Paths in Complex Analysis
Published on 8. January 2020
140 pages
978-1-5361-7058-0 (ISBN)
Description
Several scientists learn only a first course in complex analysis, and hence they are not familiar with several important properties: every polygenic function defines a congruence of clocks; the basic properties of algebraic functions and abelian integrals; how mankind arrived at a rigorous definition of Riemann surfaces; the concepts of dianalytic structures and Klein surfaces; the Weierstrass elliptic functions; the automorphic functions discovered by Poincare' and their links with the theory of Fuchsian groups; the geometric structure of fractional linear transformations; Kleinian groups; the Heisenberg group and geometry of the complex ball; complex powers of elliptic operators and the theory of spectral zeta-functions; an assessment of the Poincare' and Dieudonne' definitions of the concept of asymptotic expansion. The book is unique both for the selection of topics and for the readable access that it offers to the otherwise too large landscape of modern complex analysis.
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Giampiero Esposito
Paths in Complex Analysis
Book
01/2020
Nova Science Publishers Inc
€110.36
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Content
- Intro
- PATHS IN COMPLEX ANALYSIS
- PATHS IN COMPLEX ANALYSIS
- Contents
- Preface
- Chapter 1From Holomorphic to PolygenicFunctions
- Abstract
- 1.1. Holomorphic Functions
- 1.2. Local Theory of Pseudo-Holomorphic Functions
- 1.3. Global Theory of Pseudo-Holomorphic Functions
- 1.4. Polygenic Functions
- 1.5. Kasner Circle and Its Congruence of Clocks
- Chapter 2Algebraic Functions
- Abstract
- 2.1. Algebraic Functions and Their Critical Points
- 2.2. Branch Points
- Chapter 3From Analytic Spaces toRiemann Surfaces
- Abstract
- 3.1. Analytic Spaces
- 3.2. Riemann Surfaces
- Chapter 4Abelian Integrals
- Abstract
- 4.1. The Concept of Abelian Integral
- Chapter 5Dianalytic Structures and KleinSurfaces
- Abstract
- 5.1. Dianalytic Structures
- 5.2. Klein Surfaces
- Chapter 6Elliptic Functions
- Abstract
- 6.1. From Simply Periodic to Doubly Periodic Functions
- 6.2. The FunctionÃ
- 6.3. Non-Linear Equation forÃand the g2,g3 Invariants
- 6.4. Roots ofÃ0(u)
- 6.5. The Weierstrass z-Function
- 6.6. The Entire Function s
- Chapter 7Automorphic Functions
- Abstract
- 7.1. Poincar´e's Theory of Automorphic Functions
- 7.2. q-Fuchsian Functions
- 7.3. Further Theorems on Fuchsian Functions
- 7.4. z-Fuchsian Functions
- 7.5. Fuchsian Functions and Non-Linear PartialDifferential Equations
- Chapter 8Fuchsian Groups
- Abstract
- 8.1. The Group PSL(2,R)
- 8.2. Discrete Subgroups of Isom(H )
- 8.3. Subgroups of PSL(2,R)
- Chapter 9Fractional LinearTransformations
- Abstract
- 9.1. Fractional Linear Transformations
- 9.2. Fractional Reflections
- 9.3. Fixed Points of Fractional Linear Transformations
- 9.4. A Closer Look at the Geometry of FractionalLinear Transformations
- Chapter 10Kleinian Groups
- Abstract
- 10.1. Discontinuous Groups
- 10.2. Kleinian Groups
- Chapter 11The Heisenberg Group
- Abstract
- 11.1. Heisenberg Group Hn and Geometryof the Complex Ball
- 11.2. Realization of Hn as a Group of AffineHolomorphic Bijections
- Chapter 12Spectral z-Functions
- Abstract
- 12.1. Complex Powers of an Elliptic Operator
- 12.2. Results of Seeley and Discussion
- 12.3. The Analytic Group of Operators As
- 12.4. Heat Kernel and Spectral z-Function
- 12.5. Asymptotic Expansions According to Poincar´e
- 12.6. Asymptotic Expansions According to Dieudonn´e
- Appendix: Landau and Hardy Notation
- Conclusion
- References
- About the Author
- Index
- Blank Page
