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Starting with the Unit Circle : Background to Higher Analysis

Author: Loo-keng Hua
Publisher: New York, NY : Springer New York, 1981.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Printed edition:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Loo-keng Hua
ISBN: 9781461381365 1461381363 9781461381389 146138138X
OCLC Number: 840282825
Description: 1 online resource
Contents: 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.
Responsibility: by Loo-keng Hua.

Abstract:

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  Read more...

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