

Contents


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, CauchyRiemann 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, NewtonLeibnitz 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, CauchyHadamard 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 PhragmenLindelöf 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.

