Complex variables:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2011
|
| Schriftenreihe: | Pure and applied undergraduate texts
16 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | IX, 305 S. Ill., graph. Darst. |
| ISBN: | 9780821869017 |
Internformat
MARC
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| 245 | 1 | 0 | |a Complex variables |c Joseph L. Taylor |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2011 | |
| 300 | |a IX, 305 S. |b Ill., graph. Darst. | ||
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| 490 | 1 | |a Pure and applied undergraduate texts |v 16 | |
| 500 | |a Includes bibliographical references and index | ||
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| 650 | 7 | |a Functions of a complex variable -- Instructional exposition (textbooks, tutorial papers, etc.) |2 msc | |
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Datensatz im Suchindex
| _version_ | 1804148349814702080 |
|---|---|
| adam_text | Contents
Preface
vii
Chapter
1.
The Complex Numbers
1
1.1.
Definition and Simple Properties
1
1.2.
Convergence in
С
8
1.3.
The Exponential Function
14
1.4.
Polar Form for Complex Numbers
19
Chapter
2.
Analytic Functions
27
2.1.
Continuous Functions
27
2.2.
The Complex Derivative
34
2.3.
Contour Integrals
41
2.4.
Properties of Contour Integrals
47
2.5.
Cauchy s Integral Theorem for a Triangle
53
2.6.
Cauchy s Theorem for a Convex Set
60
2.7.
Properties of the Index Function
66
Chapter
3.
Power Series Expansions
75
3.1.
Uniform Convergence
75
3.2.
Power Series Expansions
82
3.3.
Liouville s Theorem
89
3.4.
Zeroes and Singularities
95
3.5.
The Maximum Modulus Principle
102
Chapter
4.
The General Cauchy Theorems
109
4.1.
Chains and Cycles
110
4.2.
Cauchy s Theorems
118
vi
Contents
4.3. Laurent
Series
124
4.4.
The Residue Theorem
131
4.5.
Rouché s
Theorem and Inverse Functions
137
4.6.
Homotopy
141
Chapter
5.
Residue Theory
153
5.1.
Computing Residues
153
5.2.
Evaluating Integrals Using Residues
159
5.3.
Fourier Transforms
166
5.4.
The Laplace and Mellin Transforms
175
5.5.
Summing Infinite Series
179
Chapter
6.
Conformai
Mappings
185
6.1.
Definition and Examples
185
6.2.
The Riemann Sphere
191
6.3.
Linear Fractional Transformations
197
6.4.
The Riemann Mapping Theorem
202
6.5.
The
Poisson
Integral
207
6.6.
The Dirichlet Problem
213
Chapter
7.
Analytic Continuation and the
Picard
Theorems
221
7.1.
The
Schwarz
Reflection Principle
222
7.2.
Continuation Along a Curve
226
7.3.
Analytic Covering Maps
232
7.4.
The
Picard
Theorems
236
Chapter
8.
Infinite Products
245
8.1.
Convergence of Infinite Products
245
8.2.
Weierstrass
Products
251
8.3.
Entire Functions of Finite Order
257
8.4.
Hadamard s Factorization Theorem
262
Chapter
9.
The Gamma and
Zeta
Functions
269
9.1. Euler s
Gamma Function
270
9.2.
The Riemann
Zeta
Function
275
9.3.
Properties of
ζ
281
9.4.
The Riemann Hypothesis and Prime Numbers
287
9.5.
A Proof of the Prime Number Theorem
291
Bibliography
299
Index
301
The text covers a broad spectrum between basic and advanced
complex variables on the one hand and between theoretical
and applied or computational material on the other hand.
With careful selection of the emphasis put on the various
sections, examples, and exercises, the book can be used in a
one- or two-semester course for undergraduate mathematics
majors, a one-semester course for engineering or physics
majors, or a one-semester course for first-year mathematics
graduate students. It has been tested in all three settings at the
University of Utah.
The exposition is clear, concise, and lively. There is a clean and modern approach
to Cauchy s theorems and Taylor series expansions, with rigorous proofs but no
long and tedious arguments. This is followed by the rich harvest of easy conse¬
quences of the existence of power series expansions.
Through the central portion of the text, there is a careful and extensive treat¬
ment of residue theory and its application to computation of integrals,
conformai
mapping and its applications to applied problems, analytic continuation, and the
proofs of the
Picard
theorems.
Chapter
8
covers material on infinite products and zeroes of entire functions.
This leads to the final chapter which is devoted to the Riemann
zeta
function, the
Riemann Hypothesis, and a proof of the Prime Number Theorem.
ISBN
978-0-8218-6901-7
9 »780821 «869017
AMSTEXT/16
For additional information
and updates on this book, visit
.ams.org/bookpages/amstext-lc
AMS
on the Web
www.ams.org
|
| any_adam_object | 1 |
| author | Taylor, Joseph L. 1941- |
| author_GND | (DE-588)1017302960 |
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| classification_rvk | SK 700 SK 750 |
| classification_tum | MAT 300f |
| ctrlnum | (OCoLC)750883239 (DE-599)BVBBV039535710 |
| dewey-full | 515/.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.9 |
| dewey-search | 515/.9 |
| dewey-sort | 3515 19 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV039535710 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:05:44Z |
| institution | BVB |
| isbn | 9780821869017 |
| language | English |
| lccn | 2011019541 |
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| physical | IX, 305 S. Ill., graph. Darst. |
| publishDate | 2011 |
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| publisher | American Math. Soc. |
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| series | Pure and applied undergraduate texts |
| series2 | Pure and applied undergraduate texts |
| spelling | Taylor, Joseph L. 1941- Verfasser (DE-588)1017302960 aut Complex variables Joseph L. Taylor Providence, RI American Math. Soc. 2011 IX, 305 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied undergraduate texts 16 Includes bibliographical references and index Functions of complex variables Functions of a complex variable -- Instructional exposition (textbooks, tutorial papers, etc.) msc Komplexe Variable (DE-588)4164905-9 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s DE-604 Komplexe Variable (DE-588)4164905-9 s Pure and applied undergraduate texts 16 (DE-604)BV035489189 16 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024387901&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024387901&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Taylor, Joseph L. 1941- Complex variables Pure and applied undergraduate texts Functions of complex variables Functions of a complex variable -- Instructional exposition (textbooks, tutorial papers, etc.) msc Komplexe Variable (DE-588)4164905-9 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4164905-9 (DE-588)4018935-1 |
| title | Complex variables |
| title_auth | Complex variables |
| title_exact_search | Complex variables |
| title_full | Complex variables Joseph L. Taylor |
| title_fullStr | Complex variables Joseph L. Taylor |
| title_full_unstemmed | Complex variables Joseph L. Taylor |
| title_short | Complex variables |
| title_sort | complex variables |
| topic | Functions of complex variables Functions of a complex variable -- Instructional exposition (textbooks, tutorial papers, etc.) msc Komplexe Variable (DE-588)4164905-9 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Functions of complex variables Functions of a complex variable -- Instructional exposition (textbooks, tutorial papers, etc.) Komplexe Variable Funktionentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024387901&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024387901&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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