This course focuses on the complex function theory. We begin with the fundamentals of analytic functions such as Cauchy-Riemann equation, Laplace equation, Integral theorem of Cauchy, and then present application topics, Taylor and Laurent series expansions and the evaluation of the integrals using the residue theorem. By combining lectures and exercises, the course enables students to understand and acquire the fundamentals of complex function.
We aim to teach fundamentals and applications of the complex function theory which is one of indispensable basic tools in mechanical engineering. While understanding the basic concept of differentiation in complex functions, students learn the relations between complex functions and the second-order partial differential equations, and application to the calculation of integrated value of real function.
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.
Linear second-order partial differential equation, Laplace equation, differentiation and integral in the complex plane, residues, Riemann surface.
✔ 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 integral path |
Class 4 | Integral theorem of Cauchy, Taylor and Laurent series expansion | Derivation of the series expansion |
Class 5 | Residue theorem, evaluation of the integrals using the residue theorem | Examples of integrals using the residue theorem |
Class 6 | Jordan's lemma, Bromwitch integral path | Application to Laplace transform |
Class 7 | Multivalued function, Riemann surface | Determination of the branches on the Riemann surface |
Class 8 | Summary and application |
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. Yamamoto、yamamoto.t.ba[at]m.titech.ac.jp、ext) 3182