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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Review: Visual Complex Analysis
User Review  Nishant Pappireddi  GoodreadsI got this book because I was promised geometrically intuitive explanations of the results in a standard Complex Analysis course, and I was not disappointed! Almost every result the author stated was ... Read full review
Review: Visual Complex Analysis
User Review  Amar Pai  GoodreadsReally great book... the closest I've come to actually 'getting' complex analysis. Basic operations like complex multiplication are clearly explained in terms of vector diagrams. Hyperbolic geometry ... Read full review
Contents
Geometry and Complex Arithmetic  1 
Eulers Formula  10 
Transformations and Euclidean Geometry  30 
Exercises  45 
Complex Functions as Transformations  55 
Power Series  64 
The Exponential Function  79 
Multifunctions  90 
Winding Numbers and Topology  338 
Polynomials and the Argument Principle  344 
Rouches Theorem  353 
The Generalized Argument Principle  363 
Exercises  369 
Cauchys Theorem  377 
The Complex Integral  383 
Conjugation  392 
The togarithm Function  98 
Exercises  111 
Mobius Transformations and Inversion  122 
Three Illustrative Applications of Inversion  136 
Basic Results  148 
Mobius Transformations as Matrices  156 
Visualization and Classification  162 
Decomposition into 2 or 4 Reflections  172 
Exercises  181 
The Amplitwist Concept  189 
Critical Points  204 
Exercises  211 
An Intimation of Rigidity  219 
Polynomials Power Series and Rational Func  226 
VIM Geometric Solution of E E  232 
Celestial Mechanics  241 
Analytic Continuation  247 
Exercises  258 
NonEuclidean Geometry  267 
Spherical Geometry  278 
Hyperbolic Geometry  293 
Exercises  328 
The Exponential Mapping  401 
Parametric Evaluation  409 
The General Formula of Contour Integration  418 
Cauchys Formula and Its Applications  427 
Calculus of Residues  434 
Annular taurent Series  442 
Physics and Topology  450 
Winding Numbers and Vector Fields  456 
Flows on Closed Surfaces  462 
Exercises  468 
Complex Integration in Terms of Vector Fields  481 
The Complex Potential  494 
Exercises  505 
Conformal Invariance  513 
The Complex Curvature Revisited  520 
Flow Around an Obstacle  527 
The Physics of Riemanns Mapping Theorem  540 
Dirichlets Problem  554 
VIM Exercises  570 
579  
Common terms and phrases
algebraic amplitwist analytic function analytic mapping angle arbitrary Argument Principle branch point Chapter complex function complex inversion complex numbers complex plane complex potential conformal mapping consider constant contour convergence critical point curvature curve deduce defined derivative dipole direct motion disc of convergence distance ellipse equal equation Euclidean Euclidean geometry example exercise fact Figure fixed points flow flux formula geometric given h,line harmonic hyperbolic geometry hyperbolic plane illustrated image points infinitesimal infinity inside integral interior intersection length line,segment linear Mobius transformation multiplication n,gon non,Euclidean obtain one,to,one origin orthogonal pole Polya vector field polynomial power series preimages pseudosphere quaternion radius real axis real number reflection region result Riemann Riemann sphere round segment simple loop singularity sphere square stereographic projection streamlines surface symmetric tangent triangle unit circle unit disc upper half,plane vanish verify vertical winding number yields