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Mobius and harmonic



Find a Mobius Transform w = (2) from D = {|2| < 1, (21-i > 1} to W = {lu| <
7, v > 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.]

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