I was studying for an exam and I found this question which has caused me a bit of trouble:

Given the Green's function that satisfies the equation

$$\Box G(\mathbf{r}-\mathbf{r}',t-t')=-4\pi\delta(\mathbf{r}-\mathbf{r}',t-t'),$$ where $\Box$ is the d'Alembertian operator

$$\Box=\Delta-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}$$ show that, given a suitable method of handling the poles, its Fourier transform is


The aim of the question is obviously to look at ways of solving the wave equation by Fourier transform, but I'm a little bit unsure how to proceed, especially since $g$ looks a bit bizarre to me. The term $\omega$ refers to the angular frequency of the wave, and $\mathbf{k}$ the wave vector. What is the simplest/best way to go about this? Finding $G$ first and then taking its Fourier transform seems like a tedious way to solve this problem.

Thanks for any help.


Generally you would Fourier transform the differential equation. The elements of the D'Alembertian transform to $\vec{k}\cdot\vec{k}$ (the Laplacian) and $-\omega^2$ (the 2nd time derivative), while the delta function transforms to $1$. Then you get something like $$ (2\pi)^4 \left(\left\lVert \vec{k}\right\lVert^2 - \frac{\omega^2}{c^2}\right) g= 4\pi $$ Solving for $g$ gives you the answer.

This is obviously greatly simplified and makes many assumptions about certain integrals converging, but it's the general approach.


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