課程名稱 |
複變函數論 Functions of a Complex Variable |
開課學期 |
103-1 |
授課對象 |
理學院 數學系 |
授課教師 |
王金龍 |
課號 |
MATH3201 |
課程識別碼 |
201 31300 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
必修 |
上課時間 |
星期二5,6(12:20~14:10)星期四5,6(12:20~14:10) |
上課地點 |
新302新302 |
備註 |
總人數上限:75人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1031MATH3201_CA1 |
課程簡介影片 |
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課程概述 |
An introduction to the theory of analytic (holomorphic) functions of one complex variable. Studying domains (open connected subsets) in the (extended) complex plane, conformal transformations of planar domains. Line integrals as functions of arcs. Cauchy's theory. Calculus of Residues, Local properties of analytic functions. Power series. Harmonic Functions. Entire functions. Normal families of analytic functions. Euler's Gamma function. Riemann's zeta function and the Prime Number Theorem.
(Optional, most likely in the second semester) Elliptic functions, Picard's theorem, linear differential equations, Analytic continuations. |
課程目標 |
Complex-valued functions on domains inside the complex plane. Basic general theory of these analytic functions. Line integrals as a tool. From the fundamental theorem of calculus to Cauchy's integral theorem. Power series as tool. Elementary functions. Special analytic functions and maps. Analytic continuations. Riemann zeta functions and its applications. Riemann's mapping theorem. |
課程要求 |
Prerequisite : Calculus with proof, namely familiarity with point set topoogy, metric spaces, and regorous definitions of real numbers, limits and integrals. |
預期每週課後學習時數 |
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Office Hours |
每週五 12:20~13:10 每週四 16:30~17:20 備註: 星期四 16:30-17:20 為導生優先. |
指定閱讀 |
Stein: Complex Analysis |
參考書目 |
Ahlfors: Complex analysis
Whittaker and Watson: Modern Analysis |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
期中考 |
40% |
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2. |
期末考 |
40% |
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3. |
作業 |
20% |
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週次 |
日期 |
單元主題 |
第1週 |
9/16,9/18 |
Ch.1 Holomorphic functions. |
第2週 |
9/23,9/25 |
Ch.2 Cauchy's theorem and integral formula. |
第3週 |
9/30,10/02 |
Ch.2 Applications. |
第4週 |
10/07,10/09 |
Ch.3 Meromorphic functions and residue. |
第5週 |
10/14,10/16 |
Ch.3 The argument principle. |
第6週 |
10/21,10/23 |
Ch.4 Fourier transform. |
第7週 |
10/28,10/30 |
Ch.5 Growth of functions and infinite products. |
第8週 |
11/04,11/06 |
Ch.5 Factorizations of entire functions |
第9週 |
11/11,11/13 |
Review. 11/13 midterm exam. |
第10週 |
11/18,11/20 |
Self-study break |
第11週 |
11/25,11/27 |
Ch.6 Gamma and zeta. |
第12週 |
12/02,12/04 |
Ch.7 Prime number theorem. |
第13週 |
12/09,12/11 |
Ch.8 Conformal mappings and Schwaartz lemma. |
第14週 |
12/16,12/18 |
Ch.8 Riemann mapping theorem. |
第15週 |
12/23,12/25 |
Ch.9 Elliptic integrals and elliptic functions. |
第16週 |
12/30,1/01 |
Ch.9 Addition theorem. |
第17週 |
1/06,1/08 |
Concluding remarks. 1/08 Final exam. |
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