What is Dirac delta: Definition and 333 Discussions

In mathematics, the Dirac delta function (δ function) is a generalized function or distribution, a function on the space of test functions. It was introduced by physicist Paul Dirac. It is called a function, although it is not a function R → C.
It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. No function has these properties, such that the computations made by theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.
In engineering and signal processing, the delta function, also known as the unit impulse symbol, may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable. The convolution of a (theoretical) signal with a Dirac delta can be thought of as a stimulation that includes all frequencies. This leads to a resonance with the signal, making the theoretical signal "real" (i.e. causal). The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

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  1. T

    I Understanding the Dirac Delta Identity: A Guide to Fetter and Walecka's Formula

    Hi all, I'm trying to verify the following formula (from Fetter and Walecka, just below equation (12.38)) but it doesn't quite make sense to me: where and The authors are using the fact that ##\delta(ax) = |a|^{-1}\delta(x)## but to me, it seems like the...
  2. G

    Showing that a certain summation is equal to a Dirac delta?

    I'm studying Quantum Field Theory for the Gifted Amateur and feel like my math background for it is a bit shaky. This was my attempt at a derivation of the above. I know it's not rigorous, but is it at least conceptually right? I'll only show it for bosons since it's pretty much the same for...
  3. Delerion24

    Help with the derivative of the Dirac delta

    My goal is to develop the equation 21. You should asume that \delta(r_2-r_1)^2 =0. These is named renormalization. Then my question is , do my computes are correct with previous condition ?
  4. D

    Evaluate ##\int_{-\infty}^{\infty} e^{-|x|}\delta(x^2 +2x -3) dx##

    Hi, Is it correct to say that the dirac delta function is equal to 0 except if the argument is 0? Thus, ##x^2 +2x -3## must be equal to 0. Then, we have x = 1 or -3. What does that means? ##\int_{-\infty}^{\infty} e^{-|x|}\delta(x^2 +2x -3) dx = e^{-1}## and/or ##e^{-3}## ? Thank you
  5. H

    Sifting property of a Dirac delta inverse Mellin transformation

    Hi, I have to verify the sifting property of ##\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} e^{-sa}e^{st} ds## which is the inverse Mellin transformation of the Dirac delta function ##f(t) = \delta(t-a) ##. let ##s = iw## and ##ds = idw## ##\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iwa}e^{iwt}...
  6. H

    Mellin transform of Dirac delta function ##\delta(t-a)##

    Hi, I found Laplace transform of this Dirac delta function which is ##F(s) = e^{-st}## since ##\int_{\infty}^{-\infty} f(t) \delta (t-a) dt = f(a)## and that ##\delta(x) = 0## if ##x \neq 0## Then the Mellin transform ##f(t) = \frac{1}{2 \pi i} \int_{\gamma - i \omega}^{\gamma +i \omega}...
  7. Salmone

    A Prove a formula with Dirac Delta

    Why is the Laplacian of ##1/r## in spherical coordinates proportional to Dirac's Delta, namely: ##\left(\frac{\partial^2 }{\partial r^2}+\frac{2}{r}\frac{\partial }{\partial r}\right)\left(\frac{1}{r}\right)=-\frac{\delta(r)}{r^2}## I get that the result is zero.
  8. P

    Scattered State Solutions of a Repulsive Dirac Delta Potential

    I feel that this problem can be directly answered from the E>0 case of the attractive Dirac delta potential -a##\delta##(x), with the same reflection and transmission coefficients. Can someone confirm this hunch of mine?
  9. Leo Liu

    I What is the Definition of the Delta Function?

    I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept...
  10. F

    Conservation of charge with Dirac delta

    Hello, I was reviewing a part related to electromagnetism in which the charge and current densities are defined by the Dirac delta: ##\rho(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x} - \underline{x}_n(t))## ##\underline{J}(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x} -...
  11. U

    A question on the Dirac delta distribution

    Is it correct to say that $$\int e^{-i(k+k’)x}\,\mathrm{d}x$$ is proportional to ##\delta(k+k’)##?
  12. A

    Writing the charge density in the form of the Dirac delta function

    Hey guys! Sorry if this is a stupid question but I'm having some trouble to express this charge distribution as dirac delta functions. I know that the charge distribution of a circular disc in the ##x-y##-plane with radius ##a## and charge ##q## is given by $$\rho(r,\theta)=qC_a...
  13. Haorong Wu

    I Could this function be approximated by Dirac delta function?

    hi, there. I am doing some frequency analysis. Suppose I have a function defined in frequency space $$N(k)=\frac {-1} {|k|} e^{-c|k|}$$ where ##c## is some very large positive number, and another function in frequency space ##P(k)##. Now I need integrate them as $$ \int \frac {dk}{2 \pi} N(k)...
  14. M

    I Probability: why can we use the Dirac delta function for a conditional pdf?

    Hi, I have a quick question about something which I have read regarding the use of dirac delta functions to represent conditional pdfs. I have heard the word 'mask' thrown around, but I am not sure whether that is related or not. The source I am reading from states: p(x) = \lim_{\sigma \to...
  15. redtree

    I Integrating with the Dirac delta distribution

    Given \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y) \end{split} \end{equation} where ##\epsilon > 0## Is the following also true as ##\epsilon \rightarrow 0## \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon}...
  16. Antarres

    A Dirac delta function confusion

    As a part of a bigger problem, I was trying to evaluate the D'Alambertian of ##\epsilon(t)\delta(t^2-x^2-y^2-z^2)##, where ##\epsilon(t)## is a sign function. This term appears in covariant commutator function, so I was checking whether I can prove it solves Klein-Gordon equation. Since there's...
  17. rannasquaer

    MHB Dirac Delta and Fourier Series

    A beam of length L with fixed ends, has a concentrated force P applied in the center exactly in L / 2. In the differential equation: \[ \frac{d^4y(x)}{dx^4}=\frac{1}{\text{EI}}q(x) \] In which \[ q(x)= P \delta(x-\frac{L}{2}) \] P represents an infinitely concentrated charge distribution...
  18. giveortake

    Engineering Dirac Delta Function in an Ordinary Differential Equation

    1.) Laplace transform of differential equation, where L is the Laplace transform of y: s2L - sy(0) - y'(0) + 9L = -3e-πs/2 = s2L - s+ 9L = -3e-πs/2 2.) Solve for L L = (-3e-πs/2 + s) / (s2 + 9) 3.) Solve for y by performing the inverse Laplace on L Decompose L into 2 parts: L =...
  19. Arman777

    Solve $$\int_{∞}^{∞}dxf(x)\delta((x-x_1))$$: Dirac Delta Function

    If the question was $$ \int_{∞}^{∞}dxf(x)δ((x - x_1)) = ? $$ The answer would be ##f(x_1)## So the delta function has two roots, I searched the web and some books but I am not sure what approach should I use here. I guess there's sometihng happens when ##x_1 = -x_2##. So I am not sure what...
  20. JorgeM

    I Is this Dirac delta function integral correct?

    I have to integrate this expression so I started to solve the delta part from the fact that when n=0 it results equals to 1. And the graph is continuous in segments I thought as the sumation of integers $$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx $$ From the fact that actually $$ δ(sin(x))=...
  21. SisypheanZealot

    Dirac Delta using periodic functions

    I know it is something simple that I am missing, but for the life of me I am stuck. So, I used the identity ##sin(a)sin(b)+cos(a)cos(b)=cos(a-b)## which gives me $$\int^{\infty}_{-\infty}dx\:f(x)\delta(x-y)=\int^{\infty}_{-\infty}dx\:f(x)\frac{1}{2L}\sum^{\infty}_{n=-\infty}\lbrace...
  22. mishima

    I Dirac Delta, higher derivatives with test function

    Hi, I am curious about: $$x^m \delta^{(n)}(x) = (-1)^m \frac {n!} {(n-m)!} \delta^{(n-m)}(x) , m \leq n $$ I understand the case where m=n and m>n but not this. Just testing the left hand side with m=3 and n=4 and integrating by parts multiple times, I get -6. With the same values, the...
  23. amjad-sh

    Dirac delta function of a function of several variables

    Form solid state physics, we know that the volume of k-space per allowed k-value is ##\triangle{\mathbf{k}}=\dfrac{8\pi^3}{V}## ##\sum_{\mathbf{k}}F(\mathbf{k})=\dfrac{V}{(2\pi)^3}\sum_{\mathbf{k}}F(\mathbf{k})\triangle{\mathbf{k}}##...
  24. hilbert2

    Delta potential in classical mechanics

    In quantum mechanics, there exist some systems where the potential energy of some particle is a Dirac delta function of position: ##V(x) = A\delta (x-x_0 )##, where ##A## is a constant with proper dimensions. Is there any classical mechanics application of this? It would seem that if I...
  25. Another

    I Delta Kronecker Vs Dirac delta

    I want to know if these functions are related? for example. I can write Dirac delta in term Delta Kronecker from? Where can I learn these?
  26. A

    I Liouville equation with Dirac delta as probability density

    I would like to know the solution to Liouville equation ∂ρ/∂t=-{ρ,H} given the initial condition ρ(t=0)=δ(q,p) where δ(q,p) is a dirac delta centered in some point (q,p) in phase space. I have the feeling, but I'm not sure, that the solution is of the form ρ(t)=δ(q(t),p(t)) where q(t) and...
  27. Telemachus

    Correct numerical modeling of the 3D Dirac Delta function

    Hi. I was trying to test a code for the diffusion equation, using the analytical solution for infinite media, with a Dirac delta source term: ##q(\mathbf{r},t)=\delta (\mathbf{r}) \delta (t)##. The code is not giving the analytical solution, and there might be several reasons why this is so...
  28. Cathr

    I Distributions (generalised functions) basics

    I started studying distribution theory and I am struggling with the understanding of some basic concepts. I would hugely appreciate any help, made as simple as possible, because by now I'm only familiar with the formalism, but not all the meaning behind. The concepts I am struggling with are...
  29. D

    I Show that the integral of the Dirac delta function is equal to 1

    Hi, I am reading the Quantum Mechanics, 2nd edition by Bransden and Joachain. On page 777, the book gives an example of Dirac delta function. $\delta_\epsilon (x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$ I am wondering how I can show $\lim_{x\to 0+} \int_{a}^{b} \delta_\epsilon (x) dx$...
  30. P

    Calculate the Dirac delta function integral

    https://1drv.ms/w/s!Aip12L2Kz8zghV6Cnr8jPcRTpqTX https://1drv.ms/w/s!Aip12L2Kz8zghV6Cnr8jPcRTpqTX My question is in the above link
  31. P

    Dirac delta; fourier representation

    Homework Statement I know that we can write ## \int_{-\infinity}^{\infinity}{e^{ikx}dx}= 2\pi \delta (k) ## But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L? Homework...
  32. S

    I Question about the Dirac delta function

    Hi, if I have an interval on the x-axis, defined by the parameter L, can this, interval be transformed to a Dirac delta function instead, on the x-axis? Thanks!
  33. I

    I Meaning of Dirac Delta function in Quantum Mechanics

    If I have a general (not a plain wave) state $$|\psi\rangle$$, then in position space : $$\langle \psi|\psi\rangle = \int^{\infty}_{-\infty}\psi^*(x)\psi(x)dx$$ is the total probability (total absolute, assuming the wave function is normalized) So if the above is correct, does that mean...
  34. L

    I Lebesgue Integral of Dirac Delta "function"

    Is the "function" R->R f(x) = +oo, if x =0 (*) 0, if x =/= 0 Lebesgue measureable? Does its Lebesgue Integral exist? If yes, how much is it? (*) Certainly we shoud give a convenient meaning to that writing. -- lightarrow
  35. RJLiberator

    Valid Representation of Dirac Delta function

    Homework Statement Show that this is a valid representation of the Dirac Delta function, where ε is positive and real: \delta(x) = \frac{1}{\pi}\lim_{ε \rightarrow 0}\frac{ε}{x^2+ε^2} Homework Equations https://en.wikipedia.org/wiki/Dirac_delta_function The Attempt at a Solution I just...
  36. K

    I What happens when we replace the Dirac Delta function with a sine function?

    If we were to replace δ(x), the orginal Dirac Delta, with δ(sin(ωx)), what would be the result? Would we have an infinite spike everywhere on the graph of sinx where x is a multiple integer of π/ω? and 0 everywhere else? I apologize in advance if I had posted in the wrong category.
  37. U

    Functional Derivative with respect to Dirac Spinors

    Homework Statement I am currently working on an exercise list where I need to calculate the second functional derivative with respect to Grassmann valued fields. $$ \dfrac{\overrightarrow{\delta}}{\delta \psi_{\alpha} (-p)} \left( \int_{x} \widetilde{\bar{\psi}}_{\mu} (x) i \partial_{s}^{\mu...
  38. D

    Charge density using the Dirac delta funciton

    Currently, I am reading this article which introduces electromagnetism. It gives a function for the charge density as: $$\rho = q\delta(x-r(t))$$ The paper states that "the delta-function ensures that all the charge sits at a point," but how does it do that? Also, if ##r(t)## is the trajectory...
  39. Isaac0427

    I Questions about the Dirac delta

    Hi, Consider this definition of the Dirac delta: $$\delta (x-q)=\lim_{a \rightarrow 0}\frac{1}{a\sqrt \pi}e^{-(x-q)^2/a^2}$$ First, this would make a normalized position eigenfunction $$\psi (x)=\lim_{a \rightarrow 0}\frac{1}{\sqrt{a\sqrt \pi}}e^{-x^2/2a^2}$$ right? If that is so, why do...
  40. A

    How Can I Integrate with Dirac Delta in this Expression?

    Homework Statement I need to integrate this expression : P(k, w) = A * δ(w-k*v) * f(k, w) A is constant and δ, Dirac Delta.Homework Equations [/B] There is double integration : I = ∫0∞ dk ∫0∞ P(k,w) dw = A ∫∫0∞ δ(w-k*v) * f(k, w) dw dk The Attempt at a Solution [/B] I'm confused with...
  41. H

    I Square of Dirac delta function

    Is the square of a Dirac delta function, ##(\delta(x))^2##, still a Dirac delta function, ##\delta(x)##? A Dirac delta function peaks at one value of ##x##, say 0. If it is squared, it still peaks at the same value, so it seems like the squared Dirac delta function is still a Dirac delta...
  42. Pushoam

    Dirac delta function in spherical cordinates

    Homework Statement Calculate ##\int_{r=0}^\inf δ_r (r -r_0)\,dr## Homework Equations ##\int_V \delta^3(\vec{r} - \vec{r}') d\tau = 1## The Attempt at a Solution $$\int_V \delta^3(\vec{r} - \vec{r}') d\tau = \int_V \frac {1}{r^2 sinθ}\delta_r(r-r_0) \delta_θ (θ-θ_0) \delta_Φ (Φ-Φ_0) r^2...
  43. redtree

    I Understanding the Dirac Delta function

    I just want to make sure that I am understanding the Dirac Delta function properly. Is the following correct?: For two variables ##x## and ##y##: \begin{equation} \begin{split} \delta(x-y) f(x) &= f(y) \end{split} \end{equation} And: \begin{equation} \begin{split} \delta(x-x) f(x) &=...
  44. D

    A Dirac Delta and Residue Calculus

    I'm an undergraduate student, so I understand that it may be difficult to provide an answer that I can understand, but I have experience using both the Dirac delta function and residue calculus in a classroom setting, so I'm at least familiar with how they're applied. Whether you're integrating...
  45. Dopplershift

    Solving 3-D Dirac Delta Function Homework Question?

    Homework Statement \begin{equation} \int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau \end{equation} where: \begin{equation} \vec{r}' = 3\hat{x} + 2\hat{y} + \hat{z} \end{equation} Where d $\tau$ is the volume element, and V is a solid sphere with radius 4, centered...
  46. J

    I Evaluating 2D Delta Function Integral - Any Help Appreciated

    I am quite new here, and was wondering if anybody knows how this 2D integral is evaluated. $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \delta(k_1 x-k_2y)\,dx\,dy$$Any help is greatly appreciated! Thanks!
  47. arpon

    I Dirac Delta using Fourier Transformation

    We know, $$\delta(x) = \begin{cases} \infty & \text{if } x = 0 \\ 0 & \text{if } x \neq 0 \end{cases}$$ And, also, $$\int_{-\infty}^{\infty}\delta(x)\,dx=1$$ Using Fourier Transformation, it can be shown that, $$\delta(x)=\lim_{\Omega \rightarrow \infty}\frac{\sin{(\Omega x)}}{\pi x}$$ Let's...
  48. P

    Dirac Delta Function (x+a)

    Homework Statement hi i have to find the result of ##\int_{0}^{\pi}[\delta(cos\theta-1)+ \delta(cos\theta+1)]sin\theta d\theta## Homework Equations i know from Dirac Delta Function that ##\int \delta(x-a)dx=1## if the region includes x=a and zero otherwise The Attempt at a Solution first i...
  49. I

    I Is the Gradient of Dirac Delta Independent of the Coordinate System?

    Dear all, I have a quick question, is the following statement true? $$\nabla_\textbf{x'} \delta(\textbf{x}-\textbf{x'}) = \nabla_\textbf{x} \delta(\textbf{x}-\textbf{x'})?$$ I thought I have seen this somewhere before, but I could not remember where and why. I know the identity ##d/dx...