Visual Complex Analysis

Front Cover
Clarendon Press, 1998 - Mathematics - 592 pages
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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绝对好书。

Review: Visual Complex Analysis

User Review  - James Van alstine - Goodreads

Very unique take on complex analysis. This is a great read for a physicist. That is, it is seriously lacking in mathematical rigor. I would recommend this to anyone who wants to develop an intuitive ... Read full review

Contents

Geometry and Complex Arithmetic
1
Eulers Formula
10
The Exponential Function
14
Transformations and Euclidean Geometry
30
Exercises
45
Complex Functions as Transformations
55
Power Series
64
XI
74
A Topological Argument Principle
350
The Generalized Argument Principle
363
Exercises
369
Cauchys Theorem
377
The Complex Integral
383
Conjugation
395
The Exponential Mapping
401
Parametric Evaluation
409

Cosine and Sine
84
Multifunctions
90
The Logarithm Function
98
Exercises
111
Möbius Transformations and Inversion
122
Complex Inversion
124
The Riemann Sphere
139
Basic Results
148
Möbius Transformations as Matrices
156
Vll Visualization and Classification
162
VIIl Decomposition into 2 or 4 Reflections
172
Exercises
181
The Amplitwist Concept
189
Some Simple Examples
199
The CauchyRiemann Equations
207
Further Geometry of Differentiation
216
Ill Visual Differentiation of logz
222
Visual Differentiation of the Power Function
229
NonEuclidean Geometry
267
Winding Numbers and Topology
338
Polynomials and the Argument Principle
344
The General Formula of Contour Integration
418
Cauchys Formula and its Applications
427
387
433
Physics and Topology
450
Vector Fields and Complex Integration
472
Flows and Harmonic Functions
508
Conformal Invariance of the Laplacian
515
Flow Around an Obstacle
527
The Physics of Riemanns Mapping Theorem
540
Dirichlets Problem
554
Exercises
570
79
575
Index
579
102
581
267
583
450
584
414
587
560
588
328
589
494
590
Copyright

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About the author (1998)


Tristan Needham is Associate Professor of Mathematics at the University of San Francisco. For part of the work in this book, he was presented with the Carl B. Allendoerfer Award by the Mathematical Association of America.

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