Visual Complex Analysis

Name: Visual Complex Analysis
Availability: InStock
ISBN: 9780198534464

ISBN-10: 0198534469

ISBN-13: 9780198534464

Edition: 1998

Authors: Tristan Needham

List price: $86.00

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This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to…

Book details

List price: $86.00
Copyright year: 1998
Publisher: Oxford University Press, Incorporated
Publication date: 2/18/1999
Binding: Paperback
Pages: 616
Size: 6.14" wide x 9.21" long x 1.26" tall
Weight: 2.244




Geometry and Complex Arithmetic



Introduction



Euler's Formula



Some Applications



Transformations and Euclidean Geometry*



Exercises



Complex Functions as Transformations



Introduction



Polynomials



Power Series



The Exponential Function



Cosine and Sine



Multifunctions



The Logarithm Function



Averaging over Circles*



Exercises



Mobius Transformations and Inversion



Introduction



Inversion



Three Illustrative Applications of Inversion



The Riemann Sphere



Mobius Transformations: Basic Results



Mobius Transformations as Matrices*



Visualization and Classification*



Decomposition into 2 or 4 Reflections*



Automorphisms of the Unit Disc*



Exercises



Differentiation: The Amplitwist Concept



Introduction



A Puzzling Phenomenon



Local Description of Mappings in the Plane



The Complex Derivative as Amplitwist



Some Simple Examples



Conformal = Analytic



Critical Points



The Cauchy-Riemann Equations



Exercises



Further Geometry of Differentiation



Cauchy-Riemann Revealed



An Intimation of Rigidity



Visual Differentiation of log(z)



Rules of Differentiation



Polynomials, Power Series, and Rational Functions



Visual Differentiation of the Power Function



Visual Differentiation of exp(z)



Geometric Solution of E' = E



An Application of Higher Derivatives: Curvature*



Celestial Mechanics*



Analytic Continuation*



Exercises



Non-Euclidean Geometry*



Introduction



Spherical Geometry



Hyperbolic Geometry



Exercises



Winding Numbers and Topology



Winding Number



Hopf's Degree Theorem



Polynomials and the Argument Principle



A Topological Argument Principle*



Rouche's Theorem



Maxima and Minima



The Schwarz-Pick Lemma*



The Generalized Argument Principle



Exercises



Complex Integration: Cauchy's Theorem



Introduction



The Real Integral



The Complex Integral



Complex Inversion



Conjugation



Power Functions



The Exponential Mapping



The Fundamental Theorem



Parametric Evaluation



Cauchy's Theorem



The General Cauchy Theorem



The General Formula of Contour Integration



Exercises



Cauchy's Formula and Its Applications



Cauchy's Formula



Infinite Differentiability and Taylor Series



Calculus of Residues



Annular Laurent Series



Exercises



Vector Fields: Physics and Topology



Vector Fields



Winding Numbers and Vector Fields*



Flows on Closed Surfaces*



Exercises



Vector Fields and Complex Integration



Flux and Work



Complex Integration in Terms of Vector Fields



The Complex Potential



Exercises



Flows and Harmonic Functions



Harmonic Duals



Conformal Invariance



A Powerful Computational Tool



The Complex Curvature Revisited*



Flow Around an Obstacle



The Physics of Riemann's Mapping Theorem



Dirichlet's Problem



Exercises


References


Index