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課程名稱 |
複分析導論
Introduction to Complex Analysis |
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開課學期 |
107-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
張志中 |
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課號 |
MATH5230 |
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課程識別碼 |
221 U6560 |
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班次 |
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學分 |
4.0 |
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全/半年 |
半年 |
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必/選修 |
必帶 |
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上課時間 |
星期二6,7(13:20~15:10)星期四6,7(13:20~15:10) |
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上課地點 |
新405新302 |
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備註 |
此課程研究生選修不算學分。
限學士班學生
總人數上限：75人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1071MATH5230_CpxAna |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
Contents
• analytic functions of a complex variable and power series,
• Cauchy's integral theorem,
• maximum modulus principle and open mapping theorem,
• singularities of analytic functions and Laurent series,
• residue theorem and its applications: argument principle, Rouche's theorem and the evaluation of integrals,
• analytic continuation,
• conformal mapping (on basic domains) and Schwarz lemma,
• Weierstrass infinite products,
• harmonic functions and the Dirichlet problem. |
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課程目標 |
The goal of this introductory course is to enable students to understand the concept of complex analyticity, to use residue calculus for evaluation of integrals, and to learn some additional topics. |
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課程要求 |
Completion of introduction to mathematical analysis 1 and 2. |
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預期每週課後學習時數 |
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Office Hours |
備註： to be determined |
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指定閱讀 |
to be assigned |
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參考書目 |
1. Stein, E.M., Shakarchi, R., “Complex analysis”(textbook)
2. Ahlfors, L., “Complex analysis”
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評量方式
(僅供參考) |
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週次 |
日期 |
單元主題 |
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第1週 |
9/11,9/13 |
Chapter 1 except prop. 2.3 and theorem 2.4 |
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第2週 |
9/18,9/20 |
9/18: Proposition 2.3 and theorem 2.4
9/20: 2.1-2.3 (Cauchy's theorem) |
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第3週 |
9/25,9/27 |
2.4 Cauchy's integral formulas, Cauchy inequalities, and part of 3.2 the residue formula |
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第4週 |
10/02, 10/04 |
10/2: Analyticity of holo. functions, and Liouville's theorem
10/4: the rest of 2.4 and 3.1 zeroa and poles |
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第5週 |
10/09,10/11 |
10/09: 3.2 The residue formula
10/11: More examples on the application of residue formula |
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第6週 |
10/16,10/18 |
10/16: 13:20- 15:10 quiz 1 (up to homework 4)
10/18: 5.1, 5.2, and 5.3 of chapter 2 |
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第7週 |
10/23,10/25 |
10/23: symmetry principle (5.4 of chapter 2)
10/25: Schwarz reflection principle, analytic continuation, and a few remarks |
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第8週 |
10/30,11/01 |
10/30: Analyticity of gamma function and Runge's approximation theorem 演習課發 quiz 1 考卷
11/01: Proof of Runge's approximation theorem
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第9週 |
11/06,11/08 |
11/06: Homework 6 延至今日繳交。Mittag-Leffler's theorem (p. 156, ex. 5.16)
11/08: Theorem 3.1, corollary 3.2, and Laurent series expansions of chapter 3 |
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第10週 |
11/13,11/15 |
11/13: 僅 14:20 - 15:10 上課。3.3 Laurent series, characterization of isolated singularities, and Casorati-Weierstrass theorem.
11/15 校慶，自主學習
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第11週 |
11/20, 11/22 |
11/20: 僅 13:20 - 14:10 上演習課。
11/22: Quiz 2 |
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第12週 |
11/27,11/29 |
11/27: 3.6 the complex logarithm
11/29: 3.4 the winding number and the first version of the argument principle |
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第13週 |
12/04,12/06 |
12/4: the argument principle and Rouche's theorem
12/6: the open mapping theorem and the max modulus principle
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第14週 |
12/11,12/13 |
12/11: Riemann surfaces and more applications of the argument principle
12/13: 8.2 the Schwarz lemma and 5.1 Jensen's formula
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第15週 |
12/18,12/20 |
12/18: 5.3 Infinite products, Lemma 1.2 and 5.2 Functions of finite order
12/20: 5.2 Functions of finite order, and ex. 3.15, 5.13, 5.14, and 5.12. |
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第16週 |
12/25,12/27 |
12/25: 5.4 Canonical factors and infinite products
12/27: 5.5 (determination of the order of growth of canonical products and Hadamard's factorization theorem) |
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第17週 |
1/01,1/03 |
1/3: Proof of Hadamard's factorization theorem. (Hand in your term paper today)
1/8: Quiz 3 |
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