Fourier transform for Helmholtz equation 
The Helmholtz equation takes the form,
  $$u_{xx} + u_{yy} + k^
2u = f(x, y),$$
  for $−∞ < x < ∞$, $−∞ < y < ∞$.
i) Assuming that the functions $u(x, y)$ and $f(x, y)$ have Fourier transforms
  show that the solution to this equation can formally be written:
$$u(x, y) = −
\frac{1}{4\pi^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
e^{−i(λ(x−ζ)+µ(y−η))} \frac{f(ζ, η)}{λ^2 + µ^2 − k^2}dλ  dµ d ζ dη$$
ii) Is this result consistent with the special case where $f(x, y) = 0$?

I tried taking fourier transform with respect to $x$. ($ F(f^{''}(x)) = -s^2F(f))$ and obtained the inhomogenous equation. 
$\frac{\partial^2F(u)}{\partial y^2} + (k^2 - s^2) F(u)=F(f)$ . Not quite sure if taking right direction or how to solve this equation correctly. Thanks.
 A: You want to take the 2d FT over $x$ and $y$ (integration limits are $-\infty\to\infty$ in all integrals below):
$$\hat{g}(k_x,k_y) \equiv \iint g(x',y')e^{-ik_xx'-ik_yy'}\,{\rm d}x'\,{\rm d}y' .$$
and with this convention the inverse transform is given by
$$g(x,y) \equiv \frac{1}{(2\pi)^2}\iint \hat{g}(k_x,k_y)e^{ik_xx + ik_yy}\,{\rm d}k_x\,{\rm d}k_y .$$
This transforms your equation $u_{xx} + u_{yy} + k^2u = f(x, y)$ to
$$\hat{u}(k_x,k_y)(-k_x^2-k_y^2+k^2) = \iint f(x',y')e^{-ik_xx' - ik_yy'}\,{\rm d}x'\,{\rm d}y'$$
Now dividing by $(-k_x^2-k_y^2+k^2)$ and taking the inverse transform you end up with the answer in the book (up to a renaming of the integration labels $x'\to \zeta$, $y'\to \eta$ etc.). Just be careful to not reuse an integration label already in use (this is a very common mistake and is why I used $x',y'$ instead of $x,y$ in defining the transform above).

When deriving this solution I made the implicit assumption that the FT exists (in the sense of a Riemann integral). When we take $f = 0$ we get $u = 0$ and this is indeed a solution. However this is only one of the solutions (and the same goes with the solution above). For example $u(x,y) = \sin(kx)$ is also a solution. The most general solution is obtained by adding the solution of the homogeneous equation $u_{xx} + u_{yy} + k^2u = 0$. None of the solutions of this equation has a FT (e.g. $\int_{-\infty}^\infty \sin(kx)e^{-ik_xx}{\rm d}x$ does not converge) in the traditional sense. The FT can be extended to a larger class of functions which has a FT in what we call a distribution (for which Dirac delta is one example) and we could make the derivation above work to produce these solutions, but that is a longer story (what changes is that an equation like $k\hat{u}(k) = \hat{f}$ no longer implies $\hat{u} = \frac{\hat{f}}{k}$ but rather $\hat{u} = \frac{\hat{f}}{k} + C\delta(k)$). Anyway separation of variables is a better way of solving the homogeneous equation if needed.
