# Complex Analysis (MATH552) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Complex Analysis MATH552 Elective Courses 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English Elective Courses Ph.D. 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 An ability to access, analyze and evaluate the knowledge in chemistry and comprehend the literature.
2 An ability to define an advanced scientific problem in the required field and finding an alternative for the solution.
3 An ability to design and conduct scientific and technological experiments in the lab- and pilot-scale, and to analyze and interpret their results.
4 An ability to perform independent research.
5 An ability to give a presentation at national and international scientific conferences and to publish scientific publications in international journals.
6 An ability to achieve the necessary knowledge to follow current developments in science and technology, and do scientific research or developing projects in the field of chemistry.
7 An ability to work in a multi-disciplinary environment and to work as a part of a team.
8 An understanding of the professional and occupational responsibilities. Awareness and responsibility about professional, legal, ethical and social issues in the required field.

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