Analysis for Electrical and Electronic Engineers focuses on the fundamentals of mathematical methodology of complex functions and differential equations.
Topics covered in this course will include the following: basic arithmetics of complex number, fundamentals of complex functions, differential calculus, regular function, Cauchy–Riemann equations, integration, Cauchy's theorem, Cauchy's integral expression, Goursat's theorem, residue theorem, 1st and 2nd order ordinary differential equation, partial differential equation, etc. Problem solving is essential for mastery of calculus and mathematical insight.
Firstly, complex number and its arithmetic calculation will be defined. Expression of complex plane will help your deep understandings of the relation between real and complex number. After introduction of power of complex and nth root, exponential function, trigonometric function and power series, etc. will be defined as an expansion of real functions. Subsequently, derivation of complex function will be introduced. Concept of differentiability of complex function and its condition will be given, which enables the expression of differential equation widely used in science and engineering field. Integral of complex function will be defined as a line integral in complex plane. Cauchy's integration formula will be explained, with which students shall acquire the calculation method of integration.
Latter half of this lecture will be started by the theory of solving the 1st order differential equation using variable separation method and variation constants method. To solve 2nd order differential equation, characteristic equation will be introduced. Relation between differential equation and electrical circuit will be pointed out. Finally, partial differential equations will be introduced and wave equation will be focused as an important application of electrical and electronic engineering. Through this lecture, students will acquire the mathematical principle and methodology for solving the problems in wide range of engineering field.
By the end of this course, students will be able to:
1) Understand the definition of complex number and carry out algebratic calculation of complex.
2) Understand the definition of power series, nth root, trigonometric function, exponential function, etc. and carry out calculation with those functions.
3) Understand the definition of derivation of complex function and the concept of differentiable function. Also explain harmonic function.
4) Define integration of complex function and carry out integral calculation using Cauchy's integration formula.
5) Understand and solve 1st order differential equations using variable separation method and variation of constants method.
6) Understand and solve 2nd order differential equations using characteristic equation. And understand the mathematical relation between electrical circuit and differential equations.
7) Understand and solve differential equations combining with variation of constants method, method of Lagrange multiplier and power series expansion.
8) Understand and solve the basic partial differential equations including Wave equations and Diffusion equation.
Corresponding educational goals are:
(1) Specialist skills Fundamental specialist skills
(6) Firm fundamental specialist skills on electrical and electronic engineering, including areas such as electromagnetism, circuits, linear systems, and applied mathematics
Former half: Complex, complex function, regular function, complex differential, Cauchy-Riemann equation, the complex line integral, series, radius of convergence, Laurent expansion, singularity, residue, principal value integration
Latter half: ordinary differential equations (ODF), separation of variable, homogeneous, 1st-order linear ODF, 2nd-order ODF, homogeneous linear ODF, non-homogeneous linear ODF, variation of parameters, circuit analysis, simultaneous linear ODF, partial differential equations
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
At the beginning of every lecture, the answer of the previous exercises will be explained as a review. At the end of the lecture, we will question the exercises related to contents of the day. Required learning should be completed outside of the classroom for preparation and review purposes.
|Course schedule||Required learning|
|Class 1||Complex number, complex plane, polar expression, De Moivre's formula, n-th root, the complex plane figure||Explain the definition of complex number with imaginary unit as well as complex plane and polar expression. Understand and carry out calculation of nth root of complex. Derive the expression of plane figures on complex plane.|
|Class 2||Complex function, mapping, linear transformation, exponential, trigonometric, hyperbolic, logarithmic, and power functions||Understand the definitions and carry out calculations of complex function, mapping, linear transformation, exponential, trigonometric, hyperbolic, logarithmic, and power functions.|
|Class 3||Differential, regularity, Cauchy-Riemann equations, harmonic function, the complex potential||Understand differential and regularity of complex functions, Cauchy-Riemann equations, harmonic function, and the complex potential.|
|Class 4||Line integral, integral formula, the integration path, parametric expression of the integration path, complex line integration, integration of integer power function||Understand the derivation and carry out calculation with line integral, and integral formula of complex functions, the integration path, parametric expression of the integration path, complex line integration, and integration of integer power function.|
|Class 5||Cauchy's integral theorem, change of the path of integration, Cauchy's integral formula, the derivative||Understand Cauchy's integral theorem, change of the path of integration, Cauchy's integral formula, and the derivative of complex functions.|
|Class 6||Series, radius of convergence, Taylor expansion, power series expression, Laurent expansion, singularity||Understand definitions and carry out calculation with series of complex, radius of convergence, Taylor expansion, power series expression, Laurent expansion, and singularity of complex functions.|
|Class 7||Residue, how to obtain a residue, residue theorem, the definite integral of trigonometric functions and rational functions, the principal value of integration, etc.||Understand and carry out calculation of residue and how to obtain a residue of complex functions, residue theorem, the definite integral of trigonometric functions and rational functions, and the principal value of integration, etc.|
|Class 8||Confirmation of understanding of contents of the lectures up to this week and commentary||Confirmation is carried out up to the previous lectures.|
|Class 9||Basic matters of differential equations and differential equations with separation of variables and homogeneous differential equations.||Understand what the ordinary differential equations (ODF) are, and do exercise of 1st order ODF's with separation of variables and 1st-order homogeneous ones.|
|Class 10||First-order linear homogeneous equations, first-order linear non-homogeneous equations, and first-order exact ODFs.||Understand and solve 1st-order linear homogeneous ODFs, 1st-order linear non-homogeneous ODFs, and 1st-order exact ODFs.|
|Class 11||Circuit analysis (LR, CR circuit), the structure of solutions of the second-order linear differential equations, and Second-order linear homogeneous differential equations with constant coefficients.||Understand and solve circuit analysis (LR, CR circuit) as ODF, the structure of the solution of the second floor linear equations, and Second-order linear homogeneous differential equations with constant coefficient.|
|Class 12||Second-order linear non-homogeneous equations and method of variation of parameters. Also, interim confirmation of understanding of contents of the lectures up to this week and commentary.||Understand and solve 2nd-order linear non-homogeneous ODFs and method of variation of parameters. Also, confirmation is carried out up to the previous lectures of differential equations.|
|Class 13||Circuit analysis (LCR, RLC circuit), series expansion method||Understand and solve circuit analysis (LCR, RLC circuit) as ODFs, and series expansion method.|
|Class 14||Simultaneous first-order linear ordinary differential equations with constant coefficients.||Understand and solve simultaneous 1st-order linear ODFs with constant coefficients.|
|Class 15||Prospects to partial differential equations||Understand and solve problems to have prospects to partial differential equations.|
Former half: "Complex analysis for students in standard engineering course", Koichi Yasuoka and Jiro Hirokawa, Kodansha
Latter half: "Differntial equations for students in standard engineering course, Jiro Hirokawa and Koichi Yasuoka, Kodansha
Nobuyuki Naito "Fundamental mathematics for electrical and electronic engineering", Institute of Electrical and Electronic Engineers, Japan
【Score】 Summation of 20 points for exercises and 20 points × 4 times examination (in Lecture 4, 8 , 12 and final examination).
Both the understanding of mathematical principle and the capability of calculation will be assessed.
To have mastered calculus and linear algebra of 1st-year undergraduate level.