DEGREE PROGRAMS  Associate's Degree (Short Cycle)  Bachelor’s Degree (First Cycle)  Master’s Degree (Second Cycle)

Course Description Course Name : Theory of Complex Functions Course Code : MT 334 Course Type : Compulsory Level of Course : First Cycle Year of Study : 3 Course Semester : Spring (16 Weeks) ECTS : 8 Name of Lecturer(s) : Prof.Dr. NAİME EKİCİ Learning Outcomes of the Course : Establishes one-to-one correspondence between real plane and complex numbers. lnvestigates the existence of derivatives of complex functions and differentiates functions of a complex variable. Evaluates contour integrals in complex planes. Evaluates real and complex integrals using the Cauchy´s Theorem and Cauchy integral formula. Classifies singular points of complex functions. Determines whether complex functions are analytic. Finds Taylor and Laurent series of complex functions. Evaluates complex integrals using the residue theorem. Evaluates some real integrals using complex integration technique. Finds images of certain sets under complex linear functions and some elementary functions. Mode of Delivery : Face-to-Face Prerequisites and Co-Prerequisites : None Recommended Optional Programme Components : None Aim(s) of Course : The aim of this course is to acquaint the student with the theory of the calculus of a function of a complex variable and then to introduce the basic theory and ideas of the integration of a function of a complex variable, state the main theorems such as Cauchy’s theorem, Cauchy integral formula, and the Cauchy’s residue theorem with endowing the students with practical skills in evaluating real and complex integrals. Course Contents : Complex numbers, regions, transformations, limit, continuity, differentiation, Cauchy-Riemann equations, Analytic functions, Harmonic functions, elementary transformations, transformations by elementary functions, integrals, contour integrals, Cauchy-Goursat´s theorem, residue, applications of residue: improper integrals. Language of Instruction : Turkish Work Place : Classrooms of Arts and Sciences Faculty   Course Outline /Schedule (Weekly) Planned Learning Activities Week Subject Student's Preliminary Work Learning Activities and Teaching Methods 1 Basic properties of comlex numbers, Polar forms, powers, roots, domains. Review of the relevant pages from sources Lecture and discussion 2 Functions of a complex variable, limit and Limit theorems Review of the relevant pages from sources Lecture and discussion 3 Continuity, derivatives and the Cauchy-Riemann equations Review of the relevant pages from sources Lecture and discussion 4 Sufficient conditions for derivatives, analytic functions, harmonic functions Review of the relevant pages from sources Lecture and discussion 5 Exponential, logarithmic, trigonometric, hyperbolic, inverse trigonometric functions Review of the relevant pages from sources Lecture and discussion 6 Line integrals, upper bound for integrals, anti-derivatives Review of the relevant pages from sources Lecture and discussion 7 Cauchy-Goursat theorem , Cauchy´s integral formula, simply and multiply connected domains Review of the relevant pages from sources Lecture and discussion 8 mid-term exam Review of the topics discussed in the lecture notes and sources Written exam 9 Taylor and Laurent series Review of the relevant pages from sources Lecture and discussion 10 sums and product of the series Review of the relevant pages from sources Lecture and discussion 11 Residues, Cauchy´s residue theorem, Review of the relevant pages from sources Lecture and discussion 12 Classification of singular points, residues at poles Review of the relevant pages from sources Lecture and discussion 13 Applications of residues:evaluation of improper integrals Review of the relevant pages from sources Lecture and discussion 14 examples of improper integrals Review of the relevant pages from sources Lecture and discussion 15 solving problems Review of the relevant pages from sources Lecture and discussion 16/17 Final exam Review of the topics discussed in the lecture notes and sources Written exam   Required Course Resources Resource Type Resource Name Recommended Course Material(s)  Complex Variables and Appliations, author: J.W.Brown, R.V. Churchill  Kompleks Fonksiyonlar Teorisi , author :Turgut Başkan,  Kompleks Değişkenli Fonksiyonlar Teorisi, author:Metin Başarır Required Course Material(s)  http://math.cu.edu.tr/nekici/MT334.htm   Assessment Methods and Assessment Criteria Semester/Year Assessments Number Contribution Percentage     Mid-term Exams (Written, Oral, etc.) 1 90     Homeworks/Projects/Others 5 10 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 Is able to prove Mathematical facts encountered in secondary school. 5 2 Recognizes the importance of basic notions in Algebra, Analysis and Topology 5 3 Develops maturity of mathematical reasoning and writes and develops mathematical proofs. 4 4 Is able to express basic theories of mathematics properly and correctly both written and verbally 3 5 Recognizes the relationship between different areas of Mathematics and ties between Mathematics and other disciplines. 3 6 Expresses clearly the relationship between objects while constructing a model 4 7 Draws mathematical models such as formulas, graphs and tables and explains them 5 8 Is able to mathematically reorganize, analyze and model problems encountered. 4 9 Knows at least one computer programming language 4 10 Uses effective scientific methods and appropriate technologies to solve problems 0 11 Knows programming techniques and is able to write a computer program 1 12 Is able to do mathematics both individually and in a group. 0 13 Has sufficient knowledge of foreign language to be able to understand Mathematical concepts and communicate with other mathematicians 0 14 In addition to professional skills, the student improves his/her skills in other areas of his/her choice such as in scientific, cultural, artistic and social fields 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 5 70     Out of Class Study (Preliminary Work, Practice) 14 5 70 Assesment Related Works     Homeworks, Projects, Others 5 5 25     Mid-term Exams (Written, Oral, etc.) 1 15 15     Final Exam 1 20 20 Total Workload: 200 Total Workload / 25 (h): 8 ECTS Credit: 8