2019 Complex Function Theory A

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Academic unit or major
Undergraduate major in Mechanical Engineering
Yamamoto Takatoki  Suekane Tetsuya 
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Course description and aims

 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.

Student learning outcomes

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.

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

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

  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
Class 8 Summary and application


Ryuichi Watabe, Hiroshi Miyazaki, Shizuo Endo, "Complex Function" , Baifuukan (1980)

Reference books, course materials, etc.

To be announced

Assessment criteria and methods

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%).

Related courses

  • MEC.B213 : Partial Differential Equations

Prerequisites (i.e., required knowledge, skills, courses, etc.)

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: tsuekane[at]es.titech.ac.jp, ex) 5494

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