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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 |
109-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
蔡宜洵 |
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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/1091MATH5230_ |
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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 |
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指定閱讀 |
待補 |
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參考書目 |
1. E.M. Stein and R. Shakarchi: “Complex analysis”(textbook)
Princeton Univ Press
(科大 文化事業 2697-1353)
2. L. Ahlfors: “Complex analysis”
3. R. B. Ash and W.P. Novinger: Complex variables
PDF files can be downloaded at
https://faculty.math.illinois.edu/~r-ash/ |
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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 |
9/15,9/17 |
[以下綱要為暫定，將隨時根據課程進行調整，未必完全一項綱目一週]
Analytic functions, Cauchy-Riemann equations |
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第2週 |
9/22,9/24 |
Elementary transcendental functions, Conformal property, linear fractional maps on discs |
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第3週 |
9/29,10/01 |
Cauchy’s integral formulas (C.I.F.) (omitting “the general form of Cauchy’s theorem” in Section 4, Chap. 4 of Ahlfors) |
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第4週 |
10/06,10/08 |
Applications of C.I.F.: derivatives estimates, Liouville’s theorem, Morera’s theorem, fundamental theorem of algebra, Line integrals for complex valued functions
（第一次小考） |
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第5週 |
10/13,10/15 |
Cauchy’s theorem for a rectangle, disc |
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第6週 |
10/20,10/22 |
Residues, Laurent series, Taylor expansions, singularities, |
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第7週 |
10/27,10/29 |
（第二次小考） |
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第8週 |
11/03,11/05 |
Evaluation of definite integrals |
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第9週 |
11/10,11/12 |
open mappings
期中考 |
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第10週 |
11/17,11/19 |
Maximum modulus principle |
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第11週 |
11/24,11/26 |
argument principle, Rouche’s theorem |
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第12週 |
12/01,12/03 |
conjugate harmonic differentials, mean-value property for harmonic functions
（第三次小考） |
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第13週 |
12/08,12/10 |
Poisson’s formula, Dirichlet’s problem on a disc |
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第14週 |
12/15,12/17 |
Reflection principle, remarks on applications to elliptic functions
（第四次小考） |
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第15週 |
12/22,12/24 |
Euler’s discovery, partial fractions |
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第16週 |
12/29,12/31 |
Weierstrass Infinite products |
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第17週 |
1/05,1/07 |
Conformal mappings on basic domains
（第五次小考） |
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第18週 |
1/12,1/14 |
Selected topics (Hadamard’s factorization theorem, or Prime number theorem)
期末考 |
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