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    Mobius Transformations and Conformal Maps

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    (1) For j = 1,2 let R_j be the circle of diameter j/2 and center at (j/4)i. Also, let p(z) = 1/z be the inversion map.

    (a) If G is the region outside R_1 and inside R_2 then prove that p(G) = {z : -2 < Im z < -1}.

    (b) Prove that e^(pie*z) maps the strip {z : -2 < Im z < -1}onto the upper half-plane H_u.

    (c) Use the preceding results to find a conformal mapping of G onto the open unit disk D = {z : abs(z) < 1}.

    (2) Prove the orientation principle: Let R_1 and R_2 be 2 circles in C_oo, and let T be the mobius transformation such that T(R_1)=R_2. Let (z_1,z_2,z_3) be an orientation for R_1. Then T takes the right side and the left side of R_1 onto the right side and left side of R_2 with respect to the orientation (Tz_1,Tz_2,Tz_3).

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