課程大綱

課程資訊

課程名稱

複分析導論
Introduction to Complex Analysis

開課學期

105-1

授課對象

理學院數學系

授課教師

蔡宜洵

課號

MATH5230

課程識別碼

221 U6560

班次

學分

全/半年

半年

必/選修

必修

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

新302新302

備註

複變函數論得用221 U6560複分析導論(4學分)替代。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/1051MATH5230_

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, 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

待補

預期每週課後學習時數

Office Hours

指定閱讀

Textbook: Ahlfors, L., Complex analysis

參考書目

1. Stein, E.M., Shakarchi, R., Complex analysis"
2. Lang, S., Complex analysis", GTM 103
3. Ahlfors, L., Complex analysis"

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	Quiz	30%
2.	Midterm	35%
3.	Final	35%

課程進度

週次	日期	單元主題
第1週	9/13,9/15	9/13 Analytic functions, Cauchy-Riemann equations, Elementary transendental functions 9/15 中秋節
第2週	9/20,9/22	9/20 Conformal property, Linear fractional maps on discs, Line integrals for complex valued functions 9/22 Cauchy's theorem for a rectangle
第3週	9/27,9/29	9/27 No class 9/29 Cauchy's theorem in a disc, Cauchy's integral formulas (9/29 Quiz 1)
第4週	10/04,10/06	10/4 Applications of Cauchy's integral formula: derivatives estimates, Liouville’s theorem, Morera’s theorem, fundamental theorem of algebra; Residues 10/6 Evaluation of definite integrals, Laurent series
第5週	10/11,10/13	10/11 Laurent series, argument principle, Rouche’s theorem, Taylor expansions 10/13 Open mapping theorem, removable singularities, poles (10/13 Quiz 2)
第6週	10/18,10/20	10/18 Maximum modulus principle, Schwarz lemma, Euler’s discovery 10/20 Essential singularities, Weierstrass's Theorem
第7週	10/25,10/27	10/25 Hurwitz theorem, Taylor series, proof of an identity about π^2/sin^2(πz) 10/27 Mittag-Leffler theorem (10/27 Quiz 3)
第8週	11/01,11/03	11/01 Addition theorem and inversion of elliptic integrals, Weierstrass ℘-function 11/03 Differential equation and Laurent series of ℘-function
第9週	11/08,11/10	11/08 Weierstrass Infinite products 11/10 Midterm exam
第10週	11/15,11/17	自主學習週 11/15 校慶
第11週	11/22,11/24	11/22 Conjugate harmonic differentials, mean-value property for harmonic functions 11/24 Poisson's formula
第12週	11/29,12/01	11/29 Poisson's formula, Schwarz's theorem, Dirichlet's problem on a disc, Reflection principle 12/01 Reflection principle, remarks on applications to elliptic functions
第13週	12/06,12/08	12/06 linear fractional transformations, cross ratio 12/08 cross ratio, symmetry (12/8 Quiz 4)
第14週	12/13,12/15	12/13 Applications of symmetry principle, elementary conformal mappings 12/15 Elementary conformal mappings
第15週	12/20,12/22	12/20 The zeta function, Euler product formula, prime number theorem 12/22 integral formula for ψ_1
第16週	12/27,12/29	12/27 zeros of the zeta function 12/29 Estimates for ζ, ζ', 1/ζ (12/29 Quiz 5)
第17週	1/03,1/05	1/03 proof of the asymptotics for ψ_1, functional equation 1/05 heat kernel
第18週	1/12	1/12 Final exam