[Complex Analysis] Finding a conformal map

[nonequilibrium]

Homework Statement

I have to find a conformal map from $$\Omega = \{ z \in \mathbb C | -1 < \textrm{Re}(z) < 1 \}$$
to the unit disk D(0,1)

Homework Equations

an analytical function f is conformal in each point where the derivative is non-vanishing
specifically, we can think of linear fractionals/mobius transformations, which are conformal everywhere and determined by the image of three numbers

The Attempt at a Solution

I've busted my brain on this one, but I can't think of anything that will transform these two vertical boundary lines into something useful. Note that I do not have to transform it to the unit disk; also a half/quarter-plane, a disk with a slit cut out, half a disk, will do, as I already know (from previous exercises) conformal transformations from those sets to the unit disk

 

[Dick]

If you take the set {z : 0<Im(z)<pi} then exp(z) maps that strip into the upper half plane, right? That's a lot like your problem.

 

[nonequilibrium]

Oh jeesh I can't believe I didn't see that x_x

Let's hope I get all the stupidity out before the exam

Thank you!