# Complex Analysis (MATH552) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Complex Analysis MATH552 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English N/A Natural & Applied Sciences Master's Degree Face To Face Lecture, Question and Answer, Team/Group. This course is designed to provide necessary backgrounds and further knowledge in Complex Analysis for graduate students of Mathematics. The topics covered by this course have numerous applications in pure and applied mathematics. The students who succeeded in this course; Understand the conformality, conformal mappings and elementary Riemann surfaces. Learn the maximum principle and the calculus of residues. Understand complex integration. Learn harmonic and entire functions. Know analytic continuation. Analytic functions as mappings, conformal mappings, complex integration, harmonic functions, series and product developments, entire functions, analytic continuation, algebraic functions.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 The algebra of complex numbers. Introduction to the concept of analytic function. Elementary theory of power series. pp. 1-42
2 Elementary point set topology: sets and elements, metric spaces, connectedness, compactness, continuous functions, topological spaces. pp. 50-67
3 Conformality. Elementary conformal mappings. Elementary Riemann surfaces. pp. 68-97
4 Fundamental theorems of complex integration. Cauchy’s integral formula. pp. 101-120
5 Local properties of analytic functions: removable singularities, Taylor’s formula, zeros and poles, the local mapping, the maximum principle. pp. 124-133
6 Mid-Term Examination
7 The general form of Cauchy’s theorem. Multiply connected regions pp. 137-144
8 The calculus of residues: the residue theorem, the argument principle, evaluation of definite integrals. pp. 147-153
9 Harmonic functions. pp. 160-170
10 Power series expansions. The Laurent series. Partial fractions and factorization. pp. 173-199
11 Entire functions. pp. 205-206
12 Normal families of analytic functions. pp. 210-217
13 Analytic continuation. pp. 275-287
14 Algebraic functions. pp. 291-294
15 Picard’s theorem. pp. 297
16 Final Examination

### Sources

Course Book 1. L. V. Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York 1966. 2. A. I. Markuschevich, Theory of Functions of a Complex Variable, 1985. 3. A J. W. Brown and R. V. Churcill, Complex Variables and Applications, McGraw-Hill, New York, 2003.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 15
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 35
Toplam 8 100
 Percentage of Semester Work 65 35 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 14 3 42
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 2 10
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 7 14
Prepration of Final Exams/Final Jury 1 11 11