Mobius and harmonic

Find a Mobius Transform w = (2) from D = {|2| 1} to W = {lu| 0}, where z = a try and w =utiv. Use the strategy in Parts (a) (c) below to find o, and then use your result to answer Part (d). (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 VH = 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. (iv) Using computer software of your choice, plot the contours 0, 0.05, . .., 1 of h on D. [If using Mathematica, you might use ContourPlot [. .., RegionFunction-> Function [{x, y, z}, ...]] to restrict the region plotted.]