Complex analysis

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Cauchy integral formula for holomophic and non holomorphic functions. Solution to the non homogeneous debar equation on planar domains using Cauchy integral.
Schwarz lemma. Automorphisms of the unit disc.
Convex functions. Hadamard three-circle theorem.Phragmen-Lindelöf theorem.
Compatness and convergence in the space of holomorphic functions. Normal families. Montel's theorem. Hurwitz's theorem. Riemann mapping theorem.
Koebe's theorem. Bloch's theorem. Landau's theorem, Picards' theorem. Schottky's theorem.
Product convergence. Weierstrass factorization theorem. Runge's theorem on approximation by rational functions. Mittag-Leffler's theorem on existence of holomorphic functions with prescribed principal parts. Interpolation by holomorphic functions on discrete sets.
Schwarz reflection principle. Analytic continuation along path. Monodromy theorem.Complete analytic function. Sheaf of germs of analytic functions. Riemann surface.
Other possible topics: Harmonic and subharmonic functions. Poisson kernel and the solution of the Dirichlet problem on zhe disc. Properties of Poisson integraland connection to the Cauchy integral. Mergelyan theorem. Entire functions. The genus and the order of entire function. Hadamard factorization theorem.

Readings

L. Ahlfors: Complex Analysis, 3rd edition, McGraw-Hill, New York, 1979.
C. A. Berenstein, R. Gay: Complex Analysis and Special Topics in Harmonic Analysis, Springer, New York, 1995.
J. B. Conway: Functions of One Complex Variable I, 2nd edition, Springer, New York-Berlin, 1995.
R. Narasimhan, Y. Nievergelt: Complex Analysis in One Variable, 2nd edition, Birkhäuser, Boston, 2001.
W. Rudin: Real and Complex Analysis, 3rd edition, McGraw-Hill, New York, 1987.
T. Gamelin: Complex analysis, Springer-Verlag, New York, 2001.

Objectives and competences

Students learn some basic concepts of theory of functions of one complex variable. Elementary methods of analysis and topology are applied.
With individual presentations and team work interactions within seminar/project activities students acquire communication and social competences for successful team work and knowledge transfer.

Intended learning outcomes

Knowledge and understanding: Understanding some of the fundamental topics and techniques of complex analysis.
Application: Applications lie mainly in other parts of mathematical analysis and geometry. Conformal maps are applied to solving problems in physics and mechanics.
Reflection: Understanding the theory on the basis of examples and applications.
Transferable skills: The ability to identify, formulate and solve mathematical and non mathematical problems using methods of complex analysis. Acquiringn skills in using domestic and foreign literature. Developing the skills of independent presentation of the material in the form of seminar lectures.

Learning and teaching methods

Lectures, exercises, seminar, homeworks, consultations

Assessment

Type (homework, seminar paper, oral exam, coursework, project):
homework and seminar paper
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Miran Černe:
ČERNE, Miran, ZAJEC, Matej. Boundary differential relations for holomorphic functions on the disc. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 473-484. [COBISS-SI-ID 15710553]
ČERNE, Miran, FLORES, Manuel. Generalized Ahlfors functions. Transactions of the American Mathematical Society, ISSN 0002-9947, 2007, vol. 359, no. 2, str. 671-686. [COBISS-SI-ID 14227801]
ČERNE, Miran, FLORES, Manuel. Quasilinear [overline{partial}]-equation on bordered Riemann surfaces. Mathematische Annalen, ISSN 0025-5831, 2006, vol. 335, no. 2, str. 379-403. [COBISS-SI-ID 13970777]
Barbara Drinovec Drnovšek:
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. The Poletsky-Rosay theorem on singular complex spaces. Indiana University mathematics journal, ISSN 0022-2518, 2012, vol. 61, no. 4, str. 1407-1423. [COBISS-SI-ID 16679257]
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Holomorphic curves in complex spaces. Duke mathematical journal, ISSN 0012-7094, 2007, vol. 139, no. 2, str. 203-254. [COBISS-SI-ID 14351705]
DRINOVEC-DRNOVŠEK, Barbara. On proper discs in complex manifolds. Annales de l'Institut Fourier, ISSN 0373-0956, 2007, t. 57, fasc. 5, str. 1521-1535. [COBISS-SI-ID 14379865]
Franc Forstnerič:
FORSTNERIČ, Franc, WOLD, Erlend Fornæss. Embeddings of infinitely connected planar domains into C [sup] 2. Analysis & PDE, ISSN 2157-5045, 2013, vol. 6, no. 2, str. 499-514. [COBISS-SI-ID 16645209]
FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]
FORSTNERIČ, Franc. Noncritical holomorphic functions on Stein manifolds. Acta mathematica, ISSN 0001-5962, 2003, vol. 191, no. 2, str. 143-189. [COBISS-SI-ID 13138009]
Jasna Prezelj:
FORSTNERIČ, Franc, PREZELJ-PERMAN, Jasna. Oka's principle for holomorphic submersions with sprays. Mathematische Annalen, ISSN 0025-5831, 2002, band 322, heft 4, str. 633-666. [COBISS-SI-ID 11554649]
PREZELJ-PERMAN, Jasna. Interpolation of embeddings of Stein manifolds on discrete sets. Mathematische Annalen, ISSN 0025-5831, 2003, band 326, heft 2, str. 275-296. [COBISS-SI-ID 12518489]
PREZELJ-PERMAN, Jasna. Weakly regular embeddings of Stein spaces with isolated singularities. Pacific journal of mathematics, ISSN 0030-8730, 2005, vol. 220, no. 1, str. 141-152. [COBISS-SI-ID 13910873]