Visual Complex Analysis

Preface
Contents
1.
Geometry and Complex Arithmetic
	Introduction
		Historical Skentch
		Bombelli´s \"Wild Thought\"
		Some terminology and notation
		Practice
		Equivalence of Symbolic and geometric arithmetic
	Euler´s Formula
		Introduction
		Moving particle argument
		Power series argument
		Sine and cosine in terms of Euler´s formula
	Some applications
		Introduction
		Trigonometry
		Geometry
		Calculus
		Algebra
		Vectorial operations
	Transformations and Euclidean geometry
		Geometry through the eyes of Felix Klein
		Classifying motions
		Three reflections theorem
		Similarities and Complex arithmetic
		Spatial complex numers?
	Excercises
2.
Complex functions as transformations
	Introduction
	Polynomals
		Positive Integer Powers
		Cubics revisited *
		Cassinian Curves *
	Power series
		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*
	The exponential function
		Power series approach
		The geometry of the mapping
		Another approach
	Cosine and sine
		Definitions and identities
		Relation to hyperbolic functions
		The geometry of the mapping
	Multifunctions
		Example: Fractional powers
		Single-valued branches of a multifunction
		Relevance to power series
		An example with two branch points
	The logarithm function
		Inverse of the exponential function
		The logarithmic power series
		General powers
	Averaging over circles*
		The centroid
		Averaging over regular polygons
		Averaging over circles
	Exercises
3.
Möbius Transformations and Inversion
	Introduction
		Definition of Möbius transformations
		Connection with Einstein´s theory of relativity*
		Decomposition into simple transformations
	Inversion
		Preliminary definitions and facts
		Preservation of circles
		Construction using orthogonal circles
		Preservation of angles
		Preservation of symmetry
		Inversion in a sphere
	Three illustrative applications of inversion
		A problem on touching circles
		Quadrilaterals with orthogonal diagonals
		Ptolemy´s theorem
	The riemann sphere
		The point at infinity
		Stereografic projection
		Transferring complex functions to the sphere
		Behaviour of functions at infinity
		Stereographic formulae
	Möbius transformations:Basic results
		Preservation of circles, angles and symmetry
		Non-uniqueness of the coefficients
		The group property
		Fixed points
		Fixed points at infinity
		The cross-ratio
	Möbius transformations as matrices*
		Evidence of a link with linear algebra
		The explanation: Homogeneous coordinates
		Eigenvectors and eigenvalues
		Rotations of the sphere
	Visualization and classification
		The main idea
		Elliptic, hiperbolic, and loxodromic types
		Local geometric inerpretation of the multipler
		Parabolic transformations
		Computing the multipler
		Eingenvalue interpretation of the multipler
	Decomposition into 2 or 4 reflections
		Introduction
		Elliptic case
		Hyperbolic case
		Parabolic case
		Summary
	Automorphisms of the unit disc
		Counting derrees of freedom
		Finding the formula via the symmetry principie
		Interpreting the formula geometrically
		Introduction to Riemann´s Mapping Theorem
	Exercises
4.
Differentiation: the amplitwist concept
	Introduction
	A puzzling phenomenon
	Local description of mappings inthe plane
		Introduction
		The jacobian matrix
		The amplitwist concept
	The complex direivative as amplitwist
		The real derivative re-examined
		The complex derivative
		Analytic functions
		A brief summary
	Some simple examples
	Conformal = analytic
		Introduction
		Conformality throughout a region
		Conformality and the Riemann sphere
	Critical points
		Degrees of crushing
		Breakdown of conformality
		Branch points
	The Cauchy-Riemann equations
		Introduction
		The geometry of linear transformations
		The Cauchy-Riemann equations
	Exercises
5.
Further geometry of differentiation
	Cauchy-Riemann revealed
		Introduction
		The cartesian form
		The polar form
	An intimation of rigidity
	Visual differentiation of log(z)
	Rules of differentiation
		Composition
		Inverse functions
		Addition and multiplication
	Polynomials, power series, and rational functions
		Polynomials
		Power series
		Rational functions
	Visual differentiation of the power function
	Visual differentiation of exp(z)
	Geometric solution of E´=E
	An application fo higher derivates: curvature*
		Introduction
		Analytic transformation of curvature
		Complex curvature
	Celestial mechanics*
		Central force fields
		Two kinds of elliptical orbit
		Changing the first into the second
		The geometry of force
		An explanation
		The Kasner-Arnold´s theorem
	Analitic continuation*
		Introduction
		Rigidity
		Uniqueness
		Preservation of indentities
		Analytic continuation via reflections
	Exercises
6.
Non-Euclidean geometry
	Introduction
		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 Möbius transformations
	Spherical geometry
		The angular excess of a spherical triangle
		Motions of the sphere
		A conformal map of the sphere
		Spatial rotations as Möbius transformations
		Spatial Rotations and quaternions
	Hiperbolic geometry
		The tractix and the pseudosphere
		The constant curvature of the pseudosphere
		A conformal map of the pseudosphere
		Beltrami´s hiperbolic plane
		Hiperbolic lines and reflections
		The Bolyai-Lobachevsky formula
		The three types of direct motion
		Decomposition into two reflections
		The angular excess of a hiperbolic triangle
		The Poincare disc
		Motions of the Poincaré disc
		The hemisphere model and hyperbolic space
	Exercises
7.
Winding numbers and topology
	Winding number
		Definition
		What does \"inside\" mean?
		Finding winding numbers quickly
	Hopf´s degree theorem
		The result
		Loops as mappings of the circle*
		The explanation*
	Polynomials and the argument principie
	A topological argument principie*
		Counting preimages algebraically
		Counting preimages geometrically
		Topological characteristics of analyticity
		A topological argument principie
		Two examples
	Rouché´s theorem
		The result
		The fundamental theorem of algebra
		Brouwer´s fixed point theorem*
	Maxima and minima
		Maximum-modulus theorem
		Related results
	The Schwarz-Pick lemma*
		Schwarz´s lemma
		Liouville´s theorem
		Pick´s result
	The generalized argument principle
		Rational functions
		Poles and essential singularities
		The explanation*
	Exercises
8.
Complex integration: Cauchy´s theorem
	Introduction
	The real integral
		The Riemann sum
		The trapezoidal rule
		Geometric estimation of errors
	The complex integral
		Complex Riemann sums
		Visual Technique
		A useful inequality
		Rules of integration
	Complex inversion
		A circular arc
		General loops
		Winding number
	Conjugation
		Introduction
		Area interpretation
		General loops
	Power functions
		Integration along a circular arc
		Complex inversion as a limiting case
		General contours and the deformation theorem
		A further extension of the theorem
		Residues
	The exponential mapping
	The fundamental theorem
		Introduction
		An example
		The fundamental theorem
		The integral as antiderivate
		Logaritm as integral
	Parametric evaluation
	Cauchy´s theorem
		Some preliminaries
		The explanation
	The general Cauchy theorem
		The result
		The explanation
		A simpler explanation
	The general formula of contour integration
	Exercises
9.
Cauchy´s formula and its applications
	Cauchy´s Formula
		Introduction
		First explanation
		Gauss´mean value theorem
		General Cauchy formula
	Infinite differentiability and Taylor series
		Infinity differentiability
		Taylor series
	Calculus of residues
		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
	Annular Laurent series
		An example
		Laurent´s theorem
	Exercises
10.
Vector fields: physics and topology
	Vector fields
		Complex functions as vector fields
		Physical vector fields
		Flows and force fields
		Sources and sinks
	Winding numbers and vector fields*
		The index of a singular point
		The index according to Poincaré
		The index theorem
	Flows on closed surfaces*
		Formulation of the Poincaré-Hopf theorem
		Defining the index on a surface
		An explanation fo the Poincaré-Hopf theorem
	Exercises
11.
Vector fields and complex integration
	Flux and work
		Flux
		Work
		Local flux and local work
		Divergence and crul in geometric form*
		Divergence-free and crul-free vector fields
	Complex integration in terms of vector fields
		The Pólya 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
	The complex potential
		Introduction
		The stream function
		The gradient field
		The potential function
		The complex potential function
		Examples
	Exercises
12.
Flows and harmonic functions
	Harmonic duals
		Dual flows
		Harmonic duals
	Conformal inveriance
		Conformal invariance of harmonicity
		Conformal invariance of the Laplacian
		The meaning fo the Laplacian
	A powerful computational tool
	The complex curvature revisited*
		Some geometry of harmonic equipotentials
		The curvature of harmonic equipotentials
		Further complex curvature calculations
		Further geometry of the complex curvature
	Flow around an oblstacle
		Introduction
		An example
		The metoth of images
		Mapping one flow onto another
	The physics of Riemann´s mapping theorem
		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
	Dirichlet´s problem
		Introduction
		Schwarz´s interpretation
		Dirichlet´s problem for the disc
		The interpretations of Neumann and Böcher
		Green general formula
	Exercises
References
Index