複變函數論
Complex Analysis
學期
109-2
學分
3
學分
當期課號
1382
永久課號
DAM3409
開課單位
應用數學系
授課教師
葉立明
校區
光復
類別
必修
上課時間表
| 節 | 週一 | 週四 |
|---|---|---|
2 09:00–09:50 | 複變函數論 SA311 | |
5 13:20–14:10 | 複變函數論 SA311 2 節連堂 | |
6 14:20–15:10 |
* 根據陽明交大上課時間表所列
概述
本課程介紹複數函數的基本性質及應用。複數是實數的延伸,許多實數函數上的結果也是複數函數的結果,但複數平面並非是單純的二維的實數平面,複數函數有它自身的運算體系,它在力學以及工程科學中有著很多的應用。對有興趣工程科學與學術研究的同學,這是一門不可或缺的課。
先修科目
calculus, linear algebra
教學方式
板書
評分方式
作業 40 % 三次期中考試 60 %
週次計畫
| 週次 | 主題 |
|---|---|
| 第 1 週 | Topology of complex plane: open and closed sets, limits of functions, continuity, path, change of parameter, subpaths and sums of paths, the paving lemma. |
| 第 2 週 | Topology of complex plane: connectedness, space-filling curves Power series: sequence, series, power series, product of series |
| 第 3 週 | Differentiation: basic results, The Cauchy-Riemann equations, connected set and differentiability, hybrid functions, power series |
| 第 4 週 | The exponential function: the exponential function, real exponentials and logrithms, trigonometric functions, an analytic definition of $\pi$, the behavior of real trigonometric functions |
| 第 5 週 | The exponential function: explanation of Euler's formula, complex exponential and trigonometric functions are periodic, other trigonometric functions, hyperbolic functions |
| 第 6 週 | Integration: The real case, complex integration along a smooth path, length of a path, regular paths and curves, regular and singular points |
| 第 7 週 | Integration: countour integration, the fundamental theorem of countour integration, an integration depends on the path, Gamma function |
| 第 8 週 | Integration: The estimation lemma, consequence of the fundamental theorem Angles, Logrithms, and the widing number: radian measure of angle, the argument of a complex number, the complex logrithm, the winding number |
| 第 9 週 | Angles, Logrithms, and the widing number: the winding number as an integral, components of the complement of a path, computing the winding number |
| 第 10 週 | Cauchy's Theorem: Cauchy's Theorem for a triangle, existence of an intiderivative in a star domain, local existence of an antiderivative |
| 第 11 週 | Cauchy's Theorem: Cauchy's Theorem, applications of Cauchy's Theorem, simply connected domain Homotopy version of Cauchy's Theorem |
| 第 12 週 | Taylor series: Cauchy integral formula, Taylor series Morera's Theorem, Cauchy's estimate, zero, extension function |
| 第 13 週 | Laurent series: series involving negative powers, isolated singularities, behavior near an isolated singularity, the extended complex plane, |
| 第 14 週 | Laurent series: behavior of a differentiable function at infinity, meromorphic function Residue: Cauchy residue theorem, counting residues |
| 第 15 週 | Residue: evaluation of definite integrals, summation of series, counting zeros |
教科書
Complex Analysis (2nd edition), Ian Stewart , David Tall Cambridge University Press; 2018
Office Hours
- 地點
- SA 348
- 時間
- Wed. 4-5 pm
- 聯絡方式
- Tel: 56439 Email: liming@math.nctu.edu.tw