Sketching the image of a given function Given the upper half-plane $\cal H$ and the map $$\varphi:\cal H\rightarrow\mathbb{C},~z\mapsto\frac{z-i}{z+i}.$$
Can somebody sketch the image $\varphi(S)$ with $$S=\{z\in{\cal H}:|\text{Re}(z)|<1/2,~ |z|>1\}?$$
I know that the image is included in the unit disk. And I guess that the boundary is some kind of triangle. Does this 'triangle' touches the boundary of the unit disk? Where is the point $i\infty$ maped to? 

This being a Mobius transformation, $\varphi$ maps straight lines or circles to straight lines or circles.  A straight line or circle is determined by $3$ points, so take three points on it, find their images, and see what is the straight line or circle through those points.  
If $z$ is real, $z - i = \overline{z+i}$, so $\varphi(z) = 1$.  Thus $\varphi$ maps the real line to the unit circle.  So it's not true that the image is in the unit circle.  The upper half plane maps into the unit circle, the lower half plane outside.  
