**Graduate Studies in Mathematics**

Volume: 97;
2008;
489 pp;
Hardcover

MSC: Primary 30;

Print ISBN: 978-0-8218-4479-3

Product Code: GSM/97

List Price: $81.00

Individual Member Price: $64.80

**Electronic ISBN: 978-1-4704-1163-3
Product Code: GSM/97.E**

List Price: $81.00

Individual Member Price: $64.80

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#### Supplemental Materials

# Complex Made Simple

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*David C. Ullrich*

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.

#### Table of Contents

# Table of Contents

## Complex Made Simple

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction ix10 free
- Part 1. Complex Made Simple 114 free
- Chapter 0. Differentiability and the Cauchy-Riemann Equations 316
- Chapter 1. Power Series 922
- Chapter 2. Preliminary Results on Holomorphic Functions 1528
- Chapter 3. Elementary Results on Holomorphic Functions 3346
- Chapter 4. Logarithms, Winding Numbers and Cauchy's Theorem 5164
- Chapter 5. Counting Zeroes and the Open Mapping Theorem 7992
- Chapter 6. Euler's Formula for sin(z) 87100
- Chapter 7. Inverses of Holomorphic Maps 105118
- Chapter 8. Conformal Mappings 113126
- 8.0. Meromorphic Functions and the Riemann Sphere 113126
- 8.1. Linear-Fractional Transformations, Part I 117130
- 8.2. Linear-Fractional Transformations, Part II 120133
- 8.3. Linear-Fractional Transformations, Part III 128141
- 8.4. Linear-Fractional Transformations, Part IV: The Schwarz Lemma and Automorphisms of the Disk 130143
- 8.5. More on the Schwarz Lemma 135148

- Chapter 9. Normal Families and the Riemann Mapping Theorem 141154
- Chapter 10. Harmonic Functions 167180
- 10.0. Introduction 167180
- 10.1. Poisson Integrals and the Dirichlet Problem 171184
- 10.2. Poisson Integrals and Aut(D) 182195
- 10.3. Poisson Integrals and Cauchy Integrals 183196
- 10.4. Series Representations for Harmonic Functions in the Disk 184197
- 10.5. Green's Functions and Conformal Mappings 189202
- 10.6. Intermission: Harmonic Functions and Brownian Motion 199212
- 10.7. The Schwarz Reflection Principle and Harnack's Theorem 215228

- Chapter 11. Simply Connected Open Sets 225238
- Chapter 12. Runge's Theorem and the Mittag-Leffler Theorem 229242
- Chapter 13. The Weierstrass Factorization Theorem 245258
- Chapter 14. Caratheodory's Theorem 257270
- Chapter 15. More on Aut(D) 267280
- Chapter 16. Analytic Continuation 277290
- Chapter 17. Orientation 307320
- Chapter 18. The Modular Function 319332
- Chapter 19. Preliminaries for the Picard Theorems 337350
- Chapter 20. The Picard Theorems 357370

- Part 2. Further Results 365378
- Chapter 21. Abel's Theorem 367380
- Chapter 22. More on Brownian Motion 375388
- Chapter 23. More on the Maximum Modulus Theorem 385398
- Chapter 24. The Gamma Function 399412
- Chapter 25. Universal Covering Spaces 421434
- Chapter 26. Cauchy's Theorem for Nonholomorphic Functions 435448
- Chapter 27. Harmonic Conjugates 441454

- Part 3. Appendices 443456
- References 483496
- Index of Notations 485498 free
- Index 487500
- Back Cover Back Cover1506

#### Readership

Graduate students interested in complex analysis.

#### Reviews

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