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Definition: Let a be in C and |a|<1, then phi_a(z)=(z-a)/(1-(a`)z) is a Mobius transformation, where a` is the complex conjugate of a. phi_a is a conformal mapping self-map of the unit disk D for each a, and specifically of the boundary of the unit disk to itself. The same holds for (phi_a)^(-1)=phi_(-a).