I am looking to a formula for the conformal map from the unit disc in the interior of an ellipse centered in 0 and with semiaxes a,b>0. I know that depends on elliptic function, but I don't find any references? thanks in advance for any formula or reference.
This paper seems to provide the formula you desire (section 2.2), as well as more references apropos of its derivation.
This paper by Kanas & Sugawa seems to meander around your question. In the first section they remark that Schwarz found such a map back in the day, but so far I have been unable to locate it a source for it. Section 3 discusses an ODE which one of your desired functions satisfies.
This paper deals with mapping the boundaries of the domains you're considering. While it is not immediately useful, it may be of interest. At the end, the authors remark that a conformal map of the form you desire will ''not be able to map concentric circles to concentric ellipses''.