Visual Complex Analysis

Description

This 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 user-friendly 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. 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