Given the image, I would like to conformally map the plane minus the cut of this contour to a plane with a cut along the positive real axis say. As pointed out by @saulspatz Möbius transforms cannot be used.
I would also like my map to be injective. Any hints or help would be greatly appreciated, this has been driving me crazy!
This can't be done with a Möbius transformation. The inverse transformation is also Möbius and would carry the positive real axis to a half-line or a semicircle, not to the union of both.