| １． |
Outline |
|
We will explain various concepts including four arithmetic operations with complex numbers, power series expansions and Euler’s formula, complex functions and mapping, elementary functions of complex numbers, regular functions and Cauchy-Riemann relations, complex integrals and residues, and conformal mapping and harmonic functions. We will also examine the harmonic functions applied to fluid mechanics and heat transfer.
|
| ２． |
Objectives |
|
In these lectures, we move from the world of real numbers to the world of complex numbers, and expand this to regular functions to complex functions. It is not easy to approach complex numbers and complex functions, but graphing them on a Gaussian plane enables to visualize them on a regular Cartesian coordinate plane, which helps to understand the differentials and integrals of complex functions. The students will gain an understanding not only of various concepts related to complex functions, but also of the theory of lift acting on a wing as an example of applications in fluid mechanics.
|
| ３． |
Grading Policy |
|
Students will be evaluated based on the following two exams.
Mid-term test 25% Test and post-test commentary to help learn later.
Final exam 75%
In addition, as mid-term test's feedback, we will explain the problem after the final exam.
The mid-term test results should be returned to each person and be understood through practice questions which will be distributed separately. In addition, before and after the mid-term exam, we will give an overview of the entire course and explain the main points of learning.
|
| ４． |
Textbook and Reference |
|
Textbook: Masato Murakami "Complex Functions", Kaimeisha Co.Ltd.ISBN4-87525-206-4
|
| ５． |
Requirements (Assignments) |
|
You need to have a good understanding of operations on complex numbers, differentiation and integration of real functions.
About preparatory learning: Each lecture will advance about 20 pages of the textbook, so take approximately one and a half hours to read this range carefully, summarize the questions, etc., and attend the lecture.
About reviewing: Always read back the textbook, learn the definitions and other things that you should remember first, and work repeatedly until you can solve the textbook exercises. I think that there are individual differences, but let's review over one and a half hours every time.
As you will print and distribute the exercises at the end of the mid-term test, let's practice again and again to understand the logic and get the correct answer.
More than 36 hours are necessary for the above-mentioned preliminary review in the period concerned.
|
| ６． |
Note |
|
It sounds like a very difficult subject when it comes to complex functions, but it is important to understand and use trigonometric functions, exponential functions, knowledge of differential and integral functions, and arc degree methods such as what you learn from high school to the first year of university. Familiarize yourself with the Taylor expansion of real functions and focus on where this is used to define complex functions.
It is necessary to take notes firmly. There are many practice questions listed in the textbook, so write your answers on your notes and practice over and over again.
Related subjects: Flows around wings
Reference books: Shiga Koji "A Trip to Mathematics 7 Days" Kinokuniya Bookstore is a very good book to help you understand complex functions, but unfortunately it was out of print several years ago.
|
| ７． |
Schedule |
|
１. Imaginary numbers and complex numbers (Imaginary numbers are roots and discriminants of quadratic equations, classifications of numbers)
|
２.
Imaginary and complex numbers (addition and subtraction of complex number, complex planes, complex numbers and vectors)
|
３. Power series expansion of functions
|
４. Euler's formula and its applications (Nth root of one, de Moivre's theorem)
|
５. Complex plane and polar form
|
６. Complex function (mapping, quadratic function of complex variable)
|
７. Complex functions (power series expansion of real functions, elementary functions of complex variables)
|
８. Complex functions and derivatives (singularities, regular functions, Cauchy-Riemann relations)
|
９. Integration of complex numbers (Integration of power series, integral formula of real function, contour integral)
|
10. Complex integral (integration along a closed curve, Cauchy's integral theorem)
|
11. Intermediate examination, Explanation of the problem
|
12. Complex integral (residue, Laurent expansion, pole)
|
13. Complex integral (application to integral of real function )
|
14. Conformal mapping (conformal mapping of elementary functions, Jewowski transformation)
|
15. Harmonic function (Laplace equation, application of conformal mapping)
|
|
| １． |
Outline |
|
We will explain various concepts including four arithmetic operations with complex numbers, power series expansions and Euler’s formula, complex functions and mapping, elementary functions of complex numbers, regular functions and Cauchy-Riemann relations, complex integrals and residues, and conformal mapping and harmonic functions. We will also examine the harmonic functions applied to fluid mechanics and heat transfer.
|
| ２． |
Objectives |
|
In these lectures, we move from the world of real numbers to the world of complex numbers, and expand this to regular functions to complex functions. It is not easy to approach complex numbers and complex functions, but graphing them on a Gaussian plane enables to visualize them on a regular Cartesian coordinate plane, which helps to understand the differentials and integrals of complex functions. The students will gain an understanding not only of various concepts related to complex functions, but also of the theory of lift acting on a wing as an example of applications in fluid mechanics.
|
| ３． |
Grading Policy |
|
Students will be evaluated based on the following two exams.
Mid-term test 25% Test and post-test commentary to help learn later.
Final exam 75%
In addition, as mid-term test's feedback, we will explain the problem after the final exam.
The mid-term test results should be returned to each person and be understood through practice questions which will be distributed separately. In addition, before and after the mid-term exam, we will give an overview of the entire course and explain the main points of learning.
|
| ４． |
Textbook and Reference |
|
Textbook: Masato Murakami "Complex Functions", Kaimeisha Co.Ltd.ISBN4-87525-206-4
|
| ５． |
Requirements (Assignments) |
|
You need to have a good understanding of operations on complex numbers, differentiation and integration of real functions.
About preparatory learning: Each lecture will advance about 20 pages of the textbook, so take approximately one and a half hours to read this range carefully, summarize the questions, etc., and attend the lecture.
About reviewing: Always read back the textbook, learn the definitions and other things that you should remember first, and work repeatedly until you can solve the textbook exercises. I think that there are individual differences, but let's review over one and a half hours every time.
As you will print and distribute the exercises at the end of the mid-term test, let's practice again and again to understand the logic and get the correct answer.
More than 36 hours are necessary for the above-mentioned preliminary review in the period concerned.
|
| ６． |
Note |
|
It sounds like a very difficult subject when it comes to complex functions, but it is important to understand and use trigonometric functions, exponential functions, knowledge of differential and integral functions, and arc degree methods such as what you learn from high school to the first year of university. Familiarize yourself with the Taylor expansion of real functions and focus on where this is used to define complex functions.
It is necessary to take notes firmly. There are many practice questions listed in the textbook, so write your answers on your notes and practice over and over again.
Related subjects: Flows around wings
Reference books: Shiga Koji "A Trip to Mathematics 7 Days" Kinokuniya Bookstore is a very good book to help you understand complex functions, but unfortunately it was out of print several years ago.
|
| ７． |
Schedule |
|
１. Imaginary numbers and complex numbers (Imaginary numbers are roots and discriminants of quadratic equations, classifications of numbers)
|
２.
Imaginary and complex numbers (addition and subtraction of complex number, complex planes, complex numbers and vectors)
|
３. Power series expansion of functions
|
４. Euler's formula and its applications (Nth root of one, de Moivre's theorem)
|
５. Complex plane and polar form
|
６. Complex function (mapping, quadratic function of complex variable)
|
７. Complex functions (power series expansion of real functions, elementary functions of complex variables)
|
８. Complex functions and derivatives (singularities, regular functions, Cauchy-Riemann relations)
|
９. Integration of complex numbers (Integration of power series, integral formula of real function, contour integral)
|
10. Complex integral (integration along a closed curve, Cauchy's integral theorem)
|
11. Intermediate examination, Explanation of the problem
|
12. Complex integral (residue, Laurent expansion, pole)
|
13. Complex integral (application to integral of real function )
|
14. Conformal mapping (conformal mapping of elementary functions, Jewowski transformation)
|
15. Harmonic function (Laplace equation, application of conformal mapping)
|
|
|