MSM 452 -Quizy and Solutions -Theory of Functions of Complex Variables.pdf

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1MULUNGUSHI UNIVERSITYSCHOOL OF SCIENCE, ENGINEERING AND TECHNOLOGYDEPARTMENT OF MATHEMATICS AND SCIENCEMSM 452 -Theory of Functions of Complex Variables IIQuiz 1Instructions:(a) Answer all questions(b) Show all your working to earn full marksTIME ALLOWED: 1 Hour18 April, 2019————————————————————————————————————————–1.(a) Determine the region in thew-plane in which the rectangle bounded by the linesx= 0,y= 0,x= 2andy= 1 is mapped under the transformationw=2·e4·z. Hence, sketch the graph.(b) Show that the functionw=4ztransforms the straight linex=cin thez-plane into a circle in thew-plane.2.(a) Define(i) a conformal mapping,(ii) a linear fractional transformation.(b) Letw=f(z) =az+bcz+dwheread-bc6= 0 andc6= 0. Findlimz-→∞f(z).(c) Prove that ifc= 0, then the linear fractional transformationf(z) =az+bcz+d, can be decomposed intoa series of basic transformations consisting of translations, inversions, rotations and dilations.3. Letf(z) =1-z1+z. Setf1(z) =f(z) =1-z1+z,f2(z) =f1(f1(z)) and in generalfn(z) =fn-1(f1(z))(a) Show thatf(z) is a one-to-one function(b) Find the fixed points off(z)(c) Find the normal form off(z)(d) Find an expression forfn(z)4.(a) Define(i) the cross ratio of four distinct pointsz,z1,z2andz3in the complex plane.(ii) the symmetry of two pointszandz*in the complex plane.(b) Find the linear fractional transformation which maps the points 1,i,-1 into he points 2,i,-2respectively. Find the fixed and critical points of the transformation
2MULUNGUSHI UNIVERSITYSCHOOL OF SCIENCE, ENGINEERING AND TECHNOLOGYDEPARTMENT OF MATHEMATICS AND SCIENCEMSM 452 -Theory of Functions of Complex Variables IIQuiz 1 : SolutionsInstructions:(a) Answering all questions(b) Expected working to be shown in order to earn full marksTIME ALLOWED: 1 Hour18 April, 2019

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