Find Möbius transformation that maps $x-$axis to $y=x$, maps $y-$axis to $y=-x$, maps the unit circle to itself.
I was thinking this means that $f$ maps $0$ to $0$, and maps 1 or -1 to $1+i$ or $1-i$, and maps $i$ or $-i$ to $-1+i$ or $1-i$. And I tried to plug in the formula $$f(z)=\frac{az+b}{cz+d}$$ But since there are so many combinations I am not sure I got the idea correct or not. Since I tried one combination and the result seems wrong...
Thanks for any help~