Dirac delta Definition and 323 Threads

  1. H

    Help Understanding Dirac Delta Function in Lecture Notes

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

    Dirac Delta as Gaussian functions

    I am looking at a problem, part of which deals with expressing delta dirac as a limiting case of gaussian function. I am aware of the standard ways of doing it. In addition, I would also like to know if the following are correct - \delta(x-a) = \lim_{\sigma \rightarrow{0}} \int_{a -...
  3. F

    Multivariable Dirac Delta Functions

    Hello all. So I am trying to integrate a function of this form: \int\intF(x,y)\delta[a(Cos[x]-1)+b(Cos[y]+1)]dxdy The limits of integration for x and y are both [0,2Pi). I know that this integral is only nonzero for x=0, y=Pi. So this should really only sample one point of F(x,y)...
  4. C

    Derivation of Dirac Delta Function

    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.
  5. H

    Dirac delta integrated from 0 to infinity

    I was wondering if I integrate the dirac delta function from 0 to infinity where the function it's integrated with is the constant 1, will I get 0.5 or 1? And why? This is not homework so I decided to post this here although I asked this question in class and the teacher (assistant) wasn't...
  6. M

    Plotting Dirac Delta Function in Maple14: Troubleshooting

    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)...
  7. W

    Proofs for Dirac delta function/distribution

    [SOLVED] Proofs for Dirac delta function/distribution Homework Statement Prove that \delta(cx)=\frac{1}{|c|}\delta(x) Homework Equations \delta(x) is defined as \delta(x)=\left\{\stackrel{0 for x \neq 0}{\infty for x=0} It has the properties...
  8. V

    Integral of dirac delta function at x=0

    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
  9. L

    Please help with Quantum Mechanics Dirac Delta Problems

    These problems are from Introductory Quantum Mechanics (Liboff, 4th Ed.) Note: I'm using "D" as the dirac delta function. 3.9 (a) Show that D( sqrt(x) ) = 0 This has me stumped. It is my understanding that the Dirac function is 0, everywhere, except at x=0. So, how can I show this to be...
  10. A

    Power expansion of the Dirac Delta function?

    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.
  11. pellman

    Dividing both sides by a Dirac delta function - ok?

    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.
  12. S

    Signals - Integration of Heavyside Step & Dirac Delta Functions

    Homework Statement \int_{-\infty}^{\infty}{u(t)e^{-t}(\delta(t+1)+\delta(t-1))dt Homework Equations \int_{-\infty}^{t}{u(t)dt = \left\{\begin{array}{cc}0,&\mbox{ if } t< 0\\t, & \mbox{ if } t>0\end{array}\right. \int_{-\infty}^{\infty}{f(t)\delta(t-a)dt} = f(a) The...
  13. V

    Using Dirac Delta Function to Determine Point Mass Density

    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...
  14. K

    Dirac Delta Function: Application & Uses

    how do we apply dirac delta function?when do we apply?
  15. J

    Dirac Delta in polar coordinates

    Homework Statement Hi, I would like to know what is the right way to write continuous deltas standing in a circle of radius a? Homework Equations The Attempt at a Solution I am not sure weather it's δ(r-a) or is it δ(r-a)/|r-a| Thank you
  16. D

    Prove Idntity - Dirac Delta - Distributions

    Homework Statement The Identity to prove: Homework Equations Using Integration by parts The Attempt at a Solution I couldn't produce the denominator.
  17. A

    Ramp function, Dirac delta function and distributions

    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: \[...
  18. L

    Shankar CH1 Derivative of Dirac delta

    Hi, On p67 of shankar Principles of QM, he considers the delta functions derivative. He says: \int \delta'(x-x')f(x')dx'= \int \frac{d\delta(x-x')}{dx}f(x')dx'= \frac{d}{dx}\int \delta(x-x') f(x')dx'=\frac{df(x)}{dx} I don't understand how the second equality follows, how can the...
  19. F

    Integration of Dirac delta w/ different dimensionalities

    Hi! The Dirac delta satisfies \int dx f(x) \delta(x-a) = f(a) But how about \int d^3x f(x) \delta^{(4)}(x-a) Here, x and a are four-momenta, and the integral is over the regular 3-dim momentum. How does the delta behave here?
  20. C

    Derivative of dirac delta function

    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...
  21. D

    Finding mass with dirac delta function

    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...
  22. E

    Dirac Delta Function: Definition &amp; Samples

    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...
  23. D

    Dirac Delta Scaling: Solving the Integral Equation

    Using the defining property of the dirac delta function, \int{dx f(x) \delta(x-c)} show that \delta(ax)=\frac{1}{|a|}\delta(x) I think all I need to do is make the right u substitutions and it will come out right, but I can't think of how to make the substitutions...after a long time working...
  24. K

    Dirac delta function as the limit of a seqquence

    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?
  25. E

    Evaluating Dirac Delta Integrals: Homework Statement

    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...
  26. F

    Dominate Convergence Theorem for the Dirac delta function

    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...
  27. pellman

    Complex integral representation of Dirac delta function?

    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...
  28. K

    Integration of dirac delta composed of function of integration variable

    Hi all, I'm working through Chandrasekhar's http://prola.aps.org/abstract/RMP/v15/i1/p1_1" and can not understand the steps to progress through Eq. (66) in Chapter 1. The integral is: \prod^{N}_{j=1} \frac{1}{l^{3}_{j}|\rho|}\int^{\infty}_{0} sin(|\rho|r_{j})r_{j}\delta...
  29. F

    Product of dirac delta distributions

    I'm told that a product of distributions is undefined. See, http://en.wikipedia.org/wiki/Distribution_(mathematics)#Problem_of_multiplication where the Dirac delta function is considered a distribution. Now the Dirac delta function is defined such that, \[ \int_{ - \infty }^{ +...
  30. A

    Why is the integral of the Dirac Delta potential related to its properties?

    why in the problem of dirac delta potential, the integral \int^{\epsilon}_{-\epsilon}\phi''(x)dx is equal to \phi'(\epsilon)-\phi'(-\epsilon)? but \int^{\epsilon}_{-\epsilon}\phi(x)dx is equal to 0 if, for example\phi(x)=e^x then \phi(x)''=\phi(x) but, the firts integral is...
  31. V

    Dirac delta function evaluation

    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
  32. H

    Integral of Exp(I x) and the Dirac Delta

    I am trying to see why exactly the momentum eignenstates for a free particle are orthogonal. Simply enough, one gets: \int_{-\infty}^{\infty} e^{i (k-k_0) x} dx = \delta(k-k0) I can see why, if k=k0, this integral goes to zero. But if they differ, I don't see why it goes to zero. You have...
  33. S

    Simplifying the integral of dirac delta functions

    hello all, i am unaware of how to handle a delta function. from what i read online the integral will be 1 from one point to another since at zero the "function" is infinite. overall though i don't think i know the material well enough to trust my answer. and help on how to take the integral of...
  34. E

    How do I convert f(x) into its Fourier Transform?

    Homework Statement I am really confused in my electrodynamics class. I have the following function. f(x) = \delta (x + \alpha ) + \delta(x -\alpha) How do i convert this into Fourier Tranform ? Those are dirac delta functions on either sides of the origin. Homework Equations...
  35. H

    Understanding Dirac Delta Squares: Clarifying Doubts

    hi, may someone help me to clarify my doubts... in my work, i encounter diracdelta square \delta(x-x_1)\delta(x-x_2) i am not sure what it means... it seems if i integrate it \int dx \;\delta(x-x_1)\delta(x-x_2) = \delta(x_1-x_2) is either zero of infinity. is this correct? thanks
  36. I

    Square root of Dirac Delta function

    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...
  37. A

    Line of charge as a volume charge dist. (w/ Dirac delta fcn.)

    How would you write an infinite line charge with constant charge per unit length \lambda as a volume charge density using Dirac delta functions? Perhaps in cylindrical coordinates? I'm confused because if you integrate this charge distribution over all space, you should get an infinite amount...
  38. S

    Where Did I Go Wrong Calculating the Divergence of \(\widehat{r}/r^{2}\)?

    I know that this question was posted before but I just couldn't get it using another way around. So your comments are highly appreciated. In the textbook, \nabla\bullet\left(\widehat{r}/r^{2}\right)=4\pi\delta^{3}\left(r\right). But when I want to calculate the divergence using Catesian...
  39. A

    Dirac delta function is continuous and differential

    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?
  40. T

    Integral over a sphere with the dirac delta function

    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...
  41. T

    Integral Over a Sphere with dirac delta function

    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. \[...
  42. J

    An identity involving a 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...
  43. M

    Dirac Delta Integration Problem

    Homework Statement \int_{-\infty}^t (cos \tau)\delta(\tau) d\tau Evaluate the integral. I'm supposed to evaluate this for all t I believe, so I'm concerned with t<0, t=0, t>0. Homework Equations \int_{-\infty}^{\infty} f(x)\delta(x) dx = f(0) The Attempt at a Solution...
  44. M

    How to handle the Dirac delta function as a boundary condition

    Using perturbation theory, I'm trying to solve the following problem \frac{\partial P}{\partial \tau} = \frac{1}{2}\varepsilon^2 \alpha^2 \frac{\partial^2 P}{\partial f^2} + \rho \varepsilon^2 \nu \alpha^2 \frac{\partial^2 P}{\partial f \partial \alpha} + \frac{1}{2}\varepsilon^2 \nu^2...
  45. V

    Help with EM Fields and Dirac Delta Needed

    Hi guys. I play now a bit with EM fields and I have encountered some problems connected with Dirac delta. By coincidence I visited this forum and I thought I could find some help in here. The problem is that in order to get a potential in some point from a single charge you need to just...
  46. S

    Why is the integral of the Dirac delta distribution equal to unity?

    Previously I posted a question on the Dirac delta function and was informed it was not a true function, but rather a distribution. However, I have to admit I still did not understand why its integral (neg inf to pos inf) is unity. I've thought about this and came up with the following...
  47. J

    Property of the Dirac Delta Function

    Homework Statement How do you show that int[delta(t)]dt from negative infinity to infinity is 1? Homework Equations Dirac delta function defined as infinity if t = 0, 0 otherwise The Attempt at a Solution My teacher said that it has to do with m->infinity for the following...
  48. Z

    How can integration by parts be used to prove the Dirac delta function?

    1. The problem statement Show that: \int_{-\infty}^{\infty} f(x) \delta^{(n)}(x-a) dx = (-1)^n f^{(n)}(a) The Attempt at a Solution I am trying to understand how to prove: \int_{-\infty}^{\infty} f(x) \delta '(x) dx =- f'(x) I know that we need to use integration by parts, but I'm...
  49. C

    Can someone explain the 3D Dirac Delta Function in Griffiths' Section 1.5.3?

    Griffiths' section 1.5.3 states that the divergence of the vector function r/r^2 = 4*Pi*δ^3(r). Can someone show me how this is derived and what it means physically? Thanks in advance.
  50. S

    Starting with the definition of the Dirac delta function,

    Homework Statement Starting with the definition of the Dirac delta function, show that \delta( \sqrt{x}) um... i have looked in my book and looked online for a problem like this and i really have no clue where to start. the only time i have used the dirac delta function is in an integral...
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