Editing Green's function (section)

==Finding Green's functions==

===Units===
While it does not uniquely fix the form the Green's function will take, performing a [[dimensional analysis]] to find the units a Green's function must have is an important sanity check on any Green's function found through other means. A quick examination of the defining equation,
<math display="block"> L G(x, s) = \delta(x - s), </math>
shows that the units of <math>G</math> depend not only on the units of <math>L</math> but also on the number and units of the space of which the position vectors <math>x</math> and <math>s</math> are elements. This leads to the relationship:
<math display="block"> [[G]] = [[L]]^{-1} [[d x]]^{-1}, </math>
where <math>[[G]]</math> is defined as, "the physical units of {{nowrap|<math>G</math>"}}{{Explain | reason=What does this mean? This is ungoogleable and without a description this section is meaningless. | date = October 2024}}, and <math>dx</math> is the [[volume element]] of the space (or [[spacetime]]).

For example, if <math>L = \partial_t^2</math> and time is the only variable then:
<math display="block">\begin{align}[]
[[L]] &= [[\text{time}]]^{-2}, \\[1ex]
[[dx]] &= [[\text{time}]],\ \text{and} \\[1ex]
[[G]] &= [[\text{time}]].
\end{align}</math>
If {{nowrap|<math>L = \square = \tfrac{1}{c^2}\partial_t^2 - \nabla^2</math>,}} the [[d'Alembert operator]], and space has 3 dimensions then:
<math display="block">\begin{align}[]
[[L]] &= [[\text{length}]]^{-2}, \\[1ex]
[[dx]] &= [[\text{time}]] [[\text{length}]]^3,\ \text{and} \\[1ex]
[[G]] &= [[\text{time}]]^{-1} [[\text{length}]]^{-1}.
\end{align}</math>

===Eigenvalue expansions===
If a [[differential operator]] {{math|''L''}} admits a set of [[eigenvectors]] {{math|Ψ<sub>''n''</sub>(''x'')}} (i.e., a set of functions {{math|Ψ<sub>''n''</sub>}} and scalars {{math|''λ''<sub>''n''</sub>}} such that {{math|1=''L''Ψ<sub>''n''</sub> = ''λ''<sub>''n''</sub> Ψ<sub>''n''</sub>}}&thinsp;) that is complete, then it is possible to construct a Green's function from these eigenvectors and [[eigenvalues]].

"Complete" means that the set of functions {{math|{{mset|Ψ<sub>''n''</sub>}}}} satisfies the following [[completeness relation]],
<math display="block">\delta(x-x') = \sum_{n=0}^\infty \Psi_n^\dagger(x') \Psi_n(x).</math>

Then the following holds, 
{{Equation box 1
|indent =:
|equation = <math>G(x, x') = \sum_{n=0}^\infty \dfrac{\Psi_n^\dagger(x') \Psi_n(x)}{\lambda_n},</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where <math>\dagger</math> represents complex conjugation.

Applying the operator {{math|''L''}} to each side of this equation results in the completeness relation, which was assumed.

The general study of Green's function written in the above form, and its relationship to the [[function space]]s formed by the eigenvectors, is known as [[Fredholm theory]].

There are several other methods for finding Green's functions, including the [[method of images]], [[separation of variables]], and [[Laplace transform]]s.<ref>{{cite book |first1=K.D. |last1=Cole |first2=J.V. |last2=Beck |first3=A. |last3=Haji-Sheikh |first4=B. |last4=Litkouhi |chapter=Methods for obtaining Green's functions |title=Heat Conduction Using Green's Functions |publisher=Taylor and Francis |year=2011 |pages=101–148 |isbn=978-1-4398-1354-6}}</ref>

===Combining Green's functions===
If the differential operator <math>L</math> can be factored as <math>L = L_1 L_2</math> then the Green's function of <math>L</math> can be constructed from the Green's functions for <math>L_1</math> and {{nowrap|<math>L_2</math>:}}
<math display="block"> G(x, s) = \int G_2(x, s_1) \, G_1(s_1, s) \, ds_1. </math>
The above identity follows immediately from taking <math>G(x, s)</math> to be the representation of the right operator inverse of {{nowrap|<math>L</math>,}} analogous to how for the [[Invertible matrix#Other properties|invertible linear operator]] {{nowrap|<math>C</math>,}} defined by {{nowrap|<math>C = (AB)^{-1} = B^{-1} A^{-1}</math>,}} is represented by its matrix elements {{nowrap|<math>C_{i,j}</math>.}}

A further identity follows for differential operators that are scalar polynomials of the derivative, {{nowrap|<math>L = P_N(\partial_x)</math>.}} The [[fundamental theorem of algebra]], combined with the fact that <math>\partial_x</math> [[Commutative property|commutes with itself]], guarantees that the polynomial can be factored, putting <math>L</math> in the form:
<math display="block"> L = \prod_{i=1}^N \left(\partial_x - z_i\right),</math>
where <math>z_i</math> are the zeros of {{nowrap|<math>P_N(z)</math>.}} Taking the [[Fourier transform]] of <math>L G(x, s) = \delta(x - s)</math> with respect to both <math>x</math> and <math>s</math> gives:
<math display="block"> \widehat{G}(k_x, k_s) = \frac{\delta(k_x - k_s)}{\prod_{i=1}^N (ik_x - z_i)}. </math>
The fraction can then be split into a sum using a [[partial fraction decomposition]] before Fourier transforming back to <math>x</math> and <math>s</math> space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if <math>L = \left(\partial_x + \gamma\right) \left(\partial_x + \alpha\right)^2</math> then one form for its Green's function is:
<math display="block"> \begin{align}
G(x, s) & = \frac{1}{\left(\gamma - \alpha\right)^2}\Theta(x-s) e^{-\gamma(x-s)} - \frac{1}{\left(\gamma - \alpha\right)^2}\Theta(x-s) e^{-\alpha(x-s)} + \frac{1}{\gamma-\alpha} \Theta(x - s) \left(x - s\right) e^{-\alpha(x-s)} \\[1ex]
& = \int \Theta(x - s_1) \left(x - s_1\right) e^{-\alpha(x-s_1)} \Theta(s_1 - s) e^{-\gamma (s_1 - s)} \, ds_1.
\end{align} </math>
While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, when <math>\nabla^2</math> is the operator in the polynomial).

===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]]
|}