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Title:Princip največje absolutne vrednosti
Authors:Makovec, Edina (Author)
Jakovac, Marko (Mentor) More about this mentor... New window
Files:.pdf MAG_Makovec_Edina_2018.pdf (1,02 MB)
Work type:Master's thesis/paper (mb22)
Typology:2.09 - Master's Thesis
Organization:FNM - Faculty of Natural Sciences and Mathematics
Abstract:Magistrsko delo obravnava princip največje absolutne vrednosti, ki predstavlja pomemben rezultat v kompleksni analizi. Izraz princip največje absolutne vrednosti pravzaprav predstavlja skupek izrekov, ki govorijo o največji absolutni vrednosti holomorfne funkcije na nekem območju. Princip lahko uporabimo tudi za dokazovanje nekaterih pomembnih matematičnih izrekov, kot so osnovni izrek algebre, Schwarzova lema in mnogi drugi. V prvem delu magistrskega dela bomo predstavili nekaj osnovnih de finicij, izrekov in pomembnih rezultatov, ki jih bomo potrebovali v naslednjih poglavjih. Drugi del je posvečen predstavitvi principa največje absolutne vrednosti. V tem poglavju bomo podali več verzij izreka o največji absolutni vrednosti in jih tudi dokazali. Omenili bomo še Gaussov izrek o srednji vrednosti, ki ga bomo uporabili pri dokazu principa največje absolutne vrednosti. V tretjem in zadnjem poglavju pa bomo prikazali uporabo principa največje absolutne vrednosti pri dokazu drugih matematičnih izrekov in podali še fizikalno interpretacijo. Za konec bomo za boljše razumevanje predstavili princip največje absolutne vrednosti na konkretnem matematičnem zgledu.
Keywords:princip največje absolutne vrednosti, holomorfne funkcije, Schwarzova lema, osnovni izrek algebre, Phragmen-Lindelofov princip, Borel-Caratheodoryjev izrek, Hadamardov izrek o treh premicah.
Year of publishing:2018
Publisher:[E. Makovec]
COBISS_ID:23868168 Link is opened in a new window
License:CC BY-NC-ND 4.0
This work is available under this license: Creative Commons Attribution Non-Commercial No Derivatives 4.0 International
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Secondary language

Title:The maximum modulus principle
Abstract:This master thesis focuses on the maximum modulus principle, which is an important result in complex analysis. We use the term maximum principle for multiple theorems, concerning the maximum modulus of holomorphic functions on a region. The principle can be used for the purpose of proving important mathematical theorems, such as the fundamental theorem of algebra, the Schwarz's lemma and many others. In the first part of the master thesis, we will introduce some basic de nitions, theorems and important results which will be needed in the following chapters. The second part is dedicated to the representation of the maximum modulus principle. In this chapter we will give several theorems of the principle and also prove them. We will mention Gauss's mean value theorem, which we will use to prove the maximum modulus principle. In the third and the last part, we will show the application of the maximum modulus principle for proving other mathematical theorems and also give a physical interpretation. We will conclude with a concrete mathematical example, for better understanding of the principle.
Keywords:maximum modulus principle, holomorphic functions, Schwarz's lemma, fundamental theorem of algebra, Phragmen-Lindelof principle, Borel-Caratheodory theorem, Hadamard three-lines theorem.


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