Given U_1 = {z = x+iy \in C: y > |x|}, f_1(z) = z^2, U_2 = f_1(U_1) = {z \in C: Re(z) < 0}  (i) why U_2 = {z \in C: Re(z) < 0} = {z \in C: (|z-1|/|z+1| >1}? how to get this transformation? (ii) How to find a conformal map f_3 that maps U_1 onto C\ \overline{D}_1(0) (maps U_1 onto complex plane minus closed 1-disc centred at 0).

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