Finding a conformal map from unit disk to half-plane

Question

I'm trying to find a conformal map $f$ from the open unit disk to the set $\mathbb{C}-[-1/4,-\infty)$ (I think this means the half-plane Re$(w)>-1/4$ with the properties $f(0)=0$ and $f'(0)>0$. I know that the mapping $$f(z)=\frac{i+z}{i-z}$$ returns the right half-plane Re$(w)>0$ from the open unit disk, but subtracting 1/4 from it doesn't satisfy $f(0)=0$. I can't seem to find a lot of other examples. Are there any other conformal maps that I should try?

Answer 1

Here's an outline; I'll leave the details to you:

The map you have will send the unit disc to a half plane. To get from a half plane to all of $\mathbb{C}$ minus a ray, postcompose with $z\mapsto z^2$. Now, to get the missing ray where you want it, rotate and translate.

Lastly, look at the pre-image of $0$. You can precompose with an automorphism of the disk sending $0$ to that point. Then all that's left is to check that, when you compose all these maps, the derivative is a positive number.

Answer 2

Start with

$$L_1(z)=\frac{i}{2} \frac{z+1}{1-z}$$

Then rotate -90 degrees to get the right half plane. Then $z^2$ etc....