Lecture Notes
Karmaşık Değişkenler ve Uygulamalar: R.V.Churchill ( Arif Kaya ) MEB ,1989.
Other Sources
.
Theodore W. Gamelin, Complex Analysis, Springer, 2001.
Course Schedules
Week
Contents
Learning Methods
1. Week
1. Complex Numbers
1.1 Sums and Products
1.2 Basic Algebraic Properties
1.3 Further Properties
1.4 Vektors ve Moduli
Oral represantation
2. Week
1.5 Complex Conjugates
1.6 Exponential Form
1.7 Products and Powers in Exponential Form
1.8 Arguments of Products and Quotients
1.9 Roots of Complex Numbers
1.10 Examples
1.11 Regions in the Complex Plane
Oral represantation
3. Week
2 Analytic Functions
2.1 Functions of a Complex Variable
2.2 Mappings
2.3 Mappings by the Exponential Function
2.4 Limits
2.5 Theorems on Limits
2.6 Limits Involving the Point at Infinity
2.7 Continuity
3 Elementary Functions
3.1 The Exponential Function
3.2 The Logarithmic Function
3.3 Branches and Derivatives of Logarithms
3.4 Some Identities Involving Logarithms
3.5 Complex Exponents
3.6 Trigonometric Functions
3.7 Hyperbolic Functions
3.8 Inverse Trigonometric and Hyperbolic Functions
Oral represantation
6. Week
4 Integrals
4.1 Derivatives of Functions w(t)
4.2 Definite Integrals of Functions w(t)
4.3 Contours
4.4 Contour Integrals
4.5 Some Examples
Oral represantation
7. Week
4.6 Examples with Branch Cuts
4.7 Upper Bounds for Moduli of Contour Integrals
4.8 Antiderivatives
4.9 Proof of the Theorem
Oral represantation
8. Week
4.10 Cauchy-Goursat Theorem
4.11 Simply Connected Domains
4.12 Multiply Connected Domains
4.13 Cauchy Integral Formula
Oral represantation
9. Week
4.14 An Extension of the Cauchy Integral Formula
4.15 Some Consequences of the Extension
4.16 Liouville’s Theorem and the Fundamental Theorem of Algebra
4.17 Maximum Modulus Principle
Oral represantation
10. Week
5 Series
5.1 Convergence of Sequences
5.2 Convergence of Series
5.3 Taylor Series
5.4 Proof of Taylor’s Theorem
Oral represantation
11. Week
5.5 Examples
5.6 Laurent Series
5.7 Proof of Laurent’s Theorem
5.8 Examples
5.9 Absolute and Uniform Convergence of Power Series
Oral represantation
12. Week
5.10 Continuity of Sums of Power Series
5.11 Integration and Differentiation of Power Series
5.12 Uniqueness of Series Representations
Oral represantation
13. Week
6 Residues and Poles
6.1 Isolated Singular Points
6.2 Residues
Oral represantation
14. Week
6.3 Cauchy’s Residue Theorem
6.4 Residue at Infinity
6.5 The Three Types of Isolated Singular Points
Oral represantation
15. Week
6.6 Residues at Poles
6.7 Examples
6.8 Zeros of Analytic Functions
6.9 Zeros and Poles
6.10 Behavior of Functions Near Isolated Singular Points
Oral represantation
16. Week
Final exam
-Writing Exam
17. Week
Final exam
-Writing Exam
Assessments
Evaluation tools
Quantity
Weight(%)
Quizzes
10
30
Final Exam
1
70
Program Outcomes
PO-1
Interpreting advanced theoretical and applied knowledge in Mathematics and Computer Science.
PO-2
Critiquing and evaluating data by implementing the acquired knowledge and skills in Mathematics and Computer Science.
PO-3
Recognizing, describing, and analyzing problems in Mathematics and Computer Science; producing solution proposals based on research and evidence.
PO-4
Understanding the operating logic of computer and recognizing computational-based thinking using mathematics as a discipline.
PO-5
Collaborating as a team-member, as well as individually, to produce solutions to problems in Mathematics and Computer Science.
PO-6
Communicating in a foreign language, and interpreting oral and written communicational abilities in Turkish.
PO-7
Using time effectively in inventing solutions by implementing analytical thinking.
PO-8
Understanding professional ethics and responsibilities.
PO-9
Having the ability to behave independently, to take initiative, and to be creative.
PO-10
Understanding the importance of lifelong learning and developing professional skills continuously.
PO-11
Using professional knowledge for the benefit of the society.
Learning Outcomes
LO-1
Represent complex numbers algebraically and geometrically,
LO-2
Define and analyze limits and continuity for complex functions as well as consequences of continuity,
LO-3
Apply the concept and consequences of analyticity and the Cauchy-Riemann equations and of results on harmonic and entire functions including the fundamental theorem of algebra,
LO-4
Analyze sequences and series of analytic functions and types of convergence,
LO-5
Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula,
LO-6
Represent functions as Taylor, power and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem.
