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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.

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  • $\begingroup$ The conditions don't make sense for $x\in [-1,0].$ $\endgroup$ – zhw. Mar 10 at 18:09
  • $\begingroup$ @zhw. How come? I copied the question exactly and I do not see why that would cause a problem $\endgroup$ – Tyler6 Mar 10 at 18:26
  • $\begingroup$ Books can have mistakes. Clearly $|\psi(x,y)|<x, |x|\leq 1$ cannot hold (try $x=-1/2$) $\endgroup$ – zhw. Mar 10 at 18:39
  • $\begingroup$ @zhw. I edited the question to fix that. Do you know any way to solve it now? $\endgroup$ – Tyler6 Mar 10 at 19:24

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