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Master's Dissertation
DOI
10.11606/D.45.2018.tde-13122017-161946
Document
Author
Full name
Jéssica Laís Calado de Barros
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2016
Supervisor
Committee
Oliveira, Oswaldo Rio Branco de (President)
Martin, Paulo Agozzini
Oliveira, Ernandes Rocha de
Title in Portuguese
O teorema da aplicação de Riemann: uma prova livre de integração
Keywords in Portuguese
Abordagem Weierstrassiana
Lema de Schwarz
Princípio do argumento
Teorema da aplicação de Riemann
Teorema fundamental da álgebra
Abstract in Portuguese
Neste trabalho, seguindo a abordagem de Weierstrass, temos o objetivo de responder a seguinte questão: conhecida a equivalência entre holomorfia e analiticidade no caso complexo, quais propriedades das funções analíticas podem ser obtidas sem assumir tal equivalência? Analisando esta situação, resultados interessantes serão obtidos sem o uso de qualquer teorema de integração complexa e, para alcançar tal objetivo, nossas principais ferramentas serão a teoria de somas não ordenadas de famílias em C e propriedades do índice de caminhos fechados. Entre os resultados apresentados estão os conhecidos Teorema Fundamental da Álgebra, Lema de Schwarz, Teorema de Montel, Teorema da Série Dupla de Weierstrass, Princípio do Argumento, Teorema de Rouché, Teorema da Fatoração de Weierstrass, Pequeno Teorema de Picard e o Teorema da Aplicação de Riemann.
Title in English
The Riemann mapping theorem: an integration free proof
Keywords in English
Argument principle
Fundamental theorem of algebra
Riemann's mapping theorem
Schwarz's lemma
Weierstrassian approach
Abstract in English
In this work, following the Weierstrass's approach, we aim to answer the following question: knowing the equivalence between holomorphy and analyticity in the complex case, which properties of analytic functions can be obtained without assuming such equivalence? Through analyzing this situation, interesting results will be obtained without employing of any complex integration theorem and in order to achieve this goal, our main tools will be the theory of unordered sums in C and properties of winding numbers of closed paths. Among the proven results are the well known Fundamental Theorem of Algebra, Schwarz's Lemma, Montel's Theorem, Weierstrass's Double Series Theorem, Argument Principle, Rouché's Theorem, Weierstrass's Factorization Theorem, Picard's Little Theorem and the Riemann's Mapping Theorem.
 
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Publishing Date
2018-01-24
 
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