課程名稱 |
複分析 Complex Analysis (Honor Program) |
開課學期 |
106-1 |
授課對象 |
理學院 數學系 |
授課教師 |
莊武諺 |
課號 |
MATH5231 |
課程識別碼 |
221 U6570 |
班次 |
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學分 |
4.0 |
全/半年 |
半年 |
必/選修 |
選修 |
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 |
天數305天數305 |
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:40人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1061complexanalysis |
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, ordinary differential equations, partial differential equations and various topics in physics and engineering.
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課程目標 |
The course will equip the students the concept of complex analyticity, the residue calculus for evaluations of integrals, conformal mappings, and some additional topics such as Riemann mapping theorem, special/elliptic functions, prime number theorem, complex dynamical systems and etc. |
課程要求 |
Undergraduate calculus and analysis. |
預期每週課後學習時數 |
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Office Hours |
每週三 13:20~14:20 每週二 15:30~16:30 備註: 星期二為授課教師的office hour,地點天數403。星期三為助教的office
hour,地點天數546。 |
指定閱讀 |
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參考書目 |
1. Ahlfors, Complex analysis
2. Stein and Shakarchi, Complex analysis
3. Gamelin, Complex analysis |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
Homework |
30% |
約有12-14份作業。捨棄最低兩次做計算。作業遲交一天打八折(兩天六四折,以此類推) 。 |
2. |
Midterm |
30% |
Nov 7 2017 |
3. |
Final |
40% |
Jan 9 2018 |
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週次 |
日期 |
單元主題 |
第1週 |
9/12,9/14 |
9/12: analytic functions, Cauchy-Riemann equation, rational functions. [Ahlfors, chap.2 sec.1]
9/14: power series, Abel's theorem. [Ahlfors, chap.2 sec.2] |
第2週 |
9/19,9/21 |
9/19: Cauchy theorem for a rectangle, Cauchy integral formula. [Ahlfors, chap.4 sec.1, sec.2]
9/21: higher derivatives. [Ahlfors, chap.4 sec.2.3] |
第3週 |
9/26,9/28 |
9/26: Taylor's theorem, zeros and poles. [Ahlfors, chap4, sec.3.1 sec.3.2]
9/28: essential singularity. |
第4週 |
10/03,10/05 |
10/03: Taylor and Laurent series, open mapping theorem, maximum principle. [Ahlfors, chap.5 sec.1 and chap.4 sec.3.3 sec.3.4]
10/05: Schwarz lemma. [Ahlfors, chap.4 sec.3.4] |
第5週 |
10/10,10/12 |
10/10: no class.
10/12: general form of Cauchy theorem, residue theorem. [Ahlfors, chap4, sec.4 sec.5] |
第6週 |
10/17,10/19 |
10/17: argument principle, examples of definite integrals. [Ahlfors, chap4, sec.5.2 sec.5.3]
10/19: Mittag-Leffler theorem. [Ahlfors, chap5, sec.2] |
第7週 |
10/24,10/26 |
10/24: infinite products, Weierstrass product theorem, Gamma function. [Ahlfors, chap5, sec.2]
10/26: Stirling formula. [Ahlfors, chap5, sec.2.5] |
第8週 |
10/31,11/02 |
10/31: Stirling formula, Jensen's formula, Hadamard theorem. [Ahlfors, chap5, sec.2 sec.3]
11/02: Hadamard theorem, Riemann zeta function. [Ahlfors, chap5, sec.3 sec.4] |
第9週 |
11/07,11/09 |
11/07: midterm. 1pm-3:30pm Astro-Math 305.
11/09: Riemann zeta function, prime number theorem [Ahlfors, chap5, sec.4] [Lang, chap16] |
第10週 |
11/14,11/16 |
自主學習週 |
第11週 |
11/21,11/23 |
11/21: prime number theorem. [Lang, chap16]
11/23: normal families. [Ahlfors, chap5, sec.5] |
第12週 |
11/28,11/30 |
11/28: normal families. [Ahlfors, chap5, sec.5]
11/30: Riemann mapping theorem. [Ahlfors, chap6, sec.1] |
第13週 |
12/05,12/07 |
12/5: boundary behavior of conformal maps, conformal maps of polygons. [Ahlfors, chap6, sec.1 sec.2]
12/7: conformal maps of polygons. [Ahlfors, chap6, sec.2] |
第14週 |
12/12,12/14 |
12/12: conformal maps of rectangles, harmonic functions. [Ahlfors, chap6, sec.2 sec.3]
12/14: harmonic functions, subharmonic functions. [Ahlfors, chap6, sec.3 sec.4] |
第15週 |
12/19,12/21 |
12/14: Dirichlet problems, Perron's method. [Ahlfors, chap6, sec.4]
12/16: conformal mapping of multiply-connected regions. [Ahlfors, chap6, sec.5] |
第16週 |
12/26,12/28 |
12/26: Picard's theorems. [Gamelin, chap12]
12/28: Picard's theorems, elliptic functions. [Ahlfors, chap7] |
第17週 |
1/02,1/04 |
1/02: Elliptic functions. [Ahlfors, chap7]
1/04: Elliptic functions. [Ahlfors, chap7] |
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