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. It is strongly recommended to take "Exercises in Analysis B I", which is a complementary recitation for this course.
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". We will explain the theory of meromorphic functions and singularities. We also explain conformal mappings and present some examples of conformal mappings on domains in the complex plane. After that, we will introduce the notion of "residue". As an application of this theory, we explain the computation of integrals.
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.
4) understand the notion of meromorphic functions and isolated singularities.
5) understand the classification of isolated singularities.
6) compute integrals using the residue theorem.
Holomorphic function, Cauchy-Riemann equation, the radius of convergence, the Cauchy integral theorem, the residue theorem, meromorphic function, isolated singularity, the residue theorem, conformal mapping.
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Standard lecture course.
Course schedule | Required learning | |
---|---|---|
Class 1 | Complex numbers, the calculation of the complex numbers | Details will be provided during each class session. |
Class 2 | Derivatives of complex functions, Cauchy-Riemann equations | |
Class 3 | Fundamental properties of power series | |
Class 4 | The Riemann sphere, elementary functions | |
Class 5 | Line integrals, Cauchy's theorem | |
Class 6 | Applications of Cauchy's theorem | |
Class 7 | Cauchy's integral theorem, its applications. | |
Class 8 | The maximum principle, Schwarz lemma and exercise, comprehension check-up | |
Class 9 | Meromorphic functions, the reflection principle | |
Class 10 | Isolated singularities of meromorphic functions | |
Class 11 | Poles and residues of meromorphic functions | |
Class 12 | Conformal mappings on plane domains | |
Class 13 | The residue theorem, the computation of integrals | |
Class 14 | Applications of the residue theorem and the integrals | |
Class 15 | The argument principle |
Fukuso Kansuu Gaisetus, Yoichi Imanishi, Saiensu-sha
To be announced.
Final exam. Details will be provided during class sessions.
Students are expected to have passed [ZUA.C201 ： Advanced Calculus I] and [ZUA.C203 ： Advanced Calculus II]. It is strongly recommended to take [ZUA.C302 ： Exercises in Analysis B I].