Konform dönüşümler ve konformal modül
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Date
2007
Authors
Koçum, Ahmet Faruk
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Publisher
Uludağ Üniversitesi
Abstract
Bu çalışma esas olarak, kompleks fonksiyonlar teorisinde önemli bir yer tutan, fen ve mühendislikte bir çok uygulama alanı olan konform dönüşümleri ayrıntılı inceleme temeline kurulmuştur. Çalışmamız dört bölümden oluşmaktadır. Birinci bölümde, diğer bölümlerde kullanılacak olan temel tanım ve teoremler verildi. İkinci bölümde konform dönüşümün tanımı diffeomorfizme bağlı olarak verildi. Kompleks düzlemde bir bölgede konform olma özelliği analitiklik ve ünivalent olmaya bağlı olarak ifade edildi. Genişletilmiş kompleks düzlemde konform dönüşümlerin tipi belirlendi. Konform dönüşümlerin temel teoremi sayılan Riemann dönüşüm teoremi verildi. Üçüncü bölümde, konformal modül tanımlanarak bazı özellikleri verildi. Konformal modülden faydalanarak Riemann dönüşüm teoreminin genellemesi olan birim dairenin konform genişlemesiyle ilgili Caratheodory-Osgood teoreminin modern ispatı verildi. Son bölümde, poligonlar üzerine konform dönüşümler için Schwarz-Christoffel formülü verilerek çeşitli uygulamaları yapıldı. Ayrıca basit bağlantılı bölgeler arasında konform dönüşüm örnekleri verildi.
This work as bases is established investigation based on conformal mappings, which are taken an important place in complex analysis and which have applications on science and engineering. Our work is formed by four chapters. In first chapter, basic definition and theories, which will be used in other chapters, were given. In second chapter, definition of conformal mapping was given depending on diffeomorphisms. In the complex plane, the link between the conformal mappings and analytic univalent functions was established. In the extended complex plane, the type of the conformal mappings was determined. Riemann mapping theorem, which is fundamental theorem in the conformal mappings, was given. In third chapter, conformal modulo was defined and its some properties were given. Glasses of functions which typically real, one direction convex and star like are defined and features of these functions are examined. Upper bound for maximum modules of these functions and its derivatives were given. The modern proof of the theorem Caratheodory-Osgood which is generalization of Riemann mapping theorem was given. In last chapter, the Schwarz-Christoffel formula and its applications were given. Also, conformal mappings examples between simple connected were given.
This work as bases is established investigation based on conformal mappings, which are taken an important place in complex analysis and which have applications on science and engineering. Our work is formed by four chapters. In first chapter, basic definition and theories, which will be used in other chapters, were given. In second chapter, definition of conformal mapping was given depending on diffeomorphisms. In the complex plane, the link between the conformal mappings and analytic univalent functions was established. In the extended complex plane, the type of the conformal mappings was determined. Riemann mapping theorem, which is fundamental theorem in the conformal mappings, was given. In third chapter, conformal modulo was defined and its some properties were given. Glasses of functions which typically real, one direction convex and star like are defined and features of these functions are examined. Upper bound for maximum modules of these functions and its derivatives were given. The modern proof of the theorem Caratheodory-Osgood which is generalization of Riemann mapping theorem was given. In last chapter, the Schwarz-Christoffel formula and its applications were given. Also, conformal mappings examples between simple connected were given.
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Keywords
Konform dönüşümler, Konformal modül ve ünivalent fonksiyonlar, Conformal mappings, Conformal modulus and univalent functions
Citation
Koçum, A. F. (2007). Konform dönüşümler ve konformal modül. Yayınlanmamış yüksek lisans tezi. Uludağ Üniversitesi Fen Bilimleri Enstitüsü.