Author | Hua, Loo-keng. author |
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Title | Starting with the Unit Circle [electronic resource] : Background to Higher Analysis / by Loo-keng Hua |
Imprint | New York, NY : Springer New York, 1981 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8136-5 |
Descript | XII, 180 p. online resource |
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 Geometryโwith 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 Lavโrentievโs 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