Editing Green's function (section)

===Table of Green's functions===
{{Disputed section|Dimensional inconsistencies and wrong scaling|date=April 2025}}

The following table gives an overview of Green's functions of frequently appearing differential operators, where {{nowrap|<math display="inline"> r = \sqrt{x^2 + y^2 + z^2}</math>,}} {{nowrap|<math display="inline"> \rho = \sqrt{x^2 + y^2}</math>,}} <math display="inline"> \Theta(t)</math> is the [[Heaviside step function]], <math display="inline"> J_\nu(z)</math> is a [[Bessel function]], <math display="inline"> I_\nu(z)</math> is a [[modified Bessel function of the first kind]], and <math display="inline"> K_\nu(z)</math> is a [[modified Bessel function of the second kind]].<ref>some examples taken from {{cite book | last = Schulz | first = Hermann | title = Physik mit Bleistift: das analytische Handwerkszeug des Naturwissenschaftlers | date = 2001 | publisher = Deutsch | isbn = 978-3-8171-1661-4 | edition = 4. Aufl | location = Frankfurt am Main}}</ref> Where time ({{mvar|t}}) appears in the first column, the retarded (causal) Green's function is listed.

{| class="wikitable"
|-
! Differential operator {{math|''L''}} !! Green's function {{mvar|G}} !! Example of application
|-
| <math>\partial_t^{n+1}</math> || <math>\frac{t^n}{n!} \Theta(t)</math> ||
|-
| <math>\partial_t + \gamma </math> || <math>\Theta(t) e^{-\gamma t}</math> ||
|-
| <math>\left(\partial_t + \gamma \right)^2</math> || <math>\Theta(t)t e^{-\gamma t}</math> ||
|-
| <math>\partial_t^2 + 2\gamma\partial_t + \omega_0^2</math> where <math> \gamma < \omega_0 </math> || <math>\Theta(t) e^{-\gamma t} \, \frac{\sin(\omega t)}{\omega}</math> &nbsp; with &nbsp; <math>\omega=\sqrt{\omega_0^2-\gamma^2}</math>|| [[Harmonic oscillator#Damped harmonic oscillator|1D underdamped harmonic oscillator]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t + \omega_0^2</math> where <math> \gamma > \omega_0 </math> || <math>\Theta(t) e^{-\gamma t} \, \frac{\sinh(\omega t)}{\omega}</math> &nbsp; with &nbsp; <math>\omega = \sqrt{\gamma^2-\omega_0^2}</math>|| 1D overdamped harmonic oscillator
|-
| <math>\partial_t^2 + 2\gamma\partial_t + \omega_0^2</math> where <math> \gamma = \omega_0 </math> || <math>\Theta(t) e^{-\gamma t} t</math> || 1D critically damped harmonic oscillator
|-
| 1D Laplace operator <math> \frac {d^2}{dx^2} </math>
| <math> \left(x - s\right) \Theta(x-s) + x\alpha (s) + \beta(s) </math>
| 1D Poisson equation
|-
| [[Laplace operator#Two dimensions|2D Laplace operator]] <math>\nabla^2_{\text{2D}} = \partial_x^2 + \partial_y^2</math> || <math>\frac{1}{2 \pi}\ln \rho </math> &nbsp; with &nbsp; <math>\rho=\sqrt{x^2+y^2}</math>|| 2D Poisson equation
|-
| [[Laplace operator#Three dimensions| 3D Laplace operator]] <math>\nabla^2_{\text{3D}} = \partial_x^2 + \partial_y^2 + \partial_z^2</math> || <math>-\frac{1}{4 \pi r}</math> &nbsp; with &nbsp; <math> r = \sqrt{x^2 + y^2 + z^2} </math> || [[Poisson equation]]
|-
| [[Helmholtz equation|Helmholtz operator]] <math>\nabla^2_{\text{3D}} + k^2</math> || <math>\frac{-e^{-ikr}}{4 \pi r} = i \sqrt{\frac{k}{32 \pi r}} H^{(2)}_{1/2}(kr) = i \frac{k}{4\pi} \, h^{(2)}_{0}(kr)</math> {{br}} where <math>H_\alpha^{(2)}</math> is the [[Hankel function of the second kind]], and <math>h_0^{(2)}</math> is the [[spherical Hankel function of the second kind]] || stationary 3D [[Schrödinger equation]] for [[free particle]]
|-
|Divergence operator <math>\nabla \cdot \mathbf{v}</math>
|<math>\frac{1}{4 \pi} \frac{\mathbf{x} - \mathbf{x}_0}{\left\|\mathbf{x} - \mathbf{x}_0\right\|^3} </math>
|
|-
| <math>\nabla^2 - k^2</math> in <math>n</math> dimensions || <math>- \left(2\pi\right)^{-n/2} \left(\frac{k}{r}\right)^{n/2-1} K_{n/2-1}(kr)</math>|| [[Yukawa potential]], [[Propagator#Feynman propagator|Feynman propagator]], [[Screened Poisson equation]]
|-
| <math>\partial_t^2 - c^2\partial_x^2</math>|| <math>\frac{1}{2c} \Theta(ct - x)</math>|| 1D [[wave equation]]
|-
| <math>\partial_t^2 - c^2\,\nabla^2_{\text{2D}}</math>|| <math>\frac{\Theta(ct - \rho)}{2\pi c\sqrt{c^2t^2 - \rho^2}}</math>|| 2D [[wave equation]]
|-
| [[D'Alembert operator]] <math>\square = \frac{1}{c^2}\partial_t^2 - \nabla^2_{\text{3D}}</math>|| <math>\frac{1}{4 \pi r} \delta\left(t-\frac{r}{c}\right)</math>|| 3D [[wave equation]]
|-
| <math>\partial_t - k\partial_x^2</math>|| <math>\left(\frac{1}{4\pi kt}\right)^{1/2} \Theta(t) e^{-x^2/4kt}</math>|| 1D [[diffusion]]
|-
| <math>\partial_t - k\,\nabla^2_{\text{2D}}</math>|| <math>\left(\frac{1}{4\pi kt}\right) \Theta(t) e^{-\rho^2/4kt}</math>|| 2D [[diffusion]]
|-
| <math>\partial_t - k\,\nabla^2_{\text{3D}}</math>|| <math>\left(\frac{1}{4\pi kt}\right)^{3/2} \Theta(t) e^{-r^2/4kt}</math>|| 3D [[diffusion]]
|-
| <math>\frac{1}{c^2}\partial_t^2 - \partial_x^2+\mu^2</math>|| <math>\begin{align}
&\tfrac{1}{2} \left(1-\sin{\mu ct}\right) \left[\delta(ct-x) + \delta(ct+x)\right] \\[0.5ex]
&+\tfrac{1}{2} \mu \Theta(ct - |x|) J_0(\mu u)
\end{align} </math> {{br}} with <math> u = \sqrt{c^2 t^2 - x^2}</math>|| 1D [[Klein–Gordon equation]]
|-
| <math>\frac{1}{c^2}\partial_t^2 - \nabla^2_{\text{2D}}+\mu^2</math>|| <math>\begin{align}
&\frac{\delta(ct-\rho)}{4\pi\rho} \left(1 + \cos(\mu ct)\right) \\[0.5ex]
&+ \frac{\mu^2\Theta(ct - \rho)}{4\pi} \operatorname{sinc}(\mu u)
\end{align}</math> {{br}} with <math> u=\sqrt{c^2t^2-\rho^2} </math>|| 2D [[Klein–Gordon equation]]
|-
| <math>\square + \mu^2</math>|| <math>\frac{1}{4\pi r} \delta{\left(t - \frac{r}{c}\right)} + \frac{\mu c}{4\pi u} \Theta(ct - r) J_1{\left(\mu u\right)}</math> &nbsp; with &nbsp; <math> u = \sqrt{c^2t^2-r^2}</math>|| 3D [[Klein–Gordon equation]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t - c^2\partial_x^2</math>
| <math>\begin{align}
&\frac{e^{-\gamma t}}{2} \left[ \delta(ct - x) + \delta(ct + x) \right] \\[0.5ex]
&+ \frac{e^{-\gamma t}}{2} \Theta(ct - |x|) \left(k I_0(k u) + \frac{\gamma t}{u} I_1(k u)\right)
\end{align}</math> {{br}} with &nbsp; <math> u=\sqrt{c^2t^2-x^2}</math> and <math>k = \gamma / c </math>|| [[telegrapher's equation]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t - c^2\,\nabla^2_{\text{2D}}</math>
| <math>\begin{align}
&\frac{e^{-\gamma t}}{4\pi\rho} \delta(ct-\rho) \left(1 + e^{-\gamma t} + 3\gamma t\right) \\
&+ \frac{e^{-\gamma t}}{4\pi u^2} \Theta(ct - \rho) \left(\frac{k u^2 - 3 c t}{c u} \sinh\left(k u\right) + 3\gamma t \cosh\left(k u\right)\right)
\end{align}</math> {{br}} with &nbsp; <math> u = \sqrt{c^2 t^2 - \rho^2}</math> and <math>k = \gamma / c</math>|| 2D [[relativistic heat conduction]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t - c^2\,\nabla^2_{\text{3D}}</math>
| <math>\begin{align}
&\frac{e^{-\gamma t}}{20\pi r^2} \delta(ct - r) \left(8 - 3e^{-\gamma t} + 2\gamma t + 4\gamma^2 t^2\right) \\[0.5ex]
&+ \frac{k e^{-\gamma t}}{20 \pi u} \Theta(ct - r) \left(k I_1(k u) + \frac{4 \gamma t}{u} I_2(k u)\right)
\end{align}</math> {{br}} with &nbsp; <math> u = \sqrt{c^2 t^2 - r^2}</math> and <math>k = \gamma / c</math>|| 3D [[relativistic heat conduction]]
|}