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E-raamat: Complex Variables: A Physical Approach with Applications 2nd edition [Taylor & Francis e-raamat]

(Washington University, St. Louis, Missouri, USA)
  • Formaat: 378 pages, 1 Tables, black and white; 96 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 14-May-2019
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9780429275166
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Complex Variables: A Physical Approach with Applications 2nd editionSuurem pilt
  • Formaat: 378 pages, 1 Tables, black and white; 96 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 14-May-2019
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-13: 9780429275166
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9780429275166
The idea of complex numbers dates back at least 300 yearsto Gauss and Euler, among others. Today complex analysis is a central part of modern analytical thinking. It is used in engineering, physics, mathematics, astrophysics, and many other fields. It provides powerful tools for doing mathematical analysis, and often yields pleasing and unanticipated answers.

This book makes the subject of complex analysis accessible to a broad audience. The complex numbers are a somewhat mysterious number system that seems to come out of the blue. It is important for students to see that this is really a very concrete set of objects that has very concrete and meaningful applications.

Features:





This new edition is a substantial rewrite, focusing on the accessibility, applied, and visual aspect of complex analysis





This book has an exceptionally large number of examples and a large number of figures.





The topic is presented as a natural outgrowth of the calculus. It is not a new language, or a new way of thinking.





Incisive applications appear throughout the book.





Partial differential equations are used as a unifying theme.
Preface to the Second Edition for the Instructor xvii
Preface to the Second Edition for the Student xxi
Preface to the First Edition xxiii
1 Basic Ideas 1(16)
1.1 Complex Arithmetic
1(6)
1.1.1 The Real Numbers
1(1)
1.1.2 The Complex Numbers
1(5)
1.1.3 Complex Conjugate
6(1)
Exercises
7(1)
1.2 Algebraic and Geometric Properties
7(10)
1.2.1 Modulus of a Complex Number
7(2)
1.2.2 The Topology of the Complex Plane
9(1)
1.2.3 The Complex Numbers as a Field
9(5)
1.2.4 The Fundamental Theorem of Algebra
14(1)
Exercises
15(2)
2 The Exponential and Applications 17(10)
2.1 The Exponential Function
17(10)
2.1.1 Laws of Exponentiation
19(1)
2.1.2 The Polar Form of a Complex Number
19(2)
Exercises
20(1)
2.1.3 Roots of Complex Numbers
21(2)
2.1.4 The Argument of a Complex Number
23(1)
2.1.5 Fundamental Inequalities
24(3)
Exercises
26(1)
3 Holomorphic and Harmonic Functions 27(18)
3.1 Holomorphic Functions
27(9)
3.1.1 Continuously Differentiable and Ck Functions
27(1)
3.1.2 The Cauchy-Riemann Equations
28(2)
3.1.3 Derivatives
30(1)
3.1.4 Definition of Holomorphic Function
31(1)
3.1.5 Examples of Holomorphic Functions
31(1)
3.1.6 The Complex Derivative
32(2)
3.1.7 Alternative Terminology for Holomorphic Functions
34(1)
Exercises
34(2)
3.2 Holomorphic and Harmonic Functions
36(5)
3.2.1 Harmonic Functions
36(1)
3.2.2 Holomorphic and Harmonic Functions
37(3)
Exercises
40(1)
3.3 Complex Differentiability
41(4)
3.3.1 Conformality
41(2)
Exercises
43(2)
4 The Cauchy Theory 45(30)
4.1 Real and Complex Line Integrals
45(9)
4.1.1 Curves
45(1)
4.1.2 Closed Curves
45(2)
4.1.3 Differentiable and Ck Curves
47(1)
4.1.4 Integrals on Curves
48(1)
4.1.5 The Fundamental Theorem of Calculus along Curves
48(1)
4.1.6 The Complex Line Integral
49(2)
4.1.7 Properties of Integrals
51(1)
Exercises
52(2)
4.2 The Cauchy Integral Theorem
54(15)
4.2.1 The Cauchy Integral Theorem, Basic Form
54(2)
4.2.2 More General Forms of the Cauchy Theorem
56(2)
4.2.3 Deformability of Curves
58(4)
4.2.4 Cauchy Integral Formula, Basic Form
62(2)
4.2.5 More General Versions of the Cauchy Formula
64(3)
Exercises
67(2)
4.3 Variants of the Cauchy Formula
69(4)
4.4 The Limitations of the Cauchy Formula
73(2)
Exercises
73(2)
5 Applications of the Cauchy Theory 75(20)
5.1 The Derivatives of a Holomorphic Function
75(12)
5.1.1 A Formula for the Derivative
76(1)
5.1.2 The Cauchy Estimates
76(2)
5.1.3 Entire Functions and Liouville's Theorem
78(1)
5.1.4 The Fundamental Theorem of Algebra
79(1)
5.1.5 Sequences of Holomorphic Functions and Their Derivatives
80(1)
5.1.6 The Power Series Representation of a Holomorphic Function
81(4)
5.1.7 Table of Elementary Power Series
85(1)
Exercises
86(1)
5.2 The Zeros of a Holomorphic Function
87(8)
5.2.1 The Zero Set of a Holomorphic Function
87(1)
5.2.2 Discrete Sets and Zero Sets
88(2)
5.2.3 Uniqueness of Analytic Continuation
90(3)
Exercises
93(2)
6 Isolated Singularities 95(14)
6.1 Behavior Near an Isolated Singularity
95(5)
6.1.1 Isolated Singularities
95(1)
6.1.2 A Holomorphic Function on a Punctured Domain
95(1)
6.1.3 Classification of Singularities
96(1)
6.1.4 Removable Singularities, Poles, and Essential Singularities
96(1)
6.1.5 The Riemann Removable Singularities Theorem
97(1)
6.1.6 The Casorati-Weierstrass Theorem
98(1)
6.1.7 Concluding Remarks
98(1)
Exercises
99(1)
6.2 Expansion around Singular Points
100(9)
6.2.1 Laurent Series
100(1)
6.2.2 Convergence of a Doubly Infinite Series
100(1)
6.2.3 Annulus of Convergence
101(1)
6.2.4 Uniqueness of the Laurent Expansion
102(1)
6.2.5 The Cauchy Integral Formula for an Annulus
102(1)
6.2.6 Existence of Laurent Expansions
102(3)
6.2.7 Holomorphic Functions with Isolated Singularities
105(1)
6.2.8 Classification of Singularities in Terms of Laurent Series
106(1)
Exercises
107(2)
7 Meromorphic Functions 109(12)
7.1 Examples of Laurent Expansions
109(5)
7.1.1 Principal Part of a Function
109(1)
7.1.2 Algorithm for Calculating the Coefficients of the Laurent Expansion
110(2)
Exercises
112(2)
7.2 Meromorphic Functions
114(7)
7.2.1 Meromorphic Functions
114(1)
7.2.2 Discrete Sets and Isolated Points
115(1)
7.2.3 Definition of Meromorphic Function
115(1)
7.2.4 Examples of Meromorphic Functions
115(1)
7.2.5 Meromorphic Functions with Infinitely Many Poles
116(1)
7.2.6 Singularities at Infinity
116(1)
7.2.7 The Laurent Expansion at Infinity
117(1)
7.2.8 Meromorphic at Infinity
118(1)
7.2.9 Meromorphic Functions in the Extended Plane
119(1)
Exercises
119(2)
8 The Calculus of Residues 121(30)
8.1 Residues
121(12)
8.1.1 Functions with Multiple Singularities
121(1)
8.1.2 The Concept of Residue
122(1)
8.1.3 The Residue Theorem
123(1)
8.1.4 Residues
123(2)
8.1.5 The Index or Winding Number of a Curve about a Point
125(2)
8.1.6 Restatement of the Residue Theorem
127(1)
8.1.7 Method for Calculating Residues
127(1)
8.1.8 Summary Charts of Laurent Series and Residues
127(2)
Exercises
129(4)
8.2 Applications to the Calculation of Integrals
133(18)
8.2.1 The Evaluation of Definite Integrals
133(1)
8.2.2 A Basic Example
133(3)
8.2.3 Complexification of the Integrand
136(1)
8.2.4 An Example with a More Subtle Choice of Contour
137(3)
8.2.5 Making the Spurious Part of the Integral Disappear
140(2)
8.2.6 The Use of the Logarithm
142(3)
8.2.7 Summing a Series Using Residues
145(1)
8.2.8 Summary Chart of Some Integration Techniques
145(3)
Exercises
148(3)
9 The Argument Principle 151(18)
9.1 Counting Zeros and Poles
151(8)
9.1.1 Local Geometric Behavior of a Holomorphic Function
151(1)
9.1.2 Locating the Zeros of a Holomorphic Function
151(1)
9.1.3 Zero of Order n
152(1)
9.1.4 Counting the Zeros of a Holomorphic Function
153(1)
9.1.5 The Argument Principle
154(2)
9.1.6 Location of Poles
156(1)
9.1.7 The Argument Principle for Meromorphic Functions
157(1)
Exercises
157(2)
9.2 Local Geometry of Functions
159(5)
9.2.1 The Open Mapping Theorem
159(3)
Exercises
162(2)
9.3 Further Results on Zeros
164(5)
9.3.1 Rouche's Theorem
164(1)
9.3.2 A Typical Application of Rouche's Theorem
165(1)
9.3.3 Rouche's Theorem and the Fundamental Theorem of Algebra
166(1)
9.3.4 Hurwitz's Theorem
166(1)
Exercises
167(2)
10 The Maximum Principle 169(8)
10.1 Local and Boundary Maxima
169(3)
10.1.1 The Maximum Modulus Principle
169(1)
10.1.2 Boundary Maximum Modulus Theorem
170(1)
10.1.3 The Minimum Principle
170(1)
10.1.4 The Maximum Principle on an Unbounded Domain
171(1)
Exercises
171(1)
10.2 The Schwarz Lemma
172(5)
10.2.1 Schwarz's Lemma
172(1)
10.2.2 The Schwarz-Pick Lemma
173(1)
Exercises
174(3)
11 The Geometric Theory 177(30)
11.1 The Idea of a Conformal Mapping
177(2)
11.1.1 Conformal Mappings
177(1)
11.1.2 Conformal Self-Maps of the Plane
178(1)
Exercises
179(1)
11.2 Mappings of the Disc
179(3)
11.2.1 Conformal Self-Maps of the Disc
179(1)
11.2.2 Mobius Transformations
180(1)
11.2.3 Self-Maps of the Disc
180(1)
Exercises
181(1)
11.3 Linear Fractional Transformations
182(6)
11.3.1 Linear Fractional Mappings
182(1)
11.3.2 The Topology of the Extended Plane
183(1)
11.3.3 The Riemann Sphere
183(2)
11.3.4 Conformal Self-Maps of the Riemann Sphere
185(1)
11.3.5 The Cayley Transform
185(1)
11.3.6 Generalized Circles and Lines
186(1)
11.3.7 The Cayley Transform Revisited
186(1)
11.3.8 Summary Chart of Linear Fractional Transformations
187(1)
Exercises
187(1)
11.4 The Riemann Mapping Theorem
188(2)
11.4.1 The Concept of Homeomorphism
188(1)
11.4.2 The Riemann Mapping Theorem
188(1)
11.4.3 The Riemann Mapping Theorem: Second Formulation
189(1)
Exercises
189(1)
11.5 Conformal Mappings of Annuli
190(3)
11.5.1 Conformal Mappings of Annuli
190(1)
11.5.2 Conformal Equivalence of Annuli
190(1)
11.5.3 Classification of Planar Domains
191(1)
Exercises
191(2)
11.6 A Compendium of Useful Conformal Mappings
193(14)
12 Applications of Conformal Mapping 207(24)
12.1 Conformal Mapping
207(1)
12.1.1 The Study of Conformal Mappings
207(1)
12.2 The Dirichlet Problem
208(6)
12.2.1 The Dirichlet Problem
208(1)
12.2.2 Physical Motivation for the Dirichlet Problem
208(5)
Exercises
213(1)
12.3 Physical Examples
214(9)
12.3.1 Steady-State Heat Distribution on a Lens-Shaped Region
215(1)
12.3.2 Electrostatics on a Disc
216(2)
12.3.3 Incompressible Fluid Flow around a Post
218(4)
Exercises
222(1)
12.4 Numerical Techniques
223(8)
12.4.1 Numerical Approximation of the Schwarz-Christoffel Mapping
223(5)
12.4.2 Numerical Approximation to a Mapping onto a Smooth Domain
228(1)
Exercises
228(3)
13 Harmonic Functions 231(14)
13.1 Basic Properties of Harmonic Functions
231(4)
13.1.1 The Laplace Equation
231(1)
13.1.2 Definition of Harmonic Function
232(1)
13.1.3 Real- and Complex-Valued Harmonic Functions
232(1)
13.1.4 Harmonic Functions as the Real Parts of Holomorphic Functions
233(1)
13.1.5 Smoothness of Harmonic Functions
234(1)
Exercises
234(1)
13.2 The Maximum Principle
235(4)
13.2.1 The Maximum Principle for Harmonic Functions
235(1)
13.2.2 The Minimum Principle for Harmonic Functions
235(1)
13.2.3 The Maximum Principle
236(1)
13.2.4 The Mean Value Property
236(1)
13.2.5 Boundary Uniqueness for Harmonic Functions
237(1)
Exercises
238(1)
13.3 The Poisson Integral Formula
239(6)
13.3.1 The Poisson Integral
239(1)
13.3.2 The Poisson Kernel
239(1)
13.3.3 The Dirichlet Problem
239(1)
13.3.4 The Solution of the Dirichlet Problem on the Disc
240(1)
13.3.5 The Dirichlet Problem on a General Disc
241(1)
Exercises
242(3)
14 The Fourier Theory 245(28)
14.1 Fourier Series
245(12)
14.1.1 Basic Definitions
245(2)
14.1.2 A Remark on Intervals of Arbitrary Length
247(1)
14.1.3 Calculating Fourier Coefficients
247(1)
14.1.4 Calculating Fourier Coefficients Using Complex Analysis
248(1)
14.1.5 Steady-State Heat Distribution
249(3)
14.1.6 The Derivative and Fourier Series
252(1)
Exercises
253(4)
14.2 The Fourier Transform
257(16)
14.2.1 Basic Definitions
257(1)
14.2.2 Some Fourier Transform Examples that Use Complex Variables
258(10)
14.2.3 Solving a Differential Equation Using the Fourier Transform
268(2)
Exercises
270(3)
15 Other Transforms 273(10)
15.1 The Laplace Transform
273(4)
15.1.1 Prologue
273(1)
15.1.2 Solving a Differential Equation Using the Laplace Transform
274(1)
Exercises
275(2)
15.2 The z-Transform
277(6)
15.2.1 Basic Definitions
277(1)
15.2.2 Population Growth by Means of the Z-Transform
277(3)
Exercises
280(3)
16 Boundary Value Problems 283(22)
16.1 Fourier Methods
283(22)
16.1.1 Remarks on Different Fourier Notations
283(2)
16.1.2 The Dirichlet Problem on the Disc
285(4)
16.1.3 The Poisson Integral
289(3)
16.1.4 The Wave Equation
292(5)
Exercises
297(8)
Appendices 305(2)
Glossary 307(26)
List of Notation 333(2)
Table of Laplace Transforms 335(2)
A Guide to the Literature 337(4)
References 341(4)
Index 345
Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.
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