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mahler1

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Homework Statement .

1) Let ##T## be the transformation over the extended complex plane that sends the poins ##0,i,-i## to the points ##0,1,\infty##. Prove that the image of the circle centered at the origin and of radius ##1## under this transformation is the line ##\{re(z)=1\}##.

2) For ##\alpha \in \mathbb C## such that ##|\alpha|\neq 1##, prove that the homographic transformation ##T(z)=\dfrac{z-\alpha}{-\overline\alpha z+1}## sends the circle ##\{|z|=1\}## to itself and sends ##\alpha## to ##0##.The attempt at a solution.

1) Let ##T(z)=\dfrac{az+b}{cz+d}##. Then, ##T(0)=0##, so ##b=0##. ##T(i)=1## and from this condition we get that ##ai=ci+d##, so ##a=c-di##. From the last condition we get that ##c(-i)+d=0##, so ##d=ci##. Finally, ##T(z)=\dfrac{2z}{z+i}##.

Sorry if this is obvious but how do I show that for any ##z \in \{|z|=1\}##, ##T(z)=w## with ##w## of the form ##w=1+bi##, ##b \in \mathbb R##?.

2) I have the same problem, if ##z \in \{|z|=1\}##, I want to show that ##T(z)=w## such that if ##w=a+bi##, then ##a^2+b^2=1##.

Edition:

I could solve 2), what I did was ##{T(z)}^2=T(z) \overline {T(z)}## and I could prove that the last expression equals to ##1##.

I would appreciate some help for 1).

1) Let ##T## be the transformation over the extended complex plane that sends the poins ##0,i,-i## to the points ##0,1,\infty##. Prove that the image of the circle centered at the origin and of radius ##1## under this transformation is the line ##\{re(z)=1\}##.

2) For ##\alpha \in \mathbb C## such that ##|\alpha|\neq 1##, prove that the homographic transformation ##T(z)=\dfrac{z-\alpha}{-\overline\alpha z+1}## sends the circle ##\{|z|=1\}## to itself and sends ##\alpha## to ##0##.The attempt at a solution.

1) Let ##T(z)=\dfrac{az+b}{cz+d}##. Then, ##T(0)=0##, so ##b=0##. ##T(i)=1## and from this condition we get that ##ai=ci+d##, so ##a=c-di##. From the last condition we get that ##c(-i)+d=0##, so ##d=ci##. Finally, ##T(z)=\dfrac{2z}{z+i}##.

Sorry if this is obvious but how do I show that for any ##z \in \{|z|=1\}##, ##T(z)=w## with ##w## of the form ##w=1+bi##, ##b \in \mathbb R##?.

2) I have the same problem, if ##z \in \{|z|=1\}##, I want to show that ##T(z)=w## such that if ##w=a+bi##, then ##a^2+b^2=1##.

Edition:

I could solve 2), what I did was ##{T(z)}^2=T(z) \overline {T(z)}## and I could prove that the last expression equals to ##1##.

I would appreciate some help for 1).

Last edited: