Learning purposes : Start with the basic calculation of complex numbers and learn about the calculus of functions with complex numbers as variables (complex functions).
Course Objectives :
1. Understand the basics and properties of complex numbers.
2. Understand the properties of complex functions and their limits and derivatives.
3. Learn the integral of complex functions.
4. Understand Laurent expansion and residue theorem.
Outline:
General or Specialized : Specialized
Field of learning : Mathematics / Physics
Required, Elective, etc. : Must complete subjects
Foundational academic disciplines : Analysis, applied mathematics and related fields / basic analysis
Relationship with Educational Objectives : This class is equivalent to "(3) Acquire deep foundation knowledge of the major subject area".
Relationship with JABEE programs : The main goals of learning / education in this class are "(A), A-1".
Course outline : In complex functions, Cauchy's integral theorem and other theories that are quite different from functions that use real numbers as variables are developed, and finally the integral theorem of complex functions called the residue theorem is reached. By applying the residue theorem to the integration of real functions, it is possible to obtain the value of the integral that is difficult to calculate in real numbers.
Style:
Course method : Classes will be centered on board writing, but at the same time, as much exercise time as possible will be provided so that students can understand the content of the lecture more deeply and acquire the ability to solve problems on their own.
Grade evaluation method : Evaluate the total of 4 regular exams (60% evaluated equally) and other exams, exercises, reports, and lesson approaches (40%). Depending on the grades, etc., a re-examination may be conducted (report submission is required). The retest will be evaluated in the same way as the main test, with an upper limit of 80 points.
Notice:
Precautions on the enrollment : Students must take this class (no more than one-third of the required number of class hours missed) in order to complete the 5th year course.
Course advice : It is important to make sure to prepare and review, and to understand the lecture contents more deeply by solving the exercises on your own.
Foundational subjects : Fundamental Mathematics (1st year), Fundamental Mathematics Practice (1st), Differential and Integral Ⅰ (2nd), Fundamental Linear Algebra (2nd), Integrated Science and Technology Practice (2nd), Differential and Integral Ⅱ (3rd), Fundamental Differential Equations (3rd), Applied Mathematics Ⅰ (4th), Applied Mathematics Ⅱ (4th)
Related subjects : Mathematics in general
Attendance advice : In complex functions, various theorems are precisely combined to develop the theory. It is necessary to try to understand the interrelationships between the theorems. It is also important to understand the content of the lecture well and solve the problem by yourself. I want you to value finding a solution on your own. If you are late a lot, you may be treated as absent after giving a warning.
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Theme |
Goals |
| 1st Semester |
| 1st Quarter |
| 1st |
Guidance, complex plane |
Understand complex numbers and their calculations.
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| 2nd |
Complex plane |
Understand conjugate complex numbers and complex planes.
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| 3rd |
Complex plane |
Understand the absolute value of complex numbers and the distance between two points.
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| 4th |
Pole form |
Understand the declination and pole form of complex numbers.
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| 5th |
Pole form |
Understand the product of complex numbers in polar form, the quotient, and De Moivre's formula.
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| 6th |
Pole form |
Understand Euler's formula, nth root.
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| 7th |
1st semester mid-term exam |
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| 8th |
Return and commentary of exam answers |
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| 2nd Quarter |
| 9th |
Complex function |
Understand complex functions.
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| 10th |
Basic complex function |
Understand exponential and trigonometric functions.
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| 11th |
Limit of complex functions |
Understand the limit of complex functions and the continuity of complex functions.
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| 12th |
Cauchy-Riemann relations |
Understand the differentiability and holomorphic functions of complex functions.
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| 13th |
Cauchy-Riemann relations |
Understand Cauchy-Riemann's relational expression.
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| 14th |
Holomorphic function and its derivative |
Understand derivative formulas, exponential derivatives, and trigonometric derivatives.
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| 15th |
(1st semester final exam) |
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| 16th |
Return and commentary of exam answers |
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| 2nd Semester |
| 3rd Quarter |
| 1st |
Integral of complex function |
Understand curves and complex integrals on the complex plane.
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| 2nd |
Integral of complex function |
Understand the properties of complex integrals.
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| 3rd |
Integral of complex function |
Understand the integral along a single closed curve.
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| 4th |
Cauchy's integral theorem |
Understand Cauchy's integral theorem.
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| 5th |
Cauchy's integral theorem |
Understand Cauchy's integral theorem.
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| 6th |
Cauchy's integral display |
Understand Cauchy's integral representation.
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| 7th |
Cauchy's integral display |
Understand the extension of Cauchy's integral representation.
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| 8th |
2nd semester mid-term exam |
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| 4th Quarter |
| 9th |
Return and commentary of exam answers |
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| 10th |
series |
Understand the limit of a sequence, the convergence and divergence of a sequence, the power series, the power series and its radius of convergence, and the power series expansion of a function.
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| 11th |
Taylor expansion |
Understand the Taylor expansion of holomorphic functions.
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| 12th |
Laurent deployment |
Understand Laurent development.
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| 13th |
Residue |
Understand isolated singularities, classification of isolated singularities, residues, pole orders and residues.
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| 14th |
Resume theorem |
Understand the residue theorem and its application to real integration.
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| 15th |
(2nd semester final exam) |
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| 16th |
Return and commentary of exam answers |
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