On the Wikipedia page [here](https://en.wikipedia.org/wiki/Green%27s_function) , it states that the Green's function for 3D relativistic heat conduction (with $c=1$)


$$[\partial\_t^2 + 2\gamma\partial\_t -\Delta\_{3D}] u(t,x) = \delta(t,x) = \delta(t)\delta(x)$$


is given by 
$$u(t,x) = \frac{e^{-\gamma t}}{20\pi}\bigg( \big[8 - 3e^{-\gamma t} + 2\gamma t + 4\gamma^2t^2\big]\frac{\delta(t-|x|)}{|x|^2} + \gamma^2\Theta(t - |x|)\big[ \frac{I\_1\big(\gamma\sqrt{t^2 - |x|^2}\big)}{\sqrt{t^2 - |x|^2}} + t\frac{I\_2\big(\gamma\sqrt{t^2 - |x|^2}\big)}{t^2 - |x|^2}\big] \bigg)$$


Where $I\_1$ and $I\_2$ are [modified Bessel functions](https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions_:_I%CE%B1,_K%CE%B1) of the 1st and 2nd kind. Furthermore, $\Theta$ is the [Heaviside step function](https://en.wikipedia.org/wiki/Heaviside_step_function) . I am looking for a derivation of this or a reference to one. 


**Soft Attempt:**
Taking the Fourier transform with respect to $x$ we obtain 
$$[\partial\_t^2 + 2\gamma\partial\_t + (4\pi^2|\xi|^2)] \hat{u}(t,\xi) = \delta(t)$$


On the same Wikipedia page, this is a 1D damped harmonic oscillator and they claim the Green's function is given by
$$\hat{u}(t,\xi) = \Theta(t)e^{-\gamma t} \frac{\sin\big(t\sqrt{4\pi^2|\xi|^2 - \gamma^2}\big)}{\sqrt{4\pi^2|\xi|^2 - \gamma^2}}$$
 So I should expect that


$$u(t,x) = \int e^{2\pi i \xi\cdot x}\Theta(t) e^{-\gamma t}\frac{\sin\big(t\sqrt{4\pi^2|\xi|^2 - \gamma^2}\big)}{\sqrt{4\pi^2|\xi|^2 - \gamma^2}}d\xi$$


But I am not sure how we would get the formula above or if this is even the correct approach.



Green's Function for 3D Relativistic Heat Equation
The "relativistic" heat equation is more generally known as the [Telegrapher's equation,](https://en.wikipedia.org/wiki/Telegrapher%27s_equations)
$$\frac{\partial f}{\partial t}+\tau\frac{\partial^2 f}{\partial t^2}=\kappa\nabla^2 f.$$
The Green's function is calculated in [Application of the three-dimensional telegraph equation to cosmic-ray transport](https://arxiv.org/abs/1606.08272) (2016), see equation 10. In equation 11 it is checked that the $\tau\rightarrow 0$ limit gives the familiar Gaussian Green's function of the 3D diffusion equation.


I note that the result in that 2016 paper is different from the formula in the OP, taken from Wikipedia.


