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Information
| Code | MT334 |
| Name | Theory of Complex Functions |
| Semester | 6. Semester |
| Duration (T+A) | 5-0 (T-A) (17 Week) |
| ECTS | 8 ECTS |
| National Credit | 5 National Credit |
| Teaching Language | Türkçe |
| Level | Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. ALİ ARSLAN ÖZKURT |
Course Goal
The aim of this course is to acquaint the student with the theory of the calculus of a function of a complex variable and then to introduce the basic theory and ideas of the integration of a function of a complex variable, state the main theorems such as Cauchy s theorem, Cauchy integral formula, and the Cauchy s residue theorem with endowing the students with practical skills in evaluating real and complex integrals.
Course Content
Complex numbers, regions, transformations, limit, continuity, differentiation, Cauchy-Riemann equations, Analytic functions, Harmonic functions, elementary transformations, transformations by elementary functions, integrals, contour integrals, Cauchy-Goursattheorem, residue, applications of residue: improper integrals.
Course Precondition
None
Resources
Kompleks Fonksiyonlar Teorisi , author :Turgut Başkan, Kompleks Değişkenli Fonksiyonlar Teorisi, author:Metin Başarır
Notes
Complex Variables and Appliations, author: J.W.Brown, R.V. Churchill
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Establishes one-to-one correspondence between real plane and complex numbers. |
| LO02 | lnvestigates the existence of derivatives of complex functions and differentiates functions of a complex variable. |
| LO03 | Evaluates contour integrals in complex planes. |
| LO04 | Evaluates real and complex integrals using the Cauchy s Theorem and Cauchy integral formula. |
| LO05 | Classifies singular points of complex functions. |
| LO06 | Determines whether complex functions are analytic. |
| LO07 | Finds Taylor and Laurent series of complex functions. |
| LO08 | Evaluates complex integrals using the residue theorem |
| LO09 | Evaluates some real integrals using complex integration technique |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Comprehend the ability to prove the mathematical knowledge gained in secondary education on the basis of theoretical basis. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Understands importance of basic consepts of Algebra, Analaysis and Topology. | 3 |
| PLO03 | Yetkinlikler - Öğrenme Yetkinliği | Mathematical reasoning demonstrates the ability to develop and write mathematical proofs by gaining maturity. | 2 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to express the basic theories of mathematics both correctly. | 5 |
| PLO05 | Bilgi - Kuramsal, Olgusal | Understands the relationship between the different fields of mathematics and its relation to other disciplines. | 3 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Comprehends the ability to understand the relationships between the objects in the most understandable way while creating a model for any problem. | 4 |
| PLO07 | Bilgi - Kuramsal, Olgusal | Comprehend and explain mathematical models such as formulas, graphs, tables and schema. | 5 |
| PLO08 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to mathematically rearrange, analyze, and model the problems they encounter. | 3 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Comprehends at least one of the computer programming languages. | 1 |
| PLO10 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to use scientific methods and appropriate technologies effectively in problem solving. | 4 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Understands sufficient knowledge of foreign language to be able to understand Mathematical concepts and communicate with other mathematicians | |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | In addition to their professional development, they demonstrate their ability to continuously improve themselves by identifying their educational needs in scientific, cultural, artistic and social areas in line with their interests and abilities. | 2 |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Understands the programming techniques and shows the ability to do programming. | |
| PLO14 | Yetkinlikler - Öğrenme Yetkinliği | Demonstrates the ability to study mathematics both independently and as a group. | 4 |
| PLO15 | Bilgi - Kuramsal, Olgusal | Demonstrate an awareness of the universal and social impacts and legal consequences of mathematical applications in the field of study. | 4 |
| PLO16 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to select, use and develop effectively for contemporary mathematical applications. | 3 |
| PLO17 | Bilgi - Kuramsal, Olgusal | It has ability of lifelong learning awareness, access to information, monitoring developments in science and technology and self-renewal ability. | 3 |
| PLO18 | Bilgi - Kuramsal, Olgusal | Gains the ability to use information technologies effectively for contemporary mathematical applications. | 3 |
| PLO19 | Bilgi - Kuramsal, Olgusal | Gains the ability to design, conduct experiments, field work, data collection, analysis, archiving, text solving and / or interpretation according to mathematics fields. | 3 |
| PLO20 | Bilgi - Kuramsal, Olgusal | Gains the consciousness of prefesional ethics and responsibility. | 4 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Basic properties of comlex numbers, Polar forms, powers, roots, domains | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 2 | Functions of a complex variable, limit and Limit theorems | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 3 | Continuity, derivatives and the Cauchy-Riemann equations | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 4 | Sufficient conditions for derivatives, analytic functions, harmonic functions | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 5 | Exponential, logarithmic, trigonometric, hyperbolic, inverse trigonometric functions | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 6 | Line integrals, upper bound for integrals, anti-derivatives | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 7 | Cauchy-Goursat theorem , Cauchy s integral formula, simply and multiply connected domains | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 8 | Mid-Term Exam | Review of the topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Taylor and Laurent series | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 10 | sums and product of the series | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 11 | Residues, Cauchy s residue theorem | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 12 | Classification of singular points, residues at poles | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 13 | Applications of residues:evaluation of improper integrals | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 14 | Examples of improper integrals | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 15 | Solving problems | Review of the relevant pages from sources | Öğretim Yöntemleri: Problem Çözme |
| 16 | Term Exams | Review of the topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Review of the topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 5 | 70 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 7 | 98 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 10 | 10 |
| Final Exam | 1 | 20 | 20 |
| Total Workload (Hour) | 198 | ||
| Total Workload / 25 (h) | 7,92 | ||
| ECTS | 8 ECTS | ||