# What is Green's function: Definition and 213 Discussions

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is the linear differential operator, then

the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function;
the solution of the initial-value problem Ly = f is the convolution (G * f ), where G is the Green's function.Through the superposition principle, given a linear ordinary differential equation (ODE), L(solution) = source, one can first solve L(green) = δs, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead.
Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.

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1. ### A How to know the number of Feynman diagrams for a given order?

Let's say we want to calculate the two-point Green's function for a fermion to a given order for a two particle interaction of the form ##U(x,y)=U(y,x)##. For the first order calculation we have to do all contractions related to...
2. ### I Hartree-Fock: Feynman diagrams vs perturbation theory

I am trying to understand Green's functions in many-body theory for condensed matter. After much struggle, I managed to calculate my first diagrammatic expansion. However I am perplexed by getting more of the usual results. The Hartree–Fock energy result I know from second quantization can be...
3. ### I Green's function boundary conditions

we know that, using the Green's identity ##\iiint\limits_V (\varphi \Delta\psi -\psi \Delta\varphi)\ dV =\iint_{\partial V} (\varphi \frac {\partial \psi}{\partial n}-\psi \frac {\partial\varphi}{\partial n})\ da## and substituting ##\varphi=\phi## and ##\psi=G## here, we can write the potential...
4. ### I Green's function for 2-D Laplacian within square/rectangular boundary

From the table of Green functions on Wikipedia we can get the generic 2-D Green's function for the Laplacian operator. But how would one apply boundary conditions like u = 0 along a rectangular boundary? Would we visualize a sort of rectangle-based, tilted pyramid, with logarithmically changing...
5. ### I Linearised gravity approach to Lense Thirring metric

Doing some revision and getting confused. It's under GR but may as well be under electromagnetism or calculus because that is where the problem is. Taking a shell of mass ##\rho = M\delta(r-R)/(4\pi R^2)## and four velocity corresponding to rotation about ##z## axis i.e. ##U = (1, -\omega y...
6. ### I The last step of this Green's function proof is not clear

Here is the conclusion of the derivation in question: where ##\phi_n## are eigenfunctions of the Hamiltonian. I don't see how at the very end the ##\sum ...## becomes ##\delta (x-y)##. What do I miss?
7. ### Green’s function of Dirac operator

I started from eq(3.113) and (3.114) of Peskin and merge them with upper relation for $S_F$, as following: \begin{align} S_F(x-y) &= \theta(x^0-y^0)(i \partial_x +m) D(x-y) -\theta(y^0-x^0)(i \partial_x -m) D(y-x) \\ &= \theta(x^0-y^0)(i \partial_x +m) < 0| \phi(x) \phi(y)|0 >...
8. ### I Trouble with Product of Green's Functions

Hi all, Consider the following Green's function: where ##\Theta(t)## is the Heaviside step function and ##\tilde{\Theta}(t)## is defined as I want to understand the following calculation: More specifically, the ##\text{Im}(G(\textbf{k},t)G(\textbf{k},-t))## from the first line to the second...
9. ### How to get general solution via Green's function?

I'll start with a characterization of the Green's function as a fundamental solution to a differential operator. This theorem is given in Ordinary Differential Equations by Andersson and Böiers. ##E(t,\tau)## is known as the fundamental solution to the differential operator ##L(t,D)##, also...
10. ### I Verifying properties of Green's function

I'm reading about fundamental solutions to differential operators in Ordinary Differential Equations by Andersson and Böiers. There is a remark that succeeds a theorem that I struggle with verifying. First, the theorem: If the leading coefficient in ##(1)## is not ##1## but ##a_n(t)##, then...
11. ### Expressing Feynman Green's function as a 4-momentum integral

I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
12. ### A Puzzled about Section 3.12 of Jackson's E&M book

Hi wizards, I'm working through Jackson's book on E&M (3rd edition) and got stuck in section 3.12 on expansions of Green functions. I have three questions regarding section 3.12: First, why is Jackson trying to find a Green function that satisfies equation 3.156? To my beginner mind, it...
13. ### I 4D d'Alembert Green's function for linearised metric

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14. ### A How to work around this equation giving infinities in the numerical calculation?

The heat conduction equation for a semi-infinite slab with a boundary condition of the first kind is as follows: The problem is delta is a very small number, so the first exponential will tend to infinity. I am programming this in Fortran and it can accommodate values up to magnitude of 310...
15. ### A Green's function for Stokes equation

So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force. $$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$ $$\nabla \cdot \textbf{u}=0$$ We can apply the...
16. M

### A Green's function for Sturm-Louiville ODE

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### Mathematica Numerical integration over a Green's function

Hi PF! I'm numerically integrating over a Green's function along with a few very odd functions. What I have looks like this NIntegrate[-(1/((-1. + x)^2 (1. + x)^2 (1. + y)^2)) 3.9787262092516675*^14 (3.9999999999999907 + x (-14.99999999999903 + x (20.00000000000097` -...
18. ### I Understanding relationship between heat equation & Green's function

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19. ### A Green's function for tunneling electrons between quantum dots

Good afternoon! I am writing with such a problem, I hope to find someone who could help me. I'm almost desperate! So, there is such a thing as the Braess paradox, this is a classic paradox for roads and power grids, and there is also such an article...
20. ### A Green's Function half-space

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21. ### I Green's function for massive photon theory

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22. ### Showing that a given propagator is proportional to Green's function

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23. ### A Green's function calculation of an infinite lattice with periodicity in 1D

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24. ### Confirming Green's function for homogeneous Helmholtz equation (3D)

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25. ### A Square of an integral containing a Green's Function

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26. ### Green's function and the resistance across a Hypercube

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27. ### I Green's function for the wave equation

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28. ### Quantum Monte-Carlo calculation of Green's function

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29. ### A Studying Green's function in many body physics

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30. ### A Why "Green's function" is used more than "correlations" in QFT?

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31. ### Green's Function for a harmonic oscillator

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32. ### I Continuity of Green's function

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33. ### Finding electric potential using Green's function

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34. ### Green's Function Boundary Conditions

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38. ### I Calculating Dyadic Green's Function Expression

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39. ### A Two questions on Feynman diagram and Green's function

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40. M

### Green's Function for Linear ODE

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41. M

Find Green's function of $$K(f(x)) = (1-x^2)f''(x)-2xf'(x)+\left(2-\frac{1}{1-x^2}\right)f(x):x\in[cos(\alpha),1]$$ subject to boundary conditions: $$f|_{x=1} < \infty\\ f|_{x=\cos(\alpha)} = 0.$$ Two fundamental solutions are associated Legendre polynomials (after all, this is Legendre's...
42. ### Can not find correct green's function

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43. ### A Applying boundary conditions on an almost spherical body

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44. M

### What is the Green's function for this specific problem?

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45. M

50. ### Green's function in acoustics,method of descent

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