Visual Complex AnalysisThis 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 userfriendly 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  GoodreadsVery 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 
575  
579  
581  
583  
584  
587  
588  
589  
590  
Common terms and phrases
algebra amplitwist analytic analytic function angle apply arbitrary becomes branch called centred Chapter circle clear clearly complex numbers composition conformal connecting consider constant construction continuous convergence corresponding curvature curves deduce defined definition derivative direction disc distance draw effect equal equation Euclidean example exercise exists explain expressed fact Figure Finally fixed points flow formula function geometry given hline hyperbolic idea illustrated infinitesimal infinity inside integral intersection inversion length look loop mapping means Möbius transformation motion moves multiplier Note obtain origin orthogonal particular passing plane positive power series precisely preserves projection radius reflection region represented result rotation round sense shown simple space sphere square Suppose surface symmetric Theorem translation triangle unit circle vector field vertical write yields