课程名称︰工程数学-复变 课程性质︰电机系大二必修 课程教师︰林坤佑 开课学院：电机资讯学院 开课系所︰电机工程学系 考试日期（年月日）︰2018/6/19 考试时限（分钟）：50 试题 : 1. Smith chart is a useful graphical tool for the normalized impedence (z) and reflection coefficient (Gamma). (a) Construct a linear fractional transformation Gamma = f(z) that maps -i, 0, i of a straight line on z-plane onto -i, -1, i of circle on Gamma- plane.(15%) (b) Find the image curve of C1 = {z | Re{z} = 1} under the mapping function Gamma = f(z).(15%) 2. Use the Schwarz-Christoffel formula to find the transformation w = f(z) to map the domain D in Fig.1(a) to the domain D' in Fig.1(b).(30%) (Hint: (a) Use w1 = f(-1) = -1 and w2 = f(1) = 1, (b)∫A/(1-z^2)^(1/2)dz= sin^(-1)z) 3. Find a complex mapping from the disk |z - 1| ≦ 1 to the disk |w| ≦ 2.(10% ) 4. Find an appropriate conformal mapping and solve the Dirichlet problem as shown in Fig.2.(30%) ∂^2 ∂^2 D.E. (---- + ---- )p(x, y) = 0, R = {z | |z| < 1}, ∂x^2 ∂y^2 B.C. p(x, y) = 2 for z = e^(iθ), -π/2 < θ < π/2 p(x, y) = 2 for z = e^(iθ), π/2 < θ < 3π/2 https://i.imgur.com/I1SuXSX.jpg --

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