Make a conform map g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i) = 0$

Question

make a conform transformation g that sends the unit disk to $A = \{w: \operatorname{arg}(w) \in(\frac{\pi}{4},\frac{3\pi}{4})\}$ such that $g(2i)$.

I actually solved it by taking the inverse of $f(x) = \frac{z^2+1}{z^2-1}$, so $g(x) = \sqrt{\frac{z+1}{z-1}}$ but this fails as $g(2i) \neq 0$. I do't see how a mobius transformation would work because the lines need to intersect in 2 points. Any tips or hints?

Answer 1

The transformation is supposed to conformally map the unit disc $U$ onto the angle $A$. You realize that the point $2i\not\in U$, right? Hence the question lacks clarity.