Conform mapping - trying to learn

Lets say I have the right half of the real plane. Furthermore the inside of the unit circle. Now I want do a conform mapping of this onto the unit circle. Hmm... I'll try to express my self a bit more mathematically.(Rock)

$\displaystyle A = \displaystyle \{ z \in \mathbb{C}|~ |z| < 1; ~Re (z) > 0 \}$

$\displaystyle B = \displaystyle \{ z \in \mathbb{C}|~ |z| < 1 \}$

So a conform mapping of A onto B.

How does the proper way of writing this look, or better yet, the right way of doing it. As of now I'd just go for something like. $\displaystyle w = z^2$

(Worried)