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 Ability to expand and get in-depth information with scientific researches in the field of mechanical engineering, evaluate information, review and implement.
2 Have comprehensive knowledge about current techniques and methods and their limitations in Mechanical engineering.
3 To complete and apply knowledge by using scientific methods using uncertain, limited or incomplete data; use information from different disciplines.
4 Being aware of the new and developing practices of Mechanical Engineering and being able to examine and learn when needed.
5 Ability to define and formulate problems related to Mechanical Engineering and develop methods for solving and apply innovative methods in solutions.
6 Ability to develop new and/or original ideas and methods; design complex systems or processes and develop innovative/alternative solutions in the designs.
7 Ability to design and apply theoretical, experimental and modeling based researches; analyze and solve complex problems encountered in this process.
8 Work effectively in disciplinary and multi-disciplinary teams, lead leadership in such teams and develop solution approaches in complex situations; work independently and take responsibility.
9 To establish oral and written communication by using a foreign language at least at the level of European Language Portfolio B2 General Level.
10 Ability to convey the process and results of their studies systematically and clearly in written and oral form in national and international environments.
11 To know the social, environmental, health, security, law dimensions, project management and business life applications of engineering applications and to be aware of the constraints of their engineering applications.
12 Ability to observe social, scientific and ethical values in the stages of data collection, interpretation and announcement and in all professional activities.

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