教學大綱 Syllabus

科目名稱：複變函數論

Course Name: Complex Analysis
  • 學年學期：113-1 Fall Semester, 2024
  • 科目代碼：701026001 Course No.701026001

修別：必

Type of Credit: Required

3.0

學分數

Credit(s)

60

預收人數

Number of Students

課程資料Course Details
  • 開課單位：應數三 Course Department:Mathematical Sciences/B/3
  • 授課老師：邱普照 Instructor: KOW PU-ZHAO
  • 先修科目：無Prerequisite(N/A)
  • 上課時間：三34五3 Session: wed10-12,fri10-11

課程簡介Course Description

本課程是應用數學系的本科必修課。本課程的主題爲介紹複數，以及其對應的微積分理論，包含：

  • 解析性（複數可微分）、Cauchy-Riemann 方程以及複數冪級數之間的等價性；
  • Morera 定理；
  • Cauchy 留數定理（Cauchy 積分公式爲其中一個特例）；以及
  • 與其他數學領域之間的關係，例如：抽象代數（代數基本定理）以及組合學（Egorychev 方法）。

This is a requied undergraduate course at the Department of Mathematical Sciences. The main theme of this course is to introduce complex numbers and discuss the corresponding theory of differentiation and integration, including:

  • the equivalence of analyticity (complex differentiability), Cauchy-Riemann equation and the power series representation;
  • Morera theorems; 
  • the Cauchy residual theorem (Cauchy integral formula is a special case); and
  • some connections with other (mathematical) fields, for example: algebra (fundamental theorem of algebra) as well as combinatorics (Egorychev method).

核心能力分析圖 Core Competence Analysis Chart

能力項目說明

    課程目標與學習成效Course Objectives & Learning Outcomes

    認識複數以及其基本運算。

    Understanding complex numbers and their basic operations.

    每周課程進度與作業要求 Course Schedule & Requirements

    教學週次Course Week 彈性補充教學週次Flexible Supplemental Instruction Week 彈性補充教學類別Flexible Supplemental Instruction Type

    本大綱只供參考，課程進行中會隨時更新。
    This syllabus is just for a reference, which will be updated at any time during the course.

    期中考試（預計第9週）及期末考試日期（預計第16週）視教學進度而定。
    The date of midterm (expect the 9th week) and final exams (expect the 16th week) will depend on the actual situation.

    我們預計根據下列計劃進行教學：
    We will plan to deliver the ideas based on the followings:
    1. 複數 The complex numbers 「第1週~第2週 1st~2nd week」
      • 複平面的定義 Definition of complex plane;
      • 拓撲性質（我們只考慮歐式拓撲） topological aspects (we will only focus on Euclidean topology);
    2. 微分 Differentiation 「第1週~第2週 1st~2nd week」
      • 複數微分以及 Cauchy-Riemann 方程 Complex derivative and Cauchy-Riemann equation; 
      • 冪級數 power series; 
      • 指數函數、正弦函數以及餘弦函數 exponential, sine and cosine functions;
    3. 積分 Integration 「第3週~第4週 3rd~4th week」
      • 線積分基本定理 The fundamental theorem of line integral;
      • Cauchy 閉曲線定理 Cauchy closed curve theorem;
    4. 解析函數的性質 Properties of analytic functions 「第5週~第8週 5th~8th week」
      • Cauchy 積分公式 Cauchy integral formula
      • Liouville 定理 Liouville theorem;
      • 代數基本定理 Fundamental theorem of algebra;
      • Morera 定理 Morera theorems;
    5. Laurent 級數以及 Cauchy 留數定理 Laurent series and the Cauchy residual theorem 「第10週~第13週 10th~13th week」
      • Riemann 可去奇點法則 Riemann's principle of removable singularities;
      • Cauchy 留數定理 Cauchy residue theorem
      • 一些組合學上的應用： Egorychev 方法 Some applications in combinatorics: Egorychev method;
    6. 一些特殊的解析函數 Some special analytic functions 「第14週~第15週 14th~15th week」
      • 複對數函數 The complex logarithmic function
      • 無窮乘積以及 Weierstrass 乘積定理 Infinite products and Weierstrass product theorem;
      • Gamma 函數：階乘函數的推廣 The Gamma function: an extension of factorial function.

    每週會有3節正課以及1節助教課。建議每週花至少3小時自習。
    There will be 3 regular classes and 1 TA class per week, it is recommend to spend at least 3 hours for self-study.

    授課方式Teaching Approach

    100%

    講述 Lecture

    0%

    討論 Discussion

    0%

    小組活動 Group activity

    0%

    數位學習 E-learning

    0%

    其他： Others:

    評量工具與策略、評分標準成效Evaluation Criteria

    作業 Homework assignments: 60%

    期中考試 Midterm Exam: 20%

    期末考試 Final Exam: 20%

    特別說明：原則上本課程禁止使用生成式AI工具。如果我們懷疑作業使用了生成式AI工具（甚至抄襲），我們會暫時給該題0分，請解釋該題如何解，解釋合理就會給分。
    Note: In principle, Generative AI tools are prohibited in this course. If we suspect that AI tools have been deployed (even plagiarism), we will temporary mark zero point for that question. Please explain to us you how to solve that question, and marks will be given if your explainations are resonable.

    考試規則：

    1. 考試中不允許使用課本或其他參考資料 Textbook or other materials are not allowed to be used during exams.

    2. 考試中禁止使用任何電子設備（包含計算機、智慧型手機、平板、電腦...） All electronic devices (including calculator, smartphone, pad, computer, ...) are prohibited during exams.

    3. 考試中上述物品不允許放在桌上，請在考試開始前10分鐘放置在書包內或椅子下。

    4. 考生不允許攜帶任何其他紙張入場。助教會提供作答所需的紙張 One also not allowed to bring your own extra paper. TA will provide answer sheets

    5. 考試期間上廁所前請先告知 Before go to washroom, one must inform us before do so.

    6. 如果考生違反上述任何規則，我們會立即中止該考生的作答且該次測驗成績爲0 If you violate one of the above rule, we will immednate terminate your writing and the marks of the exam/quiz will be 0.

    7. 考生必須出示學生證或國民身份證或全民健康保險卡或護照或居留證（不接受駕照）以供查驗。考試開始前助教應提醒考生。如果在考試期間無法出示上述證件，將視爲作弊且該次測驗成績爲0 One must show student card or national identity card or national health insurance card or passport or resident certificate (driving license not accepted) for verification. Before the exam begins, TA should reminds all of you to bring it. If one fails to show it during exam, we consider this as a cheating and the marks of the quiz will be 0.

    指定/參考書目Textbook & References

    自備講義 Handouts will be provided:

    https://puzhaokow1993.github.io/homepage/teaching/Lecture_Note/ver1_Lecture_Note_Complex_Analysis.pdf

    其他參考書目 Other references:

    J. Bak and D. Newman, Complex Analysis, Undergraduate Texts in Mathematics, Springer, New York, third edition, 2010. MR2675489, Zbl:1205.30001, doi:10.1007/978-1-4419-7288-0

    E. Freitag and R. Busam, Complex Analysis, Universitext, Springer-Vergag, Berlin, second edition, 2009. MR2513384, Zbl:1167.30001, doi:10.1007/978-3-540-93983-2

    已申請之圖書館指定參考書目 圖書館指定參考書查詢 |相關處理要點

    維護智慧財產權，務必使用正版書籍。 Respect Copyright.

    課程相關連結Course Related Links

    https://puzhaokow1993.github.io/homepage/teaching/courses_pages/Complex_Analysis_701026001_2024_Fall/Complex_Analysis_701026001_2024_Fall.html

https://moodle.nccu.edu.tw/

    課程附件Course Attachments

    課程進行中，使用智慧型手機、平板等隨身設備 To Use Smart Devices During the Class

    需經教師同意始得使用 Approval

    列印