Biholomorphic function between two,spaces

Question

Find a biholomorphic function $f$ between $A= \lbrace z: 0<\arg(z)<\dfrac{\pi}{2}, 0<|z|<1 \rbrace$ and $B=D(0,1)$.

I can't find the answer, i have no ideas coming up to my mind :'(

Thanks if you can help me !

Answer 1

Let me outline an approach that works well; it'll be instructive to fill in the details.

First, the squaring map $z \mapsto z^2$ takes the region $A$ to the intersection of the unit disk with the upper half plane. Next, notice that the linear fractional transformation $$\phi(z) = \frac{z-1}{z+1}$$ is holomorphic on this region. Because linear fractional transformations send lines and circles to lines and circles, the two boundary lines in this case get mapped to rays starting at the origin. In our case, you should check the boundary piece of $A$ lying on the real axis gets mapped onto the real axis and the semicircle piece gets mapped to the imaginary axis.

From here, it should be easy. Find a conformal map $f$ from the first quadrant onto the upper half plane. Then find a conformal map from the upper half plane to the disk (if you have a textbook, this is a classic example and likely in the book). If you compose all these mappings, you'll get the desired result. You should be able to obtain an explicit formula by evaluating this composition.