Assume $f : \mathbb R → \mathbb{R}$ is a $C^\infty_c\mathbb{(R)}$ function. I was asked to use the Fourier transform (in $x$) to obtain Poisson's integral formula solution to the Laplace equation in the upper half-plane $u_{xx} + u_{yy} = 0$ for $-\infty< x < \infty, y > 0, u(x, 0) = f(x), |u| $ bounded.
Then the fourier transform of $u$ and $f$ are given by :
$\displaystyle \hat{u}(k,y)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}u(x,y)e^{-ikx}\text{d}x,$
$\displaystyle \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}\text{d}x,$
and the boundary condition becomes $\hat{u}(k,0)=\hat{f}(k)$.
Since $\hat{u}_{yy}(k,y)=k^2 \hat{u}(k,y)$, we solve Laplace's equation for $y$ and obtain $\hat{u}(k,y)=A(k)e^{ky}+B(k)e^{-ky}.$
Then, taking in account the boundary condition we find $A(k)+B(k)=\hat{f}(k)$ and applying the reverse Fourier transformation we obtain $u(x,y)$.
I am not sure that whether this is correct or wrong somewhere as it feels that there is something missing as i did not even use the half plane property. Can anyone help me how to figure it out?
Any type of help will be appreciated. Thanks in advance.