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.
Lecturer
Date
1st March ~ 27th May, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Friday | 09:50 - 11:25 | A3-2-201 | ZOOM A | 388 528 9728 | BIMSA |
Prerequisite
Undergraduate calculus, complex analysis, acquaintance with contents of the lectures of the previous semester
Syllabus
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)
Reference
[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)
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Takashi Takebe is a researcher of mathematical physics, in particular integrable systems. He worked as a professor at the faculty of mathematics of National Research University Higher School of Economics in Moscow, Russia, till August 2023 and joined BIMSA as a professor in September 2023.