Applied Mathematics Ⅰ

Course Information

College	Toyama College	Year	2024
Course Title	Applied Mathematics Ⅰ
Course Code	0066	Course Category	Specialized / Elective
Class Format	Lecture	Credits	School Credit: 1
Department	Department of Mechanical Engineering	Student Grade	3rd
Term	First Semester	Classes per Week	前期:2
Textbook and/or Teaching Materials	「基礎解析学」（矢野健太郎，石原繁），裳華房
Instructor	Shirakawa Hidemi

Course Objectives

At the completion of this course, students will be able to
1) Calculate complex numbers.
2) Understand the polar form and complex plane of complex numbers.
3) Understand the meaning of regular functions and their properties, complex plane.
4) Understand complex functions can be differentiated and Cauchy Riemann's theorem.
5) Understand complex functions can be integrated and Cauchy's theorem.
6) Understand Taylor expansion and Laurent expansion of complex functions.
7) Understand the residue theorem can be understood and solve application problems of residues.
Each item of the rubric shown below becomes the target.

Rubric

	Ideal Level of Achievement	Standard Level of Achievement	Unacceptable Level of Achievement)
Complex numbers calculation	Can calculate complex numbers.	Can understand complex numbers.	Can't calculate complex numbers.
Complex polar forms and complex planes	Can explain complex polar forms and complex planes.	Can understand complex polar forms and complex planes.	Can't understand complex polar forms and complex planes.
tThe meaning and nature of regular functions and complex function planes	Can explain the meaning and nature of regular functions and complex function planes.	Can understand the meaning and nature of regular functions and complex function planes.	Can't understand the meaning and nature of regular functions and complex function planes.
Differentiation of complex functions and Cauchy Riemann's theorem	Can explain the differentiation of complex functions and Cauchy Riemann's theorem.	Can understand the differentiation of complex functions and Cauchy Riemann's theorem.	Can't understand differentiation of complex functions and Cauchy Riemann's theorem.
The integration of complex functions and Cauchy's theorem	Can explain the integration of complex functions and Cauchy's theorem.	Can understand the integration of complex functions and Cauchy's theorem.	Can't understand the integration of complex functions and Cauchy's theorem.
The development of complex functions such as the Taylor expansion and Laurent expansion	Can explain the development of complex functions such as the Taylor expansion and Laurent expansion.	Can understand the development of complex functions such as the Taylor expansion and Laurent expansion.	Can't understand the development of complex functions such as the Taylor expansion and Laurent expansion.
Application of the residue theorem	Can solve application problems of the residue theorem.	Can solve problems of the residue theorem.	Can't solve problems of the residue theorem.

Teaching Method

Outline:

Functions of complex variables are often used for stability determination in vibration phenomena and analysis of mechanical control, and knowledge of functions of complex variables is necessary to safely control a machine. Therefore, in this subject, the objective is to understand the calculation methods and rules of the complex variable functions to be used in control engineering, etc.

Style:

Lecture and exercise. First, the calculation of complex numbers , the pole form and complex plane of complex numbers will be revivewed and understood. Next, the regular functions, the complex function plane transformed by the function with various functions, will be understood. The differentiation and integration of the complex functions are solved and the residue theorem from the expansion of the complex functions are explained. Finally, application problems using the residence theorem are solved.

Notice:

The complex function theory is often used in fluid engineering, control engineering, etc. Please review your textbooks closely, and review the fundamentals of mathematics on your own. Also, since we will give a report after class, solve the problems, and please make sure you fully understand them before submitting them. Lesson plans may be changed according to the student's degree of understanding.Can take makeup exam in need aid up to maximum of 60 points.

Course Plan

			Theme	Goals
1st Semester
	1st Quarter
		1st	Complex numbers and arithmetic operations	Conjugate complex numbers can be understood. Complex numbers can be calculated.
		2nd	Complex number plane (z plane) and vector, complex type polar form	The complex number plane (z plane) can be understood. Absolute value calculation of vectors, complex numbers, argument and polar form of complex number can be calculated.
		3rd	Functions and complex function plane (w plane)(1)	Exponential function, complex function plane, Euler's formula, de Moivre's theorem, power, n th root can be understood.
		4th	Functions and complex function plane (w plane)(2)	Trigonometric functions, hyperbolic functions, logarithmic functions can be understood.
		5th	Differentiation of complex functions (1)	Function limit, continuous function, differentiable, regular function, differentiation of power can be understood.
		6th	Differentiation of complex functions (2)	Cauchy-Riemann's theorem, derivative of regular function, harmonic function, exponential function, trigonometric function, hyperbolic function, logarithmic function can be understood.
		7th	Integration of complex functions (1)	The general definition of complex integral and integral path, integral of circular integral path, indefinite integral, complex integral can be understood.
		8th	Intermediate examination
	2nd Quarter
		9th	Explanation of answer of intermediate examination
		10th	Integration of complex functions (2)	The closed curve integral path and area, Green's theorem, Cauchy's theorem can be understood.
		11th	Integration of complex functions (3)	Cauchy's integrated display can be understood.
		12th	Expansion of complex functions	Taylor expansion (McLoughlin development), Laurent expansion (function having singularities) can be understood.
		13th	Residue	Poles and residues, residue calculation, residue theorem can be understood.
		14th	Application of residues	The integration using the residue can be understood.
		15th	Term-end examination
		16th	Explanation of answer of term-end examination and a questionnaire

Evaluation Method and Weight (%)

	Examination	Report	Total
Subtotal	70	30	100
Understanding degree	70	30	100