Complex analysis:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY ; Berlin ; Heidelberg
Springer
1997
|
| Edition: | Second edition |
| Series: | Undergraduate Texts in Mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 294 Seiten graph. Darst. |
| ISBN: | 0387947566 |
Staff View
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Record in the Search Index
| _version_ | 1804125758179770368 |
|---|---|
| adam_text | I Contents
Preface v
1. The Complex Numbers 1
Introduction 1
1.1. The Field of Complex Numbers 2
1.2. The Complex Plane 4
1.3. Tbpological Aspects of the Complex Plane 10
1.4. Stereographic Projection; The Point at Infinity 15
Exercises 17
2. Functions of the Complex Variable z 20
Introduction 20
2.1. Analytic Polynomials 21
2.2. Power Series 25
2.3. Differentiability and Uniqueness of Power Series 27
Exercises 31
3. Analytic Functions 34
3.1. Analyticity and the Cauchy Riemann Equations 34
3.2. The Functions e2, sinz, cosz 39
Exercises 41
4. Line Integrals and Entire Functions 43
Introduction 43
vii
viii Contents
4.1. Properties of the Line Integral 43
4.2. The Closed Curve Theorem for Entire Functions 50
Exercises 54
5. Properties of Entire Functions 56
5.1. The Cauchy Integral Formula and Taylor Expansion
for Entire Functions 56
5.2. Liouville Theorems and the Fundamental Theorem
of Algebra 62
Exercises 64
6. Properties of Analytic Functions 67
Introduction 67
6.1. The Power Series Representation for Functions Analytic
in a Disc 67
6.2. Analyticity in an Arbitrary Open Set 71
6.3. The Uniqueness, Mean Value,
and Maximum Modulus Theorems 73
Exercises 78
7. Further Properties of Analytic Functions 80
7.1. The Open Mapping Theorem; Schwarz Lemma 80
7.2. The Converse of Cauchy s Theorem: Morera s Theorem;
The Schwarz Reflection Principle 85
Exercises 90
8. Simply Connected Domains 92
8.1. The General Cauchy Closed Curve Theorem 92
8.2. The Analytic Function Log z 99
Exercises 101
9. Isolated Singularities of an Analytic Function 103
9.1. Classification of Isolated Singularities; Riemann s
Principle and the Casorati Weierstrass Theorem 103
9.2. Laurent Expansions 107
Exercises 113
10. The Residue Theorem 115
10.1. Winding Numbers and the Cauchy Residue Theorem 115
10.2. Applications of the Residue Theorem 121
Exercises 127
Contents ix
11. Applications of The Residue Theorem
to the Evaluation of Integrals Sums 130
Introduction 130
11.1. Evaluation of Definite Integrals
by Contour Integral Techniques 130
11.2. Application of Contour Integral Methods to Evaluation
and Estimation of Sums 139
Exercises 146
12. Further Contour Integral Techniques 149
12.1. Shifting the Contour of Integration 149
12.2. An Entire Function Bounded in Every Direction 152
Exercises 156
13. Introduction to Conformal Mapping 157
13.1. Conformal Equivalence 157
13.2. Special Mappings 163
Exercises 175
14. The Riemann Mapping Theorem 178
14.1. Conformal Mapping and Hydrodynamics 178
14.2. The Riemann Mapping Theorem 183
Exercises 188
15. Maximum Modulus Theorems for Unbounded Domains 190
15.1. A General Maximum Modulus Theorem 190
15.2. The Phragmen Lindelof Theorem 193
Exercises 199
16. Harmonic Functions 200
16.1. Poisson Formulae and the Dirichlet Problem 200
16.2. Liouville Theorems for Re f;
Zeroes of Entire Functions of Finite Order 208
Exercises 213
17. Different Forms of Analytic Functions 215
Introduction 215
17.1. Infinite Products 215
17.2. Analytic Functions Denned by Definite Integrals 224
Exercises 226
x Contents
18. Analytic Continuation; The Gamma
and Zeta Functions 228
Introduction 228
18.1. Power Series 228
18.2. The Gamma and Zeta Functions 235
Exercises 241
19. Applications to Other Areas of Mathematics 243
Introduction 243
19.1. A Partition Problem 243
19.2. An Infinite System of Equations 244
19.3. A Variation Problem 245
19.4. The Fourier Uniqueness Theorem 247
19.5. The Prime Number Theorem 249
Exercises 254
Appendices 256
I. A Note on Simply Connected Regions 256
II. Circulation and Flux as Contour Integrals 257
III. Steady State Temperatures; The Heat Equation 258
IV. Tchebychev Estimates 260
Answers 263
Bibliography 290
Index 293
|
| any_adam_object | 1 |
| author | Bak, Joseph Newman, Donald J. 1930-2007 |
| author_GND | (DE-588)14254664X |
| author_facet | Bak, Joseph Newman, Donald J. 1930-2007 |
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| building | Verbundindex |
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| callnumber-raw | QA331.7.B35 1997 |
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| callnumber-subject | QA - Mathematics |
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| dewey-full | 515/.9 515/.920 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.9 515/.9 20 |
| dewey-search | 515/.9 515/.9 20 |
| dewey-sort | 3515 19 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| indexdate | 2024-07-09T18:06:39Z |
| institution | BVB |
| isbn | 0387947566 |
| language | English |
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| spelling | Bak, Joseph Verfasser aut Complex analysis Joseph Bak, Donald J. Newman Second edition New York, NY ; Berlin ; Heidelberg Springer 1997 X, 294 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate Texts in Mathematics Analytische functies gtt Complexe variabelen gtt Fonctions analytiques Fonctions d'une variable complexe Functions of complex variables Analytic functions Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Mehrere komplexe Variable (DE-588)4169285-8 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionentheorie (DE-588)4018935-1 s Funktion Mathematik (DE-588)4071510-3 s Mehrere komplexe Variable (DE-588)4169285-8 s 1\p DE-604 Funktionalanalysis (DE-588)4018916-8 s 2\p DE-604 Newman, Donald J. 1930-2007 Verfasser (DE-588)14254664X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007555797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bak, Joseph Newman, Donald J. 1930-2007 Complex analysis Analytische functies gtt Complexe variabelen gtt Fonctions analytiques Fonctions d'une variable complexe Functions of complex variables Analytic functions Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4018935-1 (DE-588)4169285-8 (DE-588)4071510-3 (DE-588)4123623-3 |
| title | Complex analysis |
| title_auth | Complex analysis |
| title_exact_search | Complex analysis |
| title_full | Complex analysis Joseph Bak, Donald J. Newman |
| title_fullStr | Complex analysis Joseph Bak, Donald J. Newman |
| title_full_unstemmed | Complex analysis Joseph Bak, Donald J. Newman |
| title_short | Complex analysis |
| title_sort | complex analysis |
| topic | Analytische functies gtt Complexe variabelen gtt Fonctions analytiques Fonctions d'une variable complexe Functions of complex variables Analytic functions Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| topic_facet | Analytische functies Complexe variabelen Fonctions analytiques Fonctions d'une variable complexe Functions of complex variables Analytic functions Funktionalanalysis Funktionentheorie Mehrere komplexe Variable Funktion Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007555797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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