授業の目的 【英語】
Goals of the Course [ENG] | | | 【Standardized across all programs】 Complex functions appear in various parts of the natural sciences, play a fundamental role and have a wide range of applications. In particular, its calculus forms a beautiful and unified world that is completely different from that of real numbers. The goal of this course is to learn the basics of differentiation and integration of such complex functions, especially the basic properties of holomorphic functions, and to become familiar with their various treatments, which are important for applications. |
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授業の達成目標 【英語】
Objectives of the Course [ENG] | | | To introduce the basic theory of analytic functions of a single complex variable. Students will also acquire the ability to use and apply techniques of complex differentiation and integration in a variety of settings. |
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授業の内容や構成
Course Content or Plan | | Complex numbers: The complex number system, properties of the complex numbers, Cartesian and polar form.
Holomorphic functions: Complex differentiability, Cauchy–Riemann equations, analytic functions.
Complex Integrals: Line integrals, Cauchy’s theorem, Cauchy’s integral formula and Taylor expansion.
Singularities and the Residue Theorem: Laurent series, classification of singularities, the Residue Theorem and applications to real integrals. |
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履修条件・関連する科目
Course Prerequisites and Related Courses | | Solid knowledge of calculus in one and several variables (such as that covered in Calculus I and II) will be expected. A command of basic linear algebra will also be beneficial.
Prior knowledge of complex numbers will certainly be helpful, but is not necessary. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | The grade will be determined by homework assignments (50%), a midterm and final exam (50%).
Please contact the instructor to withdraw from the course. |
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教科書
Textbook | | S. Lang: Complex Analysis. Third edition. Springer-Verlag, 1993. ISBN: 978-3-540-78059-5
It is not necessary to purchase this textbook.
Lecture summary notes will also be distributed. |
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参考書
Reference Book | | E. M. Stein, R. Shakarchi: Complex analysis. Princeton University Press, 2003.
E. Freitag and R. Busam: Complex analysis. Second edition. Springer-Verlag, Berlin, 2009.
W. Fischer, I. Lieb: A Course in Complex Analysis: From Basic Results to Advanced Topics. Springer, 2012.
MIT OpenCourseWare
Complex Variables and Applications: https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/
Functions of a Complex Variable: https://ocw.mit.edu/courses/18-112-functions-of-a-complex-variable-fall-2008/ |
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課外学修等
Study Load (Self-directed Learning Outside Course Hours) | | | Several homework assignments will be distributed during the course. Students are also expected to revise the lecture content. |
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注意事項
Notice for Students | | |
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本授業に関するWebページ
Reference website for this Course | | |
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担当教員からのメッセージ
Message from the Instructor | | |
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実務経験のある教員等による授業科目（大学等における修学の支援に関する法律施行規則に基づくもの）
Courses taught by Instructors with practical experience | | |
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授業開講形態等
Lecture format, etc | | B-3) Face-to-face course (both simultaneous interactive and on-demand methods will be provided when necessary)
* Classrooms for face-to-face classes can be found in Timetable B on the ILAS website. |
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