I was wondering if anyone could let me know how to move forward on problem one:

Consider Dirichlet problem:

$$\begin{equation} u_{xx}+u_{yy}=0\end{equation}, -\infty<x<\infty, y>0$$

$$\begin{equation} u|_{y=0}=f(x)\end{equation}$$

We need to make a Fourier Transform by x and leave the solution in the form of a Fourier Integral.

What I did first was make the Fourier transform:

$$\begin{equation} \hat{u}_{yy}-\xi^2\hat{u}=0 \end{equation}$$

$$\begin{equation} \hat{u}|_{y=0}=\hat{f}(\xi) \end{equation}$$

Which has general solution:

$$\begin{equation} \hat{u}(\xi, y)=A(\xi)e^{-|\xi|y}+B(\xi)e^{|\xi|y}\end{equation}$$

and using equation (4):

$$\begin{equation}\hat{u}(\xi,0)=A(\xi)+B(\xi)=\hat{f}(\xi) \end{equation}$$

Which I am now stuck on, how do we solve for $A, B$ ?