Question 4 20 MARKS) Find a Mbius Transform w = $(2) from D= {]-[ 1} to W = {lul 0}, where 2 = x+i y and w = u +iv. Use the strategy in Parts (a)-(e) below to find 6, and then use your result to answer Part (d). (a) Sketch or plot both regions. (b) Write down the simplest Mbius Transform (with 1 in the numerator) that maps to o the common point of all three parts of the boundary of D. Explain why this is a good idea. Sketch or plot the image of D under this mapping. (C) Apply further transforms to position and scale the resulting region as required to fit W, and hence find the Mbius Transform o(z). (d) Find the harmonic function h on D subject to the boundary conditions h = 0) on the top two arcs and h = cos 2(1 - y) on the lower arc. Proceed as follows: (i) Derive the corresponding boundary conditions on W. (ii) Solve V H = 0 on W with these boundary conditions using separation of variables H(u, v) = f(u)g(v). You should also require H +0 as v + + to obtain a well-behaved solution in W. Question 4 20 MARKS) Find a Mbius Transform w = $(2) from D= {]-[ 1} to W = {lul 0}, where 2 = x+i y and w = u +iv. Use the strategy in Parts (a)-(e) below to find 6, and then use your result to answer Part (d). (a) Sketch or plot both regions. (b) Write down the simplest Mbius Transform (with 1 in the numerator) that maps to o the common point of all three parts of the boundary of D. Explain why this is a good idea. Sketch or plot the image of D under this mapping. (C) Apply further transforms to position and scale the resulting region as required to fit W, and hence find the Mbius Transform o(z). (d) Find the harmonic function h on D subject to the boundary conditions h = 0) on the top two arcs and h = cos 2(1 - y) on the lower arc. Proceed as follows: (i) Derive the corresponding boundary conditions on W. (ii) Solve V H = 0 on W with these boundary conditions using separation of variables H(u, v) = f(u)g(v). You should also require H +0 as v + + to obtain a well-behaved solution in W