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. Muthumanimaran

    Product of Dirac delta functions

    Homework Statement δ(z*-z0*)δ(z+z0)=? δ(z*+z0*)δ(z-z0)=? where 'z' is a complex variable 'z0' is a complex number Formula is just enough, derivation is not needed.
  2. Muthumanimaran

    What is the product of two Dirac delta functions

    Homework Statement What is the product of two Dirac delta functions δ(Real(z-c))δ(Img(z-c))=? 'z' and 'c' are complex numbers. This is not a problem, But I just need to use this formula in a derivation that I am currently doing. I just want the product of these two Dirac delta functions as a...
  3. Summer95

    How Does the Dirac Delta Function Solve the Differential Equation?

    Homework Statement Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. Then solve the differential equation with...
  4. GIM

    Change of the order of integration including Dirac delta

    Can somebody help me? I am studying Faddeev-Popov trick, following the Peskin and Schroeder's QFT book, but I can't understand one thing. After they inserted the Faddeev-Popov identity, $$I = \int {{\cal D}\alpha \left( x \right)\delta \left( {G\left( {{A^\alpha }} \right)} \right)\det \left(...
  5. S

    I Fourier transform of Dirac delta

    In lectures, I have learned that F(k)= \int_{-\infty}^{\infty} e^{-ikx}f(x)dx where F(k) is the Fourier transform of f(x) and the inverse Fourier transform is f(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}f(k)dk . But on the same chapter in the lecture notes, there is an example solving...
  6. X

    Question about the Dirac Delta Function

    Homework Statement Find the Fourier spectrum of the following equation Homework Equations ##F(\omega)=\pi[\delta(\omega - \omega _0)+\delta(\omega +\omega_0)]## The Attempt at a Solution Would the Fourier spectrum just be two spikes at ##+\omega _0## and ##-\omega _0## which go up to infinity?
  7. Kevin McHugh

    I Question regarding the Dirac delta function

    Given the definition: δ(x) = 0 for all x ≠ 0 ∞ for x = 0 ∫-∞∞δ(x)dx = 1 I don't understand how the integral can equal unity. The integral from -∞ to zero is zero, and the integral from 0 to ∞...
  8. Nasbah BM

    I Why we use Dirac delta function? (in 1 Dimension & 3 Dimesions)

    I want to understand why and where exactly we use dirac delta function? what is its exact use?
  9. R

    I General Solution of Dirac Delta Potential Well

    We know that the solutions of time-independent Dirac delta potential well contain bound and scattering states: $$\psi_b(x)=\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}\text{ with energy }E_b=-\frac{mu^2}{2\hbar^2}$$ and $$ \psi_k(x)= \begin{cases} A(e^{ikx}+\frac{i\beta}{1-i\beta}e^{-ikx})...
  10. R

    Dirac Delta Function in Differential Equation

    Homework Statement Find the solution to: $$\frac{d^2}{dt^2} x + \omega^2 x = \delta (t)$$ Given the initial condition that ##x=0## for ##t<0##. First find the general solution to ##t>0## and ##t<0##. Homework Equations The Attempt at a Solution This looks like a non-homogeneous second...
  11. DavideGenoa

    Commutations and delta in deriving Ampère's law

    Hi, friends! I have been able to understand, thanks to Hawkeye18, whom I thank again, that, if ##\mathbf{J}## is measurable according to the usual ##\mathbb{R}^3## Lebesgue measure ##\mu_{\mathbf{l}}## and bounded, a reasonable hypothesis if we consider it the density of current, if...
  12. A

    Bessel functions and the dirac delta

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  13. J

    A Separating the Dirac Delta function in spherical coordinates

    The following integral arises in the calculation of the new density of a non-uniform elastic medium under stress: ∫dx ρ(r,θ)δ(x+u(x)-x') where ρ is a known mass density and u = ru_r+θu_θ a known vector function of spherical coordinates (r,θ) (no azimuthal dependence). How should the Dirac...
  14. chi_rho

    Delta Function Identity in Modern Electrodynamics, Zangwill

    I am currently reading Modern Electrodynamics by Andrew Zangwill and came across a section listing some delta function identities (Section 1.5.5 page 15 equation 1.122 for those interested), and there is one identity that really confused me. He states: \begin{align*} \frac{\partial}{\partial...
  15. j3dwards

    Average of function (using dirac delta function)

    Homework Statement Compute the average value of the function: f(x) = δ(x-1)*16x2sin(πx/2)*eiπx/(1+x)(2-x) over the interval x ∈ [0, 8]. Note that δ(x) is the Dirac δ-function, and exp(iπ) = −1. Homework Equations ∫ dx δ(x-y) f(x) = f(y) The Attempt at a Solution Average of f(x) = 1/8 ∫from...
  16. Jan Wo

    Getting electron density from probability density function

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  17. ognik

    MHB Find Fourier series of Dirac delta function

    Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...
  18. C

    Time dependent three dimensional dirac delta function

    Ok so for equations of spherical wave in fluid the point source is modeled as a body force term which is given by time dependent 3 dimensional dirac delta function f=f(t)δ(x-y) x and y are vectors. so we reach an equation with ∫f(t)δ(x-y)dV(x) over the volume V. In the textbook it then says that...
  19. B

    Dirac Delta Function - Fourier Series

    1. Homework Statement Find the Fourier series of ##f(x) = \delta (x) - \delta (x - \frac{1}{2})## , ## - \frac{1}{4} < x < \frac{3}{4}## periodic outside. Homework Equations [/B] ##\int dx \delta (x) f(x) = f(0)## ##\int dx \delta (x - x_0) f(x) = f(x_0)##The Attempt at a Solution...
  20. jk22

    Exploring the Dirac Delta Function

    I consider the Dirac delta. In physics the delta squared has an infinite norm : $$\int\delta (x)^2=\infty $$ But if i look at delta being a functional i could write : $$\delta [f]=f (0) $$ hence $$\delta^2 [f]=\delta [\delta [f]]=\delta [\underbrace {f (0)}_{constant function}]=f (0)$$ Thus...
  21. N

    Rayleight scattering derivation

    Homework Statement I have a problem understanding derivation of particular equation in a textbook on Optical Waveguide Theory (Snyder, Love) - see attached. The equation 5.79 is the starting point. Explanation of 5-80 are clear and derivation of 5-81 likewise. I am stuck, however with 5-82...
  22. ddd123

    Microcanonical partition function - Dirac delta of operators

    Homework Statement Why is it that the microcanonical partition function is ##W = Tr\{\delta(E - \hat{H})\}##? As in, for example, Mattis page 62? Moreover, what's the meaning of taking the Dirac delta of an operator like ##\hat{H}##? Homework Equations The density of states at fixed energy is...
  23. C

    Proving a Dirac delta property

    Homework Statement [/B] Prove that \delta[a(x-x_1)]=\frac{1}{a}\delta(x-x_1) Homework Equations In my attempt I have used \delta(ax)=\frac{1}{a}\delta(x) but I'm not sure I'm allowed to use it in this proof. The Attempt at a Solution Some properties of Dirac delta function are proven using...
  24. Terocamo

    Confusion Surrounding Dirac Delta Comb Sampling: Why is δ(0) Infinite?

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  25. Z

    Exploring Dirac Delta Function: Using to Express 3D Charge Distributions

    Hello community, this is my first post and i start with a question about the famous dirac delta function. I have some question of the use and application of the dirac delta function. My first question is: Using Dirac delta functions in the appropriate coordinates, express the following charge...
  26. C

    Simplifying Argument of Dirac Delta for Reexpressing a Dirac Delta

    Homework Statement Show that $$\delta(k^2) \delta[(k-q_2)^2] = \delta(k^2) \delta(k^0 - \sqrt{s}/2) \frac{1}{2\sqrt{s}},$$ where ##k = (k^0, \mathbf k)## and ##s = q_2^2,## where ##q_2 = (\sqrt{s},\mathbf 0)## 2. Homework Equations I was going to use the fact that $$\delta(f(x)) = \sum...
  27. ShayanJ

    Rigorousness of this Dirac delta formula

    Is the following equation mathematically rigorous? How can you tell? ## \int_{-\infty}^\infty \delta(x-a) \delta(x-b) dx=\delta(a-b)## Thanks
  28. Coffee_

    Poisson equation with a dirac delta source.

    Consider: ##\nabla^{2} V(\vec{r})= \delta(\vec{r})## By taking the Fourier transform, the differential equation dissapears. Then by transforming that expression back I find something like ##V(r) \sim \frac{1}{r}##. I seem to have lost the homogeneous solutions in this process. Where does this...
  29. D

    Strength of Dirac Delta Potential

    When talking about the strength of a delta potential , the delta potential is multiplied by a parameter ie α but how does a delta potential have a strength ? It is zero everywhere and infinite at x = 0. The parameter makes no difference to zero or infinity.
  30. D

    Proving properties of the Dirac delta function

    I've been thinking about the properties of the Dirac delta function recently, and having been trying to prove them. I'm not a pure mathematician but come from a physics background, so the following aren't rigorous to the extent of a full proof, but are they correct enough? First I aim to...
  31. F

    Why do integrals featuring Dirac deltas equate to certain values?

    Homework Statement Having trouble understanding dirac deltas, I understand what they look like and how you can express one (i.e. from the limiting case of a gaussian) but for the life of me I can't figure out why the results of some integrals featuring dirac deltas equate to what they do...
  32. U

    Derivatives in 3D and Dirac Delta

    For a research project, I have to take multiple derivatives of a Yukawa potential, e.g. ## \partial_i \partial_j ( \frac{e^{-m r}}{r} ) ## or another example is ## \partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} ) ## I know that, at least in the first example above, there will be a...
  33. K

    How do you integrate dirac delta functions?

    Homework Statement ∫δ(x3 - 4x2- 7x +10)dx. Between ±∞. Homework EquationsThe Attempt at a Solution Well I don't really know how to attempt this. In the case where inside the delta function there is simply 2x, or 5x, I know the answer would be 1/2 or 1/5. Or for say δ(x^2-5), the answer would...
  34. G

    ODE with Dirac Delta and conditions at infinity

    I'm trying to solve the following equation (even if I'm not sure if it's well posed) \partial_{x} \, y(x) + a(x)\, y(x) = \delta(x) with ##\quad \lim_{x \rightarrow \pm \infty}y(x) = 0## It would be a classical first order ODE If it were not for the boundary conditions and the Dirac...
  35. D

    Probability distribution, a 1-D dirac delta in n-dimensions

    Hey everybody, I'm an engineering Ph.D. so my knowledge of n-dimensional Euclidean spaces is lacking to say the least. I'm wondering what sort of approach I can take to solve this problem. ##\boldsymbol{1.}## and ##\boldsymbol{ 2. }## I am given a probability distribution for a random...
  36. Legend101

    Integral involving Dirac Delta generalized function

    Homework Statement Evaluate the integrals in the attached image Homework EquationsThe Attempt at a Solution
  37. rogeralms

    Quantum Mechanics integral for Dirac Delta with abs value

    Homework Statement Break integral into positive and negative, integrate, recombine and simplify and show that it reduces to a real-valued function. (See attachments) Homework Equations See attachments The Attempt at a Solution My solution is not reducing to a real-valued function. Please see...
  38. J

    Dirac delta function proof fourier space

    Homework Statement I am having trouble understanding this: I have a Dirac Delta function $$ \delta (t_1-t_2) $$ but I want to prove that in the frequency domain (Fourier Space), it is: $$\delta(\omega_1+\omega_2) $$ Would anyone have any ideas how to go about solving this problem? I know...
  39. genxium

    Function whose 2nd order divergence is the Dirac Delta

    Homework Statement This problem came when I was learning the Poisson's equation (refer to http://farside.ph.utexas.edu/teaching/em/lectures/node31.html). when it came to the step to find the Green's function G which satisfies \nabla^2 \cdot G(\textbf{r}, \textbf{r}') =...
  40. R

    Finding the Convolution of Two Functions Using the Laplace Transform

    Homework Statement x(t) = cos(3πt) h(t) = e-2tu(t) Find y(t) = x(t) * h(t) (ie convolution) Homework Equations Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) The Attempt at a Solution L(x(t)) = [πδ(ω - 3π) + πδ(ω + 3π)] L(h(t)) = \frac{1}{s+2} Laplace Transform inverse ...
  41. S

    Dirac delta function identities

    hi deoes anyone know any online resource for proofs of Dirac delta function identities and confirming which representations are indeed DD functions Thanks a lot.
  42. R

    Integration test of dirac delta function as a Fourier integral

    Homework Statement Problem: a) Find the Fourier transform of the Dirac delta function: δ(x) b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves). c) test by integration, that the delta function represented by a Fourier integral integrates...
  43. sinaphysics

    A question about Dirac Delta Function

    For proving this equation: \delta (g(x)) = \sum _{ a,\\ g(a)=0,\\ { g }^{ ' }(a)\neq 0 }^{ }{ \frac { \delta (x-a) }{ \left| { g }^{ ' }(a) \right| } } We suppose that g(x)\approx g(a) + (x-a)g^{'}(a) Why for Taylor Expansion we just keep two first case and neglect others...
  44. P

    Function whose Fourier transform is Dirac delta

    Is there a time domain function whose Fourier transform is the Dirac delta with no harmonics? I.e. a single frequency impulse
  45. A

    The need for the Dirac delta function

    So part of the idea presented in my book is that: div(r/r3)=0 everywhere, but looking at this vector field it should not be expected. We would expect some divergence at the origin and zero divergence everywhere else. However I don't understand why we would expect it to be zero everywhere but...
  46. A

    Derivative of the Step Function: Exploring the Delta Distribution

    How can I find the derivative of a step function?
  47. S

    Solution of differential equation with Dirac Delta

    Is it possible to solve a differential equation of the following form? $$\partial_x^2y + \delta(x) \partial_x y + y= 0$$ where ##\delta(x)## is the dirac delta function. I need the solution for periodic boundary conditions from ##-\pi## to ##\pi##. I've realized that I can solve this for some...
  48. L

    Dirac delta function. Integral

    How to calculate ##\int^{\infty}_{-\infty}\frac{\delta(x-x')}{x-x'}dx'## What is a value of this integral? In some youtube video I find that it is equall to zero. Odd function in symmetric boundaries.
  49. D

    Cannot make sense of a derivation step involving dirac delta

    I am self studying the 17th Chapter of "Mathematical Methods for Physics and Engineering", Riley, Hobson, Bence, 3rd Edition. It is about eigenfunction methods for the solution of linear ODEs. Homework Statement On page 563, it states: "As noted earlier, the eigenfunctions of a...
  50. K

    Computation about Gaussian and Dirac Delta Function

    I have a Gaussian distribution about t, say, N(t; μ, σ), and a a Dirac Delta Function δ(t). Then how can I compute: N(t; μ, σ) * δ(t > 0) Any clues? Or recommender some materials for me to read? Thanks!
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