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Geometry and Complex Arithmetic | |
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Introduction | |
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Euler's Formula | |
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Some Applications | |
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Transformations and Euclidean Geometry* | |
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Exercises | |
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Complex Functions as Transformations | |
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Introduction | |
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Polynomials | |
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Power Series | |
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The Exponential Function | |
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Cosine and Sine | |
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Multifunctions | |
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The Logarithm Function | |
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Averaging over Circles* | |
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Exercises | |
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Mobius Transformations and Inversion | |
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Introduction | |
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Inversion | |
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Three Illustrative Applications of Inversion | |
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The Riemann Sphere | |
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Mobius Transformations: Basic Results | |
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Mobius Transformations as Matrices* | |
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Visualization and Classification* | |
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Decomposition into 2 or 4 Reflections* | |
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Automorphisms of the Unit Disc* | |
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Exercises | |
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Differentiation: The Amplitwist Concept | |
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Introduction | |
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A Puzzling Phenomenon | |
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Local Description of Mappings in the Plane | |
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The Complex Derivative as Amplitwist | |
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Some Simple Examples | |
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Conformal = Analytic | |
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Critical Points | |
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The Cauchy-Riemann Equations | |
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Exercises | |
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Further Geometry of Differentiation | |
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Cauchy-Riemann Revealed | |
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An Intimation of Rigidity | |
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Visual Differentiation of log(z) | |
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Rules of Differentiation | |
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Polynomials, Power Series, and Rational Functions | |
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Visual Differentiation of the Power Function | |
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Visual Differentiation of exp(z) | |
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Geometric Solution of E' = E | |
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An Application of Higher Derivatives: Curvature* | |
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Celestial Mechanics* | |
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Analytic Continuation* | |
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Exercises | |
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Non-Euclidean Geometry* | |
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Introduction | |
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Spherical Geometry | |
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Hyperbolic Geometry | |
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Exercises | |
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Winding Numbers and Topology | |
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Winding Number | |
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Hopf's Degree Theorem | |
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Polynomials and the Argument Principle | |
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A Topological Argument Principle* | |
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Rouche's Theorem | |
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Maxima and Minima | |
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The Schwarz-Pick Lemma* | |
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The Generalized Argument Principle | |
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Exercises | |
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Complex Integration: Cauchy's Theorem | |
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Introduction | |
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The Real Integral | |
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The Complex Integral | |
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Complex Inversion | |
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Conjugation | |
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Power Functions | |
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The Exponential Mapping | |
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The Fundamental Theorem | |
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Parametric Evaluation | |
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Cauchy's Theorem | |
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The General Cauchy Theorem | |
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The General Formula of Contour Integration | |
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Exercises | |
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Cauchy's Formula and Its Applications | |
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Cauchy's Formula | |
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Infinite Differentiability and Taylor Series | |
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Calculus of Residues | |
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Annular Laurent Series | |
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Exercises | |
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Vector Fields: Physics and Topology | |
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Vector Fields | |
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Winding Numbers and Vector Fields* | |
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Flows on Closed Surfaces* | |
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Exercises | |
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Vector Fields and Complex Integration | |
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Flux and Work | |
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Complex Integration in Terms of Vector Fields | |
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The Complex Potential | |
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Exercises | |
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Flows and Harmonic Functions | |
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Harmonic Duals | |
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Conformal Invariance | |
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A Powerful Computational Tool | |
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The Complex Curvature Revisited* | |
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Flow Around an Obstacle | |
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The Physics of Riemann's Mapping Theorem | |
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Dirichlet's Problem | |
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Exercises | |
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References | |
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Index | |