Consider Laplace equation in the half-plane
\begin{equation*}
u_{xx} +u_{yy}=0 \qquad x>0, \ -\infty<y< \infty
\end{equation*}
with the boundary conditions
\begin{equation*}
u(0,y)=e^{-|y|}
\end{equation*}
and condition $\max |u|<\infty$.

Solve using Fourier Transform; write the solution in the form of a Fourier integral.

Problem 2.
Consider Laplace equation in the half-strip
\begin{equation*}
u_{xx} +u_{yy}=0 \qquad x>0, \ -1<y< 1
\end{equation*}
with the boundary conditions
\begin{gather*}
u (x,-1)=u(x,1)=0,\\[5pt]
u(0,y)=1-|y|
\end{gather*}
and condition $\max |u|<\infty$.

Write the associated eigenvalue problem.

Find all eigenvalues and corresponding eigenfunctions.

Write the solution in the form of a series expansion.