Calculus of Complex Functions and Applications (EM8): Διαφορά μεταξύ των αναθεωρήσεων
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=== General === | === General === | ||
| Γραμμή 15: | Γραμμή 17: | ||
|- | |- | ||
! Course Code | ! Course Code | ||
| | | EM8 | ||
|- | |- | ||
! Semester | ! Semester | ||
| | | 2 | ||
|- | |- | ||
! Course Title | ! Course Title | ||
| | | Calculus of Complex Functions and Applications | ||
|- | |- | ||
! Independent Teaching Activities | ! Independent Teaching Activities | ||
| Γραμμή 27: | Γραμμή 29: | ||
|- | |- | ||
! Course Type | ! Course Type | ||
| | | Special Background | ||
|- | |- | ||
! Prerequisite Courses | ! Prerequisite Courses | ||
| Γραμμή 34: | Γραμμή 36: | ||
! Language of Instruction and Examinations | ! Language of Instruction and Examinations | ||
| | | | ||
Greek | |||
|- | |- | ||
! Is the Course Offered to Erasmus Students | ! Is the Course Offered to Erasmus Students | ||
| Yes | | Yes (in English) | ||
|- | |- | ||
! Course Website (URL) | ! Course Website (URL) | ||
| Γραμμή 49: | Γραμμή 51: | ||
! 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 | ! General Competences | ||
| | | | ||
* Adapting to new situations | |||
* Decision-making | |||
* Working independently | |||
* Team work | |||
|} | |} | ||
=== Syllabus === | === 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 === | === Teaching and Learning Methods - Evaluation === | ||
| Γραμμή 66: | Γραμμή 79: | ||
! Delivery | ! Delivery | ||
| | | | ||
In class | |||
|- | |- | ||
! Use of Information and Communications Technology | ! Use of Information and Communications Technology | ||
| | | - | ||
|- | |- | ||
! Teaching Methods | ! Teaching Methods | ||
| Γραμμή 81: | Γραμμή 93: | ||
| 39 | | 39 | ||
|- | |- | ||
| | | Self study | ||
| | | 78 | ||
|- | |- | ||
| | | Homework - Projects | ||
| | | 70.5 | ||
|- | |- | ||
| Course total | | Course total | ||
| Γραμμή 93: | Γραμμή 105: | ||
! Student Performance Evaluation | ! Student Performance Evaluation | ||
| | | | ||
* Weekly assignments | |||
* Final project | |||
|} | |} | ||
| Γραμμή 99: | Γραμμή 112: | ||
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General
| School | School of Science |
|---|---|
| Academic Unit | Department of Mathematics |
| Level of Studies | Graduate |
| Course Code | EM8 |
| 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:
|
|---|---|
| General Competences |
|
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 |
| ||||||||||
| Student Performance Evaluation |
|
Attached Bibliography
- ΜΙΓΑΔΙΚΕΣ ΣΥΝΑΡΤΗΣΕΙΣ ΚΑΙ ΕΦΑΡΜΟΓΕΣ, Κωδικός Βιβλίου στον Εύδοξο: 226, Έκδοση: 1η/2005, Συγγραφείς: CHURCHILL R., BROWN J., ISBN: 960-7309-41-3, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): ΙΔΡΥΜΑ ΤΕΧΝΟΛΟΓΙΑΣ & ΕΡΕΥΝΑΣ-ΠΑΝΕΠΙΣΤΗΜΙΑΚΕΣ ΕΚΔΟΣΕΙΣ ΚΡΗΤΗΣ
- ΜΙΓΑΔΙΚΕΣ ΜΕΤΑΒΛΗΤΕΣ, Κωδικός Βιβλίου στον Εύδοξο: 12404786, Έκδοση: 1η/2011, Συγγραφείς: ABLOWITZ MARK J., ΦΩΚΑΣ ΑΘΑΝΑΣΙΟΣ Σ., ISBN: 978-960-524-337-1, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): ΙΔΡΥΜΑ ΤΕΧΝΟΛΟΓΙΑΣ & ΕΡΕΥΝΑΣ-ΠΑΝΕΠΙΣΤΗΜΙΑΚΕΣ ΕΚΔΟΣΕΙΣ ΚΡΗΤΗΣ
- Αναλυτικές συναρτήσεις και μερικές εφαρμογές τους, Κωδικός Βιβλίου στον Εύδοξο: 12166, Έκδοση: 2η έκδ./1998, Συγγραφείς: Τερσένοβ Σάββας, ISBN: 978-960-7140-66-1, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): ΔΙΑΥΛΟΣ Α.Ε. ΕΚΔΟΣΕΙΣ ΒΙΒΛΙΩΝ
- Μιγαδικές συναρτήσεις, Κωδικός Βιβλίου στον Εύδοξο: 11116, Έκδοση: 1η έκδ./1996, Συγγραφείς: Παντελίδης Γεώργιος Ν., Κραββαρίτης Δημήτρης Χ., Νασόπουλος Β., ISBN: 960-431-358-4, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): Ζήτη Πελαγία & Σια Ι.Κ.Ε.
- Μιγαδικές συναρτήσεις, Κωδικός Βιβλίου στον Εύδοξο: 11115, Έκδοση: 1η έκδ./2008, Συγγραφείς: Ξένος Θανάσης Π., ISBN: 978-960-456-092-9, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): Ζήτη Πελαγία & Σια Ι.Κ.Ε.