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 Geometrywith 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-DimensionalPhase 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. Language: English