I have gotten stuck on a boundary value problem which I believe is to be solved using the Poisson Integral Formula. The problem is: $$\nabla^{2}\psi=0, \psi(x,0)=0, |x|>1 ; |\psi(x,y)|<|x|, |x|\leq 1$$
I know how to solve these problems when the values on the boundary are given, as this has a simple formula. However I am unsure now, since I only know it’s bounded between $-1$ and $1$. This comes up in a class on complex analysis, so I do not believe it will be some advanced PDE technique, I believe the Poisson Integral Formula will be used.