# Visual Complex Analysis by Tristan Needham

• Total voters
10

## Main Question or Discussion Point

Code:
[LIST]
[*] Geometry and Complex Arithmetic
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[*] Introduction
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[*] Historical Sketch
[*] Bombelli's "Wild Thought"
[*] Some Terminology and Notation
[*] Practice
[*] Symbolic and Geometric Arithmetic
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[*] Euler's Formula
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[*] Introduction
[*] Moving Particle Argument
[*] Power Series Argument
[*] Sine and Cosine in Terms of Euler's Formula
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[*] Some Applications
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[*] Introduction
[*] Trigonometry
[*] Geometry
[*] Calculus
[*] Algebra
[*] Vectorial Operations
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[*] Transformations and Euclidean Geometry
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[*] Geometry Through the Eyes of Felix Klein
[*] Classifying Motions
[*] Three Reflections Theorem
[*] Similarities and Complex Arithmetic
[*] Spatial Complex Numbers?
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[*] Exercises
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[*] Complex Functions as Transformations
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[*] Introduction
[*] Polynomials
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[*] Positive Integer Powers
[*] Cubics Revisited
[*] Cassinian Curves
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[*] Power Series
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[*] The Mystery of Real Power Series
[*] The Disc of Convergence
[*] Approximating a Power Series with a Polynomial
[*] Uniqueness
[*] Manipulating Power Series
[*] Finding the Radius of Convergence
[*] Fourier Series
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[*] The Exponential Function
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[*] Power Series Approach
[*] The Geometry of the Mapping
[*] Another Approach
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[*] Cosine and Sine
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[*] Definitions and Identities
[*] Relation to Hyperbolic Functions
[*] The Geometry of the Mapping
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[*] Multifunctions
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[*] Example: Fractional Powers
[*] Single-Valued Branches of a Multifunction
[*] Relevance to Power Series
[*] An Example with Two Branch Points
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[*] The Logarithm Function
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[*] Inverse of the Exponential Function
[*] The Logarithmic Power Series
[*] General Powers
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[*] Averaging over Circles
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[*] The Centroid
[*] Averaging over Regular Polygons
[*] Averaging over Circles
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[*] Exercises
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[*] Mobius Transformations and Inversion
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[*] Introduction
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[*] Definition of Mobius Transformations
[*] Connection with Einstein's Theory of Relativity
[*] Decomposition into Simple Transformations
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[*] Inversion
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[*] Preliminary Definitions and Facts
[*] Preservation of Circles
[*] Construction Using Orthogonal Circles
[*] Preservation of Angles
[*] Preservation of Symmetry
[*] Inversion in a Sphere
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[*] Three Illustrative Applications of Inversion
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[*] A Problem on Touching Circles
[*] Ptolemy's Theorem
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[*] The Riemann Sphere
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[*] The Point at Infinity
[*] Stereographic Projection
[*] Transferring Complex Functions to the Sphere
[*] Behaviour of Functions at Infinity
[*] Stereographic Formulae
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[*] Mobius Transformations: Basic Results
[LIST]
[*] Preservation of Circles, Angles, and Symmetry
[*] Non-Uniqueness of the Coefficients
[*] The Group Property
[*] Fixed Points
[*] Fixed Points at Infinity
[*] The Cross-Ratio
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[*] Mobius Transformations as Matrices
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[*] Evidence of a Link with Linear Algebra
[*] The Explanation: Homogeneous Coordinates
[*] Eigenvectors and Eigenvalues
[*] Rotations of the Sphere
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[*] Visualization and Classification
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[*] The Main Idea
[*] Elliptic, Hyperbolic, and Loxodromic types
[*] Local Geometric Interpretation of the Multiplier
[*] Parabolic Transformations
[*] Computing the Multiplier
[*] Eigenvalue Interpretation of the Multiplier
[/LIST]
[*] Decomposition into 2 or 4 Reflections
[LIST]
[*] Introduction
[*] Elliptic Case
[*] Hyperbolic Case
[*] Parabolic Case
[*] Summary
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[*] Automorphisms of the Unit Disc
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[*] Counting Degrees of Freedom
[*] Finding the Formula via the Symmetry Principle
[*] Interpreting the Formula Geometrically
[*] Introduction to Riemann's Mapping Theorem
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[*] Exercises
[/LIST]
[*] Differentiation: The Amplitwist Concept
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[*] Introduction
[*] A Puzzling Phenomenon
[*] Local Description of Mappings in the Plane
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[*] Introduction
[*] The Jacobian Matrix
[*] The Amplitwist Concept
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[*] The Complex Derivative as Amplitwist
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[*] The Real Derivative Re-examined
[*] The Complex Derivative
[*] Analytic Functions
[*] A Brief Summary
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[*] Some Simple Examples
[*] Conformal = Analytic
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[*] Introduction
[*] Conformality Throughout a Region
[*] Conformality and the Riemann Sphere
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[*] Critical Points
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[*] Degrees of Crushing
[*] Breakdown of Conformality
[*] Branch Points
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[*] The Cauchy-Riemann Equations
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[*] Introduction
[*] The Geometry of Linear Transformations
[*] The Cauchy-Riemann Equations
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[*] Exercises
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[*] Further Geometry of Differentiation
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[*] Cauchy-Riemann Revealed
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[*] Introduction
[*] The Cartesian Form
[*] The Polar Form
[/LIST]
[*] An Intimation of Rigidity
[*] Visual Differentiation of log(z)
[*] Rules of Differentiation
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[*] Composition
[*] Inverse Functions
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[*] Polynomials, Power Series, and Rational Functions
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[*] Polynomials
[*] Power Series
[*] Rational Functions
[/LIST]
[*] Visual Differentiation of the Power Function
[*] Visual Differentiation of exp(z)
[*] Geometric Solution of E' = E 232
[*] An Application of Higher Derivatives: Curvature
[LIST]
[*] Introduction
[*] Analytic Transformation of Curvature
[*] Complex Curvature
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[*] Celestial Mechanics
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[*] Central Force Fields
[*] Two Kinds of Elliptical Orbit
[*] Changing the First into the Second
[*] The Geometry of Force
[*] An Explanation
[*] The Kasner-Arnol'd Theorem
[/LIST]
[*] Analytic Continuation
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[*] Introduction
[*] Rigidity
[*] Uniqueness
[*] Preservation of Identities
[*] Analytic Continuation via Reflections
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[*] Exercises
[/LIST]
[*] Non-Euclidean Geometry
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[*] Introduction
[LIST]
[*] The Parallel Axiom
[*] Some Facts from Non-Euclidean Geometry
[*] Geometry on a Curved Surface
[*] Intrinsic versus Extrinsic Geometry
[*] Gaussian Curvature
[*] Surfaces of Constant Curvature
[*] The Connection with Mobius Transformations
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[*] Spherical Geometry
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[*] The Angular Excess of a Spherical Triangle
[*] Motions of the Sphere
[*] A Conformal Map of the Sphere
[*] Spatial Rotations as Mobius Transformations
[*] Spatial Rotations and Quaternions
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[*] Hyperbolic Geometry
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[*] The Tractrix and the Pseudosphere
[*] The Constant Curvature of the Pseudosphere
[*] A Conformal Map of the Pseudosphere
[*] Beltrami's Hyperbolic Plane
[*] Hyperbolic Lines and Reflections
[*] The Bolyai-Lobachevsky Formula
[*] The Three Types of Direct Motion
[*] Decomposition into Two Reflections
[*] The Angular Excess of a Hyperbolic Triangle
[*] The Poincare Disc
[*] Motions of the Poincare Disc
[*] The Hemisphere Model and Hyperbolic Space
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[*] Exercises
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[*] Winding Numbers and Topology
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[*] Winding Number
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[*] The Definition
[*] What does "inside" mean?
[*] Finding Winding Numbers Quickly
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[*] Hopf's Degree Theorem
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[*] The Result
[*] Loops as Mappings of the Circle
[*] The Explanation
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[*] Polynomials and the Argument Principle
[*] A Topological Argument Principle
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[*] Counting Preimages Algebraically
[*] Counting Preimages Geometrically
[*] Topological Characteristics of Analyticity
[*] A Topological Argument Principle
[*] Two Examples
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[*] Rouche's Theorem
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[*] The Result
[*] The Fundamental Theorem of Algebra
[*] Brouwer's Fixed Point Theorem
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[*] Maxima and Minima
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[*] Maximum-Modulus Theorem
[*] Related Results
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[*] The Schwarz-Pick Lemma
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[*] Schwarz's Lemma
[*] Liouville's Theorem
[*] Pick's Result
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[*] The Generalized Argument Principle
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[*] Rational Functions
[*] Poles and Essential Singularities
[*] The Explanation
[/LIST]
[*] Exercises
[/LIST]
[*] Complex Integration: Cauchy's Theorem
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[*] Introduction
[*] The Real Integral
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[*] The Riemann Sum
[*] The Trapezoidal Rule
[*] Geometric Estimation of Errors
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[*] The Complex Integral
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[*] Complex Riemann Sums
[*] A Visual Technique
[*] A Useful Inequality
[*] Rules of Integration
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[*] Complex Inversion
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[*] A Circular Arc
[*] General Loops
[*] Winding Number
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[*] Conjugation
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[*] Introduction
[*] Area Interpretation
[*] General Loops
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[*] Power Functions
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[*] Integration along a Circular Arc
[*] Complex Inversion as a Limiting Case
[*] General Contours and the Deformation Theorem
[*] A Further Extension of the Theorem
[*] Residues
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[*] The Exponential Mapping
[*] The Fundamental Theorem
[LIST]
[*] Introduction
[*] An Example
[*] The Fundamental Theorem
[*] The Integral as Antiderivative
[*] Logarithm as Integral
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[*] Parametric Evaluation
[*] Cauchy's Theorem
[LIST]
[*] Some Preliminaries
[*] The Explanation
[/LIST]
[*] The General Cauchy Theorem
[LIST]
[*] The Result
[*] The Explanation
[*] A Simpler Explanation
[/LIST]
[*] The General Formula of Contour Integration
[*] Exercises
[/LIST]
[*] Cauchy's Formula and Its Applications
[LIST]
[*] Cauchy's Formula
[LIST]
[*] Introduction
[*] First Explanation
[*] Gauss' Mean Value Theorem
[*] General Cauchy Formula
[/LIST]
[*] Infinite Differentiability and Taylor Series
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[*] Infinite Differentiability
[*] Taylor Series
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[*] Calculus of Residues
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[*] Laurent Series Centred at a Pole
[*] A Formula for Calculating Residues
[*] Application to Real Integrals
[*] Calculating Residues using Taylor Series
[*] Application to Summation of Series
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[*] Annular Laurent Series
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[*] An Example
[*] Laurent's Theorem
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[*] Exercises
[/LIST]
[*] Vector Fields: Physics and Topology
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[*] Vector Fields
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[*] Complex Functions as Vector Fields
[*] Physical Vector Fields
[*] Flows and Force Fields
[*] Sources and Sinks
[/LIST]
[*] Winding Numbers and Vector Fields
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[*] The Index of a Singular Point
[*] The Index According to Poincare
[*] The Index Theorem
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[*] Flows on Closed Surfaces
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[*] Formulation of the Poincare-Hopf Theorem
[*] Defining the Index on a Surface
[*] An Explanation of the Poincare-Hopf Theorem
[/LIST]
[*] Exercises
[/LIST]
[*] Vector Fields and Complex Integration
[LIST]
[*] Flux and Work
[LIST]
[*] Flux
[*] Work
[*] Local Flux and Local Work
[*] Divergence and Curl in Geometric Form
[*] Divergence-Free and Curl-Free Vector Fields
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[*] Complex Integration in Terms of Vector Fields
[LIST]
[*] The Polya Vector Field
[*] Cauchy's Theorem
[*] Example: Area as Flux
[*] Example: Winding Number as Flux
[*] Local Behaviour of Vector Fields
[*] Cauchy's Formula
[*] Positive Powers
[*] Negative Powers and Multipoles
[*] Multipoles at Infinity
[*] Laurent's Series as a Multipole Expansion
[/LIST]
[*] The Complex Potential
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[*] Introduction
[*] The Stream Function
[*] The Potential Function
[*] The Complex Potential
[*] Examples
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[*] Exercises
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[*] Flows and Harmonic Functions
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[*] Harmonic Duals
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[*] Dual Flows
[*] Harmonic Duals
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[*] Conformal Invariance
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[*] Conformal Invariance of Harmonicity
[*] Conformal Invariance of the Laplacian
[*] The Meaning of the Laplacian
[/LIST]
[*] A Powerful Computational Tool
[*] The Complex Curvature Revisited
[LIST]
[*] Some Geometry of Harmonic Equipotentials
[*] The Curvature of Harmonic Equipotentials
[*] Further Complex Curvature Calculations
[*] Further Geometry of the Complex Curvature
[/LIST]
[*] Flow Around an Obstacle
[LIST]
[*] Introduction
[*] An Example
[*] The Method of Images
[*] Mapping One Flow Onto Another
[/LIST]
[*] The Physics of Riemann's Mapping Theorem
[LIST]
[*] Introduction
[*] Exterior Mappings and Flows Round Obstacles
[*] Interior Mappings and Dipoles
[*] Interior Mappings, Vortices, and Sources
[*] An Example: Automorphisms of the Disc
[*] Green's Function
[/LIST]
[*] Dirichlet's Problem
[LISt]
[*] Introduction
[*] Schwarz's Interpretation
[*] Dirichlet's Problem for the Disc
[*] The Interpretations of Neumann and Bocher
[*] Green's General Formula
[/LIST]
[*] Exercises
[/LIST]
[*] References
[*] Index
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I bought it and did not like it. For one, it's not visual. Complex Analysis in my opinion is very visual and they did not in my opinion capture this visual component of the subject well at all. Sorry I can't give examples, I looked on my bookshelf and seem not to have it anymore.

This is one of my favourite mathematics books. It has a strong "style" to it that some people may not like, and it is probably not the best book to use for a first course. However, as a supplemental text or a second course, it would be awesome.

I deeply love Complex Analysis because of this book.

I haven't worked through the entirety of the text, but the section on Mobius transformations and their relationship linear projective transformations is honestly the most insightful and clear that I've ever read. The book is worth the price for that chapter alone.