Based on [Applied Mathematics for Engineers Ia] in the first quarter, this course focuses on basic part of complex analysis. First, we explain Cauchy's integral theorem and Cauchy's integral formula after reviewing complex line integrals. Then, we explain the Laurent expansion of meromorphic functions after classifying isolated singularities of complex functions. Finally, we explain the residue theorem and its application to the calculation of definite integrals.
Complex analysis is an absolutely essential mathematical basis of science and engineering. The aim of this lecture is to explain the basic theory and practical way to use of complex analysis by an efficient way.
・Students are expected to understand Cauchy's integral theorem.
・Students are expected to be familiar with the classificationof isolated singularities of complex functions.
・Students are expected to be able to calculate the Laurent expansion of basic complex functions.
・Students are expected to be able to apply the residue theorem to the calculation of definit integrals.
Cauchy's integral theorem, isolated singularities, the Laurent expansion, meromorphic functions, the residue theorem
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course mixed with recitation.
Course schedule | Required learning | |
---|---|---|
Class 1 | Cauchy's integral theorem | Details will be announced during each lecture. |
Class 2 | Cauchy's integral formula | Details will be announced during each lecture. |
Class 3 | power series expansions and its applications | Details will be announced during each lecture. |
Class 4 | isolated singularities | Details will be announced during each lecture. |
Class 5 | the Laurent expansion | Details will be announced during each lecture. |
Class 6 | meromorphic functions and the residue theorem | Details will be announced during each lecture. |
Class 7 | evaluation of integrals using the residue theorem | Details will be announced during each lecture. |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
H. Shiga, Theory of complex functions to learn in 15 weeks (Japanese), Sugakushobo, 2008
None in particular
Based on overall evaluation of the results for quizzes, report and final examination. Details will be announced during a lecture.
The prerequisite to take this course is that you have acquired the credits of "Applied Mathematics for Engineers Ia".
Without having acquired the credits of the above course, the credits of this course will not be counted as the necessary number of credits for graduation.
Students are expected to have completed [Calculus I / Recitation], [Calculus II]and [Calculus Recitation II] .
In particular, students are expected to understand partial differentiation, definite integral and multiple integral clearly.