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Mandatory course for the master study program Mathematical analysis. Advanced Complex Analysis.
Last update: Pyrih Pavel, doc. RNDr., CSc. (09.05.2022)
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The credit (zápočet) is a necessary condition for coming to examination. Students obtain the credit for giving short lectures on given topics during classes. The character of the credit does not enable its repetition. Last update: Lávička Roman, doc. RNDr., Ph.D. (07.02.2023)
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Rudin, W.: Real and complex analysis, , McGraw-Hill, New York, 1966.
Luecking, D.H., Rubel, L.A.: Complex Analysis, A Functional Analysis Approach, Springer-Verlag, Universitext, 1984
Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (13.05.2022)
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Requirements to the exam correspond to the syllabus to the extent to which topics were covered during the course. Last update: Lávička Roman, doc. RNDr., Ph.D. (07.02.2023)
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1. Meromorphic functions
Meromorphic functions, operations on then, uniqueness theorem, argument principle, Rouché theorem, multiplicity of preimages and multiplicity of roots and poles, open mapping theorem, inverse to a holomorphic funkcion (local and global) Rouché theorem for a compact 2. Functions defined on the whole complex planeInfinite products, Weierstrass factorization theorem on C, Mittag-Leffler theorem on C, Cauchyova method of decomposing a meromorphic function 3. Algebra of holomorphic functionsAlgebras C(G) a H(G) - definitions, convergence, exhausting an open set by compact subsets, seminorms and a metric on C(G) and on H(G), properties Boundedness in C(G) and in H(G), Stieltjes-Osgood theorem, compactness in H(G) continuous linear functionals on H(G) Runge theorems for a compact and for an open set, approximation by polynomials, Osgood theorem applications of Runge theorem (Mittag-Leffler theorem, functions which may not be continued) 4. Conformal mappingsPreservation of angles, conformal mappings - definition and the relationship to angle, conformal mappings on the extended complex plane and on C, Schwarz lemma, Riemann theorem 5. Harmonic functions in the plane and holomorphic functionsRelationship of harmonic and holomorphic functions, Poisson integral, mean value property, Schwarz reflexion principle Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (09.05.2022)
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Elements of complex analysis as covered by course NMMA301 Introduction to Complex Analysis Last update: Kalenda Ondřej, prof. RNDr., Ph.D., DSc. (09.05.2022)
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