Post by Admin on Feb 19, 2020 9:44:48 GMT
I have amended my answer to exercise 31 to stress the 'if and only if' nature of the proofs in parts (iii) and (iv).
Vasco/Admin
Vasco/Admin
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- General Information about the Forum
- Mathematics
- Visual Complex Analysis
- Chapter 1
- Chapter 2
- Chapter 3
- I Introduction
- II Inversion
- III Three Illustrative Applications of Inversion
- IV The Riemann Sphere
- V Möbius Transformations: Basic Results
- VI Möbius Transformations as Matrices*
- VII Visualization and Classification*
- VIII Decomposition into 2 or 4 Reflections*
- IX Automorphisms of the Unit Disc*
- X Exercises chapter 3
- Errata for chapter 3
- Chapter 4
- Chapter 5
- I Cauchy-Riemann Revealed
- II An Intimation of Rigidity
- III Visual differentiation of log(z)
- IV Rules of Differentiation
- V Polynomials, Power Series, and Rational Functions
- VI Visual Differentiation of the Power Function
- VII Visual Differentiation of exp(z)
- VIII Geometric Solution of E'=E
- IX An Application of Higher Derivatives: Curvature*
- X Celestial Mechanics*
- XI Analytic Continuation*
- XII Exercises chapter 5
- Errata for chapter 5
- Chapter 6
- I Introduction
- II Spherical Geometry
- III Hyperbolic Geometry
- 1 The Tractrix and the Pseudosphere
- 2 The constant curvature of the Pseudosphere*
- 3 A Conformal Map of the Pseudosphere
- 4 Beltrami's Hyperbolic Plane
- 5 Hyperbolic Lines and Reflections
- 6 The Bolyai-Lobachevsky Formula*
- 7 The Three Types of Direct Motion
- 8 Decomposition into Two Reflections
- 9 The Angular Excess of a Hyperbolic Triangle
- 10 The Poincaré Disc
- 11 Motions of the Poincaré Disc
- 12 The Hemisphere model and Hyperbolic Space
- IV Exercises chapter 6
- Errata for chapter 6
- Chapter 7
- Chapter 8
- I Introduction
- II The Real Integral
- III The Complex Integral
- IV Complex Inversion
- V Conjugation
- VI Power Functions
- VII The Exponential Mapping
- VIII The Fundamental Theorem
- IX Parametric Evaluation
- X Cauchy's Theorem
- XI The General Cauchy Theorem
- XII The General Formula of Contour Integration
- XIII Exercises chapter 8
- Errata for chapter 8
- Chapter 9
- Chapter 10
- Chapter 11
- Chapter 12
- Mathematics and its History
- Visual Complex Analysis
