I am looking for a conformal mapping from the unit disc to itself $F:D\to D$ characterized by
- it sends a point $a\in D$ to the origin;
- it is the identity in $\partial D$.
I am aware of the exsitance of the Möbius transformations (see this post), that send $\partial D$ to $\partial D$: $F(\partial D)=\partial D$. However, the property I am looking for is pointwise along the boundary: $F(z)=z$ for all $z\in\partial D$.
The question arises from a physics context. I want to find a mapping that sends the potential lines of a point charge at $a\in D$ to the potential lines of a point charge at the origin, provided that the potential vanishes in $\partial D$.