Visual Complex Analysis 0198534477, 9780198534471

Table of contents :
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