Starting with the Unit Circle, Softcover reprint of the original 1st ed. 1981 Background to Higher Analysis

It is with great pleasure that I am writing the preface for my little book, "Starting with the Unit Circle", in the office of Springer Verlag in Heidel­ berg. This is symbolic of the fact that I have once again joined in the main­ stream of scientific exchange between East and West. Since the establishment of the People's Republic of China, I have written "An Introduction to Number Theory" for the young people studying Number Theory: for the young people studying algebra, Prof. Wan Zhe-xian (Wan Che-hsien) and I have written "Classical Groups"; for those studying the theory of functions of several complex variables, I have written "Har­ monic Analysis of Functions of Several Complex Variables in the Classical Domains", * and for university students I have written "Introduction to Higher Mathematics". The present volume had been written for those who were beginning to engage in research at the Chinese University of Science and Technology and at the Guangdong Zhongshan University. Its purpose is none other than to make the students see the crucial ideas in their simplest manifestations, so that when they go on to the more complex parts of modem mathematics, they will not be without guidance. For example, in the first chapter when I point out that the Poisson kernel is just the Jacobian of some transformation, I am merely revealing the source of one of the main tools in my work on harmonic analysis in the classical domains.

1 The Geometric Theory of Harmonic Functions.- 1.1 Remembrance of Things Past.- 1.2 Real Forms.- 1.3 The Geometry of the Unit Ball.- 1.4 The Differential Metric.- 1.5 A Differential Operator.- 1.6 Spherical Coordinates.- 1.7 The Poisson Formula.- 1.8 What Has the Above Suggested?.- 1.9 The Symmetry Principle.- 1.10 The Invariance of the Laplace Equation.- 1.11 The Mean Value Formula for the Laplace Equation.- 1.12 The Poisson Formula for the Laplace Equation.- 1.13 A Brief Summary.- 2 Fourier Analysis and the Expansion Formulas for Harmonic Functions.- 2.1 A Few Properties of Spherical Functions.- 2.2 Orthogonality Properties.- 2.3 The Boundary Value Problem.- 2.4 Generalized Functions on the Sphere.- 2.5 Harmonic Analysis on the Sphere.- 2.6 Expansion of the Poisson Kernel of Invariant Equations.- 2.7 Completeness.- 2.8 Solving the Partial Differential Equation ?2M? = ??.- 2.9 Remarks.- 3 Extended Space and Spherical Geometry.- 3.1 Quadratic Forms and Generalized Space.- 3.2 Differential Metric, Conformal Mappings.- 3.3 Mapping Spheres into Spheres.- 3.4 Tangent Spheres and Chains of Spheres.- 3.5 Orthogonal Spheres and Families of Spheres.- 3.6 Conformal Mappings.- 4 The Lorentz Group.- 4.1 Changing the Basic Square Matrix.- 4.2 Generators.- 4.3 Orthogonal Similarity.- 4.4 On Indefinite Quadratic Forms.- 4.5 Lorentz Similarity.- 4.6 Continuation.- 4.7 The Canonical Forms of Lorentz Similarity.- 4.8 Involution.- 5 The Fundamental Theorem of Spherical Geometry—with a Discussion of the Fundamental Theorem of Special Relativity.- 5.1 Introduction.- 5.2 Uniform Linear Motion.- 5.3 The Geometry of Hermitian Matrices.- 5.4 Affine Transformations Which Leave Invariant the Unit Sphere in 3-Dimensional Space.- 5.5 Coherent Subspaces.- 5.6 Phase Planes (or 2-Dimensional Phase Subspaces).- 5.7 Phase Lines.- 5.8 Point Pairs.- 5.9 3-Dimensional Phase Subspaces.- 5.10 Proof of the Fundamental Theorem.- 5.11 The Fundamental Theorems of Spacetime Geometry.- 5.12 The Projective Geometry of Hermitian Matrices.- 5.13 Projective Transformations and Causal Relations.- 5.14 Remarks.- 6 Non-Euclidean Geometry.- 6.1 The Geometric Properties of Extended Space.- 6.2 Parabolic Geometry.- 6.3 Elliptical Geometry.- 6.4 Hyperbolic Geometry.- 6.5 Geodesics.- 7 Partial Differential Equations of Mixed Type.- 7.1 Real Projective Planes.- 7.2 Partial Differential Equations.- 7.3 Characteristic Curves.- 7.4 The Relationship Between this Partial Differential Equation and Lav’rentiev’s Equation.- 7.5 Separation of Variables.- 7.6 Some Examples.- 7.7 Convergence of Series.- 7.8 Functions Without Singularities Inside the Unit Circle (Analogues of Holomorphic Functions).- 7.9 Functions Having Logarithmic Singularities Inside the Circle.- 7.10 The Poisson Formula.- 7.11 Functions with Prescribed Values on the Type-Changing Curve.- 7.12 Functions Vanishing on a Characteristic Line.- 8 Formal Fourier Series and Generalized Functions.- 8.1 Formal Fourier Series.- 8.2 Duality.- 8.3 Significance of the Generalized Functions of Type H.- 8.4 Significance of the Generalized Functions of Type S.- 8.5 Annihilating Sets.- 8.6 Generalized Functions of Other Types.- 8.7 Continuation.- 8.8 Limits.- 8.9 Addenda.- Appendix: Summability.