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.
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
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
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|
|Class 1||Complex variables and their operations||Review the definition of complex variables and define their operations|
|Class 2||Complex plane and its polar form representation, Euler's formula, roots of the n-th order eqations||Characterize polar form representation on the complex plane and obtain the roots of the nth-order equations|
|Class 3||Complex functions||Understand the basics on complex functions and Cauchy-Riemann equations|
|Class 4||Derivative of complex functions and analyticity||Define the differentiability of complex functions and extend the notion to analyticity|
|Class 5||Complex integral: Cauchy-Goursat's theorem and Cauchy's integral formula||Characterize the types of singular points and derive a way of calculating integrals|
|Class 6||Residue theorem||Discuss Laurent series expansion and obtain residue|
|Class 7||Application of complex integral to real-valued integral||Be able to differentiate poles, essential singular points, and removable singular points|
|Class 8||Conformal mapping and linear fractional transformation||Go over the conformal mapping and linear fractional transformation|
|Class 9||Basics of Laplace transform||Define Laplace transform and see its application to fundamental functions|
|Class 10||Application of Laplace transform to differential equations: partial fraction expansion||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|
To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to textbooks and other course material.
Erwin Kreyszig著 『Advanced Engineering Mathematics』 Wiley
No English reference
Student's knowledge of complex analysis, Laplace transform associated with the theory of differential equations, Fourier analysis, and their application to physical problems will be assessed. Attendance of the lectures counts.
Mid-term exam 35%, final exam 45% and exercise problems 25%.
Students must have successfully completed both Calculus I and Calculus II, or have equivalent knowledge.
Contact by e-mail in advance to schedule an appointment.