Delta function Definition and 366 Threads

I understand that the Dirac delta function can be taken as a distribution. And that one can calculate the Shannon entropy or information content of any distribution. So what is the information content of the Dirac delta function? I think it is probably identically zero, but I'd like to see the...

Hi, I am looking for approximations to the delta functoin which I can use on a computer. Although I will never get an exact delta function, I can make an approximation that it can be improved as much as I like. Would you help me to find the approximation of the delta function so that I can...

Homework Statement a.) Given \delta_n=\frac{ne^{-{n^2}{x^2}}}{\pi} Show: x{\frac{d}{dt}\delta_n}=-\delta_n b.) For the finite interval (\pi,-\pi) expand the dirac delta function \delta(x-t) in sines and cosines, sinnx, cosnx, n=1,2,3... They are not orthogonal, they are normalized to...

Hopefully people are still prowling the forums this close to christmas :) I want to show that sin(ax)/x is a representation of a delta function in the limit a->infinty i.e 1) It equals 0 unless x=0 2) integrated from plus minus infinity it equals 1 and 3) multiplying by an arbitrary...

Homework Statement Let's say we have a wire of finite length L with total charge Q evenly spread along the wire so that lambda=Q/L, linear charge density, is constant. The wire is shaped in x-y plane in some well behaved curve y = f(x). Find the surface charge density sigma(x,y). Homework...

I don't understand in the first paragraph of the attached lecture notes. Could anyone help?

What is the way to show that \lim_{y\rightarrow0}\frac{y}{x^2+y^2}=\pi\delta(x) ?

Hello, My question is about how dirac-delta function is derived by using this integral, \frac{1}{2\pi }\int_{-\infty}^{\infty}e^{ikx}dk=\delta (x) I couldn't solve this integral. Please help me. Thanks for all of your helps.

Homework Statement Pro #2 if you click on this link. http://s1104.photobucket.com/albums/h332/richard78931/?action=view&current=hw4.jpg Homework Equations , The Attempt at a Solution Click here http://s1104.photobucket.com/albums/h332/richard78931/?action=view&current=2a.jpg...

Hi there! I have a problem with one of the questions given to us in the signals and systems course. If anyone could help me I would greatly appreciate it! Homework Statement integral(from -infinity to +infinity) of u0(t) * cos(t) dt u(t) is a step function. Homework Equations...

Homework Statement Hi All. I am given this integral: \int_{-\infty}^{\infty}A\Theta e^{i\omega t}dt I need to show that it's equal to the following: =A(\pi \delta(\omega)+\frac{i}{\omega})Homework EquationsTheta is the Heavyside step function. The Attempt at a Solution The step function...

Homework Statement I want to plot the following function into Maple14. \vec{v}=frac{1}{\vec{r^{2}}} \hat{r} **In case the latex is screwed this says v=r^(-2) *r-hat The Attempt at a Solution My code for Maple is the following, but it doesn't seem to work.restart; with(LinearAlgebra)...

Hi Can somebody help me with this... Is is correct to say that, Integral(delta(0)) = 1 (limits are from -infinity to +infinity) I don't know latex and sorry for the inconvenience in readability. Thanks, VS

Hi, I hope this is the right place to ask this Is it possible to expand the Dirac delta function in a power series? \delta(x)=\sum a_n x^n If so, what's the radius of convergence or how can I find it? Thanks.

Suppose I wind up with the relation f(x)\delta (x-x')=g(x)\delta (x-x') true for all x'. Can I safely conclude that f(x) = g(x) (for all x)? Or am I overlooking something? this is a little too close to dividing both sides by zero for comfort.

I'm trying to evaluate the energy shift in a scalar field described by the Klein-Gordon equation caused by adding two time-independent point sources. In Zee's Quantum Field Theory in a Nutshell, he shows the derivation for a (3+1)-dimensional universe, and I'm trying to do the same for an...

I'm curious about the use of the Dirac Delta function. I am familiar with the function itself, but have never really seen in used in actual problems. The only problems I've worked with the function are those specifically about the function (ie. Evaluate the Dirac Delta function at x=3). My...

how do we apply dirac delta function?when do we apply?

r(x) = x if x \geq 0 and r(x) = 0 if x<0 I have to show that: 1-\[ \int_{- \infty}^{+ \infty} r(x) \varphi ''(x) dx = \varphi(0) \] And 2- that the second derivative of r is the Dirac delta. And I managed to do this by integrating by parts. Howver, I don't get why I can't just write: \[...

Hi, In Srednicki's chapter on cross sections, when he calculating the probability of a particular process from the overlap \langle f\mid i\rangle he comes across: [(2\pi)^4\delta^4(k_{in}-k_{out})]^2 He states this is can be equated as follows: [(2\pi)^4\delta^4(k_{in}-k_{out})]^2=...

Homework Statement show that the charge distribution (|\vec{r}|\equiv r ) \rho(r) = Z\delta^3(\vec{r})-\frac{Ze^{-r/R}}{4\pi R^2r} has zero net charge for any Z and R. Explain the meaning of Z. Homework Equations none given, but divergence (gauss) theorem and poisson's equation may...

Homework Statement Prove that \delta(at)=\frac{1}{abs(a)}\delta(t) Hint: Show that \int\phi(t)\delta(at)dt=\frac{1}{abs(a)}\phi(0) (the limits of integration are from -inf to +inf btw, I couldn't find how to put them in..) Homework Equations The Attempt at a Solution Ok. I...

Can someone tell me how to express a line of charge of charge per unit length \lambda as a delta function volume charge density in cylindrical coordinates?

I tried to calculate the volume of a simplex, but got an integral I couldn't do. For simplicity take a 2-simplex (the volume of a 2-simplex is 1/6) V=\int da \int dx \int dy \int dz \mbox{ } \delta(1-a-x-y-z) where the integration limits are over the 4-cube. My reasoning for this formula...

Homework Statement Using Dirac delta function in the appropriate coordinates, express the following charge distributions as three-dimensional charge densities p(x). (a) In spherical coordinates, a charge Q uniformly distributed over a spherical shell of radius a. (b) In cylindrical...

Homework Statement show x\frac{d}{dx}\delta(x)=-\delta)(x) using the gaussian delta sequence (\delta_n) and treating \delta(x) and its derivative as in eq. 1.151. Homework Equations the gaussian delta sequence given in the book is \delta_n=\frac{n}{\sqrt{\pi}}e^{-n^2x^2} and eq...

Homework Statement Distribution of matter is given in cylindrical coordinates: \rho(\vec{r})=\frac{1}{\rho}\delta(\rho^2-10\rho+9)\delta\left(\frac{z^2-a^2}{z^2+a^2}\right)\delta(\cot(\phi)) where a>0 is a constant. Find the complete mass of the object. Homework Equations The...

Hello, Dirac Delta Function is defined as the function that its amplitude is zero everywhere except at zero where its amplitude is infinitely large such that the area under the curve is unity. Sometimes it is used to describe a function consists of a sequence of samples such as...

Dirac delta function as the limit of a sequence Hi.. If I have a sequence which in some limit tends to infinity for x=0 and goes to zero for x\neq0, then can I call the limit as a dirac delta function? If not, what are the additional constraints to be satisfied?

Claim: \nabla \cdot \frac{\hat{e}_r}{r^2}=4\pi\delta^3(\vec{x}) Anyone know of a proof of this? (or a reference which covers it?) We need to show that \frac{1}{4\pi}\int_0^R{(\nabla \cdot \frac{\hat{e}_r}{r^2})f(r)dr=f(0). The claimed identity can be seen in the solution for...

Homework Statement Evaluate the following integrals: \int^{+\infty}_{-\infty}\delta[f(x)]dx and \int^{+\infty}_{-\infty}\delta[f(x)]g(x)dx Homework Equations \int^{+\infty}_{-\infty}\delta(x)dx=1 \int^{+\infty}_{-\infty}\delta(x)f(x)dx=f(0) \int^{+\infty}_{-\infty}\delta(x-a)f(x)dx=f(a)The...

I know I haven't entered the formulae with the proper syntax, but I'm extremely exhausted at the time of posting, so please just read it and give advice, forgiving me this once for not using proper form (it's basically in latex code format). Homework Statement Show...

I'm trying to understand the multiple limit processes involved with the Dirac delta function. Does it matter which process you do first the integral or the delta parameter that approaches zero? The closest theorem I found that addresses the order of taking limits is the Dominate Convergence...

We all know that \frac{1}{2\pi}\int{e^{ik(x-x')}dk=\delta(x-x'). i am working a problem which appears to depend on the statement \int e^{z^*(z-w)}dz^*\propto\delta(z-w) Does anyone know if this is valid? \delta(z-w) is defined in the usual way so that...

Claim: if \psi is a variable grassmann number, then \delta(\psi)=\psi is a Dirac delta function for integrals over \psi. I'm not seeing this. A general function of a grassmann number can be written f(\psi)=a+b\psi because anti-commutativity requires \psi^2=0. According to the wikipedia...

Hello, I have just integrated over one variable, x and have now got a delta function \delta(m) where m = constant * (s-s') now I have to integrate over either s or s' but I am a bit confused since if I integrate over say s then the delta function depends on s. Hope I have explained clearly...

I do not know how to execute the problem with the 2x in the problem. Evaluate the integral: \int_{-4}^{4} (x^2+2x+1) \delta(2x) dx

Homework Statement Given f(x,y) = DeltaFunction(y - x*tan(theta)) a) Plot function. b) Take Fourier transform. c) Plot resulting transform. Homework Equations Delta function condition non-zero condition DeltaFunction(0) = Infinity Sifting property of delta functions The...

First of all, let me copy the standard solution from Griffiths, section 2.5, just for the sake of clarity. PotentialV(x) = - \alpha \delta (x) The bound state eigenfunction: \psi (x) = \left\{ \begin{array}{l} B{e^{\kappa x}}{\rm{ (}}x \le 0{\rm{)}} \\ B{e^{ - \kappa x}}{\rm{...

I am trying to integrate a charge density over a volume in order to obtain a total charge, but there is a delta function involved and I am not entirely sure how to treat it. \rho = q* \delta (\textbf{r})- \frac {q\mu^{2} Exp(- \mu r)} {4 \pi r} Q = \int \rho (\textbf{r})d^{3}r...

Homework Statement What is the (volume) charge density of a ring of radius r_0 and uniform charge density \lambda? Homework Equations The Dirac Delta Function The Attempt at a Solution I've done a few line charge densities of straight wires along an axis (usually z, but on x as...

Homework Statement I wonder how to deal with the square root of Dirac Delta function, \sqrt{\delta(x)}. Actually, this comes from a problem which asking readers to calculate the wave function of a free particle and of a harmonic oscillator at time t, provided that the wave function at time...

I am supposed to prove that δ'(ax) = (1/a)*(1/|a|)*δ'(x) but I cannot figure out where the (1/a) term comes from. Using the scaling theorem I know that δ(ax) = (1/|a|)*δ(x), but how does this apply to the first derivative and does it explain where the (1/a) comes from?

The problem is to prove that δ'(ax) = (1/a)*(1/a)*δ'(x), where a is a constant. I tried applying the scaling theorem with the formal definition of δ'(x) but I can not get the second (1/a) term. Does anyone have some insight on this problem? Thank you...

the question is , can a delta function /distribution \delta (x-a) solve a NOnlinear problem of the form F(y,y',y'',x) the question is that in many cases you can NOT multiply a distribution by itself so you could not deal with Nonlinear terms such as (y)^{3} or yy'

since dirac delta function is not a literally a function but a limit of function,does it mean that dirac delta function is continuous and differentiable through out the infinity? is there any example of dirac delta function if yes then give meeeeeeee?

Homework Statement \[ \underset{\left|\underline{\xi}\right|=1}{\int}\delta_{0}\left(\underline{\xi}\cdot\underline{z}\right)dS_{\xi}=\intop_{0}^{2\pi}d\varphi\intop_{-r}^{+r}\delta_{0}\left(\varsigma\right)\frac{d\varsigma}{r}=\frac{2\pi}{r}\] The \delta_{0} is the dirac delta function.the...

Hi, I am not really sure whether its over the surface of the sphere or the Volume, the problem and the solution are given below, I want to know how it has been solved. The \delta_{0} is the dirac delta function. \[...

I have been reading papers for my research and I came across this equation twice: \lim_{\eta\to 0+}\frac{1}{x+i \eta} = P\left(\frac{1}{x}\right) - i \pi \delta(x) Where P is the pricipal part. It has been quite a while since I have had complex variables, but might it come from the...

Hi. How do we argue that \nabla^2\frac{1}{r} is a three dimensional delta function? I have seen some people do it using the divergence theorem, i.e. saying that \int_V \nabla\cdot\nabla\frac{1}{r}dv=-\oint_S \nabla\frac{1}{r}\cdot ds=-4\pi if S is a surface containing the origin, but I...