2016 Mathematics for Systems and Control A

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Academic unit or major
Undergraduate major in Systems and Control Engineering
Hayakawa Tomohisa 
Class Format
Lecture / Exercise     
Media-enhanced courses
Day/Period(Room No.)
Tue2-4(W935)  Fri2-4(W371)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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Course description and aims

This course covers fundamental topics on complex analysis, Laplace transform, and Fourier analysis, which are essential in the field of systems and control engineering. Specifically, it is important to be aware of the connection between time domain and frequency domain characterizations.
In this course, we focus on linear systems and go over calculus of complex functions, residue theorem, deriving solutions of differential equations with Laplace transform, periodic functions and Fourier series, aperiodic functions and Fourier transform, and convolution theorems.

Student learning outcomes

By the end of this course, students will be able to:
1) Understand fundamental facts in complex analysis
2) Compute complex integral for real-valued functions
3) Explain characteristics on signal and systems and characterize connections between time domain and frequency domain
4) Acquire fundamentals of Laplace transform and apply it to solve linear dynamical systems
5) Expand periodic functions in Fourier series and characterize aperiodic functions using Fourier transform


Complex variables, Complex functions, Cauchy-Riemann equations, Complex integral, Cauchy's integral formula, Residue theorem, Conformal mapping, Laplace transform, Partial fraction expansion, Final value theorem, Fourier series, Fourier integral, Fourier transform

Competencies that will be developed

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

Class flow

At the beginning of each class, some of the important points learned in the last class are reviewed. Then the main topics for the day is covered in detail. Students are advised to solve exercise problems at home.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Complex variables and their operations Review of the definition of complex variables and their operations
Class 2 Complex plane and its polar form representation Characterize polar form representation on the complex plane and obtain the roots of the nth-order equations
Class 3 Complex functions Basics on complex functions and Cauchy-Riemann equations
Class 4 Derivative of complex functions and analyticity Differentiability of complex functions and extend the notion to analyticity
Class 5 Complex integral and Cauchy's integral formula Characterize the types of singular points and derive a way of calculating integrals
Class 6 Residue theorem and application to real-valued integral Discuss Laurent series expansion and obtain residue
Class 7 Conformal mapping and linear fractional transformation Go over the conformal mapping and linear fractional transformation
Class 8 Basics of Laplace transform Define Laplace transform and see its application to fundamental functions
Class 9 Properties of Laplace transform Derive some useful facts for Laplace transform
Class 10 Application of Laplace transform to differential equations Derive solutions of linear differential equations using Laplace transform
Class 11 Laplace transform of special functions and final value theorem Consider Laplace transform of step and impulse functions
Class 12 Fourier series Characterize Fourier series expansion for periodic functions
Class 13 Fourier integral Extend the notion of Fourier series expansion to aperiodic functions
Class 14 Fourier transform Consider Fourier transform for some basic functions and its properties
Class 15 Application of Fourier analysis See some practical application of Fourier analysis


Erwin Kreyszig著 『Advanced Engineering Mathematics』 Wiley

Reference books, course materials, etc.

No English reference

Assessment criteria and methods

Student's knowledge of complex analysis, Laplace transform, and Fourier analysis, and their ability to apply them to problems will be assessed.
Mid-term exam 35%, final exam 45% and exercise problems 25%.

Related courses

  • SCE.A202 : Mathematics for Systems and Control B
  • SCE.I201 : Introduction to Measurement Engineering

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

Students must have successfully completed both Calculus I and Calculus II, or have equivalent knowledge.

Contact information (e-mail and phone)    Notice : Please replace from "[at]" to "@"(half-width character).

hayakawa[at]mei.titech.ac.jp, 03-5734-2762

Office hours

Contact by e-mail in advance to schedule an appointment.

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