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 joined BIMSA as a professor in September 2023. Before coming to BIMSA he worked as a professor at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow, Russia, from April 2009 till August 2023, as an associate professor at the Department of Mathematics, the Faculty of Science, of Ochanomizu University in Tokyo, Japan, from April 1999 till March 2009, and as an assistant professor at the Graduate School of Mathematical Sciences of the University of Tokyo, Japan, from October 1991 till March 1999.
He also performed his research at the University of California, Berkeley, USA, from September 1995 till August 1997, supported by the Overseas Research Fellowships of the Japan Society for the Promotion of Science (JSPS).
He graduated the Department of Mathematics, the Faculty of Science, of the University of Tokyo, Japan in March 1987 and was a graduate student of the Graduate School of Sciences from April 1987 till September 1991. He got the degree of Ph.D. (Mathematical Sciences) in 1995.
He also studied in Leningrad, the USSR, from October 1990 till September 1991 as an exchange student between the USSR and Japan.
He also performed his research at the University of California, Berkeley, USA, from September 1995 till August 1997, supported by the Overseas Research Fellowships of the Japan Society for the Promotion of Science (JSPS).
He graduated the Department of Mathematics, the Faculty of Science, of the University of Tokyo, Japan in March 1987 and was a graduate student of the Graduate School of Sciences from April 1987 till September 1991. He got the degree of Ph.D. (Mathematical Sciences) in 1995.
He also studied in Leningrad, the USSR, from October 1990 till September 1991 as an exchange student between the USSR and Japan.