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I try to solve Laplace's equation $\Delta u=0$ in the half plane: $${d^2u\over dx^2}+{d^2u\over dy^2}\:=\:0 \quad\text{where } -\infty<x<\infty \text{ and }y>0 $$ and the boundary condition $$\left.u\right|_{y=0}\:=\:\frac{x^2-1}{(x^2+1)^2} = \frac 1{x^2+1} - \frac{2}{(x^2+1)^2}$$ We can separate variables $u(x,y)=X(x)Y(y)$ and get ${X''\over X} = -{Y''\over Y} = k$, but what to do next?
These conditions will be enough or not, maybe I have to use polar coordinates and the Poisson integral?

Hanno
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  • Just use Poisson's integral formula for the upper half plane: http://mathfaculty.fullerton.edu/mathews/c2003/poissonintegralplane/PoissonIntegralTheorem.1.pdf – Alex R. Nov 20 '17 at 22:00
  • Now you can solve $X'' = kX$ and $Y'' = -kY$, which gives $X = Ae^{-\sqrt{k}x}+Be^{\sqrt{k}x}$ and $Y = C \cos(\sqrt{k}x) + D \sin(\sqrt{k}x)$. You can then determine $A,B,C,D$ from the other conditions. – Joel Nov 20 '17 at 22:42

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