2021 Faculty Courses School of Science Undergraduate major in Mathematics
Complex Analysis III
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Ege Fujikawa
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue (H111) / 3-4 Fri (H111)
- Class
- -
- Course Code
- MTH.C331
- Number of credits
- 200
- Course offered
- 2021
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The goal of this course is to outline the new epoch of classical complex analysis.
At the beginning, we will introduce the hyperbolic geometry in the upper half plane. After discussing the normal family, we will show Riemann's mapping theorem which has many applications in the complex analysis. We will explain Riemann surfaces. The theory of Riemann surfaces provides a new foundation for complex analysis on a higher level. As in elementary complex analysis, the subject matter is analytic functions. But the notion of an analytic function will have now a broader meaning as we show.
Course description and aims
By the end of this course, students will be able to:
1) understand the hyperbolic geometry.
2) obtain the notion of normal family and its applications.
3) know Riemann's mapping theorem and its applications.
4) 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 | Harmonic functions and their properties |
Details will be provided during each class session. |
| Class 2 | Dirichlet problem |
Details will be provided during each class session. |
| Class 3 | The linear fractional transformations : classification and properties |
Details will be provided during each class session. |
| Class 4 | Biholomorphic functions |
Details will be provided during each class session. |
| Class 5 | Normal family |
Details will be provided during each class session. |
| Class 6 | Montel's theorem and its applications |
Details will be provided during each class session. |
| Class 7 | Riemann's mapping theorem |
Details will be provided during each class session. |
| Class 8 | The hyperbolic plane and the Poincare disk |
Details will be provided during each class session. |
| Class 9 | Analytic continuation |
Details will be provided during each class session. |
| Class 10 | Modular function |
Details will be provided during each class session. |
| Class 11 | Covering and lifts of maps |
Details will be provided during each class session. |
| Class 12 | The definition of Riemann surfaces and a construction |
Details will be provided during each class session. |
| Class 13 | Functions on Riemann surfaces, degree and genus |
Details will be provided during each class session. |
| Class 14 | Differential form |
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.
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
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.