課程資訊
課程名稱
 複分析
Complex Analysis (Honor Program) 
開課學期
 104-1 
授課對象
 理學院  數學系  
授課教師
 蔡忠潤 
課號
 MATH5231 
課程識別碼
 221 U6570 
班次
  
學分
 4 
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304 
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/1041CPX 
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.

In this course, we shall cover the basic theory of analytic function, globally defined (over the whole complex plane) analytic functions, and the theory in the viewpoint of maps between domains of the complex plane.
 

課程目標
 Understand the concept of complex analyticity, to use residue calculus for evaluation of integrals, to understand conformal mappings, as well as some additional topics. 
課程要求
 calculus (with proof) 
預期每週課後學習時數
  
Office Hours
 每週二 15:30~16:20
每週一 13:20~14:10 
指定閱讀
 L. Ahlfors, Complex analysis  
參考書目
 1. Stein and Shakarchi, Complex analysis

2. Nevanlinna and Paatero, Introduction to complex analysis

3. Conway, Functions of one complex variable 
評量方式
(僅供參考)
    
課程進度
週次
日期
單元主題
第1週
 9/15,9/17  basics of analytic functions: Cauchy--Riemann equation, rational function, power series. Reference: [A, §1 and §2 of ch.2] 
第2週
 9/22,9/24  complex integration: Cauchy--Goursat theorem, Cauchy integral formula. Reference: [A, §1 and §2 of ch.4] 
第3週
 10/01  local structure: Taylor's theorem, zeros and poles. (Due to Typhoon Dujuan, there is no class on Tuesday, and we will have two hours lecture on Thursday.) Reference: [A, §2.3, §3.1 and §3.2 of ch.4] 
第4週
 10/06,10/08  local structure: essential singularity, Taylor and Laurent series, open mapping theorem, maximum principle. Reference: [A, §1 of ch.5 and §3.3, 3.4 of ch.4] 
第5週
 10/13,10/15  residue and argument principle. Reference: [A, §4, 5 of ch.4] 
第6週
 10/20,10/22  sums, products and Gamma function. Reference: [A, §2 of ch.5] 
第7週
 10/27,10/29  Gamma function (continued), entire function: Jensen's formula. Reference: [A, §2.5 and §3 of ch.5] 
第8週
 11/03,11/05  entire function: Hadamard factorization theorem. Reference: [A, §3.2 of ch.5];
Riemann zeta function. Reference: [A, §4 of ch.5]. 
第9週
 11/10,11/12  Prime number theorem Reference: [Lang, Complex analysis. 4th edition, ch.XVI] 
第10週
 11/17,11/19  normal family. Reference: [A, §5 of ch.5] 
第11週
 11/24,11/26  fall break 
第12週
 12/01,12/03  Riemann mapping theorem, boundary behavior of conformal mapping. Reference: [A, §1 of ch.6] 
第13週
 12/08,12/10  conformal mapping to polygons, more on harmonic functions. Reference: [A, §2 and §3 of ch.6] 
第14週
 12/15,12/17  Dirichlet problem, subharmonic function and Perron's method, conformal mapping of non-simply-connected regions. Reference: [A, §4 and §5 of ch.6] 
第15週
 12/22,12/24  conformal mapping of multiply-connected regions, application of normal family. Reference: [A, §5 of ch.6], [G, ch. XII] 
第16週
 12/29,12/31  Picard's theorems, introduction to complex dynamics. Reference: [G, ch. XII] 
第17週
 1/05,1/07  basic properties of Julia set and Mandelbrot set, introducing Riemann surfaces. Reference: [G, ch. XII] 
第18週
 1/14  Final Exam