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E-raamat: Complex Analysis

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(University of Warwick), (University of Warwick)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 23-Aug-2018
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781108527606
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Complex AnalysisSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 23-Aug-2018
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781108527606
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This new edition of a classic textbook develops complex analysis from the established theory of real analysis by emphasising the differences that arise as a result of the richer geometry of the complex plane. Key features of the authors' approach are to use simple topological ideas to translate visual intuition to rigorous proof, and, in this edition, to address the conceptual conflicts between pure and applied approaches head-on. Beyond the material of the clarified and corrected original edition, there are three new chapters: Chapter 15, on infinitesimals in real and complex analysis; Chapter 16, on homology versions of Cauchy's theorem and Cauchy's residue theorem, linking back to geometric intuition; and Chapter 17, outlines some more advanced directions in which complex analysis has developed, and continues to evolve into the future. With numerous worked examples and exercises, clear and direct proofs, and a view to the future of the subject, this is an invaluable companion for any modern complex analysis course.

This new edition of a classic textbook develops complex analysis using simple topological ideas to translate visual intuition into formal arguments. With numerous examples and exercises, clear and direct proofs, and three new chapters including a view to the future of the subject, this is an invaluable companion for any complex analysis course.

Muu info

A new edition of a classic textbook on complex analysis with an emphasis on translating visual intuition to rigorous proof.
Preface to the Second Edition xi
Preface to the First Edition xiv
0 The Origins of Complex Analysis, and Its Challenge to Intuition 1(12)
0.1 The Origins of Complex Numbers
1(4)
0.2 The Origins of Complex Analysis
5(1)
0.3 The Puzzle
6(1)
0.4 Is Mathematics Discovered or Invented?
7(3)
0.5 Overview of the Book
10(3)
1 Algebra of the Complex Plane 13(11)
1.1 Construction of the Complex Numbers
13(2)
1.2 The x + iy Notation
15(1)
1.3 A Geometric Interpretation
16(1)
1.4 Real and Imaginary Parts
17(1)
1.5 The Modulus
17(1)
1.6 The Complex Conjugate
18(1)
1.7 Polar Coordinates
19(1)
1.8 The Complex Numbers Cannot be Ordered
20(1)
1.9 Exercises
21(3)
2 Topology of the Complex Plane 24(35)
2.1 Open and Closed Sets
26(1)
2.2 Limits of Functions
27(3)
2.3 Continuity
30(5)
2.4 Paths
35(3)
2.4.1 Standard Paths
35(2)
2.4.2 Visualising Paths
37(1)
2.4.3 The Image of a Path
37(1)
2.5 Change of Parameter
38(1)
2.5.1 Preserving Direction
39(1)
2.6 Subpaths and Sums of Paths
39(4)
2.7 The Paving Lemma
43(3)
2.8 Connectedness
46(6)
2.9 Space-filling Curves
52(3)
2.10 Exercises
55(4)
3 Power Series 59(16)
3.1 Sequences
59(4)
3.2 Series
63(3)
3.3 Power Series
66(3)
3.4 Manipulating Power Series
69(2)
3.5 Products of Series
71(1)
3.6 Exercises
72(3)
4 Differentiation 75(21)
4.1 Basic Results
75(3)
4.2 The Cauchy-Riemann Equations
78(4)
4.3 Connected Sets and Differentiability
82(1)
4.4 Hybrid Functions
83(1)
4.5 Power Series
84(3)
4.6 A Glimpse Into the Future
87(5)
4.6.1 Real Functions Differentiable Only Finitely Many Times
87(1)
4.6.2 Bad Behaviour of Real Taylor Series
88(1)
4.6.3 The Blancmange function
89(2)
4.6.4 Complex Analysis is Better Behaved
91(1)
4.7 Exercises
92(4)
5 The Exponential Function 96(15)
5.1 The Exponential Function
96(2)
5.2 Real Exponentials and Logarithms
98(1)
5.3 Trigonometric Functions
99(1)
5.4 An Analytic Definition of π
100(1)
5.5 The Behaviour of Real Trigonometric Functions
101(2)
5.6 Dynamic Explanation of Euler's Formula
103(1)
5.7 Complex Exponential and Trigonometric Functions are Periodic
104(1)
5.8 Other Trigonometric Functions
105(1)
5.9 Hyperbolic Functions
106(1)
5.10 Exercises
107(4)
6 Integration 111(38)
6.1 The Real Case
112(1)
6.2 Complex Integration Along a Smooth Path
113(4)
6.3 The Length of a Path
117(5)
6.3.1 Integral Formula for the Length of Smooth Paths and Contours
119(3)
6.4 If You Took the Short Cut...
122(1)
6.5 Further Properties of Lengths
122(2)
6.5.1 Lengths of More General Paths
123(1)
6.6 Regular Paths and Curves
124(3)
6.6.1 Parametrisation by Arc Length
126(1)
6.7 Regular and Singular Points
127(3)
6.8 Contour Integration
130(3)
6.8.1 Definition of Contour Integral
131(2)
6.9 The Fundamental Theorem of Contour Integration
133(3)
6.10 An Integral that Depends on the Path
136(1)
6.11 The Gamma Function
137(3)
6.11.1 Known Properties of the Gamma Function
139(1)
6.12 The Estimation Lemma
140(3)
6.13 Consequences of the Fundamental Theorem
143(3)
6.14 Exercises
146(3)
7 Angles, Logarithms, and the Winding Number 149(20)
7.1 Radian Measure of Angles
150(1)
7.2 The Argument of a Complex Number
151(2)
7.3 The Complex Logarithm
153(2)
7.4 The Winding Number
155(4)
7.5 The Winding Number as an Integral
159(1)
7.6 The Winding Number Round an Arbitrary Point
159(1)
7.7 Components of the Complement of a Path
160(1)
7.8 Computing the Winding Number by Eye
161(3)
7.9 Exercises
164(5)
8 Cauchy's Theorem 169(18)
8.1 The Cauchy Theorem for a Triangle
171(2)
8.2 Existence of an Antiderivative in a Star Domain
173(2)
8.3 An Example - the Logarithm
175(1)
8.4 Local Existence of an Antiderivative
176(1)
8.5 Cauchy's Theorem
177(3)
8.6 Applications of Cauchy's Theorem
180(3)
8.6.1 Cuts and Jordan Contours
181(2)
8.7 Simply Connected Domains
183(1)
8.8 Exercises
184(3)
9 Homotopy Versions of Cauchy's Theorem 187(20)
9.1 Informal Description of Homotopy
187(2)
9.2 Integration Along Arbitrary Paths
189(2)
9.3 The Cauchy Theorem for a Boundary
191(4)
9.4 Formal Definition of Homotopy
195(2)
9.5 Fixed End Point Homotopy
197(1)
9.6 Closed Path Homotopy
198(3)
9.7 Converse to Cauchy's Theorem
201(1)
9.8 The Cauchy Theorems Compared
202(2)
9.9 Exercises
204(3)
10 Taylor Series 207(18)
10.1 Cauchy Integral Formula
208(1)
10.2 Taylor Series
209(3)
10.3 Morera's Theorem
212(1)
10.4 Cauchy's Estimate
213(1)
10.5 Zeros
214(3)
10.6 Extension Functions
217(2)
10.7 Local Maxima and Minima
219(1)
10.8 The Maximum Modulus Theorem
220(1)
10.9 Exercises
221(4)
11 Laurent Series 225(18)
11.1 Series Involving Negative Powers
225(5)
11.2 Isolated Singularities
230(2)
11.3 Behaviour Near an Isolated Singularity
232(2)
11.4 The Extended Complex Plane, or Riemann Sphere
234(2)
11.5 Behaviour of a Differentiable Function at Infinity
236(1)
11.6 Meromorphic Functions
237(2)
11.7 Exercises
239(4)
12 Residues 243(25)
12.1 Cauchy's Residue Theorem
243(3)
12.2 Calculating Residues
246(2)
12.3 Evaluation of Definite Integrals
248(10)
12.4 Summation of Series
258(3)
12.5 Counting Zeros
261(2)
12.6 Exercises
263(5)
13 Conformal Transformations 268(21)
13.1 Measurement of Angles
268(3)
13.1.1 Real Numbers Modulo 2π
268(1)
13.1.2 Geometry of R/2π
269(1)
13.1.3 Operations on Angles
270(1)
13.1.4 The Argument Modulo 2π
270(1)
13.2 Conformal Transformations
271(5)
13.3 Critical Points
276(2)
13.4 M6bius Maps
278(3)
13.4.1 Mobius Maps Preserve Circles
278(1)
13.4.2 Classification of Mobius Maps
279(2)
13.4.3 Extension of Mobius Maps to the Riemann Sphere
281(1)
13.5 Potential Theory
281(3)
13.5.1 Laplace's Equation
281(2)
13.5.2 Design of Aerofoils
283(1)
13.6 Exercises
284(5)
14 Analytic Continuation 289(26)
14.1 The Limitations of Power Series
289(2)
14.2 Comparing Power Series
291(2)
14.3 Analytic Continuation
293(3)
14.3.1 Direct Analytic Continuation
293(2)
14.3.2 Indirect Analytic Continuation
295(1)
14.3.3 Complete Analytic Functions
296(1)
14.4 Multiform Functions
296(3)
14.4.1 The Logarithm as a Multiform Function
297(1)
14.4.2 Singularities
298(1)
14.5 Riemann Surfaces
299(3)
14.5.1 Riemann Surface for the Logarithm
299(1)
14.5.2 Riemann Surface for the Square Root
300(1)
14.5.3 Constructing a General Riemann Surface by Gluing
301(1)
14.6 Complex Powers
302(2)
14.7 Conformal Maps Using Multiform Functions
304(1)
14.8 Contour Integration of Multiform Functions
305(6)
14.9 Exercises
311(4)
15 Infinitesimals in Real and Complex Analysis 315(35)
15.1 Infinitesimals
316(2)
15.2 The Relationship Between Real and Complex Analysis
318(4)
15.2.1 Critical Points
320(2)
15.3 Interpreting Power Series Tending to Zero as Infinitesimals
322(1)
15.4 Real Infinitesimals as Variable Points on a Number Line
323(1)
15.5 Infinitesimals as Elements of an Ordered Field
324(3)
15.6 Structure Theorem for any Ordered Extension Field of R
327(1)
15.7 Visualising Infinitesimals as Points on a Number Line
328(3)
15.8 Complex Infinitesimals
331(2)
15.9 Non-standard Analysis and Hyperreals
333(3)
15.10 Outline of the Construction of Hyperreal Numbers
336(1)
15.11 Hypercomplex Numbers
337(4)
15.12 The Evolution of Meaning in Real and Complex Analysis
341(5)
15.12.1 A Brief History
341(1)
15.12.2 Non-standard Analysis in Mathematics Education
342(2)
15.12.3 Human Visual Senses
344(1)
15.12.4 Computer Graphics
345(1)
15.12.5 Summary
346(1)
15.13 Exercises
346(4)
16 Homology Version of Cauchy's Theorem 350(24)
16.0.1 Outline of
Chapter
351(2)
16.0.2 Group-theoretic Interpretation
353(1)
16.1 Chains
354(2)
16.2 Cycles
356(2)
16.2.1 Sums and Formal Sums of Paths
357(1)
16.3 Boundaries
358(2)
16.4 Homology
360(2)
16.5 Proof of Cauchy's Theorem, Homology Version
362(6)
16.5.1 Grid of Rectangles
363(2)
16.5.2 Proof of Theorem 16.2
365(1)
16.5.3 Rerouting Segments
366(2)
16.5.4 Resumption of Proof of Theorem 16.2
368(1)
16.6 Cauchy's Residue Theorem, Homology Version
368(2)
16.7 Exercises
370(4)
17 The Road Goes Ever On... 374(8)
17.1 The Riemann Hypothesis
374(3)
17.2 Modular Functions
377(1)
17.3 Several Complex Variables
378(1)
17.4 Complex Manifolds
379(1)
17.5 Complex Dynamics
379(2)
17.6 Epilogue
381(1)
References 382(1)
Index 383
Ian Stewart, FRS, is Emeritus Professor of Mathematics at the University of Warwick. He is author or co-author of over 190 research papers and is the bestselling author of over 120 books, from research monographs and textbooks to popular science and science fiction. His awards include the Royal Society's Faraday Medal, the Institute of Mathematics and its Applications (IMA) Gold Medal, the American Association for the Advancement of Science (AAAS) Public Understanding of Science Award, the London Mathematical Society (LMS)/IMA Zeeman Medal, the Lewis Thomas Prize, and the Euler Book Prize. He is an honorary wizard of the Discworld's Unseen University. David Tall is Emeritus Professor of Mathematical Thinking at the University of Warwick and is known internationally for his contributions to mathematics education. He is author or co-author of over 200 papers and 40 books and educational computer software, covering all levels from early childhood to research mathematics.
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