Complex Functions II (MAE712)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Undergraduate
Course Code	ΜΑΕ712
Semester	7
Course Title	Complex Functions II
Independent Teaching Activities	Lectures (Weekly Hours: 3, Credits: 6)
Course Type	Special Background
Prerequisite Courses	None.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	No
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course deepens further into the study of the properties of complex, and in particular holomorphic and meromorphic, functions, aiming to derive characteristic results for them which distinguish them from real functions. The students apply the results and techniques they obtained from the introductory course in order to derive more involved results on the one hand within Complex Analysis and on the other hand in relation to its connections with other areas of Mathematics, as for instance Geometry, Topology and Partial Differential Equations, and are trained in the composition of simpler results in order to derive deeper ones.

General Competences

Αnalyse and combine data and information using various technologies. Working independently and in groups. Free, creative, analytic, and conclusive thinking. Decision making.

Syllabus

The course is a continuation of the introductory compulsory course Complex Functions I. It considers classical theoretical results which are characteristic of Complex Analysis and that highlight its connections with other areas of Mathematics. The following topics are mentioned indicatively: Conformal mappings. Harmonic Functions. Homotopy. Analytic Continuation. Homologically simply connected domains. Generalization of Cauchy’s Integral Theorem. Maximum Principle. Schwarz’ Lemma. Convergence theorems for sequences of holomorphic functions. Partial fraction decomposition. Infinite Products. Riemann Mapping Theorem.

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	Activity Semester Workload Study in class 39 Other activities determined by the teaching professor 111 Course total 150
Student Performance Evaluation	xxx

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:

• L. V. Ahlfors. Complex Analysis. Third Edition. McGraw-Hill, 1979.
• K. Jaenich. Funktionentheorie. Eine Einfuehrung. Sechste Auflage. Springer, 2011.
• S. Lang. Complex Analysis. Fourth Edition. Springer, 1999.
• Σ. Κ. Μερκουράκης, Τ. Ε. Χατζηαφράτης. Εισαγωγή στη Μιγαδική Ανάλυση. Εκδόσεις Συμμετρία, 2005.
• R. Remmert. Theory of Complex Functions. Springer, 1998.
• R. Remmert. Classical Topics in Complex Function Theory. Springer, 1998.