課程資訊

Functions of a Complex Variable

103-1

MATH3201

201 31300

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1031MATH3201_CA1

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.

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.

Prerequisite : Calculus with proof, namely familiarity with point set topoogy, metric spaces, and regorous definitions of real numbers, limits and integrals.

Office Hours

Ahlfors: Complex analysis
Whittaker and Watson: Modern Analysis

Stein: Complex Analysis

(僅供參考)

 No. 項目 百分比 說明 1. 期中考 40% 2. 期末考 40% 3. 作業 20%

 課程進度
 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.