Abel's Theorem, Allied Theory and Theta Functions by Henry Baker
Abel's Theorem, Allied Theory and Theta Functions - Table of Contents
1. The Subject of Investigation
2. The Fundamental Functions on a Riemann Surface
3. The Infinities of Rational Functions
4. Specification of a General Form of Riemann’s Integrals
5. Certain Forms of the Fundamental Equation of the Riemann Surface
6. Geometrical Investigations
7. Coordination of Simple Elements
8. Abel’s Theorem
9. Jacobi’s Inversion Problem
10. Riemann’s Theta Functions
11. The Hyperelliptic Case of Riemann’s Theta Functions
12. A Particular Form of Riemann Surface
13. Radical Functions
14. Factorial Functions
15. Relations Concerning Products of Theta Functions
16. A Direct Method of Obtaining the Equations Relating Theta Functions
17. Theta Relations with Certain Groups of Characteristics
18. Transformation of Periods
19. On Systems of Periods and on General Jacobian Functions
20. Transformation of Theta Functions
21. Complex Multiplication of Theta Functions
22. Degenerate Abelian Integrals
Appendix 1: On Algebraic Curves in Space
Appendix 2: On Matrices
What You Will Learn in Abel's Theorem, Allied Theory and Theta Functions
"Abel's Theorem and the Allied Theory, Including the Theory of the Theta Functions" by "Henry F. Baker" is a classic text that explores the deep connection between algebraic curves and complex analysis. The book focuses on "Abelian functions", providing a systematic introduction to their properties and applications. At its core is "Abel’s theorem", which relates sums of integrals on algebraic curves to algebraic relations, forming the foundation for solving the inversion problem — expressing coordinates of curves in terms of analytic functions.
Baker also develops the theory of "theta functions", which serve as essential tools for analyzing multi-periodic behavior in complex variables. He explains how these functions are connected to "Riemann surfaces", period matrices, and the Jacobian of curves. The text provides detailed treatments of hyperelliptic cases, transformations of periods, and functional relations, demonstrating how analytic methods reveal the geometric structure of algebraic curves.
This text remains influential for its rigorous and classical approach to complex analysis and algebraic geometry. It bridges analytic and geometric perspectives, providing tools for studying multi-variable functions and their integrals. Mathematicians and researchers studying "integrable systems", Abelian functions, or the historical foundations of modern algebraic geometry still rely on Baker’s insights for foundational understanding and advanced applications.
Book Details & Specifications
Title:
Abel's Theorem, Allied Theory and Theta Functions by Henry Baker
Publisher:
University press
Year:
1897
Pages:
714
Type:
PDF
Language:
English
ISBN-10 #:
1023835150
ISBN-13 #:
978-1023835152
License:
Public Domain Work
Amazon:
Amazon
About the Author: Henry Frederick Baker
The author Henry Frederick Baker
(1866–1956) was a renowned British mathematician specializing in "algebraic geometry", complex analysis, and "Abelian functions". A Cambridge scholar and Fellow of the Royal Society, he contributed significantly to bridging classical 19th-century mathematics with modern approaches, emphasizing rigorous and systematic methods. Baker’s work, including "Abel’s Theorem, Allied Theory and Theta Functions", explored the connections between "Riemann surfaces", algebraic curves, and multi-variable analytic functions. His studies on theta functions, period matrices, and integral inversion problems remain foundational, influencing modern research in complex analysis, integrable systems, and the geometric understanding of algebraic curves.
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