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: 1st and 2nd order ordinary diﬀerential equation, partial diﬀerential equation, etc., 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. Problem solving is essential for mastery of calculus and mathematical insight.
Firstly, former 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.
Secondly, 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.
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 and solve 1st order diﬀerential equations using variable separation method and variation of constants method.
2) Understand and solve 2nd order diﬀerential equations using characteristic equation. And understand the mathematical relation between electrical circuit and diﬀerential equations.
3) Understand and solve diﬀerential equations combining with variation of constants method, method of Lagrange multiplier and power series expansion.
4) Understand and solve the basic partial diﬀerential equations including Wave equations and Diﬀusion equation.
5) Understand the deﬁnition of complex number and carry out algebratic calculation of complex.
6) Understand the deﬁnition of power series, nth root, trigonometric function, exponential function, etc. and carry out calculation with those functions.
7) Understand the deﬁnition of derivation of complex function and the concept of diﬀerentiable function. Also explain harmonic function.
8) Deﬁne integration of complex function and carry out integral calculation using Cauchy's integration formula.
Former 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
Latter 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
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
|✔ ・Fundamental specialist skills on EEE|
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||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 2||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 3||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 4||Second-order linear non-homogeneous equations and method of variation of parameters.||Understand and solve 2nd-order linear non-homogeneous ODFs and method of variation of parameters.|
|Class 5||Circuit analysis (LCR, RLC circuit), series expansion method.||Understand and solve circuit analysis (LCR, RLC circuit) as ODFs, and series expansion method.|
|Class 6||Simultaneous ﬁrst-order linear ordinary diﬀerential equations with constant coeﬃcients.||Understand and solve simultaneous 1st-order linear ODFs with constant coeﬃcients.|
|Class 7||Prospects to partial diﬀerential equations.||Understand and solve problems to have prospects to partial diﬀerential equations.|
|Class 8||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 9||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 10||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 11||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 12||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 13||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 14||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.|
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Former half: "Diﬀerntial equations for students in standard engineering course, Jiro Hirokawa and Koichi Yasuoka, Kodansha
Latter half: "Complex analysis for students in standard engineering course", Koichi Yasuoka and Jiro Hirokawa, Kodansha
Nobuyuki Naito "Fundamental mathematics for electrical and electronic engineering", Institute of Electrical and Electronic Engineers, Japan
【Score】 Summation of 20 points for ordinary exercises as homework and 20 × 4 times points for understanding confirmation exercises (4th, 7th, 11th, and 14th).
As an understanding confirmation exercise, 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.