Hardcover ISBN:  9780821844793 
Product Code:  GSM/97 
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Electronic ISBN:  9781470411633 
Product Code:  GSM/97.E 
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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 oftenoverlooked original proof of the Big Picard Theorem.
This book is suitable for a firstyear 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.

Table of Contents

Part 1. Complex made simple

Chapter 0. Differentiability and the CauchyRiemann 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 MittagLeffler 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 CauchyRiemann equations revisited


Additional Material

Reviews

This is an excellent book for a firstyear 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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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 oftenoverlooked original proof of the Big Picard Theorem.
This book is suitable for a firstyear 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 CauchyRiemann 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 MittagLeffler 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 CauchyRiemann equations revisited

This is an excellent book for a firstyear 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