Faculty of Engineering and Natural Sciences · Electrical & Electronics Engineering (English 30%) · Undergraduate
Course Objective
The main aim of this course is to learn: Complex Numbers and Complex Plane, Complex Functions and Mappings, Analytic and Harmonic Functions, Cauchy-Riemann Equations, Elementary Functions: Complex Exponential, Logarithmic, Trigonometric and Inverse Trigonometric Functions, Integration in the Complex Plane, Counter Integral, Cauchy Theorem, Homotopy and Simple Connected Regions, Cauchy-Goursat Theorem, Cauchy Integral Formulas and Their Conseqeuences, Complex Sequences and Series, Taylor Series of Analytic Functions, Laurent Series, Singular and Isolated Singular Points, Zeros and Poles, Residue and Residue Theorem
Course Content
Complex Numbers and Complex Plane, Complex Functions and Mappings, Analytic and Harmonic Functions, Cauchy-Riemann Equations, Elementary Functions: Complex Exponential, Logarithmic, Trigonometric and Inverse Trigonometric Functions, Integration in the Complex Plane, Counter Integral, Cauchy Theorem, Homotopy and Simple Connected Regions, Cauchy-Goursat Theorem, Cauchy Integral Formulas and Their Conseqeuences, Complex Sequences and Series, Taylor Series of Analytic Functions, Laurent Series, Singular and Isolated Singular Points, Zeros and Poles, Residue and Residue Theorem
Course Learning Outcomes
- 1. Work with complex numbers
- 2. Be familiar with complex plane and complex functions.
- 3.Study on differentiability and analyticity of complex functions
- 4. Understand a necessary conditions analyticity: Cauchy-Riemann equations
- 5.Understand the similarities and differences between the real and complex elementary functions
- 6.. Set up and directly evaluate contour integrals and complex integrals
- 7.Evaluate contour integrals using the Cauchy Integral Theorem.
- 8.Understand the complex sequences and series
- 9.Find Taylor or Laurent series for simple functions and show understanding of the convergence regions for each type of series.
- 10.Identify and classify zeros and singular points of functions, compute residues and use residues to evaluate various contour integrals.
- 11. Evaluate improper integrals and definite integrals involving sines and cosines
Core Area Distribution
Teaching Methods
Assessment & Evaluation
ECTS / Workload
| Activity | Quantity | Duration (h) | Total Workload |
|---|---|---|---|
| Course Duration (Including Exam Week) | 16 | 3 | 48 |
| Out of Class Study Period | 16 | 3 | 48 |
| Midterm | 1 | 2 | 2 |
| Quiz | 10 | 1 | 10 |
| Assignment | 3 | 5 | 15 |
| Practice | 0 | 0 | 0 |
| Final | 1 | 2 | 2 |
Course Schedule
| Week | Subject | Preparation |
|---|---|---|
| 1 | • Complex Numbers and Their Properties • Polar Representation for Complex Numbers • de Moivre’s formula | Chartesian plane |
| 2 | Complex Powers and Roots • Sets of Points in the Complex Plane • Applications | Polar Coordintes |
| 3 | • Complex Functions • Complex Functions as Mappings | Functions and Their Graphs |
| 4 | • Limits and Continuity of Complex Functions • Analytic Functions | Limits and Continuity |
| 5 | • Cauchy-Riemann Equations • Harmonic Functions | Differentiation of Real Functions |
| 6 | • Elementary Functions: I. Complex Exponential II. Complex Trigonometric III. Complex Logarithmic Functions IV. Complex Inverse Trigonometric Functions | Elementary Functions and Their Properties |
| 7 | • Complex Integral • Countour Integral • Cauchy-Goursat Theorem | Line Integral |
| 8 | ||
| 9 | • Cauchy’s Integral Formulas and Their Consequences | Line integral |
| 10 | • Cauchy’s Integral Formula’s Recitation | Complex Integral • Countour Integral • Cauchy-Goursat Theorem |
| 11 | • Complex Sequences and Series • Series and Convergence Theorems with Complex Variables | Real Sequences, Series and Their Convergence Theorems |
| 12 | • Complex Power Series • Integration and Differentiation of Power Series | Series and Convergence Theorems with Complex Variables |
| 13 | • Taylor Series of Analytic Functions • Laurent Series | Taylor and Maclaurin Series |
| 14 | • Zeros and Poles • Residues • Cauchy’s Residue Theorem | Taylor Series of Analytic Functions • Laurent Series |
| 15 | Review for Final Exam | Review for Final Exam |