# Coming up with specific conformal maps

I have to find a conformal map from $$G := \Bbb D \setminus\{ai\, \vert\, a>0\}$$ onto

• (a) $$\Bbb D$$ and

• (b) $$H := \{w∈\Bbb C\, \vert\, \vert Re(w)\vert<1\}$$.

I don't really know where to start since this seems like kind of a guessing game. I obviously know that the concatenation of conformal maps is again conformal, I just have no idea what the individual steps should be. I should say that we have defined a map to be conformal if it is holomorphic and the derivative doesn't vanish.

For (a) I had the following idea (which I would have to do in reverse order, of course): Start by mapping $$\Bbb D$$ to the right half plane with the Möbius Transformation $$z↦\frac{-z+i}{z+i}$$ and then $$z↦z^2$$ should map the right half plane to the entire complex plane without the negative real axis. We can use a rotation to get the complex plane without the positive imaginary axis and then I would need another Möbius Transformation to map this to just the unit disk without the positive imaginary axis. I don't know how to do this last step though. For (b), I have absolutely no clue where to start.

Edit: As pointed out, the previous MT I had written here mapped the unit disk to the left half plane. I hope this one is correct now.

• $\frac{z+1}{z-1}$maps $\mathbb{D}$ to left half plane Commented Oct 29, 2017 at 11:08
• Thanks, changed it now. Do you have any idea on what the last MT would have to be? Is there even such a Möbius Transformation? Commented Oct 29, 2017 at 12:10
• Also $\frac{z+i}{z-i}$maps $\mathbb{D}$ to left half plane. Commented Oct 29, 2017 at 12:12
• See math.stackexchange.com/questions/264865/… may helps. Commented Oct 29, 2017 at 12:14

For the second part, we observe $w=\log z$ which maps circles $|z|=r$ to segments $u=\log r$ with $-\pi<v\leq\pi$, and all rays $\theta=\theta_0$ to horizontal lines $v=\theta_0$. This means $w$ maps right half plane to horizontal strip $-\pi<v<\pi$. So if you map $\mathbb{D}$ to right half plane with $\dfrac{1+z}{1-z}$ and then with $\log z$ to horizontal strip $-\pi<v<\pi$, then with a rotation, you find the map $\dfrac{2i}{\pi}\log\dfrac{1+z}{1-z}:\mathbb{D}\to H$. With the first part make desire map from $G$ to $H$.