課程資訊
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
  
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
 
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.
 

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
  
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
  
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 
 
課程進度
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]