2024 Students Enrolled in or before 2015 School of Science Mathematics
Complex Analysis I
- Academic unit or major
- Mathematics
- Instructor(s)
- Ege Fujikawa
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon / 5-6 Mon
- Class
- -
- Course Code
- ZUA.C301
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 1-2Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
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 introduce the notion of "residue". As an application of this theory, we explain the computation of integrals.
Course description and aims
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, harmonic function
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction to complex numbers and 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 and Cauchy-Riemann equations | |
| Class 4 | Line integrals | |
| Class 5 | Cauchy's theorem and its applications | |
| Class 6 | Properties of holomorphic functions | |
| Class 7 | The maximum principle, Schwarz lemma and exercise, comprehension check-up | |
| Class 8 | Meromorphic functions and Laurent expansion | |
| Class 9 | Isolated singularities of meromorphic functions | |
| Class 10 | Poles and residues of meromorphic functions | |
| Class 11 | The residue theorem | |
| Class 12 | Applications of the residue theorem and the integrals | |
| Class 13 | The argument principle | |
| Class 14 | Harmonic functions |
Study advice (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)
None
Reference books, course materials, etc.
Kawahira Tomoki, An Introduction to Complex Analysis, Shokabo
Noguchi Junjiro, Introduction to Complex Analysis, Shokabo
Evaluation methods and criteria
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
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].