Mobius transformations and harmonic functions

Find a Mobius Transform w = o(2) from D = {|z| 1} to W = {lu| 0}, where z = x +y and w = utiv. Use the strategy in Parts (a)-(c) below to find o, and then use your result to answer Part (d). (a) Sketch or plot both regions. (b) Write down the simplest Mobius Transform (with 1 in the numerator) that maps to oo 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 Mobius 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. Hint: (i) Derive the corresponding boundary conditions on W. (ii) Solve V2H = 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 - too to obtain a well-behaved solution in W. (iii) Hence, using H, find the harmonic function h(x, y) on D satisfying the required boundary conditions