Complex made simple:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, R.I.
American Mathematical Society
2008
|
| Schriftenreihe: | Graduate studies in mathematics
97 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 489 S. graph. Darst. |
| ISBN: | 9780821844793 |
Internformat
MARC
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| 100 | 1 | |a Ullrich, David C. |d 1954- |e Verfasser |0 (DE-588)114710087X |4 aut | |
| 245 | 1 | 0 | |a Complex made simple |c David C. Ullrich |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2008 | |
| 300 | |a XI, 489 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 97 | |
| 650 | 4 | |a Fonctions d'une variable complexe | |
| 650 | 4 | |a Functions of complex variables | |
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Datensatz im Suchindex
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| adam_text | Contents
Introduction ix
Part 1. Complex Made Simple
Chapter 0. Differentiability and the Cauchy-Riemann Equations 3
Chapter 1. Power Series 9
Chapter 2. Preliminary Results on Holomorphic Functions 15
Chapter 3. Elementary Results on Holomorphic Functions 33
Chapter 4. Logarithms, Winding Numbers and Cauchy s Theorem 51
Chapter 5. Counting Zeroes and the Open Mapping Theorem 79
Chapter 6. Euler s Formula for sin(z) 87
6.0. Motivation 87
6.1. Proof by the Residue Theorem 89
6.2. Estimating Sums Using Integrals 96
6.3. Proof Using Liouville s Theorem 99
Chapter 7. Inverses of Holomorphic Maps 105
Chapter 8. Conformal Mappings 113
8.0. Meromorphic Functions and the Riemann Sphere 113
v
vi Contents
8.1. Linear-Fractional Transformations, Part I 117
8.2. Linear-Fractional Transformations, Part II 120
8.3. Linear-Fractional Transformations, Part III 128
8.4. Linear-Fractional Transformations, Part IV: The Schwarz
Lemma and Automorphisms of the Disk 130
8.5. More on the Schwarz Lemma 135
Chapter 9. Normal Families and the Riemann Mapping Theorem 141
9.0. Introduction 141
9.1. Quasi-Metrics 143
9.2. Convergence and Compactness in C(D) 149
9.3. Montel s Theorem 155
9.4. The Riemann Mapping Theorem 159
9.5. Montel s Theorem Again 164
Chapter 10. Harmonic Functions 167
10.0. Introduction 167
10.1. Poisson Integrals and the Dirichlet Problem 171
10.2. Poisson Integrals and Aut(D) 182
10.3. Poisson Integrals and Cauchy Integrals 183
10.4. Series Representations for Harmonic Functions in the Disk 184
10.5. Green s Functions and Conformal Mappings 189
10.6. Intermission: Harmonic Functions and Brownian Motion 199
10.7. The Schwarz Reflection Principle and Harnack s Theorem 215
Chapter 11. Simply Connected Open Sets 225
Chapter 12. Runge s Theorem and the Mittag-Leffler Theorem 229
Chapter 13. The Weierstrass Factorization Theorem 245
Chapter 14. Caratheodory s Theorem 257
Chapter 15. More on Aut(D) 267
15.0. Classification of Elements of Aut(D) 267
15.1. Functions Invariant under Group Elements 271
Chapter 16. Analytic Continuation 277
16.0. Introduction 277
Contents vii
16.1. Continuation along Curves 280
16.2. The Complete Analytic Function 287
16.3. Unrestricted Continuation — the Monodromy Theorem 288
16.4. Another Point of View 297
Chapter 17. Orientation 307
Chapter 18. The Modular Function 319
Chapter 19. Preliminaries for the Picard Theorems 337
19.0. Holomorphic Covering Maps 337
19.1. Examples of Holomorphic Covering Maps 343
19.2. Two More Automorphism Groups 345
19.3. Normalizers of Covering Groups 349
19.4. Covering Groups and Conformal Equivalence 353
Chapter 20. The Picard Theorems 357
Part 2. Further Results
Chapter 21. Abel s Theorem 367
Chapter 22. More on Brownian Motion 375
Chapter 23. More on the Maximum Modulus Theorem 385
23.0. Theorems of Hadamard and Phragmen-Lindelof 385
23.1. An Application: The Hausdorff-Young Inequality 390
Chapter 24. The Gamma Function 399
Chapter 25. Universal Covering Spaces 421
Chapter 26. Cauchy s Theorem for Nonholomorphic Functions 435
Chapter 27. Harmonic Conjugates 441
Part 3. Appendices
Appendix 1. Complex Numbers 445
Appendix 2. Complex Numbers, Continued 449
Appendix 3. Sin, Cos and Exp 455
viii Contents
Appendix 4. Metric Spaces 461
Appendix 5. Convexity 473
Appendix 6. Four Counterexamples 475
Appendix 7. The Cauchy-Riemann Equations Revisited 479
References 483
Index of Notations 485
Index 487
|
| adam_txt |
Contents
Introduction ix
Part 1. Complex Made Simple
Chapter 0. Differentiability and the Cauchy-Riemann Equations 3
Chapter 1. Power Series 9
Chapter 2. Preliminary Results on Holomorphic Functions 15
Chapter 3. Elementary Results on Holomorphic Functions 33
Chapter 4. Logarithms, Winding Numbers and Cauchy's Theorem 51
Chapter 5. Counting Zeroes and the Open Mapping Theorem 79
Chapter 6. Euler's Formula for sin(z) 87
6.0. Motivation 87
6.1. Proof by the Residue Theorem 89
6.2. Estimating Sums Using Integrals 96
6.3. Proof Using Liouville's Theorem 99
Chapter 7. Inverses of Holomorphic Maps 105
Chapter 8. Conformal Mappings 113
8.0. Meromorphic Functions and the Riemann Sphere 113
v
vi Contents
8.1. Linear-Fractional Transformations, Part I 117
8.2. Linear-Fractional Transformations, Part II 120
8.3. Linear-Fractional Transformations, Part III 128
8.4. Linear-Fractional Transformations, Part IV: The Schwarz
Lemma and Automorphisms of the Disk 130
8.5. More on the Schwarz Lemma 135
Chapter 9. Normal Families and the Riemann Mapping Theorem 141
9.0. Introduction 141
9.1. Quasi-Metrics 143
9.2. Convergence and Compactness in C(D) 149
9.3. Montel's Theorem 155
9.4. The Riemann Mapping Theorem 159
9.5. Montel's Theorem Again 164
Chapter 10. Harmonic Functions 167
10.0. Introduction 167
10.1. Poisson Integrals and the Dirichlet Problem 171
10.2. Poisson Integrals and Aut(D) 182
10.3. Poisson Integrals and Cauchy Integrals 183
10.4. Series Representations for Harmonic Functions in the Disk 184
10.5. Green's Functions and Conformal Mappings 189
10.6. Intermission: Harmonic Functions and Brownian Motion 199
10.7. The Schwarz Reflection Principle and Harnack's Theorem 215
Chapter 11. Simply Connected Open Sets 225
Chapter 12. Runge's Theorem and the Mittag-Leffler Theorem 229
Chapter 13. The Weierstrass Factorization Theorem 245
Chapter 14. Caratheodory's Theorem 257
Chapter 15. More on Aut(D) 267
15.0. Classification of Elements of Aut(D) 267
15.1. Functions Invariant under Group Elements 271
Chapter 16. Analytic Continuation 277
16.0. Introduction 277
Contents vii
16.1. Continuation along Curves 280
16.2. The Complete Analytic Function 287
16.3. Unrestricted Continuation — the Monodromy Theorem 288
16.4. Another Point of View 297
Chapter 17. Orientation 307
Chapter 18. The Modular Function 319
Chapter 19. Preliminaries for the Picard Theorems 337
19.0. Holomorphic Covering Maps 337
19.1. Examples of Holomorphic Covering Maps 343
19.2. Two More Automorphism Groups 345
19.3. Normalizers of Covering Groups 349
19.4. Covering Groups and Conformal Equivalence 353
Chapter 20. The Picard Theorems 357
Part 2. Further Results
Chapter 21. Abel's Theorem 367
Chapter 22. More on Brownian Motion 375
Chapter 23. More on the Maximum Modulus Theorem 385
23.0. Theorems of Hadamard and Phragmen-Lindelof 385
23.1. An Application: The Hausdorff-Young Inequality 390
Chapter 24. The Gamma Function 399
Chapter 25. Universal Covering Spaces 421
Chapter 26. Cauchy's Theorem for Nonholomorphic Functions 435
Chapter 27. Harmonic Conjugates 441
Part 3. Appendices
Appendix 1. Complex Numbers 445
Appendix 2. Complex Numbers, Continued 449
Appendix 3. Sin, Cos and Exp 455
viii Contents
Appendix 4. Metric Spaces 461
Appendix 5. Convexity 473
Appendix 6. Four Counterexamples 475
Appendix 7. The Cauchy-Riemann Equations Revisited 479
References 483
Index of Notations 485
Index 487 |
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| indexdate | 2024-07-09T21:22:17Z |
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| isbn | 9780821844793 |
| language | English |
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| spelling | Ullrich, David C. 1954- Verfasser (DE-588)114710087X aut Complex made simple David C. Ullrich Providence, R.I. American Mathematical Society 2008 XI, 489 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 97 Fonctions d'une variable complexe Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1163-3 Graduate studies in mathematics 97 (DE-604)BV009739289 97 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016771380&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ullrich, David C. 1954- Complex made simple Graduate studies in mathematics Fonctions d'une variable complexe Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4123623-3 |
| title | Complex made simple |
| title_auth | Complex made simple |
| title_exact_search | Complex made simple |
| title_exact_search_txtP | Complex made simple |
| title_full | Complex made simple David C. Ullrich |
| title_fullStr | Complex made simple David C. Ullrich |
| title_full_unstemmed | Complex made simple David C. Ullrich |
| title_short | Complex made simple |
| title_sort | complex made simple |
| topic | Fonctions d'une variable complexe Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Fonctions d'une variable complexe Functions of complex variables Funktionentheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016771380&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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