I need a Möbius transformation from the set of $z$s contained in the unit circle centered at the origin with real part greater than $0$ to
the infinite sector bounded by the lines making $45$ degrees with the real axis, i.e. $$45^{^\circ}<\arg( z)<135^{\circ}.$$
So, I need to map three points but not sure if my choice is correct.
I need a map so that
$$i \to -1+i,-i \to 1+i, 1\to \infty.$$
Will these choices do?
Thanks