Syllabus
*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 plane Infinite products, Weierstrass factorization theorem on C, Mittag-Leffler theorem on C, Cauchyova method of decomposing a meromorphic function *3. Algebra of holomorphic functions Algebras 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 mappings Preservation 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 functions Relationship of harmonic and holomorphic functions, Poisson integral, mean value property, Schwarz reflexion principle
Annotation
Mandatory course for the master study program Mathematical analysis. Advanced Complex Analysis.