I lack intuition when it comes to some conformal mappings and I'm presently looking for a conformal map taking the inside of a disk, let's say the unit disk and sending it to the inside of an ellipse.
I know that a Joukowski transformation: $$z \rightarrow z + \frac{\eta}{z}$$ would take a circles and its exterior to an ellipse and its exterior but it is ill defined in its inside (singular in zero here). So it cannot be an inverse Joulkowski transform.
I heard that may be some elliptic functions could do the job. Is it the direction I should look into?