Idea for deriving Green's functions - Helmholtz equation

Question

Let $x,y\in\mathbb{R}$. Then the Green's function for the Helmholtz equation is given by

$$\left(\Delta+\frac{\omega^{2}}{c^{2}}\right)G(x,y,\omega)=\delta(x-y).$$

Now what is the idea for deriving the Green's function here? Intuitively, I would take Fourier transforms on both sides, which would give me a convolution on the LHS and an exponential function on the RHS, but this would be very messy and I think I am missing something here. I believe the answer should be

$$G(x,y,\omega)=\frac{ic}{2\omega}e^{i\omega|x-y|/c}.$$

Apologies if this is a duplicate, but I could not find what I was looking for on the search bar.

Answer 1

I don't see where you get the convolutions on the LHS. The FT would give you (up to some multiplicative constants from FT normalisation, who cares)

$$(\omega^2/c^2-|\xi|^2) G(\xi,y,\omega ) = \exp(i y \xi),$$ which should be easy to manipulate - I think a residue theorem would do the trick.