# Talk:Green's function

The use of the adjoint operator in the first equation of the article is wrong. It should be the operator itself, not the adjoint. Anyone care to defend the current version? Holmansf (talk) 21:04, 5 February 2014 (UTC)

Actually after checking some sources I see there is some disagreement over whether the definition of a Green's function should use the adjoint or the original operator. That is whether it should be

or

(In both cases the operator acts in the first variable, apparently.) One should note however that in order for something like

to be correct the non-adjoint version should be used. If the adjoint version is used, then the previous formula should be

At any rate, I think most references use the non-adjoint version, and I think that is what should be given here.Holmansf (talk) 21:44, 5 February 2014 (UTC)

## Page name

People, let's call it a Green function. This use of Green's is killing me!

I've always heard it called a Green's function. This case seems somewhat specific, would it be possible to generalise it to a degree n? - 3mta3 01:23, 20 Jun 2005 (UTC)
It should be 'Green function': compare Hermite polynomial, Abelian Group, Grothendieck topology, etc.--88.104.215.243 11:05, 9 July 2006 (UTC)
I'm going to rename the page accordingly. AMS 108, Applied Functional Analysis by Zeidler calls it "Green function". —Ben FrantzDale 00:17, 13 November 2006 (UTC)
You've renamed the page, but left the references to Green's function in the article. Not very thorough. In my experience, most of the standard texts (e.g., Morse and Feshbach) refer to it as Green's function. When I studied the Green's function, the professor specifically mentioned the unusual use of the possessive. His explanation was that to call it the 'Green function' sounded as if you were talking about a function that was colored green in some way, which tended to confuse people, especially students.
--Tex 16:22, 8 February 2007 (UTC)
I don't care whether it is Green or Green's, all I want is to know which it SHOULD be. By that I mean, we should use whatever form was first introduced. Who introduced the term and when? Skip down to the discussion on the "title" (on this page) for more about who uses which form. TJ 19:45, 24 May 2007 (UTC)

## All functions are equal, and discontinous functions aren't less equal than others, are they?

The article says "Green's functions are distributions in general, not functions, meaning they can have discontinuities.". This cannot possibly be correct, can it? I mean, a function having discontinuities sure doesn't make it any less of a function. The author must have had something else in mind. Please rephrase. Jonas Olson 21:36, 27 September 2005 (UTC)

Umh. Not totally satisfactory formulation. More discontinous than any function, so to say, like Dirac delta function, which isn't a function. --Pjacobi 23:01, 27 September 2005 (UTC)
You raise a good point about the Green's being called stricly a distribution, and not a 'true' function. What of the case of a dimensionless operator L=1? Would not the Green's be identified with the Dirac Delta (assuming infinitesimal integration)? I.E. - it is only a function in the sense that it acts locally on said operator.

## Applications to Quantum Mechanics

I'd love to see an elaboration of the applications to QM. --User:TobinFricke

A couple of references I would suggest mentioning here are the books Green Functions for Ordered and Disordered Systems (Studies in Mathematical Physics) by Antonios Gonis (# ISBN-10: 0444889868 or # ISBN-13: 978-0444889867) and Green's Functions in Quantum Physics by Eleftherios N. Economou (# ISBN-10: 3540288384 or # ISBN-13: 978-3540288381)

The relationship between the Schrödinger Green's function and the local density of states is essentially the Plemelj-Sokhotsky equations (for the resolvent).

Anyway, it would be nice to have a better expert than I to fill in these details.

"we leave it to the reader to fill in the in-between steps" Is something contained in bad textbooks and bad lectures. Can anyone fill in the blanks here?

That sentence has been replaced by in-between steps --Wolfch 04:53, 5 September 2007 (UTC)

## Minus sign missing

shouldn't there be a minus sign in the Laplacian Green's function 1/|x-x'| (and in the expressionwith the charge density \rho)? maybe I'm confused at the moment, but I'm pretty sure there should be... If no one answers I'll change it. Dan Gluck 16:35, 10 October 2006 (UTC)

If you are going to refer to the solution to electrostatic problems them you will need to insert the proper constants to make the units correct. The sign depends on the choice of the Poisson problem that you are trying to solve. In the case of electrostatics in the CGS system you have $\nabla ^{2}\Phi =-4\pi \rho$ where $\Phi$ is the electrostatic potential electrostatic units (esu) per centimeter and $\rho$ is the charge density in esu per centimeter. However, in the MKS unit system the Poisson equation for electrostatics is given by $\nabla ^{2}\Phi =-{\frac {\rho }{\epsilon _{0}}}$ where $\Phi$ is given in terms of Coulombs per meter, $\rho$ is given in terms of Coulombs per cubic meter, and $\epsilon _{0}$ is the permittivity of free space given in terms of Farads per meter. In both cases there is a negative sign in the Poisson equation and hence the solutions to these Poisson equations will be $\Phi (x)=\int {\frac {\rho (x^{\prime })}{|x-x^{\prime }|}}d^{3}x^{\prime }$ and $\Phi (x)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho (x^{\prime })}{|x-x^{\prime }|}}d^{3}x^{\prime }$ . So the (positive) sign is correct in both unit systems and the units are correct if working in the cgs unit system. HowiAuckland 00:20, 11 September 2007 (UTC)

I agree with the change suggested by HowiAuckland, his point is correct —Preceding unsigned comment added by 190.244.173.152 (talk) 20:24, 8 July 2009 (UTC)

## More Examples

I think we all know the best way to learn stuff is Examples, Like actual problems. I think it would be great if we could get many examples from some text books or somewhere, to give readers (and hopless students) an idea of how to use the this function. this is a more General statement to the applications to Quantum Mechanics Section.24.25.211.241 04:48, 24 October 2006 (UTC)

## Green's functions in condensed matter physics

I think there's a little confusion with regards to this section. Until someone disagrees (Andywall?), I'd like to replace it with a section explaining the connections and differences between this kind of Green('s) function — the inverse of a linear operator — and what's meant by the term in quantum/statistical field theory, where it essentially just means correlation function. Stevvers 03:15, 5 November 2006 (UTC)

I have added a sentence in the introduction to mention this distinction and will add something to Correlation function (quantum field theory).Stevvers 01:16, 10 November 2006 (UTC)

## Intuition

After looking at my math text, I think I just "got" this topic, at least for the case of $-u''=f$ . I would add my understanding to the page, but am not sure it is a complete understanding. Is this right?:

The Green function, ${\mathcal {G}}(x,y)$ , can be thought of as the response of the solution variable, u, to a delta function, $\delta (x-y)$ . Because the system is linear, we can apply superposition: we can represent our given f as the sum of delta functions and expect our solution to be a sum of the responses to each of those delta functions.
For example, suppose we have a horizontal string with distributed and point weights on it and we want to know its deflection along its length. Assume small deflection (so that the problem is linear). We could hang a unit mass at every point, y, along the string and record the deflection as a function of x. This would be our ${\mathcal {G}}(x,y)$ . (Obviously we would expect a V shape centered on the point). The the solution will be
$u(x)=\int _{a}^{b}{\mathcal {G}}(x,y)f(y)\,dy$ .

Is that about right? —Ben FrantzDale 00:16, 13 November 2006 (UTC)

Yes, that sounds exactly right. If you put this into the page, make sure you explicitly write the differential equation that the string satisfies. Stevvers 21:27, 15 November 2006 (UTC)

## kernel

Currently "kernel" links to the "kernel (algebra)" article i.e. the article on the kernel of a homomorphism. I don't see the connection between the two uses of the word "kernel," and I'm wondering if the link should be removed.

Atmd 01:33, 29 January 2007 (UTC)

The usage of the term kernel in the article is correct. What second use of the word, besides the one at kernel (algebra), are you referring to ? MathMartin 18:46, 7 February 2007 (UTC)

A kernel function? I don't believe a green's function is an abelian group. I'm confused as well. Sim 05:17, 1 March 2007 (UTC)

I am confused by you confusion :) So let me try to clear this up. A green function is associated to a linear operator L. Linear operators are group morphism (among other things) and thus we can define a kernel for them which is exactly what is done at Kernel_(algebra)#Linear_operators. Perhaps you could describe more clearly what exactly in the text confuses you so we can try to clear things up ? MathMartin 15:04, 21 March 2007 (UTC)

Clarified confusion: In physics, Green's functions are often taught as a solution to a specific differential equation without much accompanying mathematical formalism. The classic example is that of a point charge.

Let u[x] be the electrostatic potential of an arbitrary charge distribution f[x]. The Green's function is the solution of a point charge (delta function). Thus the total potential u[x] can be obtained simply by convolution of the Green's function with the charge distribution f[x]. Spacially it can be thought of as the superposition of the potentials of a bunch of point charges added up (integrated).

$u(x)=\int G(x,y)f(y)dy$ It is good for us physics folks to be made aware of the deeper connections between the tools we use. Nickvence (talk) 19:33, 28 May 2008 (UTC)

I believe the confusion comes about because we are used to think of Green's function as the kernel of an Integral transform (due to its role in its main usage - solving boundary value problems). The two concepts are distinct (though both are related to linear transformations). However, the wonders of hypertext should help the reader clear things up :-) --AmitAronovitch (talk) 15:45, 27 November 2008 (UTC)

## Title

The article calls it "Green's Function" throughout (aside from a rather obscurely-written lead, which I've now changed), and all the sources given call it "Green's Function". There seems to be no good reason for us not to call it by that name. (Googling supports the "Green's" versus "Green" name.) --Mel Etitis (Talk) 09:23, 1 March 2007 (UTC)

Actually, Googling supports either the use of Green or Green's (and google results alone should not be the basis on which to settle this issue). Many famous physics text books (e.g., Morse and Fechbach, Fetter and Walecka, etc.) use the term Green's consistently, however, it is just as easy to find equally famous (and well respected) texts that use the term Green function (Jackson, Cohen-Tanoudji, Eygers, etc.). Julian Schwainger (sp?) flips back and forth indiscriminantly in his essay on Greening Quantum Theory. A 2003 Physics Today article (The Green of Green Functions) by Lawrie Challis (professor emeritus of Notingham Univ. Physics Dept., and George Green historian) uses the term Green, but acknowledges that others use Green's, without explaining which it SHOULD be. This issue is very annoying to me (and apparently others here) but will not likely be resolved by simply stating that "so-and-so" uses "Green" or "Dr. whoever" told me that it is "Green's". The most appropriate usage would be whatever was first used. That begs the questions, who was the first to coin the term, where is it published, and what form did they use? I have a feeling that it was first used by Lord Kelvin in the 1850's, but I can't prove it, as I don't have any of his documents. Can anyone here answer those questions definitively and with citations? TJ 19:38, 24 May 2007 (UTC)

Haberman's "Elementary Applied PDEs" and Evan's "PDEs" both use "Green's functions" (and not "Green functions"). This are both popular, well-known texts. Lunch 03:50, 8 August 2007 (UTC)
Jackson in his text on Electrodynamics (possibly THE most famous and widely used text book in graduate physics course on E & M), as well as the Nobel laureate Cohen-Tanoudji (in his textbook on Quantum mechanics ... another extremely popular graduate level text) both use GREEN. As stated above going back and forth about who uses Green and who uses Greens is not all that useful in finding the correct usage. I do find it odd that of all the functions that I know, the Green function is the only one that people feel the need to make possessive, (e.g., it is a Bessel function NOT Bessels function.) —Preceding unsigned comment added by 134.157.226.1 (talk) 10:00, 17 September 2007 (UTC)

This article (from Nature Physics) gives a good argument for the title "Green's function";  doi:10.1038/nphys411 Ulner 15:23, 8 November 2007 (UTC)

Since both uses are common, I don't think it matters which is "correct". Both are. Except, and by this I'm really annoyed: if it's Green's function, then how come THE Green's function?! I ate John's apple, not the John's apple. --GaborPete (talk) 09:58, 29 July 2012 (UTC)

## Merge from Green's function for the three variable Laplace equation

This very specific, (previously) uncategorized article doesn't stand on its own. It needs to move to this article (or to Laplace's equation or Poisson's equation). Lunch 03:50, 8 August 2007 (UTC)

Well, you know there are many different Green's functions for partial differential equations. Specifically in 3-dimensions there is the Green's function for the Helmholtz equation, for the diffusion (heat) equation, and for the wave equation. There are even Green's functions for the biharmonic equation. If all these articles were to be written and merged into this one, this page would get very long. It seems better to have this page be an introduction to Green's function theory with links to more specific pages on Green's functions. Just as there are different pages for the various partial differential equations, so should there be, for their corresponding Green's functions. HowiAuckland 02:15, 16 September 2007 (UTC)

## Change of introduction?

Is it possible to give a definition without distributions? This should be helpful for the general reader, I think. Ulner 15:19, 8 November 2007 (UTC)

## example

The 'example' section on this page refers to "demand-2","demand-3",and "demand-4" without any actual reference to what these entities are. Have I missed something here? Is it referring to the constraints? In any case I think it should be made more clear. Gazzo 144.32.126.14 (talk) 17:43, 3 June 2008 (UTC)

I think the equation numbering got messed up. Does someone know how to fix this? mike40033 (talk) 05:12, 20 June 2008 (UTC)

## Manifolds vs. R3

Throughout the article x and s are used as points on a manifold, so x-s, which is also used, makes no sense. Jjauregui (talk) 15:42, 16 September 2008 (UTC)

Your absolutely right, for the moment I am changing the article remove this discrepancy. Thenub314 (talk) 18:52, 17 September 2008 (UTC)

## Eq. 3

The derivation leading to (3) is not quite correct. The solution to the preceding equation can include a homogeneous part i.e. solving Lu=0 which should not be allowed here, since G has presumably been chosen so that the boundary conditions are already satisfied - i.e. without an extra homogeneous term.

The proper steps should be:

Write

$f(x)=\int \delta (x-s)f(s)ds\,$ So find G such that

$LG(x,s)=\delta (x-s)\,$ Then combine  and  to give

$f(x)=\int LG(x,s)f(s)ds=L\int G(x,s)f(s)ds\,$ Hence

$u=\int G(x,s)f(s)ds\,$ solves

$Lu=f\,$ I'll change it to this on 3rd December if no one objects.

## Convolution

The paragraph mentioning convolution is not quite right either. Presently it reads

"Convolving with a Green's function gives solutions to inhomogeneous differential-integral equations, most commonly a Sturm-Liouville problem. If G is the Green's function of an operator L, then the solution for u of the equation Lu = f is given by

$u(x)=\int {f(s)G(x,s)\,ds}$ .

This can be thought of as an expansion of f according to a Dirac delta function basis (projecting f over δ(x − s)) and a superposition of the solution on each projection. Such an integral is known as a Fredholm integral equation, the study of which constitutes Fredholm theory."

First of all, 'convolution' should be referenced. Second, it is appropriate here only if the operator L(x) is invariant under displacement - i.e. does not include an explicit reference to x. If so then

$L(x)=L(x-s)=>L(x-s)G(x,s)=\delta (x-s)=>G(x,s)->G(x-s)\,$ .

and now u is a convolution of G with f:

$u(x)=\int G(x-s)f(s)ds$ I'll change it to this on 3rd December if no one objects.

--Mike (talk) 22:04, 30 October 2008 (UTC)

## Notation And Clarity

First, why does the definition begin with "In addition..." In addition to what?

Second, why do we move from $G(x,s)$ to $G(x,x')$ ? The notation $G(x,s)$ is much less confusing considering that prime is overloaded and typically means differentiation.

Third, I know it's common to say that distributions are not properly functions, but are they not linear functionals over a vector subspace of infinitely differentiable functions? So technically, they are functions, just not functions from $\mathbb {R} ^{m}$ to $\mathbb {R} ^{n}$ . I think this perhaps could be clarified since technically it's wrong.

Fourth, what is with the hats on the $A$ and $\sigma$ in the section on Green's Theorem for the Laplacian? Is this standard in physics? It looks a bit odd and cluttered, as if $A$ was defined somewhere else and ${\hat {A}}$ is some modification of $A$ . —Preceding unsigned comment added by 129.2.175.79 (talk) 17:09, 16 August 2009 (UTC)

## Second Order

The article starts well givng general statements, but in the Framework subsection, it restricts to second order diferential equations, is there any reason for that? It exists Green's function for higher order? Paranoidhuman (talk) 00:11, 13 January 2011 (UTC)

This page has some major errors. The definition of the Green's function given is only for self-adjoint operators (fortunately all of the examples have self-adjoint operators), and will not work for other operators. It should be the adjoint of the operator that appears in the definition of the Green's function. Also taking the operator outside the integral is not ok with the assumptions given. This should be done by repeated "integration by parts" (or application of the divergence theorem) from which the definition of the adjoint operator arises.

I also can't see why there should be any linear operator without a Green's function (it might not be found analytically, but it will still exist and could be found numerically).

Source: Applied Partial Differential Equations: S. Howison, A. Lacey, A. Movchan, J. Ockendon.

I will try and make changes when I have timeAklamas (talk) 16:25, 21 January 2011 (UTC)

The introduction ends with: "Green's functions are studied largely from the point of view of fundamental solutions instead." Instead of what? I find this unclear. — Preceding unsigned comment added by 171.64.163.233 (talk) 02:41, 24 January 2012 (UTC)

## Add use in wave physics

I would propose adding the following to the end of the "Definitions and Uses" part:

--

i.e., Green's functions give the wavefield response in one place (x) as a result of an impulse-like source in another place (s).

— Preceding unsigned comment added by Frankdj100 (talkcontribs) 10:47, 26 March 2012 (UTC)

## Messy structure

The article's structure seems messy to me. Could someone (native speaker) try to restructure the whole article? (And help me, where my table of Green's functions fits the best. I added it because I suppose that's the kind of information many people are looking for.) Thanks, --PassPort (talk) 00:29, 2 April 2013 (UTC) — Preceding unsigned comment added by PassPort (talkcontribs)

Using address 125.202.66.200 I corrected the error.--Enyokoyama (talk) 13:53, 2 April 2013 (UTC)

Oh thanks, I didn't notice that the references tag was missing since there was already a references section. Anyone comments on the article's structure? --PassPort (talk) 21:42, 2 April 2013 (UTC) — Preceding unsigned comment added by PassPort (talkcontribs)

## Another missing minus sign (or just an error)

I think that there is an error in the "Table of Green's Functions" section, unless I'm reading something wrong. The claimed solution for the 2D Laplacian ($\Delta _{2D}=\partial _{x}^{2}+\partial _{y}^{2}$ ) appears to be wrong, according to this source, doing out myself, and basic logic e.g. (for physicists) it makes it look like an election and a proton will attract in 3D but repel in 2D.

Unless I hear someone with an argument not to, I'm going to change it to ${\frac {1}{2\pi }}\ln \rho$ in a few days. TChapProctor (talk) 16:04, 15 July 2013 (UTC)

## What is θ(t)?

The Table of Green's Functions uses θ(t) in several places, but as far as I can see, this notation is not defined on this page. 172.56.9.253 (talk) 16:07, 24 October 2013 (UTC)