課程資料
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2187 複變函數論
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開課學期：1101
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開課班級：
應數系 4B
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授課教師：鄭文巧
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選修
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學期課
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學分數：3.0
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大義 0405 星期四 15:10-18:00
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2187 COMPLEX ANALYSIS
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2021 Fall
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Department of Applied Mathematics 4B
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Professor：CHENG, WEN-CHIAO
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Elective
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Semester
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Credits:
3.0
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Da Yi 0405 Thursday 15:10-18:00
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發展願景
傳揚中華文化，促進跨領域創新，與時精進，邁向國際
It is our objective to promote Chinese culture, enhance cross-disciplinary innovation, seek constant advancement, and embrace global community.
辦學宗旨
秉承質樸堅毅校訓，承東西之道統，集中外之精華，研究高深學術, 培養專業人才，服務社會，致力中華文化之發揚, 促進國家發展.
Based on our motto—“Temperament, Simplicity, Strength, and Tenacity,” “inheriting the merits of the East and the West” and “absorbing the essence of Chinese and foreign cultures,” we make it our mission to pursue advanced research, develop professional talents, serve the society, promote Chinese culture and support national development.
校教育目標
校基本素養
校核心能力
院教育目標
奠定自然科學基礎培養後續學習能力
強化理論與實務並重的多元課程
推動跨領域學習
促進國際化教學提升學生競爭力
院核心能力
自然科學知識的能力
理論與實務結合的能力
國際化與團隊溝通合作的能力
多元整合的能力
系教育目標
訓練學生具備紮實的數學基本能力
依興趣選擇應用數學學群，統計科學學群，或計算機科學學群
兼顧理論與實務，讓學生得以繼續升學或直接就業
系核心能力
具備基礎科學知識能力
具計算、分析、演算法與證明等能力
使用數學或套裝軟體求解問題能力
解釋結果與表達溝通能力
課程目標
瞭解複數的計算、觀念，複變數函數的建立，及其微分，路徑積分，一些級數表現法及其在物理、工程上的各種應用。
Understand the calculation and concepts of complex numbers, the establishment of complex variable functions, and their differentiation, contour integrals. We also show some series expressions and their various applications in physics and engineering.
課程能力
具備基礎科學知識能力 (比重 30%)
具計算、分析、演算法與證明等能力 (比重 35%)
使用數學或套裝軟體求解問題能力 (比重 10%)
解釋結果與表達溝通能力 (比重 25%)
課程概述
介紹複變函數的理論和應用，內容包括複數系的介紹、複變數函數、多值函數、基本函數、複變數函數的極限連續性與其相關定理，複數數列及極限，複變數函數的導函數、可解析的函數、Cauchy-Riemann方程式、調和函數、基本函數的導函數、奇異點，複變數函數的積分，線積分的一些性質，簡單連通和複連通區域，平面上的Green定理，Cauchy定理，Morera定理，不定積分，Cauchy定理的一些結果，Cauchy積分公式，Liouville定理，代數基本定理，最大模數定理，Poisson積分公式，函數數列，函數級數，絕對收斂，一致收斂，冪級數，Laurent定理，奇異點的分類，全函數，留數，留數定理，一些定積分的求值問題。
In this course, we will discuss the theory and applications of complex variables. The contents will cover complex number systems, complex variable functions, multiple-valued functions, elementary functions, the limits of complex variable functions, some consequences of the limits, the continuities of the complex variable functions, theorems on continuities of the complex variable functions, sequences of the complex numbers and their limits , the derivatives, analytic functions, Cauchy-Riemann equations, the derivatives of elementary functions, singularities, complex line integrals, some consequences of the line integrals, simply and multiply connected regions, Green’s theorem , Cauchy’s theorem, Morera’s theorem, indefinite integrals , some consequences of the Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, fundamental theorem of algebra, maximum modulus theorem, Poisson integral formula, sequences of functions, absolute convergence, uniform convergence, power series, Laurent’s theorem, classifications of the singularities, entire functions, residues, the residue theorems, evaluation of the definite integrals.
授課內容
介紹複變函數的理論和應用，內容包括複數系的介紹、複變數函數、多值函數、基本函數、複變數函數的極限連續性與其相關定理，複數數列及極限，複變數函數的導函數、可解析的函數、Cauchy-Riemann方程式、調和函數、基本函數的導函數、奇異點，複變數函數的積分，線積分的一些性質，簡單連通和複連通區域，平面上的Green定理，Cauchy定理,Morera定理，不定積分，Cauchy定理的一些結果，Cauchy積分公式，Liouville定理，代數基本定理，最大模數定理，Poisson積分公式，函數數列，函數級數，絕對收斂，一致收斂，冪級數，Laurent定理，奇異點的分類，全函數，留數，留數定理，一些定積分的求值問題。
TheIn this course, we will discuss the theory and applications of complex variables. The contents will cover complex number systems, complex variable functions, multiple-valued functions, elementary functions, the limits of complex variable functions, some consequences of the limits, the continuities of the complex variable functions, theorems on continuities of the complex variable functions, sequences of the complex numbers and their limits , the derivatives, analytic functions, Cauchy-Riemann equations, harmonic functions, the derivatives of elementary functions, singularities, complex line integrals, some consequences of the line integrals, simply and multiply connected regions, Green’s theorem , Cauchy’s theorem, Morera’s theorem, indefinite integrals , some consequences of the Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, fundamental theorem of algebra, maximum modulus theorem, Poisson integral formula, sequences of functions, absolute convergence, uniform convergence, power series, Laurent’s theorem, classifications of the singularities, entire functions, residues, the residue theorems, evaluation of the definite integrals.
授課方式
除課堂解說外，將提供整理之資料協助同學了解授課內容。
評量方式
上課用書
（師生應遵守智慧財產權及不得非法影印）
Complex Variables and Applications, by James Ward Brown and Ruel V. Churchill; 歐亞書局
參考書目
（師生應遵守智慧財產權及不得非法影印）
1. Basic Complex Analysis, 3rd ed. ,by Jerrold E. Marsden and Michael J. Hoffman
2. Advanced Engineering Mathematics, 6th ed. , by C.R.Wylie and Louis C. Barrett
輔導時間
- 星期二 13:00-15:00
- 星期三 15:00-17:00
- 星期四 13:00-15:00
教師聯絡資訊
Email：zwq2@faculty.pccu.edu.tw
分機：25101
課程進度
| theorem of limit and continuity |
| branchs and derivatives of log function |