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.
Lecturer
Date
25th September ~ 25th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Friday | 09:50 - 11:25 | A3-1a-204 | ZOOM 05 | 293 812 9202 | BIMSA |
Prerequisite
Undergraduate calculus, complex analysis.
Syllabus
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)
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)
Audience
Advanced Undergraduate
, Graduate
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.