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| SEC 415 | Selected Topics in Complex Analysis | 4+0+0 | ECTS:8 | | Year / Semester | Fall Semester | | Level of Course | First Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | Face to face, Group study, Practical | | Contact Hours | 14 weeks - 4 hours of lectures per week | | Lecturer | -- | | Co-Lecturer | Assoc. Prof. Dr. Bahadır Özgür Güler | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | This course is built on the materials covered in MATH 331 and MATH 348. Firsly, it is given some familiar material in greater detail and then continueded on to cover basic material in complex analysis. Some functional analytic techniques will be developed and applied to prove results in complex analysis. Essential aim is to prove the Riemann mapping theorem which characterizes to be conformal equivalence between two plane regions |
| Learning Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | LO - 1 : | study some spaces of complex valued continious ,analytic and meromorphic functions | 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22 | 1,3,6 | | LO - 2 : | give the Automorphism groups for some special regions in complex plane, | 1,2,3,4,5,6,8,9,11,12,13,14,15,16,17,18,19,20,21,22 | 1,3,6 | | LO - 3 : | improve their ability to independently read materials on conformal equivalence. | 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22 | 1,3,6 | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), LO : Learning Outcome | | |
| Review of the basic properties of analytic functions; winding numbers and homotopy; Logarithms, simple connectedness and antiderivatives; Identity theorem for analytic functions, maximum modulus theorem; Open mapping theorem, argument principle, inverses of analytic functions ; Metric space structures on H (G) , C (G) and M (G) ; Boundedness and compactness in H (G) . ; Normal families; metric space structure of M (G) ; Hurwitz's theorem, the Schwarz lemma; the Riemann mapping theorem. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Vector Spaces and Complex Variables,
| | | Week 2 | C(G)and H(G) Spaces | | | Week 3 | The H(U) space on the open unit disc U ,
| | | Week 4 | The Hahn-Banach Theorem and Applications,
| | | Week 5 | The Dual of H(G) ,
| | | Week 6 | Runge's Theorem, | | | Week 7 | The Cauchy Theorem,
| | | Week 8 | Infinite products | | | Week 9 | Mid-term exam
| | | Week 10 | Ideals in The ring H(G), | | | Week 11 | The Riemann mapping Theorem,
| | | Week 12 | Carathéodory Kernels and Farrell?s Theorem, | | | Week 13 | Ring homomorphisms of H(G), | | | Week 14 | Isomorphims of H(G),Algebraic characterizations of conformally equivalence, | | | Week 15 | End-of-term exam | | | |
| 1 | Conway,J.B. 1979; Springer Graduate Texts in Mathematics, Second edition,New York | | | |
| 1 | Luecking,D.H. ,Rubel,L.A.1984; Complex Analysis: A Functional Analysis Approach,Springer Universitext ,New York | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 12/11/2015 | 2 | 50 | | End-of-term exam | 16 | 05/01/2016 | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 4 | 14 | 56 | | Sınıf dışı çalışma | 1 | 14 | 14 | | Laboratuar çalışması | 0 | 0 | 0 | | Arasınav için hazırlık | 1 | 4 | 4 | | Arasınav | 2 | 1 | 2 | | Uygulama | 0 | 0 | 0 | | Klinik Uygulama | 0 | 0 | 0 | | Ödev | 0 | 0 | 0 | | Proje | 0 | 0 | 0 | | Kısa sınav | 0 | 0 | 0 | | Dönem sonu sınavı için hazırlık | 2 | 6 | 12 | | Dönem sonu sınavı | 2 | 1 | 2 | | Diğer 1 | 0 | 0 | 0 | | Diğer 2 | 0 | 0 | 0 | | Total work load | | | 90 |
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