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Course Description Course Name : Complex Analysis Course Code : MT-505 Course Type : Optional Level of Course : Second Cycle Year of Study : 1 Course Semester : Fall (16 Weeks) ECTS : 6 Name of Lecturer(s) : Prof.Dr. DOĞAN DÖNMEZ Learning Outcomes of the Course : Knows Cauchy-Goursat Theorem Knows Cauchy integral formulas and consequences Knows Topology of C, Möbius transformations and the Riemann sphere Knows Maximumum modulus and three circle theroems Knows Open mapping property. Isolated singular points. Removable and essential singularities. Poles Knows Schwartz Lemma. Automorphisms of the unit circle and the upper half plane. Knows Conformal mapping, Dirichlet Problem, Poisson formula Knows Field of meromorphic functions. Elliptic fuctions Mode of Delivery : Face-to-Face Prerequisites and Co-Prerequisites : None Recommended Optional Programme Components : None Aim(s) of Course : To grasp fully the basic theorems of functions of one complex variable and understand their significance in algebra and topology Course Contents : Riemann sphere, analytic functions and their properties, Cauchy integral formulas and consequences, singularities. Meromorphic functions, Elliptic functions Language of Instruction : Turkish Work Place : Department Classrooms   Course Outline /Schedule (Weekly) Planned Learning Activities Week Subject Student's Preliminary Work Learning Activities and Teaching Methods 1 Topology of C and the Riemann Sphere Reading the relevant sections in the textbook and solving problems Lecturing 2 Möbius transformations and their properties Reading the relevant sections in the textbook and solving problems Lecturing 3 Differentiable and analytic functions. Cauchy Riemann conditions. Harmonic functions. Reading the relevant sections in the textbook and solving problems Lecturing 4 Line integrals, closed curves. Jordan curve theorem. Reading the relevant sections in the textbook and solving problems Lecturing 5 Cauchy-Goursat Theorem, Cauchy integral formulas. Fundamental Theorem of Algebra. Reading the relevant sections in the textbook and solving problems Lecturing 6 Open mapping property of analytic functions. Conformal mapping Reading the relevant sections in the textbook and solving problems Lecturing 7 Schwartz Lemma and corollaries. Three circle theorem. Reading the relevant sections in the textbook and solving problems Lecturing 8 Isolated singularities. Poles and essential singularities. Riemann s removable singularity theorem. Reading the relevant sections in the textbook and solving problems Lecturing 9 Midterm Exam Review of the material and problem solving Written Exam 10 Generalized Cauchy integral formula, residue theorem. Applications of the residue theorem. Reading the relevant sections in the textbook and solving problems Lecturing 11 Meromorphic functions. Laurent series. Reading the relevant sections in the textbook and solving problems Lecturing 12 Mittag-Leffler Theorem, topological properties of Meromorphic functions Weierstrass Theorem. Reading the relevant sections in the textbook and solving problems Lecturing 13 Lattices and doubly periodic functions. Properties of doubly periodic functions. Reading the relevant sections in the textbook and solving problems Lecturing 14 Weierstrass s P function, its properties, derivative and differential equation. Structure of the field of meromorphic functions. Reading the relevant sections in the textbook and solving problems Lecturing 15 Eliptic integral-elliptic function relation. Picard s theorem. Reading the relevant sections in the textbook and solving problems Lecturing 16/17 Final Exam Review of the material and problem solving Written Exam   Required Course Resources Resource Type Resource Name Recommended Course Material(s)  L.V. Ahlfors Complex Analysis Required Course Material(s)   Assessment Methods and Assessment Criteria Semester/Year Assessments Number Contribution Percentage     Mid-term Exams (Written, Oral, etc.) 1 70     Homeworks/Projects/Others 3 30 Total 100 Rate of Semester/Year Assessments to Success 40   Final Assessments 100 Rate of Final Assessments to Success 60 Total 100   Contribution of the Course to Key Learning Outcomes # Key Learning Outcome Contribution* 1 Aquires sufficient knowledge to enable one to do research over and above the undergraduate level 5 2 Learns theoretical foundations of his/her field thoroughly 4 3 Uses the knowledge in his/her field to solve mathematical problems 1 4 Proves basic theorems in different areas of Mathematics 4 5 Creates a model for the problems in his/her field and expresses the relationship between objects in a simple and comprehensive way. 1 6 Uses technical tools in his/her field 5 7 Works independently in his/her field requiring expertise 0 8 Works with other colleagues collaboratelly, takes responsibilities if necessary during the research process 0 9 Argues and analyzes knowledge in his/her field and applies them in other fields if necessary 4 10 Has sufficient knowledge to follow literature in his/her field and communicate with stakeholders 4 11 Uses the language and technology to develop knowledge in his/her field and shares these with stakeholders systematically when necessary 4 12 Knows and abides by the ethical rules in analyzing, solving problems and publishing results 0 * Contribution levels are between 0 (not) and 5 (maximum).   Student Workload - ECTS Works Number Time (Hour) Total Workload (Hour) Course Related Works     Class Time (Exam weeks are excluded) 14 3 42     Out of Class Study (Preliminary Work, Practice) 14 5 70 Assesment Related Works     Homeworks, Projects, Others 3 6 18     Mid-term Exams (Written, Oral, etc.) 1 10 10     Final Exam 1 15 15 Total Workload: 155 Total Workload / 25 (h): 6.2 ECTS Credit: 6