### 2018　Complex Analysis III

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Instructor(s)
Fujikawa Ege  Shiga Hiroshige
Course component(s)
Lecture
Day/Period(Room No.)
Tue5-6(H136)  Fri5-6(H136)
Group
-
Course number
MTH.C331
Credits
2
2018
Offered quarter
4Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
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### 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

 Intercultural skills Communication skills ✔ Specialist 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 Conformal mappings, especially on the upper half plane and the unit disk. Details will be provided during each class session.
Class 2 The linear fractional transformations : classification and properties. Details will be provided during each class session.
Class 3 The hyperbolic plane and the Poincare disk Details will be provided during each class session.
Class 4 Normal family. Details will be provided during each class session.
Class 5 Wierstrass theorem on normal families and its applications. 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 Riemann's mapping theorem and its applications. Details will be provided during each class session.
Class 9 Analytic continuation. Details will be provided during each class session.
Class 10 The definition of Riemann surfaces and a construction. Details will be provided during each class session.
Class 11 Functions on Riemann surfaces, degree and genus Details will be provided during each class session.
Class 12 Riemann surfaces as quotient spaces Details will be provided during each class session.
Class 13 Elliptic functions Details will be provided during each class session.
Class 14 Picard's theorems Details will be provided during each class session.
Class 15 Groups of Mobius transformations, comprehension check-up Details will be provided during each class session.

None.

### Reference books, course materials, etc.

J. Gilman, I. Kra and R. Rodriguez: Complex Analysis (Springer, GTM 245).
E. Freitag : Complex Analysis 2 (Springer, Universität).

### 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.