Complex analysis:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
2010
|
| Edition: | Third edition |
| Series: | Undergraduate Texts in Mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 328 Seiten graph. Darst. |
| ISBN: | 9781441972873 9781441972880 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804143503141240832 |
|---|---|
| adam_text | Contents
Preface
to the Third Edition
.........................................
v
Preface to the Second Edition
........................................
vii
1
The Complex Numbers
......................................... 1
Introduction
.................................................... 1
1.1
The Field of Complex Numbers
.............................. 1
1.2
The Complex Plane
......................................... 4
1.3
The Solution of the Cubic Equation
........................... 9
1.4
Topological Aspects of the Complex Plane
..................... 12
1.5 Stereographic
Projection; The Point at Infinity
.................. 16
Exercises
...................................................... 18
2
Functions of the Complex Variable
z
............................. 21
Introduction
.................................................... 21
2.1
Analytic Polynomials
....................................... 21
2.2
Power Series
.............................................. 25
2.3
Differentiability and Uniqueness of Power Series
................ 28
Exercises
...................................................... 32
3
Analytic Functions
............................................. 35
3.1
Analyticity and the Cauchy-Riemann Equations
................. 35
3.2
The Functions e sin z, cos
г
................................. 40
Exercises
...................................................... 41
4
Line Integrals and Entire Functions
.............................. 45
Introduction
.................................................... 45
4.1
Properties of the Line Integral
................................ 45
4.2
The Closed Curve Theorem for Entire Functions
................ 52
Exercises
...................................................... 56
x
Contents
5
Properties of Entire Functions
................................... 59
5.1
The Cauchy Integral Formula and Taylor Expansion
for Entire Functions
........................................ 59
5.2
Liouville Theorems and the Fundamental Theorem of Algebra; The
Gauss-Lucas Theorem
...................................... 65
5.3
Newton s Method and Its Application to Polynomial Equations
___ 68
Exercises
...................................................... 74
6
Properties of Analytic Functions
................................. 77
Introduction
.................................................... 77
6.1
The Power Series Representation for Functions Analytic in a Disc
.. 77
6.2
Analytic in an Arbitrary Open Set
............................. 81
6.3
The Uniqueness, Mean-Value, and Maximum-Modulus Theorems;
Critical Points and Saddle Points
.............................. 82
Exercises
...................................................... 90
7
Further Properties of Analytic Functions
......................... 93
7.1
The Open Mapping Theorem;
Schwarz
Lemma
................. 93
7.2
The Converse of Cauchy s Theorem:
Morera s
Theorem; The
Schwarz
Reflection, Principle and Analytic Arcs
................. 98
Exercises
...................................................... 104
8
Simply Connected Domains
..................................... 107
8.1
The General Cauchy Closed Curve Theorem
.................... 107
8.2
The Analytic Function logz
.................................. 113
Exercises
...................................................... 116
9
Isolated Singularities of an Analytic Function
..................... 117
9.1
Classification of Isolated Singularities; Riemann s Principle and the
Casorati-
Weierstrass
Theorem
................................ 117
9.2
Laurent Expansions
......................................... 120
Exercises
...................................................... 126
10
The Residue Theorem
.......................................... 129
10.1
Winding Numbers and the Cauchy Residue Theorem
............. 129
10.2
Applications of the Residue Theorem
.......................... 135
Exercises
...................................................... 141
11
Applications of the Residue Theorem to the Evaluation of Integrals
and Sums
..................................................... 143
Introduction
.................................................... 143
11.1
Evaluation of Definite Integrals by Contour Integral Techniques
... 143
11.2
Application of Contour Integral Methods to Evaluation
and Estimation of Sums
..................................... 151
Exercises
...................................................... 158
Contents xi
12
Further Contour Integral Techniques
............................ 161
12.1
Shifting the Contour of Integration
............................ 161
12.2
An Entire Function Bounded in Every Direction
................. 164
Exercises
...................................................... 167
13
Introduction to
Conformai
Mapping
............................. 169
13.1
Conformai
Equivalence
..................................... 169
13.2
Special Mappings
.......................................... 175
13.3
Schwarz-Christoffel Transformations
.......................... 187
Exercises
...................................................... 192
14
The Riemann Mapping Theorem
................................ 195
14.1
Conformai
Mapping and Hydrodynamics
....................... 195
14.2
The Riemann Mapping Theorem
.............................. 200
14.3
Mapping Properties of Analytic Functions on
Closed Domains
........................................... 204
Exercises
...................................................... 213
15
Maximum-Modulus Theorems
for Unbounded Domains
........................................ 215
15.1
A General Maximum-Modulus Theorem
....................... 215
15.2
The
Phragmén-Lindelöf
Theorem
............................. 218
Exercises
...................................................... 223
16
Harmonic Functions
............................................ 225
16.1
Poisson
Formulae and the Dirichlet Problem
.................... 225
16.2
Liouville Theorems for Re
ƒ ;
Zeroes of Entire Functions
of Finite Order
............................................. 233
Exercises
...................................................... 238
17
Different Forms of Analytic Functions
............................ 241
Introduction
.................................................... 241
17.1
Infinite Products
........................................... 241
17.2
Analytic Functions Defined by Definite Integrals
................ 249
17.3
Analytic Functions Defined by Dirichlet Series
.................. 251
Exercises
...................................................... 255
18
Analytic Continuation; The Gamma
and
Zeta
Functions
............................................. 257
Introduction
.................................................... 257
18.1
Power Series
.............................................. 257
18.2
Analytic Continuation of Dirichlet Series
....................... 263
18.3
The Gamma and
Zeta
Functions
.............................. 265
Exercises
...................................................... 271
xii Contents
19
Applications to Other Areas of Mathematics
...................... 273
Introduction
.................................................... 273
19.1
A Variation Problem
........................................ 273
19.2
The Fourier Uniqueness Theorem
............................. 275
19.3
An Infinite System of Equations
.............................. 277
19.4
Applications to Number Theory
.............................. 278
19.5
An Analytic Proof of The Prime Number Theorem
............... 285
Exercises
...................................................... 290
Answers
...........................................................291
References
.........................................................319
Appendices
........................................................321
Index
.............................................................325
|
| any_adam_object | 1 |
| author | Bak, Joseph Newman, Donald J. 1930-2007 |
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| author_facet | Bak, Joseph Newman, Donald J. 1930-2007 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 750 SK 780 SK 700 |
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| dewey-full | 515/.920 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.9 20 |
| dewey-search | 515/.9 20 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Third edition |
| format | Book |
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| id | DE-604.BV036808163 |
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| indexdate | 2024-07-09T22:48:42Z |
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| isbn | 9781441972873 9781441972880 |
| language | English |
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| physical | XII, 328 Seiten graph. Darst. |
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| series2 | Undergraduate Texts in Mathematics |
| spelling | Bak, Joseph Verfasser aut Complex analysis Joseph Bak, Donald J. Newman Third edition New York, NY [u.a.] Springer 2010 XII, 328 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate Texts in Mathematics Functions of complex variables Analytic functions Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Mehrere komplexe Variable (DE-588)4169285-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 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 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020724201&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 Functions of complex variables Analytic functions Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4071510-3 (DE-588)4018935-1 (DE-588)4169285-8 (DE-588)4018916-8 (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 | Functions of complex variables Analytic functions Funktion Mathematik (DE-588)4071510-3 gnd Funktionentheorie (DE-588)4018935-1 gnd Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Functions of complex variables Analytic functions Funktion Mathematik Funktionentheorie Mehrere komplexe Variable Funktionalanalysis Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020724201&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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