2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Complex Analysis I
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Ege Fujikawa
- Class Format
- Lecture/Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-6 Mon
- Class
- -
- Course Code
- MTH.C301
- Number of credits
- 110
- Course offered
- 2025
- Offered quarter
- 1Q
- Syllabus updated
- Mar 19, 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. This course will be followed by Complex Analysis II.
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".
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 property of holomorphic functions.
Keywords
Holomorphic function, Cauchy-Riemann equation, the Cauchy integral theorem.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course with exercise.
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 | Recitation | Details will be provided during each class session. |
| Class 3 | Fundamental properties of power series | Details will be provided during each class session. |
| Class 4 | Recitation | Details will be provided during each class session. |
| Class 5 | Derivatives of complex functions and Cauchy-Riemann equations | Details will be provided during each class session. |
| Class 6 | Recitation | Details will be provided during each class session. |
| Class 7 | Line integrals | Details will be provided during each class session. |
| Class 8 | Recitation | Details will be provided during each class session. |
| Class 9 | Cauchy's theorem and its applications | Details will be provided during each class session. |
| Class 10 | Recitation | Details will be provided during each class session. |
| Class 11 | Properties of holomorphic functions | Details will be provided during each class session. |
| Class 12 | Recitation | Details will be provided during each class session. |
| Class 13 | The maximum principle, Schwarz lemma and exercise | Details will be provided during each class session. |
| Class 14 | Recitation | Details will be provided during each class session. |
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
Evaluation methods and criteria
Final exam 60%, exercise 40%.
Related courses
- ZUA.C201 ： Advanced Calculus I
- ZUA.C203 ： Advanced Calculus II
- MTH.C302 ： Complex Analysis II
Prerequisites
Students are expected to have passed [ZUA.C201 ： Advanced Calculus I] and [ZUA.C203 ： Advanced Calculus II].