TITLE: Mapping half-plane to unit disk?
QUESTION [23 upvotes]: Say you have the half-plane $\{z\in\mathbb{C}:\Re(z)>0\}$. Is there a rigorous explanation why the transformation $w=\dfrac{z-1}{z+1}$ maps the half plane onto $|w|<1$?

REPLY [12 votes]: ( This is more comment then answer but I can't add images in comment ) 

Her is image and src code