|
課程名稱 |
古典橢圓函數論二
Elliptic Functions A Classical Approach (Ⅱ) |
|
開課學期 |
100-2 |
|
授課對象 |
理學院 數學系 |
|
授課教師 |
蔡宜洵 |
|
課號 |
MATH3308 |
|
課程識別碼 |
201 39540 |
|
班次 |
|
|
學分 |
3 |
|
全/半年 |
半年 |
|
必/選修 |
選修 |
|
上課時間 |
星期五7,8,9(14:20~17:20) |
|
上課地點 |
天數204 |
|
備註 |
先備知識:以修過本課程(一)者為基礎；未修過(一)者可先與老師聯繫。
總人數上限：40人 |
|
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1002EFACA |
|
課程簡介影片 |
|
|
核心能力關聯 |
核心能力與課程規劃關聯圖 |
|
課程大綱
|
|
為確保您我的權利,請尊重智慧財產權及不得非法影印
|
|
課程概述 |
課程 "古典橢圓函數論", 第二學期延續上學期的基礎; 其中
theta 函數將是重點之一. 由於此函數的發展極依賴先前的工作,
因此欲修此課者, 請先將上學期 Jacobi 橢圓函數的
基本定義, 主要性質以及微分方程 ( in Gauss' approach)
做好復習 (例如 期末考試的所有問題), 以便於這學期的銜接.
關於本課程的整體構想及參考書, 請參閱上學期已給的大綱.
以下僅就這學期課程的內容與大致進度, 條列如下 (每條目大約相應於一個星期的份量).
Chapter 1
1. Introduction to theta functions (I)
2. Introduction to theta functions (II)
3. Introduction to theta functions (III)
Chapter 2
4. Jacobi's imaginary transformation and modular transformations (I)
5. Jacobi's imaginary transformation and modular transformations (II)
Chapter 3
6. Complex modulus and group property (I)
7. Complex modulus and group property (II)
Chapter 4
8. theta functions and zeta functions
9. Jacobi's triple identity and Euler's formula
Chapter 5
10. Elliptic intergrals of the second kind (EISK)
11. theta functions in terms of EISK
12. Addition formula for EISK, Euler's discovery and Legendre's relation
Chapter 6
Topics selected from: 1) transformation theory 2) Abel's theorem (a
constructive,
original approach due to Abel's own work) 3) Jacobi's inversion problem (a
classical
approach to the problem with historical remarks). It does not
seem possible to cover
all the topics; it depends on the available time for us to proceed any
futher. |
|
課程目標 |
123 |
|
課程要求 |
13 |
|
預期每週課後學習時數 |
|
|
Office Hours |
|
|
指定閱讀 |
|
|
參考書目 |
[AE] Armigate, J.V., Eberlein, W.F., "Elliptic functions": hypothetical work
of Abel on theta fcns
[Cay] Cayley, A., "Elliptic functions": for Jacobi's work
[Mar] Markushevich, A.I., "The remarkable sine": for Lemniscatic sine
[Cox] Cox, D.A., "The arithmetic-Geometric mean of Gauss", L'Enseig. Math.
(1984) 275-330: a paper that discusses work of Gauss in connection with
elliptic functions
[G] Collected work of Gauss, especially v. 10, v. 8
[A] Collected work of Abel
[RL] Rauch, H.E., Lebowitz, A., "Elliptic functions, theta functions and
Riemann surfaces"
[Cox2] Cox, D.A., "Galois theory" Chap. 15 on Lemniscatic integrals
[Bor] Borwein, J.M., Borwein, P.B., "Pi and AGM": some topics treated
classically
[Mar2] Markushevich, A.I., "Introduction to classical theory of abelian
functions"
[Euler] Euler, L., "Introduction to analysis of the infinite": source for some
work of Euler
[LP] Laudal, O.A., Piene, R., "The legacy of N.H. Abel" see p.21-179 for
Abel's work
[KY] Kolmogorov, A.N., Yushkevich, A.P., "Mathematics of the 19th century":
historical
[Bell] Bellman, R., "A brief introduction to theta function"
[Dieu] Dieudonne, J., "Abrege d'histoire des mathematiques" 1700-1900, v. 2,
Chap. 7
[Bliss] Bliss, G.A., "Algebraic functions" Chap. 6 on Abel's theorem
[S] Siegel, C.L., "Topics in complex function theory" v. 1 on Jacobi's
inversion problem
[V] Valiron, G., "The geometric theory of ordinary differential equations and
algebraic functions" Chap. 2 on Abelian integrals and Abel's theorem
|
|
評量方式
(僅供參考) |
|
|
週次 |
日期 |
單元主題 |
|
第1週 |
2/24 |
Review the Abel's elliptic function & Jacobi's elliptic function with some important results & Introduction to theta functions in connection with Jacobi's elliptic function. |
|