COURSE INTRODUCTION AND APPLICATION INFORMATION


Course Name
Complex Analysis
Code
Semester
Theory
(hour/week)
Application/Lab
(hour/week)
Local Credits
ECTS
MATH 656
Fall/Spring
3
0
3
7.5
Prerequisites
None
Course Language
English
Course Type
Elective
Course Level
Third Cycle
Mode of Delivery -
Teaching Methods and Techniques of the Course
Course Coordinator -
Course Lecturer(s)
Assistant(s) -
Course Objectives We will first be dealing with questions of existence of holomorphic and meromorphic functions with prescribed properties on plane domains. By solving the inhomogeneous Cauchy‐Riemann equation on plane domains and using Runge's Theorem on approximating holomorphic functions, we establish two fundamental existence results: existence of meromorphic functions with prescribed principal parts (Mittag‐Leffler's Theorem) and existence of meromorphic functios with prescribed zeros and poles (Weiestrass' Theorem). We will also discuss a special class of meromorphic functions on the complex plane, viz., those that are doubly periodic with respect to a lattice. This includes the construction of elliptic functions and of theta functions, and the representation of elliptic functions in terms of translates of the sigma function.
Learning Outcomes The students who succeeded in this course;
  • will be able to define Holomorphic and meromorphic functions.
  • will be able to understand Cauchy-Riemann equation.
  • will be able to understand Runge’s theorem.
  • will be able to understand MittagLeffler’s theorem.
  • will be able to define Special class of meromorphic functions on the complex plane.
Course Description This course contains the questions of existence of holomorphic and meromorphic functions with prescribed properties on plane domains, viz., those that are doubly‐periodic with respect to a lattice. This includes the construction of elliptic functions and of theta functions, and the representation of elliptic functions in terms of translates of the sigma function.
Related Sustainable Development Goals

 



Course Category

Core Courses
X
Major Area Courses
Supportive Courses
Media and Managment Skills Courses
Transferable Skill Courses

 

WEEKLY SUBJECTS AND RELATED PREPARATION STUDIES

Week Subjects Required Materials
1 Holomorpic functions. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
2 Local properties of holomorphic functions. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
3 Meromorphic functions. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
4 The local mapping. The maximum principle. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
5 The general form of Cauchy’s theorem. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
6 Power series expansions. Weierstrass’ theorem. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
7 Partial functions and factorization. MittagLeffler theorem. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
8 The Gamma functions. Entire functions. Hadamard’ theorem. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
9 Families of holomorphic functions. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
10 Analytic continuation. The Weierstrass theory. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
11 Homotopic curves. The Monodromy theorem. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
12 Algebraic functions. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
13 Picard’s theorem. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
14 Ordinary points. Regular singular points. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
15 Riemann’s point view. Complex Analysis, 3rd Edition, by Lars V. Ahlfors. McGrawHill, New York, 1966.
16 Review of the Semester  
Course Notes/Textbooks The extracts above and exercises will be given.
Suggested Readings/Materials None

 

EVALUATION SYSTEM

Semester Activities Number Weigthing
Participation
Laboratory / Application
Field Work
Quizzes / Studio Critiques
Portfolio
Homework / Assignments
Presentation / Jury
Project
Seminar / Workshop
Oral Exam
Midterm
1
50
Final Exam
1
50
Total

Weighting of Semester Activities on the Final Grade
1
50
Weighting of End-of-Semester Activities on the Final Grade
1
50
Total

ECTS / WORKLOAD TABLE

Semester Activities Number Duration (Hours) Workload
Course Hours
(Including exam week: 16 x total hours)
16
3
48
Laboratory / Application Hours
(Including exam week: 16 x total hours)
16
Study Hours Out of Class
15
6
90
Field Work
Quizzes / Studio Critiques
Portfolio
Homework / Assignments
Presentation / Jury
Project
1
25
Seminar / Workshop
Oral Exam
Midterms
1
25
Final Exams
1
37
    Total
225

 

COURSE LEARNING OUTCOMES AND PROGRAM QUALIFICATIONS RELATIONSHIP

#
Program Competencies/Outcomes
* Contribution Level
1
2
3
4
5
1

To develop and deepen his/her knowledge on theories of mathematics and statistics and their applications in level of expertise, and to obtain unique definitions which bring innovations to the area, based on master level competencies,

X
2

To have the ability of original, independent and critical thinking in Mathematics and Statistics and to be able to develop theoretical concepts,

X
3

To have the ability of defining and verifying problems in Mathematics and Statistics,

X
4

With an interdisciplinary approach, to be able to apply theoretical and applied methods of mathematics and statistics in analyzing and solving new problems and to be able to discover his/her own potentials with respect to the application,

X
5

In nearly every fields that mathematics and statistics are used, to be able to execute, conclude and report a research, which requires expertise, independently,

X
6

To be able to evaluate and renew his/her abilities and knowledge acquired in the field of Applied Mathematics and Statistics with critical approach, and to be able to analyze, synthesize and evaluate complex thoughts in a critical way,

X
7

To be able to convey his/her analyses and methods in the field of Applied Mathematics and Statistics to the experts in a scientific way,

X
8

To be able to use national and international academic resources (English) efficiently, to update his/her knowledge, to communicate with his/her native and foreign colleagues easily, to follow the literature periodically, to contribute scientific meetings held in his/her own field and other fields systematically as written, oral and visual.

X
9

To be familiar with computer software commonly used in the fields of Applied Mathematics and Statistics and to be able to use at least two of them efficiently,

X
10

To contribute the transformation process of his/her own society into an information society and the sustainability of this process by introducing scientific, technological, social and cultural advances in the fields of Applied Mathematics and Statistics,

X
11

As having rich cultural background and social sensitivity with a global perspective, to be able to evaluate all processes efficiently, to be able to contribute the solutions of social, scientific, cultural and ethical problems and to support the development of these values,

X
12

As being competent in abstract thinking, to be able to connect abstract events to concrete events and to transfer solutions, to analyze results with scientific methods by designing experiment and collecting data and to interpret them,

X
13

To be able to produce strategies, policies and plans about systems and topics in which mathematics and statistics are used and to be able to interpret and develop results,

X
14

To be able to evaluate, argue and analyze prominent persons, events and phenomena, which play an important role in the development and combination of the fields of Mathematics and Statistics, within the perspective of the development of other fields of science,

X
15

In Applied Mathematics and Statistics, to be able to sustain scientific work as an individual or a group, to be effective in all phases of an independent work, to participate decision-making process and to make and execute necessary planning within an effective time schedule.

X

*1 Lowest, 2 Low, 3 Average, 4 High, 5 Highest