2016 Complex Function Theory

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Academic unit or major
Undergraduate major in Mechanical Engineering
Yamamoto Takatoki  Horiuti Kiyosi 
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Course description and aims

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.

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.


Linear second-order partial differential equation, Laplace equation, differentiation and integral in the complex plane, residues, Riemann surface.

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 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)

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、ext) 3182

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