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First part of mobius and harmonic


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Find a Mobius Transform w = o(z) from D = {|z| < 1, [z+1-i| > 1} to W = flu| <
7, v> 0}, where z = x +iy and w = utiv.
(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).

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