Elliptic integrals and elliptic functions II
Lecturer

Takashi Takebe (Professor)
Date

2024-03-01 ~ 2024-05-31
Schedule
Weekday    Time    Venue    Online    ID    Password
Mon,Fri    09:50-11:25    A3-2-201    ZOOM    388 528 9728    BIMSA

Prerequisite

Undergraduate calculus, complex analysis, acquaintance with contents of the lectures of the previous semester
Introduction

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.
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)
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)
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.
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