Learning objectives
The object of the course is to familiarize the students with the basic language and of some fundamental theorems in General Topology and in the theory of holomorphic finctions of one complex variable.
In particular,
1) the notions of topological spaces, subspaces, product and quotient topology, compact, connected and path-connected spaces, metric spaces and topological groups di spazio metrico will be provided;
2) the basic properties and facts of holomorphic finctions of one complex variable, as Cauchy-Riemann equations, Cauchy Theorem and Cauchy Formula, analytic continuation, Laurent series and residue theorem will be discussed.
Prerequisites
Geometry 1, Analysis 1, Algebra.
Course unit content
Topology and holomorphic functions of one complex variable.
Full programme
Topological spaces. Metric spaces. Continuos maps, homeomrphisms. Closure, interior adn boundary of a set. Neighbourhoods and basis of neighbourhoods. Countable and separation axioms. Topological subspaces. Product topology. Quotient topology. Compact, connected and path-connected spaces. Locally connected spaces. Metric spaces. Extension of continuos map. Topological groups. Matrices groups.
Complex derivative. Cauchy-Riemann equations. Cauchy Theorem. Cauchy Formula. Power series. Analytic functions. LIouville Theorem. Laurent Series. Residue Theorem.
Spazi topologici. Spazi metrici. Applicazioni continue, omeomorfismi. Chiusura, parte interna, frontiera. Basi di aperti, sistema fondamentale di intorni, assiomi di numerabilita'. Sottospazi. Topologia prodotto. Topologia quoziente. Assiomi di separazione. Quasi compattezza e compattezza. Connessione e connessione per archi. Spazi localmente connessi. Spazi metrici completi. Estensioni di funzioni continue. Metrica della convergenza uniforme. Gruppi topologici. Gruppi di matrici.
Funzioni olomorfe di una variabile complessa. Le condizioni di Cauchy-Riemann. Il Teorema di Cauchy. La formula di Cauchy. Disuguaglianze di Cauchy. Serie di potenze. Analiticita' delle funzioni olomorfe. Il teorema di Liouville. Il teorema fondamentale dell'algebra. Sviluppo in serie di Laurent. Il teorema dei residui.
Bibliography
H. Cartan, Elementary theory of analytic functions of one or several complex variables, Dover Publications, Inc., New York, 1995. 228 pp., V. Checcucci, A. Tognoli, E. Vesentini, Lezioni di topologia generale, ed. - Milano, Feltrinelli, 1977. 242 pp.
R. V. Churchill, Introduction to Complex Variables and Applications, McGraw- Hill Book Company, Inc., New York, 1948. vi+216 pp., E. Sernesi, Geometria 2, Torino Bollati Boringhieri, 1994. V. Villani, Elementi di topologia generale
Teaching methods
Geometry 1, Analysis 1, Algebra.
Assessment methods and criteria
Written and oral exams.
Other information
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