### 2019　Complex Analysis I

Font size  SML

Instructor(s)
Fujikawa Ege
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Mon3-6(H136)
Group
-
Course number
MTH.C301
Credits
2
2019
Offered quarter
1Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

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".

### Student learning outcomes

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 maximum principle, Schwarz lemma.

### Keywords

Holomorphic function, Cauchy-Riemann equation, the radius of convergence, the Cauchy integral theorem.

### 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 with exercise.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Complex numbers, the calculation of the complex numbers Details will be provided during each class session.
Class 2 Recitation Details will be provided during each class session.
Class 3 Derivatives of complex functions, Cauchy-Riemann equations Details will be provided during each class session.
Class 4 Recitation Details will be provided during each class session.
Class 5 Fundamental properties of power series Details will be provided during each class session.
Class 6 Recitation Details will be provided during each class session.
Class 7 The Riemann sphere, elementary functions Details will be provided during each class session.
Class 8 Recitation Details will be provided during each class session.
Class 9 Line integrals, Cauchy's theorem Details will be provided during each class session.
Class 10 Recitation Details will be provided during each class session.
Class 11 Applications of Cauchy's theorem Details will be provided during each class session.
Class 12 Recitation Details will be provided during each class session.
Class 13 Cauchy's integral theorem, its applications. Details will be provided during each class session.
Class 14 Recitation Details will be provided during each class session.
Class 15 The maximum principle, Schwarz lemma and exercise, comprehension check-up Details will be provided during each class session.

### Textbook(s)

Fukuso Kansuu Gaisetus, Yoichi Imanishi, Saiensu-sha

### Reference books, course materials, etc.

Details will be provided during class session.

### Assessment criteria and methods

Final exam 70%, exercise 30%.

### Related courses

• ZUA.C201 ： Advanced Calculus I
• ZUA.C203 ： Advanced Calculus II
• MTH.C302 ： Complex Analysis II

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

Students are expected to have passed [ZUA.C201 ： Advanced Calculus I] and [ZUA.C203 ： Advanced Calculus II].