Show how to map the semicircle $x^2 +y^2 = 1$, $y > 0$, onto $(x−1)^2+y^2 = 4$, $y > 0$, by a combination of $z \to z+l$ and $z \to kz$.

Question

I need some help with this one!

One can begin to understand the geometric significance of linear fractional transformations of the half plane by studying the simplest ones, $z \to z+l$ and $z \to kz$ for real $k$ and $l$.

Show how to map the semicircle $x^2 + y^2 = 1$, $y > 0$, onto the semicircle $(x−1)^2+y^2 = 4$, $y > 0$, by a combination of transformations $z \to z+l$ and $z \to kz$.

Answer 1

$z \rightarrow 2z$ will map $x^2 + y^2 = 1, y>0$ onto the circle $x^2 + y^2 = 2^2, y>0$.

Then $z \rightarrow z+1$ maps $x^2 + y^2 = 4, y>0$ onto $(x-1)^2 + y^2 = 4, y>0$.