Elliptic integrals and elliptic functions 1
This is an introductory course on elliptic integrals and elliptic functions for advanced undergraduate students and graduate students who are not familiar with the subject.
An elliptic function is defined as a doubly periodic meromorphic function on the complex plane. The study of elliptic integrals started by Fagnano, Legendre, Gauss and others in the eighteenth century was turned into the theory of elliptic functions by Abel and Jacobi. Then Riemann, Weierstrass and Liouville developed the theory further by using complex analysis.
The theory of elliptic functions thus founded is a prototype of today's algebraic geometry. On the other hand, elliptic functions and elliptic integrals appear in various problems in mathematics as well as in physics. Examples: arc length of an ellipse, arithmetic-geometric mean, solutions of physical systems (pendulum, top, skipping rope, the KdV equation), solution of quintic equations, etc.
In this course we shall put emphasis on analytic aspects and applications.
An elliptic function is defined as a doubly periodic meromorphic function on the complex plane. The study of elliptic integrals started by Fagnano, Legendre, Gauss and others in the eighteenth century was turned into the theory of elliptic functions by Abel and Jacobi. Then Riemann, Weierstrass and Liouville developed the theory further by using complex analysis.
The theory of elliptic functions thus founded is a prototype of today's algebraic geometry. On the other hand, elliptic functions and elliptic integrals appear in various problems in mathematics as well as in physics. Examples: arc length of an ellipse, arithmetic-geometric mean, solutions of physical systems (pendulum, top, skipping rope, the KdV equation), solution of quintic equations, etc.
In this course we shall put emphasis on analytic aspects and applications.
讲师
日期
2023年09月25日 至 12月25日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一,周五 | 09:50 - 11:25 | A3-1a-204 | ZOOM 05 | 293 812 9202 | BIMSA |
修课要求
Undergraduate calculus, complex analysis.
课程大纲
Details might change depending on the wishes of the audience.
1. Introduction
2. Arc length of an ellipse
3. Arc length of a lemniscate
4. Classification of elliptic integrals
5. Arithmetic-geometric mean
6. Simple pendulum
7. Jacobi's elliptic functions (definitions)
8. Jacobi's elliptic functions (properties)
9. Simple pendulum revisited
10. Shape of a skipping rope
11. Riemann surfaces
12. Analysis on Riemann surfaces
13. Elliptic curves
14. Complex elliptic integrals
15. Conformal mapping from the upper half plane to a rectangle
16. Abel-Jacobi theorem (statement and preparation of the proof)
17. Surjectivity of the Jacobi map
18. Injectivity of the Jacobi map
19. Elliptic functions on the complex plane (definition and examples)
20. Elliptic functions on the complex plane (properties)
1. Introduction
2. Arc length of an ellipse
3. Arc length of a lemniscate
4. Classification of elliptic integrals
5. Arithmetic-geometric mean
6. Simple pendulum
7. Jacobi's elliptic functions (definitions)
8. Jacobi's elliptic functions (properties)
9. Simple pendulum revisited
10. Shape of a skipping rope
11. Riemann surfaces
12. Analysis on Riemann surfaces
13. Elliptic curves
14. Complex elliptic integrals
15. Conformal mapping from the upper half plane to a rectangle
16. Abel-Jacobi theorem (statement and preparation of the proof)
17. Surjectivity of the Jacobi map
18. Injectivity of the Jacobi map
19. Elliptic functions on the complex plane (definition and examples)
20. Elliptic functions on the complex plane (properties)
参考资料
[1] T. Takebe, Elliptic integrals and elliptic functions (2023)
[2] V. Prasolov, Y. Solovyev, Elliptic functions and elliptic integrals (1997)
[3] E. T. Whittaker, G. N. Watson, A course of modern analysis (1902)
[4] D. Mumford, Tata lectures on Theta I (1983)
[2] V. Prasolov, Y. Solovyev, Elliptic functions and elliptic integrals (1997)
[3] E. T. Whittaker, G. N. Watson, A course of modern analysis (1902)
[4] D. Mumford, Tata lectures on Theta I (1983)
听众
Advanced Undergraduate
, Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Takashi Takebe 从事数学物理可积系统方向的研究。 2023年8月前，他在俄罗斯莫斯科国立研究大学高等经济学院数学系担任教授，并于2023年9月加入北京雁栖湖应用数学研究院任研究员一职。