課程名稱 |
複分析導論 Introduction to Complex Analysis |
開課學期 |
110-1 |
授課對象 |
理學院 數學系 |
授課教師 |
李庭諭 |
課號 |
MATH5230 |
課程識別碼 |
221 U6560 |
班次 |
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學分 |
4.0 |
全/半年 |
半年 |
必/選修 |
必帶 |
上課時間 |
星期二6,7(13:20~15:10)星期四6,7(13:20~15:10) |
上課地點 |
新405新302 |
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 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1101MATH5230_ |
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 |
課程大綱
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課程概述 |
Complex function theory is a valuable tool used in many branches of pure, applied mathematics and natural sciences, including geometry, number theory, partial differential equations and various topics in physics and engineering. A basic course shall enable students to understand the concept of complex analyticity, to use residue calculus for evaluation of integrals and to learn some additional topics (depending on available time) selected from Riemann mapping theorem, special functions, prime number theorem, complex dynamical systems, etc. |
課程目標 |
Contents:
1. Analytic functions of a complex variable and power series.
2. Cauchy's integral theorem.
3. Maximum modulus principle and open mapping theorem.
4. Singularities of analytic functions and Laurent series.
5. Residue theorem and its applications: argument principle, Rouche's theorem and the evaluation of integrals.
6. Analytic continuation.
7. Conformal mapping (on basic domains) and Schwarz lemma.
8. Weierstrass infinite products.
9. Harmonic functions and the Dirichlet problem.
Selected topics: Riemann mapping theorem, Schwarz-Christoffel integral, complex dynamical systems, prime number theorem, elliptic functions, etc. |
課程要求 |
已修過微積分及分析導論 |
預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Bak J. and Newman, D. J., Complex analysis, Third edition. |
參考書目 |
1. Stein, E.M., Shakarchi, R., Complex analysis"
2. Ahlfors, L., Complex analysis" |
評量方式 (僅供參考) |
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週次 |
日期 |
單元主題 |
第1週 |
9/23 |
Complex numbers and Cauchy Riemann equations |
第2週 |
9/28,9/30 |
Power series and line integrals |
第5週 |
10/19,10/21 |
Maximum and minimum modulus principle |
第6週 |
10/26,10/28 |
Open mapping theorem, Schwarz lemma, Morera's Theorem and the reflection principle |
第7週 |
11/02,11/04 |
Simply connected domains and logarithmic functions |
第8週 |
11/09,11/11 |
Isolated singularities |
第9週 |
11/16,11/18 |
midterm exam |
第10週 |
11/23,11/25 |
The residue theorem |
第11週 |
11/30,12/02 |
Applications of the residue theorem I |
第12週 |
12/07,12/09 |
Conformal mappings |
第13週 |
12/14,12/16 |
Riemann mapping theorem |
第14週 |
12/21,12/23 |
analytic continuation of power series and Dirichlet series |
第15週 |
12/28,12/30 |
Gamma and Zeta function |
第16週 |
1/04,1/06 |
Prime number theorem |
第17週 |
1/11,1/13 |
final exam |
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