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2020　Complex Analysis III

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Fujikawa Ege 
Class Format
Lecture    (ZOOM)
Media-enhanced courses
Day/Period(Room No.)
Tue3-4(H111)  Fri3-4(H111)  
Group
-
Course number
MTH.C331
Credits
2
Academic year
2020
Offered quarter
4Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

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. Finally, an introductory lecture on elliptic functions and elliptic curves will be given.

Student learning outcomes

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 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 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 Picard's theorems Details will be provided during each class session.
Class 13 The definition of Riemann surfaces and a construction Details will be provided during each class session.
Class 14 Functions on Riemann surfaces, degree and genus Details will be provided during each class session.

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)

None.

Reference books, course materials, etc.

J. Gilman, I. Kra and R. Rodriguez: Complex Analysis (Springer, GTM 245).

Assessment criteria and methods

Final exam (70%) and report (30%).

Related courses

  • MTH.C301 ： Complex Analysis I
  • MTH.C302 ： Complex Analysis II

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

Students are expected to have passed [MTH.C301 ： Complex Analysis I] and　[MTH.C302 ： Complex Analysis II].

Other

None in particular.

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