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Upper half of a complex plane

Show that transformation
W (Z) = (a Z + b) / (c Z + d)
of the upper half of a complex plane is 1-1 and onto the upper half plane if a, b, c, and d are real and satisfy condition
a d > b c

Solution Preview

We see that it is 1-1 just from inverting the transformation:
Z(W) = (d W - b) / (-c W + a).

To see that the transformation is onto we show that both the direct and the ...