Calculus of Complex Functions and Applications (EM8)

Graduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM10
Semester	2
Course Title	Calculus of Complex Functions and Applications
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

By the end of the course the student should be able to: give an account of the concepts of analytic function and harmonic function and to explain the role of the Cauchy-Riemann equations. explain the concept of conformal mapping, describe its relation to analytic functions, and know the mapping properties of the elementary functions. describe the mapping properties of Möbius transformations and know how to use them for conformal mappings. define and evaluate complex contour integrals. give an account of and use the Cauchy integral theorem, the Cauchy integral formula and some of their consequences. analyze simple sequences and series of functions with respect to uniform convergence, describe the convergence properties of a power series, and determine the Taylor series or the Laurent series of an analytic function in a given region. give an account of the basic properties of singularities of analytic functions and be able to determine the order of zeros and poles, to compute residues and to evaluate integrals using residue techniques. use the theory, methods and techniques of the course to solve mathematical problems.

General Competences

Adapting to new situations Decision-making Working independently Team work

Syllabus

Complex numbers, topology in ℂ. Functions of one complex variable, limits, continuity and differentiability. The Cauchy-Riemann equations. Analytic and harmonic functions. Conformal mappings. Elementary functions from ℂ to ℂ, in particular Möbius transformations and the exponential function. Solution of boundary value problems in the plane for the Laplace equation using conformal mappings. Complex integration. Cauchy's integral theorem. The maximum principle for analytic and harmonic functions. Poisson's integral formula. Uniform convergence and analyticity. Power series. Taylor and Laurent series with applications. Zeros and isolated singularities. Residue calculus with applications. Rouché's theorem. Briefly about connections with Fourier series and Fourier integrals. The Riemann-Hilbert problem.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project