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MAT5130Advanced Complex Analysis3+0+0ECTS:7.5
Year / SemesterFall Semester
Level of CourseSecond Cycle
Status Elective
DepartmentDEPARTMENT of MATHEMATICS
Prerequisites and co-requisitesNone
Mode of DeliveryFace to face, Group study
Contact Hours14 weeks - 3 hours of lectures per week
Lecturer--
Co-LecturerNone
Language of instructionTurkish
Professional practise ( internship ) None
 
The aim of the course:
The aim of the course is to teach the principal techniques and methods of analytic function theory. This is quite different from real analysis and has much more geometric emphasis. It also has significant applications to other fields like analytic number theory
 
Programme OutcomesCTPOTOA
Upon successful completion of the course, the students will be able to :
PO - 1 : make simple arguments concerning limits of real-valued functions; show continuity and differentiability of complex-valued functions; and make simple uses of these.1,2,3,4,5,61,6
PO - 2 : make calculate Taylor and Laurent expansions and use the calculus of residues to evaluate integrals. 1,2,3,4,5,61,6
PO - 3 : characterize the automorphism groups of some special regions in the complex plane.1,2,3,4,5,61,6
PO - 4 : give the proofs of the fundamental theorem of algebra and to study the algebraic structure of some analytic functions 1,2,3,4,5,61,6
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), PO : Learning Outcome

 
Contents of the Course
Complex numbers; Analytic functions; Cauchy's integral theorem and formula; Taylor expansion and applications, such as the maximum modulus principle; Laurent series; the calculus of residues; Basics of analytic continuation; Riemann's mapping theorem .
 
Course Syllabus
 WeekSubjectRelated Notes / Files
 Week 1Complex derivation
 Week 2Power series
 Week 3Complex line integrals
 Week 4Cauchy's Integral Theorem and Formula
 Week 5Convergence of Analytic Functions
 Week 6Elementary Properties of Analytic Functions
 Week 7 Analytic continuiation
 Week 8The Singularities of an analytic Function
 Week 9Mid-term exam
 Week 10Analtic functions on Annuli
 Week 11The Winding Number of a Curve
 Week 12The Residue Theorem and Applications
 Week 13Function Theoretic Consequences of the Residue Theorem
 Week 14The General Cauchy Integral Theorem
 Week 15Montel's Theorem ,The Riemann Mapping Theorem
 Week 16End-of-term exam
 
Textbook / Material
1Ahlfors,Lars V. 1979; Complex Analysis, McGraw-Hill,Inc.,Printed in the United States of America,third edition,
 
Recommended Reading
1Conway, John B. 1978; Functions of One Complex Variable, Second Edition, Graduate Texts inMathematics 11, Springer-Verlag, New York
2Rudin,Walter. 1987; Real and Complex Analysis, Third Edition, McGraw-Hill, New York
3Remmert, Reinhold . 1991; Theory of Complex Functions, Graduate Texts in Mathematics 122,Springer-Verlag, New York
 
Method of Assessment
Type of assessmentWeek NoDate

Duration (hours)Weight (%)
Mid-term exam 9 11/04/2018 2 30
Quiz 11 25/04/2018 1 30
End-of-term exam 16 7/06/2018 2 40
 
Student Work Load and its Distribution
Type of workDuration (hours pw)

No of weeks / Number of activity

Hours in total per term
Yüz yüze eğitim 3 14 42
Sınıf dışı çalışma 5 14 70
Laboratuar çalışması 0 0 0
Arasınav için hazırlık 4 8 32
Arasınav 2 1 2
Uygulama 0 0 0
Klinik Uygulama 0 0 0
Ödev 4 9 36
Proje 0 0 0
Kısa sınav 1 1 1
Dönem sonu sınavı için hazırlık 5 8 40
Dönem sonu sınavı 2 1 2
Diğer 1 0 0 0
Diğer 2 0 0 0
Total work load225