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 fundamental 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.
By the end of this course, students will be able to:
1) Have an understanding of overview of complex number and complex function, and gain an ability to solve basic problem.
2) Understand the advantage of complex function, and gain an ability to solve real problems in various engineering.
Complex derivative, linear second-order partial differential equation, Laplace equation, complex integration, residues.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
After devoting the classes to fundamentals, the course advances to applications. To allow students to get a good understanding of the course contents and practice application, problems related to the contents of this course are provided.
Course schedule | Required learning | |
---|---|---|
Class 1 | Differentiation and integral in the complex plane, Cauchy-Riemann equation | Derivation of Cauchy-Riemann equation |
Class 2 | Basics of second-order partial differential equation, Laplace equation | Relation to 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 | Integral theorem of Cauchy | Integration method using Caucie'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 |
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.
Ryuichi Watabe, Hiroshi Miyazaki, Shizuo Endo, "Complex Function" , Baifuukan (1980)
To be announced
Students' knowledge of basic topics of complex function, and their ability to apply them to problems will be assessed. Learning achievement is evaluated by a final exam (60%) and exercises (40%).
It is desirable to have knowledge in the partial differential equation.
T. Suekane: tsuekane[at]es.titech.ac.jp, ex) 5494
T. Yamamoto、yamamoto.t.ba[at]m.titech.ac.jp, ex) 3182