Green function Definition and 85 Threads

Homework Statement The generalized Green function is $$ G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}. $$ Show G_g satisfies the equation $$ (\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x') $$ where \delta(x - x') = \frac{2}{\ell}\sum_{n =...

The generalized Green function is \[ G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}. \] Show \(G_g\) satisfies the equation \[...

I had made a post in the past about the same problem and unfortunately I wasn't clear enough so I am trying again. I am studying an article and there I found without any proof that the solution of: Let ##g \in \mathbb{C}## and let ##u:(0,\infty)\to \mathbb{C}## $$ -u''+\lambda^2u=f\,\, on...

hello all ! my teacher told me to do a research on examples of problems that has connection with green function on solving differential equations (with programmed numerical solutions) in my final year project , can you give me such problems to work on as an undergraduate ? , thank you !

Homework Statement I would like to calculate the electric potential in all the space in the following set up: Conductor sphere of radius a whose surface is kept at a zero potential. 1 point charge ##q_1## at distance ##d_1## from the center of the sphere. 1 point charge ##q_2## at a distance...

Hi everybody. Does someone know the solution of the following differential equation in two dimensions: 1) \Big(\frac{\partial^2}{\partial\,r^2}+\frac{ \partial^2}{\partial\,z^2}\Big)G(r,z)=\delta(r-ro)\delta(z-zo) and 2)...

The solution of the problem \left(\nabla^2 + k^2 \right)\psi(\mathbf{r})=f(\mathbf{r}) is, using green function \psi(\mathbf{r})=-\int G(\mathbf{r},\mathbf{r}_1) f(\mathbf{r}) where for the tridimensional case the Green function is...

Hello there, I would like to obtain the Green function for the operator F, F [u(x)] = u '''', and the boundary conditions u(0) = u'(0) = u (1) = u' (1) = 0. I am looking for a function G ( x, s ) such that G'''' (x,s) = delta (x-s) (the apecis referring to differentiation w.r.t. x, and...

Hello, Can anyone explain to me the next settence, found http://relativity.livingreviews.org/open?pubNo=lrr-2011-7&page=articlese16.html in section 1.4: "The causal structure of the Green’s functions is richer in curved spacetime: While in flat spacetime the retarded Green’s function...

Homework Statement The problem is attached in the first picture, the provided solutions are in the second.The Attempt at a Solution I got to where they are, but aren't they missing an additional term of sin(t)*cos(∏)*f(∏) from the second integral in dx/dt ?

Homework Statement I'm trying to self study Green function and I can't follow the very last step of a demonstration in an online notes (that I attach in this post). Page 7 to 8. Basically he says that from G_{tt}(t,t')+\omega G(t,t')=0 for all t>t' with the conditions G(t,t'+\varepsilon)=0...

Homework Statement Calculate the solution to \triangle u (r, \phi )=1 in a circle of radius a with u(r=a)=0. Homework Equations Green function I think, the exercise is listed with other exercises related to Green function. So even though separation of variables works here or any other...

If we have Green function g(x,s)=exp[-\int^x_s p(z)dz] we want to think about that as distribution so we multiply it with Heaviside step function g(x,s)=H(x-s)exp[-\int^x_s p(z)dz] Why we can just multiply it with step function and tell that the function is the same. Tnx for the answer.

Please teach me this: Why the electromagnetic potential obeys the Callan-Symazik equation in renormalization group theory like propagator functions.By the way,are there any relation between classical potential and interaction Haminton(the product of different field operators). Thank you very...

In studying the scattering problem, one should know green function in free space. while in two dimension, green function is the hankel function, seen in the attached file. bu i got confused in the second equation in the attached file. could anyone give me some details about this relationship? it...

There exists very simple formula for Green function for wave equation: G(t,x,t',x') = \delta (t-t'\pm \frac{|x-x'|}{c})/|x-x'| . I wonder whether there exist similar formula for Green function for Klein-Gordon equation (with mass >0) for any boundary condition.

as a student in physics, i cannot see the usefulness of green function to me, the definition of a green function is ugly and singular we have to deal with functions that are not smooth, e.g., the derivative is not continuous at some point. How these functions can be useful in math and...

I was reading up on Classical Mechanics and the general method for solving for an undamped harmonic oscillator was given as \frac{d^{2}q}{dt^{2}} + \omega^{2}q = F(t) was solved using the Green function, G, to the equation \frac{d^{2}G}{dt^{2}} + \omega^{2}q = \delta(t-t') and then...

In my project, we enconter such kind of bessel's differential equation with stochastic source, like \Phi''+\frac{1+2\nu}{\tau}\Phi'+k^2\Phi=\lambda\psi(\tau) where we use prime to denote the derivative with \tau, \nu and \lambda are real constant parameter. how to get the green...

green function and complex integration Homework Statement By reading this paper http://arxiv.org/pdf/hep-ph/0610391v4 I cannot proof the following relation on page 9 equation (23) , by a suitable choice of a contour in the complex omega plane: \int\frac{d^3 p}{(2\pi)^2}...

Homework Statement In Jackson 3.16 we have to prove the expansion \frac{1}{\left{|}\vec{x}-\vec{x'}\right{|}}=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}dke^{im(\phi-\phi')}J_m(k\rho)J_m(k\rho')e^{-k(z_{>}-z_{<})} Homework Equations The Attempt at a Solution I tried to use the...

Hello forum, in the case we are in the time domain t, the Green function is a function of time. In the translation variant case, how do we express and separate the regular time variability of the Green function from the the time variability of its functional form? 1) What is the best...

Hello, I don't fully understand the meaning of Green function, and how one should use it. According to Jackson's "Classical Electrodynamics" - 'the method of images is a physical equivalent of the determination of the appropriate F(x, x') to satisfy the boundary conditions'. Where Green...

Hey folks, I'm trying to find the Green function for the equation -\partial_\mu \partial^\mu \phi = K where K is some source term. Its a 2D problem with the wave confined to a rectangular cavity where the cavity is located at z = 0 and z=a. This tells me that G|_0= G|_a=0 I've pretty...

Hey folks, I'm not really sure which forum to put this question in but I figured this was probably the best as it deals with issues of regularization. I'm reading Miltons 'The Casimir Effect'. In chapter one he derives the Casimir energy for a massless scalar field by employing...

find the eigenfunctions and eigen values of the next equation: d^2y/dx^2+u_n^2y=0 where y(0)=0=y(pi). Now find the green function of the above non-homogeneous equation, i.e: d^2G_{\lambda}(x,a)/dx^2-\lambda G_{\lambda}(x,a)=\delta(x-a) where a is in (0,pi) and lambda doesn't equal the...

I need to find the green function of (\frac{d^2}{dx^2}-k^2)\psi(x)=f(x) s.t it equals zero when x approaches plus and minus ifinity. Now according to my lecturer I first need to solve the homogenoues equation, i.e its solution is: psi(x)=Ae^(kx)+Be^(-kx) and G(x,x')=\sum...

Hey folks! I'm starting with the Lagrangian of a massive scalar field and have found an expression for the expectation value of the energy-momentum tensor. <T_{\mu \nu}>=(\partial_\mu \partial_\nu-\frac{1}{2}(g_{\mu \nu}(\partial_\mu \partial_\nu+m^2))G(x-x') let say I have some Green...

Hey, I am trying to find a GF for the function: y''+\frac{1}{24}y=f(x) The function is bounded by: y(0)=y(\pi)=0 I have followed a math textbook that goes through the exact process for the function: y''+k^2y=f(x) and have found a nice looking general solution...

Homework Statement Hey folks, I need to find a Green function for the equation: y'' +1/4y = f(x) With boundary conditions y(0)=y(pi) = 0 The Attempt at a Solution I tried some combination of solutions that look like sin(kx) and sin(k-pi) and looked at the strum liouville...

What're the condition for a "green function to exist? That's my question,let's suppose i define the functions: G(x,s)=exp(x-s)^{2} and R(x,s)=(e^{st}-1)^{-1} My question is, could G and R satisfy the condition (for a linear operator L) LG(x,s)=\delta (x-s) ?. My interest lies...

Hi, I have a basic ODE: y''(x)+\frac{1}{4}y'(x)=f(x) on 0<x<L With Boundary conditions: y(0)=y(L)=0 For which I would like to construct a Green Function. Rather than just plain ask for help, I'll show you what I've been thinking and maybe someone wiser can help/correct me...

I am demonstrating the mean value theorem, which says that for charge-free space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any sphere centered on that point. I have already found one way to do this, but would also like to...

Dear PF, Could you tell me what is Green function for bilocal operators? As I understand from its form G(x1y1, x2y2,...)... now the pairs of XY are considered as points instead of single X points as in normal Green function G(x1,x2,x3...). So What do we need it for? Or can it be decomposed...

Hello there, I am glad that I found this forum. Because I have a little bit trouble with theoretical physics. The problem is the Green function in theoretical electrodynamic. I try to understand the difference between the Dirichlet Condition and the Neumann Condition. I understand...