Elliptic integrals and elliptic functions II
This is a continuation of the course "Elliptic integrals and elliptic functions I" in the previous semester. We further develop the theory of elliptic integrals and elliptic functions and pursue their various properties and applications.
Examples: Weierstrass's elliptic function, theta functions, addition theorems of elliptic functions, infinite product expansion, Jacobi's elliptic functions over the complex field.
Examples: Weierstrass's elliptic function, theta functions, addition theorems of elliptic functions, infinite product expansion, Jacobi's elliptic functions over the complex field.
讲师
日期
2024年03月01日 至 05月27日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一,周五 | 09:50 - 11:25 | A3-2-201 | ZOOM A | 388 528 9728 | BIMSA |
修课要求
Undergraduate calculus, complex analysis, acquaintance with contents of the lectures of the previous semester
课程大纲
Details might change depending on the wishes of the audience.
1. Review of the lectures in the previous semester
2. Weierstrass's functions
3. Addition theorem of Weierstrass's elliptic function
4. Addition theorems of general elliptic functions
5. Characterisation of elliptic functions by addition theorems
6. Theta functions (definition)
7. Theta functions (properties; quasi-periodicity, Jacobi's relations, modular properties)
8. Theta functions (infinite product expansion)
9. Jacobi's elliptic functions over the complex field (definition)
10. Jacobi's elliptic functions over the complex field (properties)
11. Various topics (examples: arithmetic-geometric mean through theta functions, solution of a quintic equation)
1. Review of the lectures in the previous semester
2. Weierstrass's functions
3. Addition theorem of Weierstrass's elliptic function
4. Addition theorems of general elliptic functions
5. Characterisation of elliptic functions by addition theorems
6. Theta functions (definition)
7. Theta functions (properties; quasi-periodicity, Jacobi's relations, modular properties)
8. Theta functions (infinite product expansion)
9. Jacobi's elliptic functions over the complex field (definition)
10. Jacobi's elliptic functions over the complex field (properties)
11. Various topics (examples: arithmetic-geometric mean through theta functions, solution of a quintic equation)
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)
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笔记公开
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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。