2026 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Complex Analysis III
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Masaharu Tanabe
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.C331
- Number of credits
- 200
- Course offered
- 2026
- Offered quarter
- 4Q
- Syllabus updated
- Mar 5, 2026
- Language
- Japanese
Syllabus
Course overview and goals
The aim of this course is to provide an overview of the modern form of classical complex analysis. First, we discuss the normal family, and we show Riemann's mapping theorem which has many applications in the complex analysis. We also demonstrate the elementary theory of Riemann surfaces. A Riemann surface is a two-real-dimensional manifold with holomorphic coordinate transformations. The theory of Riemann surfaces has been a source of inspiration and examples for many fields of mathematics. Our goal for this topic is to understand the Hodge decomposition.
Course description and aims
By the end of this course, students will be able to:
1) obtain the notion of normal family and its applications.
2) know Riemann's mapping theorem and its applications.
3) understand Riemann surfaces.
Keywords
Normal family, Riemann's mapping theorem, Riemann surface.
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 | Biholomorphic maps |
Details will be provided during each class session. |
| Class 2 | The linear fractional transformations 1 |
Details will be provided during each class session. |
| Class 3 | Normal family |
Details will be provided during each class session. |
| Class 4 | Montel's theorem and its applications |
Details will be provided during each class session. |
| Class 5 | Riemann's mapping theorem |
Details will be provided during each class session. |
| Class 6 | The hyperbolic gerometry |
Details will be provided during each class session. |
| Class 7 | Analytic continuation |
Details will be provided during each class session. |
| Class 8 | Riemann surfaces |
Details will be provided during each class session. |
| Class 9 | Topology of Riemann surfaces |
Details will be provided during each class session. |
| Class 10 | Differential forms |
Details will be provided during each class session. |
| Class 11 | Harmonic differetials, holomorphic differetials |
Details will be provided during each class session. |
| Class 12 | Orthogonality relations |
Details will be provided during each class session. |
| Class 13 | Hodge decomposition |
Details will be provided during each class session. |
| Class 14 | Bilinear relations |
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.
M. Farkas and I. Kra: Riemann Surfaces (Springer, GTM 71).
J. Gilman, I. Kra and R. Rodriguez: Complex Analysis (Springer, GTM 245).
Junjiro Noguchi, Introduction to complex analysis, Shokabo
Evaluation methods and criteria
Final exam，Report
Related courses
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
Prerequisites
Students are expected to have passed [MTH.C301 ： Complex Analysis I] and [MTH.C302 ： Complex Analysis II].
Other
None in particular.