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Syllabus ( MATH 535 )


   Basic information
Course title: Theory Of Complex Valued Functions I
Course code: MATH 535
Lecturer: Assoc. Prof. Dr. Feray HACIVELİOĞLU
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: Turkish
Mode of delivery: Face to face
Pre- and co-requisites: None
Professional practice: No
Purpose of the course: To discuss advanced studies and applications in the theory of functions of a complex variable.
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Identif the basic properties of complex numbers and complex valued functions.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics

    Method of assessment

    1. Written exam
  2. Explain general principles of theory of analytic functions.

    Contribution to Program Outcomes

    1. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  3. Explain general principles of theory of conformal mappings.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  4. Explain Cauchy Integral Theorem, Cauchy Integral Formula and use them in various applications.

    Contribution to Program Outcomes

    1. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  5. Explain the basic theorems of complex analysis and establish relations between them.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
  6. Talk about Schwarz Lemma and its applications.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Defend research outcomes at seminars and conferences.

    Method of assessment

    1. Written exam
   Contents Up
Week 1: Complex Numbers, Complex Plane, Properties of Modules and Argument, Number Sequences and Series, Stereographic Projection, Riemann Sphere, Curves and regions in the Complex Plane.
Week 2: Complex Valued Functions and Transformation, Limits, Modulus of Continuity
Week 3: Derivative with respect to a Complex Variable, Cauchy-Riemann Conditions, Analytic Functions, Modules of derivative and Geometric meaning of Argument
Week 4: Concept of Conformal Mapping, Linear Fractional Transformations and its properties
Week 5: Linear Fractional Transformations and its properties, Exponential Functions
Week 6: Logarithmic Functions, Examples of Riemann Surface, Joukowski,Trigonometric, and Hyperbolic Functions.
Week 7: Integral with respect to a Complex Variable and its Relation with Curve Integrals, Primitive, Newton-Leibnitz formula, Passing to Limit under the Integral Sign
Week 8: Cauchy Integral Theorem. Cauchy Integral Formula (MIDTERM EXAM)
Week 9: Cauchy’s Integral Formula for Derivatives, Morera’s Theorem. Sequences and Series of Analytic Functions, Weierstrass’ theorem, Power Series, Abel’s Theorem, Cauchy-Hadamard formula
Week 10: Expansions of Analytic Functions into Power Series and Uniqueness.
Week 11: Cauchy’s Inequality.Uniqueness Theorem and Maximum Principle, Class of Analytic Functions, Order of Zero, Uniqueness Theorem
Week 12: Maximum Model Principle and Schwarz Lemma, Hadamard’s three Circles and Phragmen-Lindelöf Theorems
Week 13: Laurent Series, Convergence Region of Laurent Series, Laurent Expansion of Analytic Functions, Uniqueness of Expansion, Cauchy’s Formula and Inequality for Coefficients. Liouville Theorem, Theorem related to Removable Singular Point. Disjoint Singular points, Classification of Singular Points of Single Valued Analytic Function with respect to Laurent Series and behavior of the function, its Pole and Order, Essential Singular Point, Casorati -Weierstrass’ and Picard’s Theorems
Week 14: Residues and Principle of Argument: Residue, Cauchy Theorem about Residues, Evaluation of Residues, Logarithmic Residue
Week 15*: General review.
Week 16*: Final Exam.
Textbooks and materials:
Recommended readings: 1.A.I. Markushevich, Theory of Functions of a Complex Variable;
2. Edward B. Saff and Arthur David Snider, Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics;
3. Mithat İdemen, Kompleks Değişkenli Fonksiyonlar Teorisi.
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 8 40
Other in-term studies: 0
Project: 0
Homework: 0
Quiz: 0
Final exam: 16 60
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 15
Own studies outside class: 6 15
Practice, Recitation: 0 0
Homework: 0 0
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 10 2
Mid-term: 1 1
Personal studies for final exam: 12 2
Final exam: 3 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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