International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 2, Issue 2(February 2013), PP.25-29 www.irjes.com
Schwarz-Christoffel Transformation On A Half Plane Abdualah Ibrahim Sultan Mathematics Department of Brawijaya University Malang, Indonesia. Mathematics Department of, Faculty of Science, Khoms, Al-Mergeb University , Khoms, Libya.
ABSTRACT :The Schwarz-Christoffel transformation has been studied by many reseachers due to a huge of its application in solving Laplace's equations and studying fluid flow phenomenas ( see [3],[4]). The proof of this theorem can find in (see for example [5]). Their method depend on the tangent of đ?‘‡đ?‘— = 1 + đ?‘–0 and the rotation of đ?‘‡đ?‘— from 0 to đ?œ‹ on points (đ?‘Ľ, 0) . In this paper we would like to prove the ShwarzChristoffeltransformation directly by using the arguments identity of complex number, rotation of it from 0 to 2đ?œ‹ and conformal mapping. Keywords : Complex function, half plane, polygon, analytic function, transformation, Conformal mapping.
I.
INTRODUCTION
The Schwarz-Christoffel transformation known as a map from the upper half-plane to simply connected polygon with or without corners at infinity in the complex planes.The Schwarz-Christoffel transformation has been studied by many reseachers due to a huge of its application for example used in physical applications involving fluid movement, heat conduction and electrostatic pontential, as well asthe studying of Laplace's equations (see [3],[4]) and also from mathematical point of views ( see [1],[2],[6]). In this paper we study about the proof of the Schwarz-Christoffel theorem. The prove of this theorem can found in (see for example [5]). Their method depend on the tangent of đ?‘‡đ?‘— = 1 + đ?‘–0 and the rotation of đ?‘‡đ?‘— from 0 to đ?œ‹ on points (đ?‘Ľ, 0). In this paper we would like to prove the Shwarz-Christoffel transformation directly by using the arguments identity of complex number, rotation of it from 0 to 2đ?œ‹ and conformal mapping.
II.
PRELIMINARIES
We need to review some important definitions and theorems which will be used later from complex analysis to understand the Schwarz-Christoffeltheorem.These definitions and theorems below can be found in the refences ( see [3], [4] ). Definition 2.1 A complex number is a number of the form đ?‘§ = đ?‘Ž + đ?‘–đ?‘? where the imaginary unit is defined as đ?‘– = −1andđ?‘Ž is the real part of đ?‘§, đ?‘Ž = đ?‘…đ?‘’(đ?‘§), and đ?‘? is the imaginary part of đ?‘§ , đ?‘? = đ??źđ?‘š(đ?‘§). The set of all complex number is written by đ??ś. Definition 2.2. Arguments of complex number Let r and θ be polar coordinates of the point đ?‘Ž, đ?‘? that corresponds to a nonzero complex number đ?‘§ = đ?‘Ž + đ?‘–đ?‘?. Since đ?‘Ž = cos đ?œƒand đ?‘? = sin đ?œƒ, the number đ?‘§can be written in polar form as đ?‘§ = r (cos đ?œƒ + đ?‘– sin đ?œƒ) = đ?‘§ đ?‘’ đ?‘–đ?œƒ , where đ?‘§ is a positive real number called the modulus of đ?‘§, and đ?œƒ is real number called the argument of đ?‘§, and written by arg đ?‘§.The real numberđ?œƒ represents the angle and measured in radians. Theorem 2.1. Law of arguments of products . Let đ?‘?, đ?‘? ∈ đ??ś. Then (1.1) arg đ?‘?đ?‘? = arg b + arg c. Theorem 2.2. Law of arguments of power Let đ?‘?, đ?‘? ∈ đ??ś. Then (1.2) arg đ?‘? đ?‘? = đ?‘? arg đ?‘?. Definition 2.3.Complex Function. A complex function is a function đ?‘“ whose domain and range are subsets of the set đ??ś of complex numbers. Definition 2.4.Conformal mapping. Let đ?‘¤ = đ?‘“ (đ?‘§) be a complex mapping defined in a domain đ??ˇ and let đ?‘§0 be a point in đ??ˇ. Then we say that đ?‘¤ = đ?‘“ (đ?‘§) is conformal at đ?‘§0 if for every pair of smoothoriented curvesđ??ś1 and đ??ś2 in đ??ˇ intersecting at đ?‘§0 the angle between đ??ś1 and đ??ś2 atđ?‘§0 is equal tothe angle between the image curves đ??ś ′ 1 and đ??ś ′ 2 at đ?‘“ (đ?‘§0 ) in both magnitude and sense.
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