Laplace Fourier Transform S5.3.P Q1

[Sebastian Lech]

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 $A, B$ ?

[Victor Ivrii]

There was an explanation why one of the solutions in half-plane should be rejected (does not satisfy condition at infinity)

[Zengyue Lin]

So for this question why we don’t use cos and sin for the general solution?

[MikeMorris]

What condition is given at infinity? I don't see any condition given in the question for the behaviour of $u$ at infinity.

[Zengyue Lin]

I think we need to make assumption that the Fourier transformation u will be 0 as y goes to infinity,that’s what my TA did in tutorial

[Victor Ivrii]

I think we need to make assumption that the Fourier transformation u will be 0 as y goes to infinity,that’s what my TA did in tutorial

Or, at least, does not grow exponentially as $|k|\to \infty$

[shuxian]

Here is my solution for solving A.