As the last step we use the Riemann mapping theorem, to map the area biholomorphically to the upper halfplane, which maps the boundary to the real axis. By some reason (to be explained) the corresponding function indeed satisfies the Abel equation.

But we do that in two steps first we map to the unit disk via and then we map to the upper (open) halfplane via .

The existence of is guarantied by the Riemann mapping theorem. The translation maps to without having a fixed point, hence the corresponding mapping in , maps the unit disk into the unit disk without a fixed point. It is well known that such a function needs to be linear fractional, i.e. of the form . Kneser shows that must be parabolic, that means it has only one fixed point at the boundary of .

To map the unit disk to the upper half plane we can again use a linear fractional transformation . There are enough parameters to chose it such that is mapped to infinity and such that .

Now we can define

which has the property

.

and is biholomorphic on the interior of .

The interior of is mapped to some area bordering on the real line. The boundary on the real axis is hence mapped to the real axis. More precisely the interval is mapped to the real axis. By the Schwarz reflection principle it can be continued to the complex conjugate of especially it is analytic on and can from there be continued to a vicinity of the whole real axis by and .

But we do that in two steps first we map to the unit disk via and then we map to the upper (open) halfplane via .

The existence of is guarantied by the Riemann mapping theorem. The translation maps to without having a fixed point, hence the corresponding mapping in , maps the unit disk into the unit disk without a fixed point. It is well known that such a function needs to be linear fractional, i.e. of the form . Kneser shows that must be parabolic, that means it has only one fixed point at the boundary of .

To map the unit disk to the upper half plane we can again use a linear fractional transformation . There are enough parameters to chose it such that is mapped to infinity and such that .

Now we can define

which has the property

.

and is biholomorphic on the interior of .

The interior of is mapped to some area bordering on the real line. The boundary on the real axis is hence mapped to the real axis. More precisely the interval is mapped to the real axis. By the Schwarz reflection principle it can be continued to the complex conjugate of especially it is analytic on and can from there be continued to a vicinity of the whole real axis by and .