
Paths in Complex Analysis
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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
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