I am trying to understand the problem defined by $$\phi_{xx} + \phi_{yy} = 0, in -\infty \lt x \lt \infty, y \gt 0$$ $$\phi = f(x) \space as \space y \to 0, \phi = 0 \space as \space y \to \infty$$
The first thing that I did was assign $$\phi(x,y) = \int_{-\infty}^\infty \phi(k,y)e^{-ikx}dk$$ which allowed me to solve for $$\phi_{yy} -k^2 \phi = 0$$ The part that I am struggling with is where I am told to use the boundry condition as $ y \to \infty$ to show that the solution to $\phi_{yy} -k^2 \phi = 0$ is $\phi(k,y) = C(k)e^{-|k|y}$. I understand everything past this point, how to solve for C(k) and $\phi(x,y)$. I don't understand where $C(k)e^{-|k|y}$ comes from.
My only thoughts on this is that $\phi = \frac{\phi_{yy}}{k^2}$.
You're almost there! Your last equation $\phi_{kk}=k^2 \phi$ is solved by an exponential, either $e^{-ky}$ or $e^{+ky}$. What combination of those two solutions satisfies $\phi\rightarrow 0$ as $y\rightarrow \infty$? (Hint: It's not really a combination.)