# What is Dirac delta function: Definition and 197 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. ### Dirac Delta Function identity

I need help to understand how equation (27) in this paper has been derived. The definition of P(k) (I discarded in the question ##\eta## or the integration with respect for it) is given by (26) and the definition of h(k) and G(k) are given by Eq. (25) and Eq. (24) respectively. In my...
2. ### Plotting the approximation of the Dirac delta function

Hi, I am not sure if I have solved the following task correctly I did the plotting in mathematica and got the following Would the plots be correct? What is meant by check for normalization, is the following meant? For the approximation for ##\epsilon > 0##, does it mean that for the...
3. ### I Dirac delta function confusions

I have read lots but still, there're some really unproductive explanations of dirac delta function. So hopefully, you can explain it by following my arguments and not formal definition because I've read it all. It's shown to be as ##\delta (x) = 0## when ##x \neq 0## and ##\delta (x) =...
4. ### I Dirac delta function integrated on a finite interval

Which is correct: $$\int_{-1}^1 \delta (x-x_0) \, dx =\begin{cases} 1, -1\leq x_0 \leq 1 \\ 0, \text { otherwise} \end{cases}$$ or $$\int_{-1}^1 \delta (x-x_0) \, dx =\begin{cases} 1, -1< x_0 < 1 \\ 0, \text { otherwise} \end{cases}$$ ?

19. ### 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}}##...
20. ### Energy Difference with a Two Delta Function Potential

Homework Statement Consider a particle of mass m moving in a one-dimensional double well potential $$V(x) = -g\delta(x-a)-g\delta(x+a), g > 0$$ This is an attractive potential with ##\delta##-function dips at x=##\pm a##. In the limit of large ##\lambda##, find a approximate formula for the...
21. ### Divergence of the E field at a theoretical Point Charge

I've been thinking about this problem and would like some clarification regarding the value of the divergence at a theoretical point charge. My logic so far: Because the integral over all space(in spherical coordinates) around the point charge is finite(4pi), then the divergence at r=0 must be...
22. ### 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...
23. ### 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 $\lim_{x\to 0+} \int_{a}^{b} \delta_\epsilon (x) dx$...
24. ### 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
25. 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!
26. ### 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...
27. ### 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...
28. ### I Checking My Understanding: Lagrangian & Path Integral Formulation

I note the following: \begin{split} \langle \vec{x}| \hat{U}(t-t_0) | \vec{x}_0 \rangle&=\langle \vec{x}| e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} | \vec{x}_0 \rangle \\ &=e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} \delta(\vec{x}-\vec{x}_0)...
29. ### 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...

45. ### Bessel functions and the dirac delta

Homework Statement Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
46. ### 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...
47. ### 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...
48. ### 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...
49. ### 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)...
50. ### 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...