2023 Faculty Courses School of Engineering Undergraduate major in Mechanical Engineering
Complex Function Theory
- Academic unit or major
- Undergraduate major in Mechanical Engineering
- Instructor(s)
- Takatoki Yamamoto / Tetsuya Suekane
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Fri (M-178(H1101))
- Class
- -
- Course Code
- MEC.B212
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 1Q
- Syllabus updated
- Jul 8, 2025
- Language
- Japanese
Syllabus
Course overview and goals
A complex number is a combination of real and imaginary numbers and is an indispensable mathematical tool to understand the various phenomena in mechanical engineering. In this lecture, you study the fundamentals and application of the complex function theory, which is required by a mechanical engineer. In addition to understanding the basics of the differentiation of complex functions, you acquire mathematical skills to solve various engineering problems by studying the second-order partial differential equations and the application to the integral calculation of real functions.
The lecture is focused on the following points.
1. By learning the calculus of the complex function in comparison with that of the real function, you understand the characteristics of the complex function and acquire the calculating ability.
2. You learn that many of the important techniques in the integral evaluation of complex functions are developed from Cauchy's integral theorem and Cauchy's integral formula by learning the Cauchy-Riemann equation, Laplace equation, Cauchy's integral theorem.
3. Learn about the advanced integral methods using Taylor and Laurent series expansion and the use of residue theory.
Course description and aims
By the end of this course, students will be able to:
1) Have an understanding of an overview of a complex number and complex function, and gain an ability to solve the basic problem.
2) Understand the advantage of complex functions, and gain an ability to solve real problems in various engineering.
Keywords
Complex derivative, linear second-order partial differential equation, Laplace equation, complex integration, residues.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
After studying the basic content, advanced and applied subjects will be studied. Exercises will be conducted as necessary to ensure the understanding of the lecture content and to develop the ability to apply it.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Differentiation and integral in the complex plane, Cauchy-Riemann equation | Derivation of Cauchy-Riemann equation |
| Class 2 | Basics of the second-order partial differential equation, Laplace equation | Relation to an elliptic second-order partial differential equation |
| Class 3 | Integral in the complex plane, Integral theorem of Cauchy, | Set-up of the integration path in the integration of complex functions |
| Class 4 | The integral theorem of Cauchy | Integration method using Cauchy's integral formula |
| Class 5 | Taylor and Laurent series expansion | Derivation of series expansion |
| Class 6 | Residue theorem, evaluation of the integrals using the residue theorem | Examples of integrals using the residue theorem |
| Class 7 | Application to Real function integration | Examples of the integral of a real function |
Study advice (preparation and review)
In order to improve the effectiveness of learning, students should study the textbook, handouts, and other relevant materials for approximately 100 minutes each for preparation and review (including assignments) of the class.
Textbook(s)
Same as the Japanese textbook.
Reference books, course materials, etc.
To be announced
Evaluation methods and criteria
Final examination (70%)
Assessment and report (30%)
Related courses
- MEC.B213 ： Partial Differential Equations
Prerequisites
It is desirable to have knowledge in the partial differential equation.
Contact information (e-mail and phone) Notice : Please replace from ”[at]” to ”@”(half-width character).
T. Yamamoto、yamamoto.t.ba[at]m.titech.ac.jp, ex) 3182
T. Suekane: suekane.t.aa[at]m.titech.ac.jp, ex) 5494
Other
In-person