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Visual Complex Analysis [Mīkstie vāki]

Tristan Needham (Associate Professor, University of San Francisco)

Formāts: Paperback / softback, 616 pages, height x width x depth: 234x156x32 mm, weight: 924 g, 501 line figures
Izdošanas datums: 26-Nov-1998
Izdevniecība: Oxford University Press
ISBN-10: 0198534469
ISBN-13: 9780198534464

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Formāts: Paperback / softback, 616 pages, height x width x depth: 234x156x32 mm, weight: 924 g, 501 line figures
Izdošanas datums: 26-Nov-1998
Izdevniecība: Oxford University Press
ISBN-10: 0198534469
ISBN-13: 9780198534464

Citas grāmatas par šo tēmu:

Permanent link: https://www.kriso.lv/db/9780198534464.html

Keywords:

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.

Recenzijas

[ Needham's] highly praised massive book Visual Complex Analysis may still be resounding in the minds of those who have read it. The original approach and the numerous graphics must have left a lasting impression. * Adhemar Bultheel, Mathematical Association of America Reviews * ... 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 * 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. * Roger Penrose * 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 2x2 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 2x2 matricesbut his are interesting. * Ian Stewart, New Scientist, 11 October 1997 * ... 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 * 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, founder of Pixar *

Geometry and Complex Arithmetic

(54)

Introduction

(9)

Historical Sketch

(2)

Bombelli's ``Wild Thought''

(3)

Some Terminology and Notation

(1)

Practice

(1)

Symbolic and Geometric Arithmetic

(2)

Euler's Formula

(4)

Introduction

(1)

Moving Particle Argument

(2)

Power Series Argument

(2)

Sine and Cosine in Terms of Euler's Formula

(1)

Some Applications

(16)

Introduction

(1)

Trigonometry

(2)

Geometry

(4)

Calculus

(2)

Algebra

(5)

Vectorial Operations

(3)

Transformations and Euclidean Geometry*

(15)

Geometry Through the Eyes of Felix Klein

(4)

Classifying Motions

(3)

Three Reflections Theorem

(2)

Similarities and Complex Arithmetic

(4)

Spatial Complex Numbers?

(2)

Exercises

(10)

Complex Functions as Transformations

(67)

Introduction

(2)

Polynomials

(7)

Positive Integer Powers

(2)

Cubics Revisited*

(1)

Cassinian Curves*

(4)

Power Series

(15)

The Mystery of Real Power Series

(3)

The Disc of Convergence

(3)

Approximating a Power Series with a Polynomial

(1)

Uniqueness

(1)

Manipulating Power Series

(2)

Finding the Radius of Convergence

(3)

Fourier Series*

(2)

The Exponential Function

(5)

Power Series Approach

(1)

The Geometry of the Mapping

(1)

Another Approach

(3)

Cosine and Sine

(6)

Definitions and Identities

(2)

Relation to Hyperbolic Functions

(2)

The Geometry of the Mapping

(2)

Multifunctions

(8)

Example: Fractional Powers

(2)

Single-Valued Branches of a Multifunction

(3)

Relevance to Power Series

(1)

An Example with Two Branch Points

(2)

The Logarithm Function

(4)

Inverse of the Exponential Function

(2)

The Logarithmic Power Series

100

(1)

General Powers

101

(1)

Averaging over Circles*

102

(9)

The Centroid

102

(3)

Averaging over Regular Polygons

105

(3)

Averaging over Circles

108

(3)

Exercises

111

(11)

Mobius Transformations and Inversion

122

(67)

Introduction

122

(2)

Definition of Mobius Transformations

122

(1)

Connection with Einstein's Theory of Relativity*

122

(1)

Decomposition into Simple Transformations

123

(1)

Inversion

124

(12)

Preliminary Definitions and Facts

124

(2)

Preservation of Circles

126

(2)

Construction Using Orthogonal Circles

128

(2)

Preservation of Angles

130

(3)

Preservation of Symmetry

133

(1)

Inversion in a Sphere

133

(3)

Three Illustrative Applications of Inversion

136

(3)

A Problem on Touching Circles

136

(1)

Quadrilaterals with Orthogonal Diagonals

137

(1)

Ptolemy's Theorem

138

(1)

The Riemann Sphere

139

(9)

The Point at Infinity

139

(1)

Stereographic Projection

140

(3)

Transferring Complex Functions to the Sphere

143

(1)

Behaviour of Functions at Infinity

144

(2)

Stereographic Formulae*

146

(2)

Mobius Transformations: Basic Results

148

(8)

Preservation of Circles, Angles, and Symmetry

148

(1)

Non-Uniqueness of the Coefficients

149

(1)

The Group Property

150

(1)

Fixed Points

151

(1)

Fixed Points at Infinity

152

(2)

The Cross-Ratio

154

(2)

Mobius Transformations as Matrices*

156

(6)

Evidence of a Link with Linear Algebra

156

(1)

The Explanation: Homogeneous Coordinates

157

(1)

Eigenvectors and Eigenvalues*

158

(3)

Rotations of the Sphere*

161

(1)

Visualization and Classification*

162

(10)

The Main Idea

162

(2)

Elliptic, Hyperbolic, and Loxodromic types

164

(2)

Local Geometric Interpretation of the Multiplier

166

(2)

Parabolic Transformations

168

(1)

Computing the Multiplie*

169

(1)

Eigenvalue Interpretation of the Multiplier*

170

(2)

Decomposition into 2 or 4 Reflections*

172

(4)

Introduction

172

(1)

Elliptic Case

172

(1)

Hyperbolic Case

173

(1)

Parabolic Case

174

(1)

Summary

175

(1)

Automorphisms of the Unit Disc*

176

(5)

Counting Degrees of Freedom

176

(1)

Finding the Formula via the Symmetry Principle

177

(1)

Interpreting the Formula Geometrically*

178

(2)

Introduction to Riemann's Mapping Theorem

180

(1)

Exercises

181

(8)

Differentiation: The Amplitwist Concept

189

(27)

Introduction

189

(1)

A Puzzling Phenomenon

189

(2)

Local Description of Mappings in the Plane

191

(3)

Introduction

191

(1)

The Jacobian Matrix

192

(1)

The Amplitwist Concept

193

(1)

The Complex Derivative as Amplitwist

194

(5)

The Real Derivative Re-Examined

194

(1)

The Complex Derivative

195

(2)

Analytic Functions

197

(1)

A Brief Summary

198

(1)

Some Simple Examples

199

(1)

Conformal = Analytic

200

(4)

Introduction

200

(1)

Conformality Throughout a Region

201

(2)

Conformality and the Riemann Sphere

203

(1)

Critical Points

204

(3)

Degrees of Crushing

204

(1)

Breakdown of Conformality

205

(1)

Branch Points

206

(1)

The Cauchy-Riemann Equations

207

(4)

Introduction

207

(1)

The Geometry of Linear Transformations

208

(1)

The Cauchy-Riemann Equations

209

(2)

Exercises

211

(5)

Further Geometry of Differentiation

216

(51)

Cauchy-Riemann Revealed

216

(3)

Introduction

216

(1)

The Cartesian Form

216

(1)

The Polar Form

217

(2)

An Intimation of Rigidity

219

(3)

Visual Differentiation of log(z)

222

(1)

Rules of Differentiation

223

(3)

Composition

223

(1)

Inverse Functions

224

(1)

Addition and Multiplication

225

(1)

Polynomials, Power Series, and Rational Functions

226

(3)

Polynomials

226

(1)

Power Series

227

(1)

Rational Functions

228

(1)

Visual Differentiation of the Power Function

229

(2)

Visual Differentiation of exp(z)

231

(1)

Geometric Solution of E'= E

232

(2)

An Application of Higher Derivatives: Curvature*

234

(7)

Introduction

234

(1)

Analytic Transformation of Curvature

235

(3)

Complex Curvature

238

(3)

Celestial Mechanics

241

(6)

Central Force Fields

241

(1)

Two Kinds of Elliptical Orbit

241

(2)

Changing the First into the Second

243

(1)

The Geometry of Force

244

(1)

An Explanation

245

(1)

The Kasner-Arnol'd Theorem

246

(1)

Analytic Continuation

247

(11)

Introduction

247

(2)

Rigidity

249

(1)

Uniqueness

250

(1)

Preservation of Identities

251

(1)

Analytic Continuation via Reflections

252

(6)

Exercises

258

(9)

Non-Euclidean Geometry

267

(71)

Introduction

267

(11)

The Parallel Axiom

267

(2)

Some Facts from Non-Euclidean Geometry

269

(1)

Geometry on a Curved Surface

270

(3)

Intrinsic Versus Extrinsic Geometry

273

(1)

Gaussian Curvature

273

(2)

Surfaces of Constant Curvature

275

(2)

The Connection with Mobius Transformations

277

(1)

Spherical Geometry

278

(15)

The Angular Excess of a Spherical Triangle

278

(1)

Motions of the Sphere

279

(4)

A Conformal Map of the Sphere

283

(3)

Spatial Rotation as Mobius Transformations

286

(4)

Spatial Rotations and Quaternions

290

(3)

Hyperbolic Geometry

293

(35)

The Tractrix and the Pseudosphere

293

(2)

The Constant Curvature of the Pseudosphere

295

(1)

A Conformal Map of the Pseudosphere

296

(2)

Beltrami's Hyperbolic Plane

298

(3)

Hyperbolic Lines and Reflections

301

(4)

The Bolyai-Lobachevsky Formula

305

(1)

The Three Types of Direct Motion

306

(5)

Decomposition into Two Reflections

311

(2)

The Angular Excess of a Hyperbolic Triangle

313

(2)

The Poincare Disc

315

(4)

Motions of the Poincare Disc

319

(3)

The Hemisphere Model and Hyperbolic Space

322

(6)

Exercises

328

(10)

Winding Numbers and Topology

338

(39)

Winding Number

338

(3)

The Definition

338

(1)

What does ``inside'' mean?

339

(1)

Finding Winding Numbers Quickly

340

(1)

Hopf's Degree Theorem

341

(3)

The Result

341

(1)

Loops as Mappings of the Circle

342

(1)

The Explanation

343

(1)

Polynomials and the Argument Principle

344

(2)

A Topological Argument Principle

346

(7)

Counting Preimages Algebraically

346

(1)

Counting Preimages Geometrically

347

(2)

Topological Characteristics of Analyticity

349

(1)

A Topological Argument Principle

350

(2)

Two Examples

352

(1)

Rouche's Theorem

353

(2)

The Result

353

(1)

The Fundamental Theorem of Algebra

354

(1)

Brouwer's Fixed Point Theorem

354

(1)

Maxima and Minima

355

(2)

Maximum-Modulus Theorem

355

(2)

Visual Complex Analysis [Mīkstie vāki]

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