Changes

<math> \frac{d^2}{dt^2} \left(\frac{1}{4\pi^2} G(\mathbf{k},\omega)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right)-c^2\frac{d^2}{dx^2}\left(\frac{1}{4\pi^2} G(\mathbf{k},\omega)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right) = \frac{1}{16\pi^4} e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}</math>

<math> \frac{d^2}{dt^2} \left(\frac{1}{4\pi^2} G(\mathbf{k},\omega)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right)-c^2\frac{d^2}{dx^2}\left(\frac{1}{4\pi^2} G(\mathbf{k},\omega)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right) = \frac{1}{16\pi^4} e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}</math>

This is a complicated equation, but it can be solved for G. Once G is calculated, we can apply an inverse Fourier transform and find g; we can then plug this into

This is a complicated equation, but it can be solved for G.

 

 

Once G is calculated, we can apply an inverse Fourier transform and find g; we can then plug this into

<math>\int{dt_i} \int{f(x_i,t_i)g(\Delta x, \Delta t) dx_i dy_i}</math>  

<math>\int{dt_i} \int{f(x_i,t_i)g(\Delta x, \Delta t) dx_i dy_i}</math>