[Math] How to find transformation from the upper half plane into the right half plane

Question

Find the general form of the linear transformation which transforms

the upper half plane into the right half plane.

In my notes I have a Mobius transformation from the upper half plane to the unit circle $T(z)=e^{i\theta_0}\frac{z-z_0}{z-\overline z_0}$.

Also another transformation from the unit circle to the upper half plane $T(z)=(1-i)\frac{z-i}{z-1}$.

But I do not know how to construct the possible composition transformation from upper half plane into the right half plane.

Could someone help please?

Any hint?

Answer 1

Simple, rotating 90 degrees to the right:

$$ T(z) = -iz $$