2021 Complex Analysis I

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Academic unit or major
Mathematics
Instructor(s)
Yanagida Eiji  Fujikawa Ege 
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Mon3-4(W321)  Mon5-6(H137)  
Group
-
Course number
ZUA.C301
Credits
2
Academic year
2021
Offered quarter
1-2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
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Course description and aims

In this course, complex analysis, we address the theory of complex-valued functions of a single complex variable. This is necessary for the study of many current and rapidly developing areas of mathematics. It is strongly recommended to take "Exercises in Analysis B I", which is a complementary recitation for this course.

At the beginning of the course, we will explain the Cauchy-Riemann equation which is a key to extend the concept of differentiability from real-valued functions of a real variable to complex-valued functions of a complex variable. A complex-valued function of a complex variable that is differentiable is called holomorphic or analytic, and this course is a study of the many equivalent ways of understanding the concept of analyticity. Many of the equivalent ways of formulating the concept of an analytic function exist and they are summarized in so-called "Cauchy theory". We will explain the theory of meromorphic functions and singularities. We also explain conformal mappings and present some examples of conformal mappings on domains in the complex plane. After that, we will introduce the notion of "residue". As an application of this theory, we explain the computation of integrals.

Student learning outcomes

By the end of this course, students will be able to:
1) understand the complex derivative and the Cauchy-Riemann equations.
2) understand the Cauchy integral theorem and its applications.
3) understand the maximum principle, Schwarz lemma.
4) understand the notion of meromorphic functions and isolated singularities.
5) understand the classification of isolated singularities.
6) compute integrals using the residue theorem.

Keywords

Holomorphic function, Cauchy-Riemann equation, the radius of convergence, the Cauchy integral theorem, the residue theorem, meromorphic function, isolated singularity, the residue theorem, conformal mapping.

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Convergence of complex functions Details will be provided during each class session.
Class 2 Fundamental properties of power series
Class 3 Derivatives of complex functions, Cauchy-Riemann equations
Class 4 Line integrals
Class 5 Applications of Cauchy's theorem
Class 6 Properties of holomorphic functions
Class 7 The maximum principle, Schwarz lemma and exercise, comprehension check-up
Class 8 Meromorphic functions, the reflection principle
Class 9 Isolated singularities of meromorphic functions
Class 10 Poles and residues of meromorphic functions
Class 11 Conformal mappings on plane domains
Class 12 The residue theorem, the computation of integrals
Class 13 Applications of the residue theorem and the integrals
Class 14 The argument principle

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

Introduction to complex function, Kawahira Tomoki, Shokabo

Reference books, course materials, etc.

To be announced.

Assessment criteria and methods

Final exam. Details will be provided during class sessions.

Related courses

  • MTH.C302 : Complex Analysis II
  • MTH.C301 : Complex Analysis I
  • ZUA.C302 : Exercises in Analysis B I

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed [ZUA.C201 : Advanced Calculus I] and [ZUA.C203 : Advanced Calculus II]. It is strongly recommended to take [ZUA.C302 : Exercises in Analysis B I].

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