cauchy’s theorem
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Chapter 5. Counting zeroes and the open mapping theorem
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Chapter 6. Euler’s formula for $\sin (z)$
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Chapter 7. Inverses of holomorphic maps
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Chapter 8. Conformal mappings
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Chapter 9. Normal families and the Riemann mapping theorem
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Chapter 10. Harmonic functions
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Chapter 11. Simply connected open sets
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Chapter 12. Runge’s theorem and the Mittag-Leffler theorem
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Chapter 13. The Weierstrass factorization theorem
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Chapter 14. Carathéodory’s theorem
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Chapter 15. More on $\mathrm {Aut}(\mathbb {D})$
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Chapter 16. Analytic continuation
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Chapter 17. Orientation
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Chapter 18. The modular function
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Chapter 19. Preliminaries for the Picard theorems
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Chapter 20. The Picard theorems
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Part 2. Further results
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Chapter 21. Abel’s theorem
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Chapter 22. More on Brownian motion
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Chapter 23. More on the maximum modulus theorem
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Chapter 24. The Gamma function
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Chapter 25. Universal covering spaces
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Chapter 26. Cauchy’s theorem for nonholomorphic functions
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Chapter 27. Harmonic conjugates
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Part 3. Appendices
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Appendix 1. Complex numbers
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Appendix 2. Complex numbers, continued
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Appendix 3. Sin, cos and exp
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Appendix 4. Metric spaces
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Appendix 5. Convexity
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Appendix 6. Four counterexamples
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Appendix 7. The Cauchy-Riemann equations revisited
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Additional Material
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Reviews
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This is an excellent book for a first-year graduate student doing a course in complex analysis. ...students will enjoy and profit from Ullrichs careful explanation of why the theorems work the way they do and also sometimes why seemingly nice ideas that promised to work do not (but often can be patched so that they do). ... In short, Ullrich has managed to write a book about a classical subject that is unusual because its exposition is aimed directly at students, not instructors. I strongly recommend this book to everyone.
MAA Reviews -
In general, the entire exposition stands out by its particular didactic features, by its expository mastery, and by its lucid style helping students grasp both the matter and the beauty of complex function theory profoundly. The prerequisites are kept to minimum, or recalled in the appendices, whereas the scope of the book is remarkably wide. Altogether, the current book offers a nearly irresistible invitation to the fascinating subject of complex analysis.
Zentralblatt MATH
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- Book Details
- Table of Contents
- Additional Material
- Reviews
- Requests
Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from?
In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work.
Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains. Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem.
This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis.
Graduate students interested in complex analysis.
-
Part 1. Complex made simple
-
Chapter 0. Differentiability and the Cauchy-Riemann equations
-
Chapter 1. Power series
-
Chapter 2. Preliminary results on holomorphic functions
-
Chapter 3. Elementary results on holomorphic functions
-
Chapter 4. Logarithms, winding numbers and Cauchy’s theorem
-
Chapter 5. Counting zeroes and the open mapping theorem
-
Chapter 6. Euler’s formula for $\sin (z)$
-
Chapter 7. Inverses of holomorphic maps
-
Chapter 8. Conformal mappings
-
Chapter 9. Normal families and the Riemann mapping theorem
-
Chapter 10. Harmonic functions
-
Chapter 11. Simply connected open sets
-
Chapter 12. Runge’s theorem and the Mittag-Leffler theorem
-
Chapter 13. The Weierstrass factorization theorem
-
Chapter 14. Carathéodory’s theorem
-
Chapter 15. More on $\mathrm {Aut}(\mathbb {D})$
-
Chapter 16. Analytic continuation
-
Chapter 17. Orientation
-
Chapter 18. The modular function
-
Chapter 19. Preliminaries for the Picard theorems
-
Chapter 20. The Picard theorems
-
Part 2. Further results
-
Chapter 21. Abel’s theorem
-
Chapter 22. More on Brownian motion
-
Chapter 23. More on the maximum modulus theorem
-
Chapter 24. The Gamma function
-
Chapter 25. Universal covering spaces
-
Chapter 26. Cauchy’s theorem for nonholomorphic functions
-
Chapter 27. Harmonic conjugates
-
Part 3. Appendices
-
Appendix 1. Complex numbers
-
Appendix 2. Complex numbers, continued
-
Appendix 3. Sin, cos and exp
-
Appendix 4. Metric spaces
-
Appendix 5. Convexity
-
Appendix 6. Four counterexamples
-
Appendix 7. The Cauchy-Riemann equations revisited
-
This is an excellent book for a first-year graduate student doing a course in complex analysis. ...students will enjoy and profit from Ullrichs careful explanation of why the theorems work the way they do and also sometimes why seemingly nice ideas that promised to work do not (but often can be patched so that they do). ... In short, Ullrich has managed to write a book about a classical subject that is unusual because its exposition is aimed directly at students, not instructors. I strongly recommend this book to everyone.
MAA Reviews -
In general, the entire exposition stands out by its particular didactic features, by its expository mastery, and by its lucid style helping students grasp both the matter and the beauty of complex function theory profoundly. The prerequisites are kept to minimum, or recalled in the appendices, whereas the scope of the book is remarkably wide. Altogether, the current book offers a nearly irresistible invitation to the fascinating subject of complex analysis.
Zentralblatt MATH
