Maple ile kompleks analiz
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Date
2010-03-12
Authors
Dökmen, Mehmet Zekeriya
Journal Title
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Volume Title
Publisher
Uludağ Üniversitesi
Abstract
Bu çalışma esas olarak, kompleks analizin temel konularına ait uygulamaların Maple 9 programıyla çözümü üzerine kurulmuştur. Konular Mathews 2001'deki sırayı takip ederek, Maple 9 programını uygulamak için Basri 2004 ve http://math.fullerton.edu/ mathews/c2001/contents.html adlı kaynaklardan önemli ölçüde yararlanılmıştır. Normalde çok uzun işlemler sonunda çözülebilen kompleks analizin bazı problemleri, program yardımıyla çok daha kısa sürede çözülebilir hale gelmektedir.Çalışmamızın birinci bölümünde, kompleks sayıların cebir ve geometrisi, kutupsal gösterimi ve bir kompleks sayının n. kuvvetten kökü ile ilgili bilgi verilerek, bu kavramlara ait örnekler program yardımıyla çözüldü.İkinci bölümde kompleks değişkenli fonksiyonların dönüşüm özellikleri, limit ve sürekliliği incelenerek, program yardımıyla lineer, n. kuvvet ve n. kuvvetten kök fonksiyonları altında birim daire ve eksenlere paralel doğruların oluşturduğu kümenin resimleri bulundu.Üçüncü bölümde, kompleks değişkenli bir fonksiyonun türevi ve türev özellikleri, Cauchy-Riemann denklemleri ve harmonik fonksiyonlar üzerinde durularak, örneklerde programdan yararlanıldı.Dördüncü bölümde kompleks sayılarda tanımlı üstel, logaritma, trigonometrik ve ters trigonometrik gibi kompleks değerli fonksiyonlar tanımlandı. Daha sonra bu fonksiyonların bazı özellikleri ve belli kümelerin bu fonksiyonlar altındaki resimleri Maple programı kullanılarak gösterildi.Beşinci ve altıncı bölümlerde kompleks integral hesabında, kompleks fonksiyonların, Taylor ve Laurent açılımında, aykırı noktaların bulunmasında ve bu noktalardaki rezidülerin hesabında, ayrıca bazı kompleks ve reel integrallerin hesabında Maple 9 programı kullanıldı.
This study is mainly based on applications belonging to the central issues of complex analysis which solved by Maple 9 program. Issues by following the order in Mathews 2001, is benefited from the sources named Basri 2004 and http://math.fullerton.edu/ mathews/c2001/contents.html to implement the program Maple 9 significantly. Normally some problems of complex analysis that can be solved at the end of a long process, can be solved by the program much more quickly.In the first part of our study, by giving information related to algebra and geometry of complex numbers, polar representation of complex numbers, and n. root of a complex number, from the root, samples corresponding to these concepts have been solved by the help of the program.In the second part, by examining the transformation properties of functions of complex variable, limit and continuity, mappings of the sets were formed under the linear, exponential and n. root functions by unit circle and lines parallel to the axis were found.In the third section, in the examples emphasizing on the derivative and properties of derivative of a complex function, Cauchy-Riemann equations and harmonic functions was benefited from the program.In the fourth section complex valued functions like exponential, logarithm, trigonometric and inverse trigonometric functions which are defined in the complex numbers will be defined. Then some properties of these functions and images of certain sets under these functions were shown using the Maple program.In the fifth and sixth chapters; The Maple 9 Program was used in complex integral calculus, Taylor and Laurent expansions of complex functions, in finding singularities and residues at these points and also evaluations of some complex and real integrals.
This study is mainly based on applications belonging to the central issues of complex analysis which solved by Maple 9 program. Issues by following the order in Mathews 2001, is benefited from the sources named Basri 2004 and http://math.fullerton.edu/ mathews/c2001/contents.html to implement the program Maple 9 significantly. Normally some problems of complex analysis that can be solved at the end of a long process, can be solved by the program much more quickly.In the first part of our study, by giving information related to algebra and geometry of complex numbers, polar representation of complex numbers, and n. root of a complex number, from the root, samples corresponding to these concepts have been solved by the help of the program.In the second part, by examining the transformation properties of functions of complex variable, limit and continuity, mappings of the sets were formed under the linear, exponential and n. root functions by unit circle and lines parallel to the axis were found.In the third section, in the examples emphasizing on the derivative and properties of derivative of a complex function, Cauchy-Riemann equations and harmonic functions was benefited from the program.In the fourth section complex valued functions like exponential, logarithm, trigonometric and inverse trigonometric functions which are defined in the complex numbers will be defined. Then some properties of these functions and images of certain sets under these functions were shown using the Maple program.In the fifth and sixth chapters; The Maple 9 Program was used in complex integral calculus, Taylor and Laurent expansions of complex functions, in finding singularities and residues at these points and also evaluations of some complex and real integrals.
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Keywords
Kompleks analiz, Maple ile kompleks analiz, Maple 9, Complex analysis, Complex analysis with maple
Citation
Dökmen, M. Z. (2010). Maple ile kompleks analiz. Yayınlanmamış yüksek lisans tezi. Uludağ Üniversitesi Fen Bilimleri Enstitüsü.