Complex Functions I (MAY611)
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General
| School |
School of Science |
|---|---|
| Academic Unit |
Department of Mathematics |
| Level of Studies |
Undergraduate |
| Course Code |
MAΥ611 |
| Semester | 6 |
| Course Title |
Complex Functions I |
| Independent Teaching Activities |
Presentations, exercises, lectures (Weekly Teaching Hours: 5, Credits: 7.5) |
| Course Type |
General Background |
| Prerequisite Courses | - |
| Language of Instruction and Examinations |
Greek |
| Is the Course Offered to Erasmus Students |
Yes |
| Course Website (URL) | See eCourse, the Learning Management System maintained by the University of Ioannina. |
Learning Outcomes
| Learning outcomes |
The course introduces the students to the subject of Complex Analysis. More precisely, the course introduces the field of complex numbers, as an extension of the field of real numbers, and its representation through the complex plane and its algebraic, topological and geometric properties are studied. With this as a basis, the notion of a complex function is introduced, as well as the limits of such functions at and to a point or infinity and their continuity, and the exponential and logarithmic functions are introduced as the most fundamental complex functions. Emphasis is given to the central notions of complex differentiability and holomorphy, to the special form of such functions as two-dimensional vector fields and to their importance as geometric transformations of the plane. Then, the study of power series follows, their holomorphy is proved and the notion of an analytic function is introduced, as a function that can be expanded locally into a power series. The power series expansions of the most fundamental complex functions are presented. The main body of the course concludes with the Integral Theory of Cauchy, which next to showing that the notions of holomorphic and analytic functions are identical, leads to first classical important properties of holomorphic functions and points out their structural difference to smooth real functions but also to smooth two-dimensional vector fields. The course concludes with the notions of isolated singularities, Laurent series and residues, and examples are given of their application to the study of generalized integrals of real functions. The course develops emphatically the ability of the combined application of results from different areas of Mathematics and highlights exemplarily their deeper interconnectedness and the usefulness of such a holistic point of view. The course shows also the way Mathematics evolve and trains the students in recalling and applying the knowledge they obtained at a previous stage to a subject that is novel to them. Finally, as one of the last compulsory courses of the undergraduate program, the course helps in a repetition of many notions and results the students learned up to then and in gaining a more tight overview over these. |
|---|---|
| General Competences |
|
Syllabus
The complex plane, Roots, Lines, Topology, Convergence, Riemann sphere, analytic properties of complex functions, Power series, elementary functions (rational, exp, log, trigonometric functions, hyperbolic, functions), line integrals, curves, conformal mappings, homotopic curves, local properties of complex functions, basic theorems, rotation index, General results, singularities, Laurent series, Residuum, Cauchy Theorem, Applications.
Teaching and Learning Methods - Evaluation
| Delivery |
Face-to-face | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Use of Information and Communications Technology |
Use of ICT for the presentation and communication for submission of the exercises | ||||||||||
| Teaching Methods |
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| Student Performance Evaluation |
Greek. Written exam (100%) on the theory and solving problems. |
Attached Bibliography
See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:
- Γιαννούλης, Ι. (2024). Μιγαδική Ανάλυση [Προπτυχιακό εγχειρίδιο]. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις. http://dx.doi.org/10.57713/kallipos-408
- R. Remmert. Theory of Complex Functions. Springer, 1998.
- S. Lang. Complex Analysis. Fourth Edition. Springer, 1999.