Konformalna preslikavanja

master's thesis

• Author: Antonija Pehar
• Mentor: Marcela Hanzer (mentor)
• Committee member: Marcela Hanzer (predsjednik povjerenstva)
• Committee member: Boris Guljaš (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Najvažniji rezultat ovog diplomskog rada je Riemannov teorem o preslikavanju. Prije njegovog dokaza, u početnom poglavlju definirali smo sama konformalna preslikavanja. Nakon toga smo proučili razlomljene linearne transformacije i njihovu povezanost sa tim preslikavanjima. Važnu ulogu u dokazu Riemannovog teorema o preslikavanju imale su normalne familije koje smo definirali u trećem poglavlju i dokazali sljedeće: ako je \(F \subset H(\Omega)\) i \(F\) je uniformno ograničena na svakom kompaktnom podskupu područja \(\Omega\), onda je \(F\) normalna familija. Nakon sto smo definirali i konformalno ekvivalentna područja, mogli smo dokazati Riemannov teorem o preslikavanju. U petom poglavlju definirali smo klasu \(S\) koju čine sve \(f \in H(U)\) koje su injekcije na \(U\) i za koje vrijedi \(f(0)=0, f'(0)=1\), te smo pokazali neka svojstva te klase. U poglavlju "Neprekidnost na rubovima" definirali smo jednostavne rubne točke i dokazali sljedeću tvrdnju: ako je \(\Omega\) ograničeno jednostavno povezano područje u kompleksnoj ravnini i ako je svaka rubna točka od \(\Omega\) jednostavna, tada se svako konformalno preslikavanje iz \(\Omega\) na \(U\) može proširiti na homeomorfizam iz \(\overline{\Omega}\) na \(\overline{U}\). U posljednjem poglavlju bavili smo se konformalnim preslikavanjima kružnih vijenaca i dokazali zanimljivo svojstvo: kružni vijenci \(A(r_1, R_1)\) i \(A(r_2, R_2)\) su konformalno ekvivalentni ako i samo ako je \(R_1/r_1=R_2/r_2\).

Abstract (english)

The most important result of this diploma thesis is the Riemann mapping theorem. Before we could prove it, we defined the conformal mappings in the first chapter. After that we studied linear fractional transformations and their connection with those mappings. The important part of proving the Riemann mapping theorem are normal families which we defined in the third chapter. We also proved that if \(F \subset H(\Omega)\) and \(F\) is uniformly bounded on every compact subset of the region \(\Omega\); then \(F\) is a normal family. After we defined conformally equivalent regions, we could prove the Riemann mapping theorem. In the fifth chapter we defined the class \(S\) ; which is the class of all \(f \in H(U)\) which are one-to-one in \(U\) and satisfy \8f(0)=0, f'(0)=1\), and proved some of its properties. In the chapter "Continuity at the boundary" we defined simple boundary points and proved the next statement: if \8\Omega\) is a bounded simply connected region in the plane and if every boundary point of \(\Omega\) is simple, then every conformal mapping of \(\Omega\) onto \(U\) extends to a homeomorphism off \(\Omega\) onto \(U\). In the last chapter we studied the conformal mapping of an annulus and proved this interesting property: \(A(r_1, R_1)\) and \(A(r_2, R_2)\) are conformally equivalent if and only if \(R_1/r_1=R_2/r_2\).

• Keywords: konformalna preslikavanja; Riemannov teorem o preslikavanju; konformalno ekvivalentna područja
• Keywords (english): conformal mappings; Riemann mapping theorem; conformally equivalent regions
• Language: croatian
• URN:NBN: urn:nbn:hr:217:692618
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-29 11:28:35