単位数: 2. 担当教員: 田中 和之. 開講年度: 2024. 科目ナンバリング: TEI-MAT201J. 開講言語: 日本語.
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Google Classroomのクラスコードは工学部Webページにて確認すること．
学部シラバス・時間割(https://www.eng.tohoku.ac.jp/edu/syllabus-ug.html)
１．目的 フーリエ解析と複素解析について，それらの基礎を学習・理解し，計算力と応用力を身につける.
２．概要 工学に現れる現象の解明に重要な役割をはたす応用数学の一部であるフーリエ解析と工学に応用される解析学の基礎をなす複素解析について，それらの基礎を学習する.
３．達成目標等 フーリエ級数，フーリエ変換を理解して，それらの計算とその応用ができるようになること。複素変数の初等関数の扱いになれ，複素関数を用いて実定積分が計算できるようになること.
今年度の本講義は google classroom 上のGoogle Meets によるオンライン授業として行います．
本講義のクラスコードは工学部Webページのhttps://www.eng.tohoku.ac.jp/edu/syllabus-ug.htmlの中の【Google Classroomコード一覧】で「電子情報システム・応物系」を選択した上で各自確認してください．
Object and summary: Students learn and understand the concepts of Fourier analysis and complex analysis and develop relevant calculation and application abilities. Both Fourier analysis and complex analysis are important mathematical methods in engineering sciences. In this class, Fourier series and Fourier transformations are first introduced. Next, elementary functions and the concept of analytic function in the complex plane are explained in terms of the differentiability of complex functions. And by using these concepts, complex integrals, Laurent series and isolated singular points are explained. Finally, students learn how to calculate some definite and improper integrals by using complex analysis and how to solve some linear differential and integral equations by using Fourier transformations and complex analysis.
Goal: Students will develop the abilities necessary in calculating Fourier series and Fourier transformation and in applying them to solve some linear differential and integral equations which play important roles in some fundamental problems in engineering sciences.Students will understand some important concepts of complex functions and their isolated singular points in the complex plane. They will be able to calculate some definite and improper integrals by using some methods of complex analysis.
The lectures of the present class is provided as online from Google Meets in the google classroom of the present class.
The google class code of the google classroom for the present class should be confirmed in Class Code List of Google Classroom for each department in the Japanese website of the School of Engineering by students themselves:
https://www.eng.tohoku.ac.jp/edu/syllabus-ug.html
履修要望科目：全学教育科目の解析学Ａ，Ｂ，Ｃ（常微分方程式），線形代数学Ａ，Ｂ，数学物理学演習I, II，
Mathematical and Physical Practice I, II, Linear Algebra A, B and Analysis A, B, C including ordinary differential equations are relevalt as other subjects
１．フーリエ級数（１）
２．フーリエ級数（２）
３．フーリエ積分
４．フーリエ変換（１）
５．フーリエ変換（２）
６．複素数と複素関数
７．初等関数
８．微分可能性と正則関数
９．複素積分
10．コーシーの積分定理とコーシーの積分公式
11．複素数列とテイラー展開
12．ローラン展開と孤立特異点
13．留数と留数定理
14．実定積分への応用
15．フーリエ変換・複素積分を用いた微分方程式・積分方程式の解法
1. Fourier series (1)
2. Fourier series (2)
3. Fourier integral
4. Fourier tranformation (1)
5. Fourier transformation (2)
6. Complex number and complex function
7. Elementary function
8. Differentiability and analytic function
9. Complex integral
10. Cauchy integral theory and Cauchy integral formula
11. Complex sequence and Taylor expansion
12. Laurent expansion and isolated singular point
13. Residue and residue theory
14. Computation of definite and improper integrals by residue theory and Jordan lemma
15. Solutions of differential and integral equations based on Fourier transformations and complex integrals
授業時間は限られているので，自主学習が重要になる．毎回の授業に対して，予習（２時間）と復習（２時間）は最低限必要である.手を動かして式を導きながら教科書を何回も熟読すること
The session time is limited and therefore self-directed learning is important.
Students should be required preparetions (at least for 2 hours) and reviews (at least for 2 hours) in each class.Students should read repeatedly and derive equations in the textbook by themselves,
出題された課題のレポート，授業中に実施された小テスト等から評価した平常点により成績評価を行う．定期試験・再試験は実施しない．詳細は各クラスの担当教員からの指示にしたがうこと．
Final evaluation is performed by only normal points (for example, reports, short tests in each class, etc.).
The final examination as well as its re-examination of the present class will not be done.
The details will be announced in each class.
電子メールにてアポイントをとった上で来室すること.
Students should visit my office after taking an appointment by e-mail.