Solve using (partial) Fourier transform with respect to $y$

\begin{align}

&\Delta u:=u_{xx}+u_{yy}=0, &&x>0,\label{7-1}\\

&u|_{x=0}= g(y),\label{7-2}\\

&\max |u|<\infty\label{7-3}

\end{align}

with $g(y)=\frac{2}{y^2+1}$.

**Hint.** Fourier transform of $g(y)$ is $\hat{g}=e^{-|\eta|}$.