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. E

    Dirac Delta Function: Is delta(x-y) the Same as delta(y-x)?

    Sorry if the question seems naive but if we have the Dirac delta function delta(x-y) is it the same as delta(y-x)?? Or there are opposite in sign? And why ? Thank you for your time
  2. M

    Transition Rates / Squared Dirac Delta

    I am not understanding something from my textbook. This is related to Fermi's Golden rule. It's about what happens when the matrix element of the perturbation H' ends up being a Dirac delta for chosen normalization. Here is Fermi's Golden rule. \Gamma_{ba} = 2\pi \left|\langle b \mid H'\mid a...
  3. M

    What is the use of Dirac delta function in quantum mechanics?

    If you ask me define Dirac delta function, i can easily define it and prove its properties using laplacian or complex analysis method. But still i don't understand what is the use of DIRAC DELTA FUNCTION in quantum mechanics. As i have done some reading Quantum mechanics from Introduction to...
  4. E

    Dirac Delta source for vectorial equation

    Hello! By manipulating Maxwell's equation, with the potential vector \mathbf{A} and the Lorentz' gauge, one can obtain the following vector wave equation: ∇^2 \mathbf{A}(\mathbf{r}) + k^2 \mathbf{A}(\mathbf{r}) = -\mu \mathbf{J}(\mathbf{r}) The first step for the solution is to consider a...
  5. X

    Proving that the Dirac Delta is the limit of Gaussians

    Homework Statement I need to prove for arbitrary functions φ(x) that: \lim_{\lambda \to 0} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx = \varphi(0), which, in the sense of distributions is basically the delta...
  6. F

    Dirac Delta Function Help: Justifying Translation/Sifting Property

    I have been reviewing some details on the Dirac Delta function and I've hit a little bit of a road block with trying to wrap my head around how the Translation/Sifting property of the function is justified. Now according to my text the overall definition is the generalized function with the...
  7. L

    Inner product of dirac delta function

    Homework Statement Find the inner product of f(x) = σ(x-x0) and g(x) = cos(x) Homework Equations ∫f(x)*g(x)dx Limits of integration are -∞ to ∞ The Attempt at a Solution First of all, what is the complex conjugate of σ(x-x0)? Is it just σ(x-x0)? And I'm not sure how to...
  8. N

    Simplifying equations involving Dirac Delta (Analog Signal Processing)

    Homework Statement I'm specifically having trouble with taking the Fourier transform of f(t) in order to sketch F(w) and also to move on with the rest of the problem. Homework Equations f(t) = (5+rect(t/4))cos(60pi*t) mixed_signal = cos(60pi*t) The Attempt at a Solution I attempted to...
  9. A

    Relation between residue integration and the Dirac Delta function

    Homework Statement OK so I'm doing a course on Signals and Systems and I'm taking inverse z transforms using residue integration. One particular formula in complex integration made me think a bit. \oint{\frac{f(z)}{z-z_0} dz} = 2\pi jf(z_0) This looks eerily similar to the definition...
  10. L

    Dirac Delta Integrals: How to Solve for the Argument of the Delta Function?

    Homework Statement This is just an example, not a specific problem. So if I have ∫σ(sinx), for example, and my limits of integration are, for example, 1 to 10, what I need to do to solve that is to find a value of x that would make the argument of the delta function 0. So for sinx, 0 makes...
  11. A

    Dirac Delta function and Divergence

    Homework Statement The Potential V(r) is given: A*e^(-lambda*r)/r, A and lambda are constants From this potential, I have to calculate: E(r), Rho(r) -- charge density, and Q -- total charge. Homework Equations The Attempt at a Solution I know that E(r) is simply minus...
  12. skate_nerd

    Proving a property of the dirac delta function

    Homework Statement Prove this theorem regarding a property of the Dirac Delta Function: $$\int_{-\infty}^{\infty}f(x)\delta'(x-a)dx=-f'(a)$$ (by using integration by parts) Homework Equations We know that δ(x) can be defined as...
  13. F

    Residue of Dirac delta function?

    Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?
  14. M

    How Does Dirac Delta Substitution Relate to Helmholtz's Decomposition Theorem?

    Hi All, I found (Wikipedia page on Helmotz's decomposition theorem) the follwoing equality, which puzzles me: $$\delta(x-y) = - (4 \pi)^{-1} \nabla^{2} \frac{1}{\vert x - y \vert}$$ I am not sure I understand, the r.h.s seems to me a proper function. The page mentions this a sa position...
  15. Philosophaie

    Dirac Delta Function: What It Does & How to Evaluate It

    What does the Dirac Delta Function do? ##\delta^3(\vec{r})## How do you evaluate it? What are its values from -inf to +inf?
  16. Vahsek

    Dirac Delta Function: Definition & Mathematics

    It's been quite some time now since I decided to stop self-studying physics and to pay more attention to the math behind. I'm working towards gaining an understanding of 100% rigorous mathematics for now. One thing that has always bothered me is the Dirac delta function. What I want to know...
  17. P

    Is δ(x+y)=δ(x-y) for Dirac Delta Function?

    Homework Statement Good day. May I know, for Dirac Delta Function, Is δ(x+y)=δ(x-y)? The Attempt at a Solution Since δ(x)=δ(-x), I would say δ(x+y)=δ(x-y). Am I correct?
  18. S

    The nature of the dirac delta function

    From what I can tell, it seems that 1/x + δ(x) = 1/x because if we think of both 1/x and the dirac delta function as the following peicewise functions: 1/x = 1/x for x < 0 1/x = undefined for x = 0 1/x = 1/x for x > 0 δ(x) = 0 for x < 0 δ(x) = undefined for x = 0 δ(x) = 0 for x > 0...
  19. P

    The Double Dirac Delta Function Potential wave functions

    Homework Statement Consider the double Dirac delta function V(x) = -α(δ(x+a) + δ(x-a)). Using this potential, find the (normalized) wave functions, sketch them, and determine the # of bound states. Homework Equations Time-Independent Schrodinger's Equation: Eψ = (-h^2)/2m (∂^2/∂x^2)ψ +...
  20. P

    A question about Dirac Delta Potential Well solution

    In Griffith's Introduction to Quantum Mechanics, on page 56, he says that for scattering states (E > 0), the general solution for the Dirac delta potential function V(x) = -aδ(x) (once plugged into the Schrodinger Equation), is the following: ψ(x) = Ae^(ikx) + Be^(-ikx), where k = (√2mE)/h...
  21. Y

    Question about Dirac Delta function

    In page 555, Appendix B of Intro to electrodynamics by D Griffiths: \nabla\cdot \vec F=-\nabla^2U=-\frac{1}{4\pi}\int D\nabla^2\left(\frac{1}{\vec{\vartheta}}\right)d\tau'=\int D(\vec r')\delta^3(\vec r-\vec r')d\tau'=D(\vec r) where ##\;\vec{\vartheta}=\vec r-\vec r'##. Is it supposed to be...
  22. Y

    Question on Dirac Delta function

    I want to proof (1)##\delta(x)=\delta(-x)## and (2) ## \delta(kx)=\frac{1}{|k|}\delta(x)## (1) let ##u=-x\Rightarrow\;du=-dx## \int_{-\infty}^{\infty}f(x)\delta(x)dx=(0) but \int_{-\infty}^{\infty}f(x)\delta(-x)dx=-\int_{-\infty}^{\infty}f(-u)\delta(u)du=-f(0) I cannot proof (1) is equal as I...
  23. Y

    Question on Dirac Delta function in Griffiths

    My question is in Griffiths Introduction to Electrodynamics 3rd edition p48. It said Two expressions involving delta function ( say ##D_1(x)\; and \;D_2(x)##) are considered equal if: \int_{-\infty}^{\infty}f(x)D_1(x)dx=\int_{-\infty}^{\infty}f(x)D_2(x)dx\;6 for all( ordinary) functions f(x)...
  24. D

    Dirac Delta Function: Explanation & Usage

    I know this probably belongs in one of the math sections, but I did not quite know where to put it, so I put it in here since I am studying Electrodynamics from Griffiths, and in the first chapter he talks about Dirac Delta function. From what I've gathered, Dirac Delta function is 0 for...
  25. R

    How can boundary conditions be written for a DEQ with Dirac delta?

    Hi All, so I'm trying to tackle this DEQ: f''[x] = f[x] DiracDelta[x - a] - b, with robin boundary conditions f'[0] == f[0], f'[c] == f[c] where a,b, and c are constants. If you're curious, I'm getting this because I'm trying to treat steady state in a 1D diffusion system where...
  26. H

    Divergence of inverse square field and Dirac delta

    \nabla \cdot \frac{\mathbf{r}}{|r|^3}=4 \pi \delta ^3(\mathbf{r}) What's the proof for this, and what's wrong with the following analysis? The vector field \frac{\mathbf{r}}{|r|^3}=\frac{1}{r^2}\hat{r} can also be written \mathbf{F}=\frac{x}{\sqrt{x^2+y^2+z^2}^3}\hat{x}+...
  27. ShayanJ

    Is Dirac Delta a Function or a Distribution?

    In texts about dirac delta,you often can find sentences like "The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin". If we take into account the important property of dirac delta: \int_\mathbb{R} \delta(x) dx=1 and the fact that it is zero...
  28. B

    Quick question about Dirac delta functions

    What does the square of a Dirac delta function look like? Is the approximate graph the same as that of the delta function?
  29. F

    Taylor series expansion of Dirac delta

    I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...
  30. L

    Limit of integral lead to proof of convergence to dirac delta

    Hi, I try to prove, that function f_n = \frac{\sin{nx}}{\pi x} converges to dirac delta distribution (in the meaning of distributions sure). On our course we postulated lemma, that guarantee us this if f_n satisfy some conditions. So I need to show, that \lim_{n\rightarrow...
  31. J

    A question about Dirac delta function

    Hello, Is this correct: \int [f_j(x)\delta (x-x_i) f_k(x)\delta (x-x_i)]dx = f_j(x_i)f_k(x_i) If it is not, what must the left hand side look like in order to obtain the right handside, where the right hand side multiplies two constants? Thanks!
  32. J

    Simple equations in Dirac Delta function terms

    Hi there, I'm trying to comprehend Dirac Delta functions. Here's something to help me understand them; let's say I want to formulate Newton's second law F=MA (for point masses) in DDF form. Is this correct: F_i = \int [m_i\delta (x-x_i) a_i\delta (x-x_i)]dx Or is it this: F_i = [\int...
  33. B

    Integrating the Dirac Delta function

    Homework Statement I am trying to integrate the function \int _{-\infty }^{\infty }(t-1)\delta\left[\frac{2}{3}t-\frac{3}{2}\right]dt Homework Equations The Attempt at a Solution I think the answer should be \frac{5}{4} because \frac{2}{3}t-\frac{3}{2}=0 when t=9/4. then (9/4-1) = 5/4...
  34. elfmotat

    Dirac Delta Function: Does it Have a Residue?

    Does the Dirac delta fuction have a residue? Given the close parallels between the sifting property and Cauchy's integral formula + residue theory, I feel like it should. Unfortunately, I have no idea how they tie together (if they do at all).
  35. G

    Dirac Delta Integral: Compute \int_ {-\infty}^\infty e^{ikx}δ(k^2x^2-1)dx

    Homework Statement compute the integral: \int_ {-\infty}^\infty \mathrm{e}^{ikx}\delta(k^2x^2-1)\,\mathrm{d}xHomework Equations none that I have The Attempt at a Solution I don't actually have any work by hand done for this because this is more complex than any dirac delta integral I have...
  36. G

    Dirac delta function how did they prove this?

    Hi all, I'm familiar with the fact that the dirac delta function (when defined within an integral is even) Meaning delta(x)= delta(-x) on the interval -a to b when integral signs are present I want to prove this this relationship but I don't know how to do it other than with a limit...
  37. Z

    Is Dirac's Delta to the fourth power equal to 0.5?

    hello, Please attached snapshot. Does the integral 7.9 equal to 0.5 \inthμσ dzμ/dτ dzσ/dτdτ ? I'm confused as to the fourth power of Dirac's Delta. Then where does the derivative on x go ? For more on this, see MTW pg180 thanks
  38. F

    Understanding Dirac Delta Function: Time Derivative & Hankel Transformation

    Hi All, I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution. My question are: 1. How can I find the time derivative...
  39. F

    Is there a coordinate independent Dirac delta function?

    I have been wondering exactly how one would express the Dirac delta in arbitrary spaces with curvature. And that leads me to ask if the Dirac delta function has a coordinate independent expression. Is there an intrinsic definition of a Dirac delta function free of coordinates and metrics? Or as...
  40. F

    Integrating with a dirac delta function

    Homework Statement I have to integrate: \int_0^x \delta(x-y)f(y)dy Homework Equations The Attempt at a Solution I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity: \delta(0)=\infty I have to express the integral in terms of function...
  41. N

    Laplace transform of the dirac delta function

    Homework Statement L[t^{2} - t^{2}δ(t-1)] Homework Equations L[ t^{n}f(t)] = (-1^{n}) \frac{d^{n}}{ds^{n}} L[f(t)] L[δ-t] = e^-ts The Attempt at a Solution My teacher wrote \frac{2}{s^{3}} -e^{s} as the answer. I got \frac{2}{s^{3}} + \frac{e^-s}{s} + 2 \frac{e^-s}{s^2} + \frac{2e^-s}{s^3}
  42. F

    Laplace transform, sum of dirac delta

    Homework Statement Homework Equations I really wish they existed in my notes! *cry*. All I can think of is that integrating or in other words summing the dirac delta functions for all t, would be infinite? None the less the laplace transform exist since its asked for in the question and i...
  43. F

    Help with heat equation dirac delta function?

    Homework Statement The question was way too long so i took a snap shot of it http://sphotos-h.ak.fbcdn.net/hphotos-ak-snc7/397320_358155177605479_1440801198_n.jpg Homework Equations The equations are all included in the snapshotThe Attempt at a Solution So for question A I've done what the...
  44. S

    Dirac delta function / Gibbs entropy

    Homework Statement This is an issue I'm having with understanding a section of maths rather than a coursework question. I have a stage of the density function on the full phase space ρ(p,x); ρ(p,x) = \frac {1}{\Omega(E)} \delta (\epsilon(p,x) - E) where \epsilon(p,x) is the...
  45. F

    Can the Dirac Delta Function be Applied in Curved Space?

    The Dirac delta function is defined as: \int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1 Or more generally the integral is, \int_{ - \infty }^{ + \infty } {\delta (\int_{{x_0}}^x {dx'} )dx} But if the metric varies with x, then the integral becomes, \int_{ - \infty }^{ + \infty }...
  46. O

    Proof involving Dirac Delta function

    Prove that x \frac{d}{dx} [\delta (x)] = -\delta (x) this is problem 1.45 out of griffiths book by the way. Homework Equations I attempted to use integration by parts as suggest by griffiths using f = x , g' = \frac{d}{dx} This yields x [\delta (x)] - \int \delta (x)dx next I tried...
  47. P

    Derivative of Dirac Delta - Fourier Transform - Time Differentitation Property

    Homework Statement I am using the time differentiation property to find the Fourier transform of the following function: Homework Equations f(t)=2r(t)-2r(t-1)-2u(t-2) The Attempt at a Solution f'(t)=2u(t)-2u(t-1)-2δ(t-2) f''(t)=2δ(t)-2δ(t-1)-?? Can somebody explain what the...
  48. B

    Dirac delta function, change of variable confusion

    The Dirac delta "function" is often given as : δ(x) = ∞ | x = 0 δ(x) = 0 | x \neq 0 and ∫δ(x)f(x)dx = f(0). What about δ(cx)? By u=cx substitution into above integral is, ∫δ(cx)f(x)dx = ∫δ(u)f(u/c)du = 1/c f(0). But intuitively, the graph of δ(cx) is the same as the graph of...
  49. F

    Must the Fourier transform exist for Dirac delta functions?

    I originally asked this in the Calculus & Analysis forum. But perhaps this is better suited as a question in Abstract algebra. For the set of all Dirac delta functions that have a difference for an argument, we have the property that: \int_{ - \infty }^\infty {{\rm{\delta (x -...
  50. S

    Is the delta in the commutation relations of QFT a dirac delta or a kronecker?

    If it's a dirac delta doesn't it mean it's infinite when x=y? Or is it a sort of kronecker where it's equal to one but the indices x and y are continuous? I'm confused.
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