1. The Mobius transformation \(T_1(z)=\dfrac{z+b}{z+d}\) maps the line \(\operatorname{Im}(z)=\operatorname{Re}(z)+3\) onto the unit circle in such a way that the region above the line is mapped to the interior of the circle and that 5i is mapped to the Origin. Find the value of d. 

2. The Mobius transformation \(T_2(z)=\dfrac{az+b}{z-1}\) maps the unit circle onto the line \(\operatorname{Re}(z)=3\) in such a way that the interior of the circle is mapped to the left of the line and that the origin gets mapped to 2+i. Find a.

3. Let \(T_1\) and \(T_2\) be as in the previous two parts, and let \(T_3(z)=T_2\circ T_1(z)\). Then we can consider \(T_3\) to be the composition of a translation followed by a dilation followed by a rotation. By what factor does \(T_3\) dilate? 

All help is appreciated!

 Jan 21, 2021

1. d = 1 + 2i


2. a = 1 - i


3. T_3 dilates by a factor of sqrt(3)/2.

 Jan 23, 2021

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