I would like to find a holomorphic map that transforms the lower half unit-disk into the upper half-plane, with two constraints:
- the image of the boundary [-1,1] of the half-disk is $\infty$
- the image of the boundary point -i is 0.
Thank you if someone has ideas.
Edit: (some precisions on background)
My question comes from analysis of elliptic PDEs.
I know an explicit formula for a harmonic function $u$ in the upper half-plane $\mathbb{R} \times \mathbb{R}_+^\star$, that satisfies a certain Neumann boundary condition on the boundary $\mathbb{R} \times \{ 0 \}$. The boundary condition is precisely $\frac{\partial u}{\partial\nu} = -\frac{1}{2}\sin(2(u-g))$ where $g$ is given by $\nu=ie^{ig}$, $\nu$ being the outer unit normal on the boundary of the domain.
Now, I would like to find an explicit formula for a harmonic function in the lower half-disk, that satisfies the same Neumann boundary condition as before, on a boundary that remains to find (but it is not the question here).
My strategy is to transform the upper half-plane into the lower half-disk using a biholomorphism $\phi$, because I will then be sure that the composition $u \circ \phi^{-1}$ is harmonic in the lower half-disk (as a composition of a harmonic function and a holomorphic function).
If I represent the vectors $(\cos u,\sin u)$ in the upper half-plane, the point $0$ is "repulsive". As I want additionally to keep the "behaviour" of these vectors for the composition $u \circ \phi^{-1}$, I ask two conditions for $\phi^{-1}$: $\phi^{-1}(-i)=0$ and $\phi^{-1}(x)=\infty$ for $x\in [-1,1]$.