Hint for Q 3 p. 97 Ahlfors continued:

The problem asks you to conformally map the plane with a half-circle arc cut out, to the complement of the unit disk.

Map F_1: Find this LFT that maps {-1,i,1} to {-1,0,1} in that order, and thus the arc segment given to the line segment [-1,1]. This gives a conformal equivalence between their respective complements in the sphere.

Map F_2: Use the inverse of Example 7, without the sign, in Stein-Shakarchi, to map the slit plane to the unit disc as described in Ahlfors p. 94.

Map G_3: If finally you compose with the map G_3 given by f(z)=1/z, the function G_3(F_2(F_1(z))) takes the complement of the arc to the outside of the unit circle. But the problem is the point at infinity is not fixed. You can rectify this is via:

Map F_2.5: An automorphism of the disk that takes F_2(F_1(infinity)) to 0

Map F_3: Now compose with f(z)=1/z.

The function F_3(F_2.5(F_2(F_1(z)))) takes the arc complement to the disk complement and the point at infinity to itself.