The wedge is given by $0< \arg(z) < \alpha \pi$, for $0 < \alpha <1$.
If I use the complex logarithm, principal branch $-\pi < 0 \le \pi$, then
$$z \mapsto w = \operatorname{Log}(z)$$
$$ = \ln|z|+ i\arg(z)$$
$$= u+iv$$
so that $0<v<\alpha\pi$, which shows that the wedge maps onto a semi-infinite, horizontal half-strip, with height ranging from $0$ to $\alpha \pi$.
What can I do next? Is the Logarithm mapping even a good start? I usually like it, when I see a circular region that I have to start with, but not for very great reasons other than that I know what the mapping does to a circular region…
Thanks,
Best Answer
This is a standard problem in beginning complex variable, as I learned from Ahlfors. First you apply the function $z\mapsto z^{1/a}$. It’s pretty clear that $a$ must be not too big, I guess less than $2$, and positive, I’m sure. You see that this sends your wedge to the upper half plane. Then any good map like $z\mapsto (iz+1)/(z+i)$ will get the UHP to the disk. To define $z^{1/a}$, you have to use a suitable branch of the logarithm, and that’s why you want $0<a<2$.