MATH555 THEO. OF FUNC. OF A COMPLEX VARIABLE
| Course Code: | 2360555 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Assoc.Prof.Dr. ÖZCAN YAZICI |
| Offered Semester: | Fall Semesters. |
Course Objectives
At the end of the course, students are expected to learn
- basic properties of analytic functions: Cauchy theory, power series, analytic continuation.
- singularities: Laurent series, classification of singular points, Residue theorem.
- zeros of analytic functions: argument principle, Rouche’s theorem, Hurwitz’s theorem, open mapping theorem
- global properties of analytic functions: Schwarz lemma, conformal mappings, Riemann mapping theorem, reflection principle
- compact families of functions: Montel and Picard theorems. Approximation theorems: Runge theorem, Mittag-Lefler theorem, infinite products.
Course Content
Analytic functions. Singular points and zeros. The argument principle. Conformal mappings. Riemann Mapping Theorem. Mittag-Lefler Theorem. Infinite products. Canonical products. Analytical continuation. Elementary Riemann surfaces.
Course Learning Outcomes
After successfully completing this course, students will be able to:
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define and explain the foundational concepts of complex analysis, including analytic functions, complex differentiability, and the Cauchy-Riemann equations.
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understand conformal mappings.
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evaluate contour integrals and derive properties of analytic functions.
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construct and analyze power series, Laurent series, and Taylor series expansions of complex functions.
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classify isolated singularities (removable, poles, essential) and analyze the behavior of functions near singularities using Laurent expansions.
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use the Residue Theorem to evaluate real and complex integrals.
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apply principles of analytic continuation and monodromy to extend the domains of analytic functions.
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understand and use concepts from the argument principle, Rouche’s Theorem to analyze zeros and poles of complex functions.
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use Schwarz lemma, conformal mappings, Riemann mapping theorem, reflection principle to analyze global properties of analytic functions.
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use Montel Theorem, Runge theorem, Mittag-Lefler theorem, infinite products to analyze compact families of functions.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution