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Visual Complex Analysis: 25th Anniversary Edition [Pehme köide]

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(Professor of Mathematics, University of San Francisco), Foreword by
  • Formaat: Paperback / softback, 720 pages, kõrgus x laius x paksus: 254x178x33 mm, kaal: 1518 g
  • Ilmumisaeg: 28-Feb-2023
  • Kirjastus: Oxford University Press
  • ISBN-10: 0192868926
  • ISBN-13: 9780192868923
Teised raamatud teemal:
Visual Complex Analysis: 25th Anniversary EditionSuurem pilt
  • Formaat: Paperback / softback, 720 pages, kõrgus x laius x paksus: 254x178x33 mm, kaal: 1518 g
  • Ilmumisaeg: 28-Feb-2023
  • Kirjastus: Oxford University Press
  • ISBN-10: 0192868926
  • ISBN-13: 9780192868923
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780192868923.html
Märksõnad:
Complex Analysis is the powerful fusion of the complex numbers (involving the 'imaginary' square root of -1) with ordinary calculus, resulting in a tool that has been of central importance to science for more than 200 years.

This book brings this majestic and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. The 501 diagrams of the original edition embodied geometrical arguments that (for the first time) replaced the long and often opaque computations of the standard approach, in force for the previous 200 years, providing direct, intuitive, visual access to the underlying mathematical reality.

This new 25th Anniversary Edition introduces brand-new captions that fully explain the geometrical reasoning, making it possible to read the work in an entirely new way—as a highbrow comic book!

Arvustused

Visual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis. * Sir Roger Penrose * ...it is comparable with Feynman's Lectures on Physics. At every point it asks "why" and finds a beautiful visual answer. * Newsletter of the European Mathematical Society * Newton would have approved... a fascinating and refreshing look at a familiar subject... essential reading for anybody with any interest at all in this absorbing area of mathematics. * Times Higher Education Supplement * One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual tradition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices--but his are interesting. * Ian Stewart, New Scientist * an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas from a geometric point of view. The style is lucid, informal, reader-friendly, and rich with helpful images (e.g. the complex derivative as an "amplitwist"). A truly unusual and notably creative look at a classical subject. * Paul Zorn, American Mathematical Monthly * If your budget limits you to only buying one mathematics book in a year then make sure that this is the one that you buy. * Mathematical Gazette * I was delighted when I came across Visual Complex Analysis. As soon as I thumbed through it, I realized that this was the book I was looking for ten years ago. * Ed Catmull, former president of Pixar and Disney Animation Studios * The new ideas and exercises bring together a body of information potentially invaluable to researchers in fields from topology to number theory... this is only the beginning of a long list of famous facts for which Needham offers attractive visual proofs: Cauchy's theorem is a satisfying example: you can see the contribution to the integral from each infinitesimal square vanish before your eyes. * Frank Farris, American Mathematical Monthly * This informal style is excellently judged and works extremely well. Many of the arguments presented will be new even to experts, and the book will be of great interest to professionals working in either complex analysis or in any field where complex analysis is used. * David Armitage, Mathematical Reviews * The arguments constructed are highly innovative; even veterans of the field will find new ideas here. This is a special book. Tristan Needham has not only completely rethought a classical field of mathematics, but has presented it in a clear and compelling way. Visual Complex Analysis is worthy of the accolades it has received * MAA Reviews * This new edition of Visual Complex Analysis applies Newton's geometrical methods from the Principia and his concept of ultimate equality to Complex Analysis. * MathSciNet * VCA is sumptuously produced and written in a compelling and wholly engaging style. There is so much to savour and enjoy on every page and the reward of a fresh insight in every figure. * Nick Lord, The Mathematical Gazette *

1 Geometry and Complex Arithmetic
1(60)
1.1 Introduction
1(10)
1.1.1 Historical Sketch
1(3)
1.1.2 Bombelli's "Wild Thought"
4(2)
1.1.3 Some Terminology and Notation
6(2)
1.1.4 Practice
8(1)
1.1.5 Equivalence of Symbolic and Geometric Arithmetic
8(3)
1.2 Euler's Formula
11(5)
1.2.1 Introduction
11(1)
1.2.2 Moving Particle Argument
12(1)
1.2.3 Power Series Argument
13(2)
1.2.4 Sine and Cosine in Terms of Euler's Formula
15(1)
1.3 Some Applications
16(17)
1.3.1 Introduction
16(1)
1.3.2 Trigonometry
16(2)
1.3.3 Geometry
18(4)
1.3.4 Calculus
22(3)
1.3.5 Algebra
25(5)
1.3.6 Vectorial Operations
30(3)
1.4 Transformations and Euclidean Geometry*
33(18)
1.4.1 Geometry Through the Eyes of Felix Klein
33(5)
1.4.2 Classifying Motions
38(3)
1.4.3 Three Reflections Theorem
41(3)
1.4.4 Similarities and Complex Arithmetic
44(4)
1.4.5 Spatial Complex Numbers?
48(3)
1.5 Exercises
51(10)
2 Complex Functions as Transformations
61(76)
2.1 Introduction
61(3)
2.2 Polynomials
64(7)
2.2.1 Positive Integer Powers
64(2)
2.2.2 Cubics Revisited*
66(1)
2.2.3 Cassinian Curves*
67(4)
2.3 Power Series
71(17)
2.3.1 The Mystery of Real Power Series
71(4)
2.3.2 The Disc of Convergence
75(3)
2.3.3 Approximating a Power Series with a Polynomial
78(1)
2.3.4 Uniqueness
79(2)
2.3.5 Manipulating Power Series
81(2)
2.3.6 Finding the Radius of Convergence
83(3)
2.3.7 Fourier Series*
86(2)
2.4 The Exponential Function
88(6)
2.4.1 Power Series Approach
88(1)
2.4.2 The Geometry of the Mapping
89(1)
2.4.3 Another Approach
90(4)
2.5 Cosine and Sine
94(6)
2.5.1 Definitions and Identities
94(1)
2.5.2 Relation to Hyperbolic Functions
95(2)
2.5.3 The Geometry of the Mapping
97(3)
2.6 Multifunctions
100(10)
2.6.1 Example: Fractional Powers
100(3)
2.6.2 Single-Valued Branches of a Multifunction
103(3)
2.6.3 Relevance to Power Series
106(2)
2.6.4 An Example with Two Branch Points
108(2)
2.7 The Logarithm Function
110(5)
2.7.1 Inverse of the Exponential Function
110(2)
2.7.2 The Logarithmic Power Series
112(1)
2.7.3 General Powers
113(2)
2.8 Averaging over Circles*
115(10)
2.8.1 The Centroid
115(3)
2.8.2 Averaging over Regular Polygons
118(3)
2.8.3 Averaging over Circles
121(4)
2.9 Exercises
125(12)
3 Mobius Transformations and Inversion
137(76)
3.1 Introduction
137(2)
3.1.1 Definition and Significance of Mobius Transformations
137(1)
3.1.2 The Connection with Einstein's Theory of Relativity*
138(1)
3.1.3 Decomposition into Simple Transformations
139(1)
3.2 Inversion
139(14)
3.2.1 Preliminary Definitions and Facts
139(3)
3.2.2 Preservation of Circles
142(2)
3.2.3 Constructing Inverse Points Using Orthogonal Circles
144(3)
3.2.4 Preservation of Angles
147(2)
3.2.5 Preservation of Symmetry
149(1)
3.2.6 Inversion in a Sphere
150(3)
3.3 Three Illustrative Applications of Inversion
153(4)
3.3.1 A Problem on Touching Circles
153(1)
3.3.2 A Curious Property of Quadrilaterals with Orthogonal Diagonals
154(2)
3.3.3 Ptolemy's Theorem
156(1)
3.4 The Riemann Sphere
157(11)
3.4.1 The Point at Infinity
157(1)
3.4.2 Stereographic Projection
158(4)
3.4.3 Transferring Complex Functions to the Sphere
162(1)
3.4.4 Behaviour of Functions at Infinity
163(2)
3.4.5 Stereographic Formulae*
165(3)
3.5 Mobius Transformations: Basic Results
168(9)
3.5.1 Preservation of Circles, Angles, and Symmetry
168(1)
3.5.2 Non-Uniqueness of the Coefficients
169(1)
3.5.3 The Group Property
170(1)
3.5.4 Fixed Points
171(1)
3.5.5 Fixed Points at Infinity
172(2)
3.5.6 The Cross-Ratio
174(3)
3.6 Mobius Transformations as Matrices*
177(7)
3.6.1 Empirical Evidence of a Link with Linear Algebra
177(1)
3.6.2 The Explanation: Homogeneous Coordinates
178(2)
3.6.3 Eigenvectors and Eigenvalues*
180(2)
3.6.4 Rotations of the Sphere as Mobius Transformations*
182(2)
3.7 Visualization and Classification*
184(10)
3.7.1 The Main Idea
184(2)
3.7.2 Elliptic, Hyperbolic, and Loxodromic Transformations
186(3)
3.7.3 Local Geometric Interpretation of the Multiplier
189(1)
3.7.4 Parabolic Transformations
190(1)
3.7.5 Computing the Multiplier*
191(1)
3.7.6 Eigenvalue Interpretation of the Multiplier*
192(2)
3.8 Decomposition into 2 or 4 Reflections*
194(5)
3.8.1 Introduction
194(1)
3.8.2 Elliptic Case
194(2)
3.8.3 Hyperbolic Case
196(1)
3.8.4 Parabolic Case
197(1)
3.8.5 Summary
198(1)
3.9 Automorphisms of the Unit Disc*
199(6)
3.9.1 Counting Degrees of Freedom
199(1)
3.9.2 Finding the Formula via the Symmetry Principle
200(1)
3.9.3 Interpreting the Simplest Formula Geometrically*
201(3)
3.9.4 Introduction to Riemann's Mapping Theorem
204(1)
3.10 Exercises
205(8)
4 Differentiation: The Amplitwist Concept
213(32)
4.1 Introduction
213(1)
4.2 A Puzzling Phenomenon
213(3)
4.3 Local Description of Mappings in the Plane
216(4)
4.3.1 Introduction
216(1)
4.3.2 The Jacobian Matrix
217(1)
4.3.3 The Amplitwist Concept
218(2)
4.4 The Complex Derivative as Amplitwist
220(5)
4.4.1 The Real Derivative Re-examined
220(1)
4.4.2 The Complex Derivative
221(2)
4.4.3 Analytic Functions
223(1)
4.4.4 A Brief Summary
224(1)
4.5 Some Simple Examples
225(2)
4.6 Conformal = Analytic
227(4)
4.6.1 Introduction
227(1)
4.6.2 Conformality Throughout a Region
228(2)
4.6.3 Conformality and the Riemann Sphere
230(1)
4.7 Critical Points
231(3)
4.7.1 Degrees of Crushing
231(1)
4.7.2 Breakdown of Conformality
232(1)
4.7.3 Branch Points
233(1)
4.8 The Cauchy-Riemann Equations
234(5)
4.8.1 Introduction
234(1)
4.8.2 The Geometry of Linear Transformations
235(2)
4.8.3 The Cauchy-Riemann Equations
237(2)
4.9 Exercises
239(6)
5 Further Geometry of Differentiation
245(58)
5.1 Cauchy--Riemann Revealed
245(4)
5.1.1 Introduction
245(1)
5.1.2 The Cartesian Form
245(2)
5.1.3 The Polar Form
247(2)
5.2 An Intimation of Rigidity
249(3)
5.3 Visual Differentiation of log(z)
252(2)
5.4 Rules of Differentiation
254(3)
5.4.1 Composition
254(1)
5.4.2 Inverse Functions
255(1)
5.4.3 Addition and Multiplication
256(1)
5.5 Polynomials, Power Series, and Rational Functions
257(3)
5.5.1 Polynomials
257(1)
5.5.2 Power Series
257(2)
5.5.3 Rational Functions
259(1)
5.6 Visual Differentiation of the Power Function
260(2)
5.7 Visual Differentiation of exp(z)
262(2)
5.8 Geometric Solution of E' = E
264(2)
5.9 An Application of Higher Derivatives: Curvature*
266(8)
5.9.1 Introduction
266(1)
5.9.2 Analytic Transformation of Curvature
267(3)
5.9.3 Complex Curvature
270(4)
5.10 Celestial Mechanics*
274(7)
5.10.1 Central Force Fields
274(1)
5.10.2 Two Kinds of Elliptical Orbit
274(3)
5.10.3 Changing the First into the Second
277(1)
5.10.4 The Geometry of Force
278(1)
5.10.5 An Explanation
279(1)
5.10.6 The Kasner--Arnol'd Theorem
280(1)
5.11 Analytic Continuation*
281(12)
5.11.1 Introduction
281(2)
5.11.2 Rigidity
283(1)
5.11.3 Uniqueness
284(2)
5.11.4 Preservation of Identities
286(1)
5.11.5 Analytic Continuation via Reflections
286(7)
5.12 Exercises
293(10)
6 Non-Euclidean Geometry*
303(82)
6.1 Introduction
303(13)
6.1.1 The Parallel Axiom
303(2)
6.1.2 Some Facts from Non-Euclidean Geometry
305(2)
6.1.3 Geometry on a Curved Surface
307(2)
6.1.4 Intrinsic versus Extrinsic Geometry
309(2)
6.1.5 Gaussian Curvature
311(2)
6.1.6 Surfaces of Constant Curvature
313(2)
6.1.7 The Connection with Mobius Transformations
315(1)
6.2 Spherical Geometry
316(17)
6.2.1 The Angular Excess of a Spherical Triangle
316(1)
6.2.2 Motions of the Sphere: Spatial Rotations and Reflections
317(4)
6.2.3 A Conformal Map of the Sphere
321(4)
6.2.4 Spatial Rotations as Mobius Transformations
325(4)
6.2.5 Spatial Rotations and Quaternions
329(4)
6.3 Hyperbolic Geometry
333(41)
6.3.1 The Tractrix and the Pseudosphere
333(2)
6.3.2 The Constant Negative Curvature of the Pseudosphere*
335(1)
6.3.3 A Conformal Map of the Pseudosphere
336(3)
6.3.4 Beltrami's Hyperbolic Plane
339(3)
6.3.5 Hyperbolic Lines and Reflections
342(5)
6.3.6 The Bolyai--Lobachevsky Formula*
347(1)
6.3.7 The Three Types of Direct Motion
348(5)
6.3.8 Decomposing an Arbitrary Direct Motion into Two Reflections
353(4)
6.3.9 The Angular Excess of a Hyperbolic Triangle
357(2)
6.3.10 The Beltrami--Poincare Disc
359(4)
6.3.11 Motions of the Beltrami--Poincare Disc
363(4)
6.3.12 The Hemisphere Model and Hyperbolic Space
367(7)
6.4 Exercises
374(11)
7 Winding Numbers and Topology
385(44)
7.1 Winding Number
385(3)
7.1.1 The Definition
385(1)
7.1.2 What Does "Inside" Mean?
386(1)
7.1.3 Finding Winding Numbers Quickly
387(1)
7.2 Hopf's Degree Theorem
388(5)
7.2.1 The Result
388(2)
7.2.2 Loops as Mappings of the Circle*
390(1)
7.2.3 The Explanation*
391(2)
7.3 Polynomials and the Argument Principle
393(1)
7.4 A Topological Argument Principle*
394(9)
7.4.1 Counting Preimages Algebraically
394(2)
7.4.2 Counting Preimages Geometrically
396(2)
7.4.3 What's Topologically Special About Analytic Functions?
398(1)
7.4.4 A Topological Argument Principle
399(2)
7.4.5 Two Examples
401(2)
7.5 Rouche's Theorem
403(2)
7.5.1 The Result
403(1)
7.5.2 The Fundamental Theorem of Algebra
404(1)
7.5.3 Brouwer's Fixed Point Theorem*
404(1)
7.6 Maxima and Minima
405(2)
7.6.1 Maximum-Modulus Theorem
405(2)
7.6.2 Related Results
407(1)
7.7 The Schwarz--Pick Lemma*
407(7)
7.7.1 Schwarz's Lemma
407(2)
7.7.2 Liouville's Theorem
409(2)
7.7.3 Pick's Result
411(3)
7.8 The Generalized Argument Principle
414(6)
7.8.1 Rational Functions
414(2)
7.8.2 Poles and Essential Singularities
416(3)
7.8.3 The Explanation*
419(1)
7.9 Exercises
420(9)
8 Complex Integration: Cauchy's Theorem
429(56)
8.1 Introduction
429(1)
8.2 The Real Integral
430(6)
8.2.1 The Riemann Sum
430(2)
8.2.2 The Trapezoidal Rule
432(2)
8.2.3 Geometric Estimation of Errors
434(2)
8.3 The Complex Integral
436(6)
8.3.1 Complex Riemann Sums
436(3)
8.3.2 A Visual Technique
439(1)
8.3.3 A Useful Inequality
440(1)
8.3.4 Rules of Integration
441(1)
8.4 Complex Inversion
442(5)
8.4.1 A Circular Arc
442(2)
8.4.2 General Loops
444(2)
8.4.3 Winding Number
446(1)
8.5 Conjugation
447(3)
8.5.1 Introduction
447(1)
8.5.2 Area Interpretation
448(2)
8.5.3 General Loops
450(1)
8.6 Power Functions
450(7)
8.6.1 Integration along a Circular Arc
450(2)
8.6.2 Complex Inversion as a Limiting Case*
452(1)
8.6.3 General Contours and the Deformation Theorem
453(1)
8.6.4 A Further Extension of the Theorem
454(1)
8.6.5 Residues
455(2)
8.7 The Exponential Mapping
457(1)
8.8 The Fundamental Theorem
458(8)
8.8.1 Introduction
458(1)
8.8.2 An Example
459(1)
8.8.3 The Fundamental Theorem
460(2)
8.8.4 The Integral as Antiderivative
462(3)
8.8.5 Logarithm as Integral
465(1)
8.9 Parametric Evaluation
466(1)
8.10 Cauchy's Theorem
467(5)
8.10.1 Some Preliminaries
467(2)
8.10.2 The Explanation
469(3)
8.11 The General Cauchy Theorem
472(3)
8.11.1 The Result
472(1)
8.11.2 The Explanation
473(1)
8.11.3 A Simpler Explanation
474(1)
8.12 The General Formula of Contour Integration
475(3)
8.13 Exercises
478(7)
9 Cauchy's Formula and Its Applications
485(26)
9.1 Cauchy's Formula
485(4)
9.1.1 Introduction
485(1)
9.1.2 First Explanation
486(1)
9.1.3 Gauss's Mean Value Theorem
487(1)
9.1.4 A Second Explanation and the General Cauchy Formula
488(1)
9.2 Infinite Differentiability and Taylor Series
489(4)
9.2.1 Infinite Differentiability
489(2)
9.2.2 Taylor Series
491(2)
9.3 Calculus of Residues
493(8)
9.3.1 Laurent Series Centred at a Pole
493(1)
9.3.2 A Formula for Calculating Residues
494(1)
9.3.3 Application to Real Integrals
495(2)
9.3.4 Calculating Residues using Taylor Series
497(1)
9.3.5 Application to Summation of Series
498(3)
9.4 Annular Laurent Series
501(5)
9.4.1 An Example
501(1)
9.4.2 Laurent's Theorem
502(4)
9.5 Exercises
506(5)
10 Vector Fields: Physics and Topology
511(26)
10.1 Vector Fields
511(7)
10.1.1 Complex Functions as Vector Fields
511(2)
10.1.2 Physical Vector Fields
513(2)
10.1.3 Flows and Force Fields
515(1)
10.1.4 Sources and Sinks
516(2)
10.2 Winding Numbers and Vector Fields*
518(7)
10.2.1 The Index of a Singular Point
518(4)
10.2.2 The Index According to Poincare
522(1)
10.2.3 The Index Theorem
523(2)
10.3 Flows on Closed Surfaces*
525(7)
10.3.1 Formulation of the Poincare--Hopf Theorem
525(2)
10.3.2 Defining the Index on a Surface
527(2)
10.3.3 An Explanation of the Poincare--Hopf Theorem
529(3)
10.4 Exercises
532(5)
11 Vector Fields and Complex Integration
537(40)
11.1 Flux and Work
537(10)
11.1.1 Flux
537(2)
11.1.2 Work
539(3)
11.1.3 Local Flux and Local Work
542(1)
11.1.4 Divergence and Curl in Geometric Form*
543(2)
11.1.5 Divergence-Free and Curl-Free Vector Fields
545(2)
11.2 Complex Integration in Terms of Vector Fields
547(15)
11.2.1 The Polya Vector Field
547(2)
11.2.2 Cauchy's Theorem
549(1)
11.2.3 Example: Area as Flux
550(1)
11.2.4 Example: Winding Number as Flux
551(2)
11.2.5 Local Behaviour of Vector Fields*
553(2)
11.2.6 Cauchy's Formula
555(1)
11.2.7 Positive Powers
556(1)
11.2.8 Negative Powers and Multipoles
557(3)
11.2.9 Multipoles at Infinity
560(1)
11.2.10 Laurent's Series as a Multipole Expansion
561(1)
11.3 The Complex Potential
562(12)
11.3.1 Introduction
562(1)
11.3.2 The Stream Function
563(2)
11.3.3 The Gradient Field
565(2)
11.3.4 The Potential Function
567(2)
11.3.5 The Complex Potential
569(3)
11.3.6 Examples
572(2)
11.4 Exercises
574(3)
12 Flows and Harmonic Functions
577(76)
12.1 Harmonic Duals
577(6)
12.1.1 Dual Flows
577(3)
12.1.2 Harmonic Duals
580(3)
12.2 Conformal Invariance
583(4)
12.2.1 Conformal Invariance of Harmonicity
583(1)
12.2.2 Conformal Invariance of the Laplacian
584(2)
12.2.3 The Meaning of the Laplacian
586(1)
12.3 A Powerful Computational Tool
587(3)
12.4 The Complex Curvature Revisited*
590(8)
12.4.1 Some Geometry of Harmonic Equipotentials
590(1)
12.4.2 The Curvature of Harmonic Equipotentials
590(4)
12.4.3 Further Complex Curvature Calculations
594(1)
12.4.4 Further Geometry of the Complex Curvature
595(3)
12.5 Flow Around an Obstacle
598(15)
12.5.1 Introduction
598(1)
12.5.2 An Example
599(5)
12.5.3 The Method of Images
604(7)
12.5.4 Mapping One Flow Onto Another
611(2)
12.6 The Physics of Riemann's Mapping Theorem
613(17)
12.6.1 Introduction
613(2)
12.6.2 Exterior Mappings and Flows Round Obstacles
615(3)
12.6.3 Interior Mappings and Dipoles
618(2)
12.6.4 Interior Mappings, Vortices, and Sources
620(4)
12.6.5 An Example: Automorphisms of the Disc
624(2)
12.6.6 Green's Function
626(4)
12.7 Dirichlet's Problem
630(19)
12.7.1 Introduction
630(1)
12.7.2 Schwarz's Interpretation
631(3)
12.7.3 Dirichlet's Problem for the Disc
634(3)
12.7.4 The Interpretations of Neumann and Bocher
637(6)
12.7.5 Green's General Formula
643(6)
12.8 Exercises
649(4)
Bibliography 653(8)
Index 661
Tristan Needham (son of the distinguished social anthropologist Rodney Needham) grew up in Oxford, England. He studied physics at Merton College, Oxford, before moving to the Mathematical Institute, where he enjoyed the great privilege of studying black holes under the supervision of Sir Roger Penrose. Tristan received his DPhil in 1987 and joined the faculty of the University of San Francisco in 1989. His continuing mission is to seek out new intuitive forms of understanding, and new visualizations.