Visual Complex Analysis

ISBN-10: 0198534469

ISBN-13: 9780198534464

Edition: 1998

Authors: Tristan Needham

List price: $80.00 Buy it from $42.00
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Description:

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 a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.
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Book details

List price: $80.00
Copyright year: 1998
Publisher: Oxford University Press, Incorporated
Publication date: 2/18/1999
Binding: Paperback
Pages: 616
Size: 6.00" wide x 9.00" long x 1.00" tall
Weight: 2.222
Language: English

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
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