Suppose T1 is inversion in the circle |z|=r1, and T2 is inversion in the circle

Question

Suppose T1 is inversion in the circle |z|=r1, and T2 is inversion in the circle |z|=r2, where r1,r2>0. Prove that T2 composed with T1 is a dilation. Conversely, show any dilation is the composition of two inversions. Suppose T1 is inversion in the circle |z|=r1, and T2 is inversion in the circle |z|=r2, where r1,r2>0. Prove that T2 composed with T1 is a dilation. Conversely, show any dilation is the composition of two inversions. Suppose T1 is inversion in the circle |z|=r1, and T2 is inversion in the circle |z|=r2, where r1,r2>0. Prove that T2 composed with T1 is a dilation. Conversely, show any dilation is the composition of two inversions.

Explanation / Answer

Let T1 and T2 are two inversion Mobius transformation in circle |z|=r1 and |z|=r2 respectively.

So T1(z)=1/z and T2(z)=1/z

Now T1(T2(z))=T1(1/z)=z which is a dilation with a=1

Also T2(T1(z))=T2(1/z)=z which is also a dilation with a=1

Because T(z)= az is called dilation.

The converse part is not true any dilation can't be written as a composition of two inversion .

Actually if a is not equal to 1 in dilation then we can not write it as composition of two inversion.

For example if we take dilation T(z)= z/2 then it can't be written as composition of two inversion T1(z)=1/z and T2(z)=1/z

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