Riemannov upodobitveni izrek : diplomsko delo

Potočar, Nina (author), Slapar, Marko (mentor)

V prvem delu diplomske naloge najprej predstavimo osnovne definicije in lastnosti kompleksne ravnine. V nadaljevanju se osredotočimo na kompleksne funkcije in lomljene linearne transformacije. Eno izmed bolj pomembnih poglavij je tudi Schwarzova lema in avtomorfizmi kroga. V drugem delu predstavimo Riemannov upodobitveni izrek, ki nam pove, da je vsako enostavno povezano območje v ₵, razen cele kompleksne ravnine, biholomorfno ekvivalentno enotskemu disku. Nato razložimo, zakaj izrek ne velja za celotno kompleksno ravnino. Na koncu sledijo primeri uporabe Riemannovega upodobitvenega izreka.

• Language: Slovenian
• Keywords: holomorfne funkcije, lomljene linearne preslikave, Schwarzova lema, Riemannov upodobitveni izrek
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Publisher: [N. Potočar]
• Year: 2017
• Number of pages: IV, 21 str.
• PID: 20.500.12556/RUL-95146
• UDC: 51(043.2)
• COBISS.SI-ID: 11694153
• Publication date in RUL: 19.09.2017

POTOČAR, Nina, 2017, Riemannov upodobitveni izrek : diplomsko delo [online]. Bachelor’s thesis. N. Potočar. [Accessed 11 April 2025]. Retrieved from: http://pefprints.pef.uni-lj.si/id/eprint/4651

Secondary language

• Language: English
• Title: Riemann mapping theorem

Abstract

In the first part of the thesis, we introduce the basic definitions and properties of the complex plane. We then focus on complex functions and linear fractional transformations. One of the main chapters of the thesis is the one regarding the Schwarz lemma and the automorphisms of the unit disk. In the second part of the diploma thesis we present the Riemann mapping theorey, which states that every simply connected domain in ₵, except for the whole plane, is biholomorphically equivalent to the unit disc. We then explain why the theorey does not apply to the whole complex plane. The final part of the thesis contains examples of the Riemann mapping theorey.

• Keywords: mathematics, matematika