What is Delta function: Definition and 378 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.

View More On Wikipedia.org
Recent contents Top users
  1. J

    I Calculating Delta Function Ordering: Position and Momentum Operators Explained

    I want to calculate $$\langle x|XP|y \rangle$$ where X is the position operator and P the momentum operator, and the states are position eigenstates. But I get two different answers depending on if I insert a complete set of states. First way: $$\langle x|XP|y \rangle=x\langle x|P|y...
  2. J

    A Delta function in continuation back to Minkowski space

    The Green's function for a scalar field in Euclidean space is $$(2\pi)^4\delta^4(p+k) \frac{1}{p^2+m^2}$$ however when I continue to Minkowski space via GMin(pMin)=GE(-i(pMin)) there's seems to be a sign error: $$(2\pi)^4\delta^4(-i (p+k)) \frac{1}{-p^2+m^2}=(2\pi)^4\delta^4(p+k)...
  3. S

    A Integration using delta function and step function

    I would like to evaluate the following integral: ##\displaystyle{\int_{-\infty}^{\infty} dp^{0}\ \delta(p^{2}-m^{2})\ \theta(p^{0})}## ##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\ \delta[(p^{0})^{2}-\omega^{2}]\ \theta(p^{0})}## ##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\...
  4. P

    How Does the Dirac Delta Function Apply to Trigonometric Integrals?

    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...
  5. Arnd Obert

    I Density of states with delta function

    Hello, I'm stuck with this exercise, so I hope anyone can help me. It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by $$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$ is also representable as...
  6. 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...
  7. 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?
  8. 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 ∞...
  9. ELB27

    Product of a delta function and functions of its arguments

    Homework Statement I am trying to determine whether $$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$ where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions. Homework Equations The defining equation of a delta function...
  10. DrPapper

    Finding the Delta Function of a Thin Ring

    Homework Statement [/B] A very thin plastic ring (radius R) has a constant linear charge density, and total charge Q. The ring spins at angular velocity \omega about its center (which is the origin). What is the current I, in terms of given quantities? What is the volume current density J in...
  11. 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?
  12. R

    Delta Function Graphs: Formula, Derivative, and Anti-Derivative with Examples

    Homework Statement For each of these sketch and provide a formula for the function (i.e. in terms of ##u(t)##, ##\delta(t)##) and its derivative and anti-derivative. Denote the ##\delta## function with a vertical arrow of length 1. (a) ##f(t)=\frac{|t|}{t}## (b) ##f(t)=u(t) exp(-t)##...
  13. Kara386

    Tricky integration (maybe delta function)

    Homework Statement I need to integrate ##\frac{A}{2a\sqrt{2\pi}} \int_{-\infty}^{\infty} \frac{e^{ik(x+x')}}{(b^2+k^2)}dk## I have tried substitution and integration by parts and that hasn't worked. I can see that part of it is the delta function, but I don't really know how to use that fact...
  14. 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...
  15. debajyoti datta

    A What is the divergence of 1/r^n for positive integer n in physics?

    I have read in Griffiths electrodynamics that divergence of 1/r^2 is delta function and I thought it was the only special case...I have understood the logic there... but a question came in mind...what would happen in general if the function is 1/r^n ...where n is positive integer>0...because the...
  16. 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...
  17. 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...
  18. KostasV

    Delta function and dirac notation

    Hello there ! I found this discussion http://physics.stackexchange.com/questions/155304/how-do-we-normalize-a-delta-function-position-space-wave-function about dirac notation and delta function . The one that answers to the problem says that ##<a|a>=1## and ##<a|-a>=0## . As far as i know: 1)...
  19. 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...
  20. B

    Delta Function Integration: Justified or Fudging?

    Hello, I feel like I am fudging these integrals a bit and would like some concrete guidance about what's going on. 1. Homework Statement Evaluate ##I = \int_{-1}^{1} dx \delta'(x)e^3{x} ## Homework EquationsThe Attempt at a Solution [/B] I use integration by parts as follows, ##u =...
  21. 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)...
  22. 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...
  23. 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...
  24. 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...
  25. grandpa2390

    Evaluating integral with delta function (Fourier Transform)

    Homework Statement Evaluate the Following integrals 1. http://www4b.wolframalpha.com/Calculate/MSP/MSP10141fif9b428c5bab0b00005dc489hi851d28h7?MSPStoreType=image/gif&s=37&w=164.&h=35. Homework Equations...
  26. M

    Understanding the Use of the Delta Function in Probability Density Functions

    Homework Statement So I have an issue evaluating the integral for a joint probability distribution given by: Pr(R) = \displaystyle \int_{0}^{r_{max}}\int_0^{2\pi}\int_0^{\pi}\sin\theta \delta(R-r\sin\theta)d\theta d\phi dr where I know the relationship between r and R is given by...
  27. H

    A simple delta function properties, sifting property

    I don't know why this is possible To use delta function properties( sifting property) integral range have to (-inf ,inf) or at least variable s should be included in [t_0,t_0+T] but there is no conditions at all (i.e. t_0 < s < t_0+T) am I wrong?
  28. avikarto

    Fortran [FORTRAN] FFT of delta function, issue w/ MKL & Intel compiler

    I am trying to program something using a backwards FFT, and am attempting to feed it a delta function as a test condition since this result is known. However, my results are nonsense compared to what is expected. It should be the case that if we have...
  29. Z

    What are some applications for the Dirac delta function in classical mechanics?

    Hey community, are there some application for the Dirac delta function in classical mechanics? Im interessted in some application of the famous delta function. If there applications can someone explain it? Greetings :)
  30. 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...
  31. 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...
  32. D

    Why can't there be an antisymmetric excited state in a delta function potential?

    I have a potential which is zero everywhere except at -2a , -a , 0 , a , 2a on the x-axis where there is an attractive delta potential at each of the 5 points. I know there is a maximum of 5 bound states. I know there can be no nodes for |x| > 2a and a maximum of one node between each delta...
  33. D

    How to make a delta function signal in a circuit?

    Basically I want to test my analog circuit using a forcing function that has a form of a delta function. The function generator I use outputs sine wave, triangular wave and square wave (+ve and -ve output in one period). Are there any ways to produce a square wave that has an output for like 5%...
  34. blue_leaf77

    Proving the Stability of Sin and Cos through the FT Relation of Delta Function

    Starting from FT relation of delta function, I can write the followings: $$ \int_{-\infty}^{\infty} \cos{\alpha x} dx = 0 $$ $$ \int_{-\infty}^{\infty} \sin{\alpha x} dx = 0 $$ The question is how am I supposed to prove those equations, sin and cos are stable oscillating functions.
  35. 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...
  36. L

    Sinc^2 as a delta function representation?

    Hi, it's actually not homework but a part of my research. I intuitively see that: \lim_{t \rightarrow \infty} \frac{sin^2[(x-a)t]}{(x-a)^2} \propto \delta(x-a) I know it's certainly true of sinc, but I couldn't find any information about sinc^2. Could someone give me a hint on how I could...
  37. Logan Rudd

    Determining bound states for delta function potential

    I'm working on a problem out of Griffith's Intro to QM 2nd Ed. and it's asking to find the bound states for for the potential ##V(x)=-\alpha[\delta(x+a)+\delta(x-a)]## This is what I'm doing so far: $$ \mbox{for $x\lt-a$:}\hspace{1cm}\psi=Ae^{\kappa a}\\ \mbox{for $-a\lt x\lt...
  38. 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.
  39. C

    Integrating a delta function with a spherical volume integral

    Homework Statement Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity. Homework EquationsThe Attempt at a Solution This integral actually came up in a homework problem for my E&M class and I'm...
  40. 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...
  41. 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...
  42. S

    Validity of integral involving delta function

    Hi, Is the following integral well defined? If it is, then what does it evaluate to? \int_{-1}^{1} \delta(x) \Theta(x) \mathrm{d}x where \delta(x) is the dirac delta function, and \Theta(x) is the the Heaviside step function. What about if I choose two functions f_k and g_k, which are...
  43. T

    Integrating exponent to get delta function

    Something i ran into while doing hw Homework Statement starting with \int{dx} e^{-ikx}\delta(x) = 1 we conclude by Fourier theory that \int{dk} e^{+ikx} = \delta(x) Now, i try to compute \int{dk} e^{-ikx} (I've dropped the normalization factors of 2\pi. I believe no harm is done by...
  44. 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...
  45. A

    Derivative of the Step Function: Exploring the Delta Distribution

    How can I find the derivative of a step function?
  46. Hepth

    Mathematica [Mathematica] Bug in Integrate with derivatives of a delta function

    Integrate[f[qs] DiracDelta'[qs (1 - 1/x)], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] Integrate[f[qs] DiracDelta'[qs - qs/x], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] This is on Mathematica 8 for windows. The results differ by a sign. They are effective...
  47. kq6up

    Delta Function Potential Barrier

    Homework Statement Background: The problem is to find the uncertainty relationship for the wave equation for a delta function potential barrier where ##V(x)=\alpha\delta(x)##. Check the uncertainty principle for the wave function in Equation 2.129 Hint: Calculating ##\left< p^2 \right> ##...
  48. kq6up

    Properties of the Delta Function

    Homework Statement Delta functions said to live under the integral signs, and two expressions (##D_1(x)## and ##D_2(x)##) involving delta functions are said to be equal if: ##\int _{ -\infty }^{ \infty }{ f(x)D_{ 1 }(x)dx } =\int _{ -\infty }^{ \infty }{ f(x)D_{ 2 }(x)dx }## (a)...
  49. 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.
  50. O

    How to numerically solve a PDE with delta function boundary condition?

    I have a PDE of the following form: f_t(t,x,y) = k f + g(x,y) f_x(t,x,y) + h(x,y) f_y(t,x,y) + c f_{yy}(t,x,y) \\ \lim_{t\to s^+} f(t,x,y) = \delta (x-y) Here k and c are real numbers and g, h are (infinitely) smooth real-valued functions. I have been trying to learn how to do this...
Back
Top