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.

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  1. R

    Dirac delta function proof help

    [SOLVED] Dirac delta function Homework Statement Prove that \delta(cx)=\frac{1}{|c|}\delta(x) Homework Equations The Attempt at a Solution For any function f(x), \int_{-\infty}^{\infty}f(x)\delta(cx) dx = \frac{1}{c}\int_{-\infty}^{\infty}f(t/c)\delta(t) dt where I have...
  2. J

    Dirac delta function with complex arguments

    This is probably a silly question to some, but I've been struggling to understand how the delta function behaves when given a complex argument, that is \delta(z), z \in C. I guess the basic definition is the same that the integral over all space is 1, but I'm looking for a more detailed guide on...
  3. P

    Dirac delta function and Heaviside step function

    [SOLVED] Dirac delta function and Heaviside step function In Levine's Quantum Chemistry textbook the Heaviside step function is defined as: H(x-a)=1,x>a H(x-a)=0,x<a H(x-a)=\frac{1}{2},x=a Dirac delta function is: \delta (x-a)=dH(x-a) / dx Now, the integral: \int...
  4. B

    Fourier transform of a function such that it gives a delta function.

    [SOLVED] Fourier transform of a function such that it gives a delta function. ok say, if you Fourier transform a delta function G(x- a), the transform will give you something like ∫[-∞ ∞]G(x-a) e^ikx dx a is a constant to calculate, which gives you e^ka (transformed into k space)...
  5. C

    Dirac Delta Function question(s)

    OK, so my basic understanding of Dirac Delta Function is that it shows the probability of finding a point at (p,q) at time t. Dirac Delta is 0 everywhere except for (p_{0},q_{0}). So my question comes Is it possible that a point enters the (p_{0},q_{0}) and stays there (for some period of...
  6. M

    Delta function defined for complez values

    is there a form to define the dirac delta function for complex values ? i mean \delta (x-a-bi) or \delta (-ix) using 'test functgions' i get that they converge nowhere (always infinite) which makes no sense at all, using scalling properties we could define \delta (ix) = \delta(x)...
  7. P

    Convolution of a dirac delta function

    Alright...so I've got a question about the convolution of a dirac delta function (or unit step). So, I know what my final answer is supposed to be but I cannot understand how to solve the last portion of it which involves the convolution of a dirac/unit step function. It looks like this: 10 *...
  8. M

    Dirac Delta Function Potential (One Dimension)

    Alright, I'm in my first QM course right now, and one of the topics we've looked at is solving the one-dimensional time-independent Schrodinger equation for various potentials, such as the harmonic oscillators, infinite and finite square wells, free particles, and last, but not least, the dirac...
  9. P

    Proving the Delta Function Property: \delta(ax) = {\delta(x) \over {|a|}}

    Homework Statement I would like to prove that \delta(ax)={\delta(x) \over {|a|}}. My problem is that I don't know how the absolute value brackets arise. Homework Equations \int_{-\infty}^{\infty} \delta(x)dx = 1The Attempt at a Solution I start from \int_{-\infty}^{\infty} \delta (ax) dx, and...
  10. E

    Delta Function Limits: Solving Integrals from 0 to 1

    with limits from 0 to 1 \int delta(x) * cos(x) dx does this delta function integral even make sense from 0 to 1?
  11. G01

    Quick Question on the Dirac Delta Function

    The Dirac delta function, \delta (x) has the property that: (1) \int_{-\infty}^{+\infty} f(x) \delta (x) dx = f(0) Will this same effect happen for the following bounds on the integral: (2) \int_{0}^{+\infty} f(x) \delta (x) dx = f(0)...
  12. Y

    Prove a delta function identity

    Hi, I'm stuck with the last proof I need to do Homework Statement I need to prove that f(x)delta(g(x)) = f(x) delta (x-x0)/abs(g'(x)) By delta I mean the Dirac delta function here. (I'm new to this forum, so i don't know how to write it all nicely like so many of you do!) Homework...
  13. C

    Dirac Delta function and charge density.

    I have a line charge of length L and charge density /lambda on the Z-axis. I need to express the charge density in terms of the Dirac Delta function of theta and phi. How would I go about doing this?
  14. O

    Integrating the Dirac Delta Function

    I am trying to evaluate the following integral. \int_{-\infty}^{\infty}{\delta(2t-3)\sin(\pi t) dt} where delta represents the Dirac delta function. I am told that the answer is -1. However, when I evaluate it in MATLAB and Maple 11, I get an answer of -1/2. What is the correct way...
  15. A

    Proving Delta Function: Help with Homework Equation | Attachment Included

    Homework Statement Can u pls help me to prove delta function. Here is the problem in attachment. Homework Equations The Attempt at a Solution
  16. S

    Properties of Dirac delta function

    Homework Statement I'm trying to prove that \delta'(y)=-\delta'(-y). Homework Equations The Attempt at a Solution I'm having trouble getting the LHS and the RHS to agree. I've used a test function f(y) and I am integrating by parts. For the LHS, I have...
  17. J

    What is the integral of a square of Dirac delta function?

    Homework Statement Hi there, I'm stuck at a problem where I have (sorry i don't know how to use mathtype so I'll try my best at making this clear) the integral of a dirac delta function squared: int[delta(x*-x)^2] between minus infinity and infinity (x*=constant) I know that the function...
  18. L

    Delta function in spherical coords

    Homework Statement If we have a delta function in cartesian coords, how do we convert it into spherical. for example : delta (r) = delta(x-x0) delta(y-y0) delta(z-z0) Homework Equations The Attempt at a Solution I used delta (r) = delta(r-r0) delta(cos{theta}-cos{theta0}) delta...
  19. S

    Solving Dirac Delta Function Beam Problem

    1. The ProblemHomework Statement 4 Parts to the Assignment. Finding the Displacement of a beam assuming w to be constant. 1. Cantilever beam, free at one end. Length =l, Force P applied concentrated at a point distance rl from the clamped end. Boundary Conditions y(0)=0, y'(0)=0, y"(l)=0, and...
  20. S

    Contour integral and delta function

    I have two related questions. First of all, we have the identity: \int_{-\infty}^{\infty} e^{ikx} dk = 2 \pi \delta(x) I'm wondering if it's possible to get this by contour integration. It's not hard to show that the function is zero for x non-zero, but the behavior at x=0 is bugging...
  21. J

    Solving simple dirac delta function

    [b]1. Homework Statement \int x[delta(x)-delta(x/3+4)] dx Homework Equations so I'm supposed to use this principle: \int f(x)delta(x-xo)dx=f(xo) The Attempt at a Solution So it seems simple but I just want to make sure that I'm applying the above principle correctly. I...
  22. D

    Delta Function Well and Uncertainty Principle

    [SOLVED] Delta Function Well and Uncertainty Principle Homework Statement Griffiths Problem 2.25. I need to calculate < p^{2}> for the Delta Function Well. The answer given is: < p^{2}> = (m\alpha/\hbar)^2 The wave function given by the book is...
  23. J

    Help converting dirac delta function

    Homework Statement SO I'm given a dirac delta function, also known as a unit impulse function. d(t-t'_=(1/P) sum of e^[in(t-t')], for n from negative to positive infinity. I need to graph this. Homework Equations I understand that at t', there is a force made upon the system which...
  24. P

    Understanding the Delta Function Well in Quantum Mechanics

    So I'm studying that part right now. I only get parts of it though, it seems. The first thing the book goes over (This is intro to QM by Griffiths) is a potential that has the form -A*deltafunction. Okay, that's just something he plucked for simplicity. But then if the potential is lower...
  25. E

    Delta function potential problem

    Homework Statement Why does it make sense that a negative delta function potential represents a highly localized attractive force and a positive delta function potential represents a highly localized repulsive force? How do you explain that using -dV/dx = f(x) ? I guess I am confused about...
  26. 1

    Can DiracDelta Functions Be Plotted and Transformed in 2D?

    I'm trying to plot the function f(x,y) = DiracDelta[r-r0]and then take the Fourier transform. Is this a radial delta function? I'm having trouble understanding the significance of this "function" . Thanks!
  27. Z

    Convolution of delta function

    for linear time invariant system, y(t)=h(t)*x(t) where y(t) is the output , x(t) is the input and h(t) is the impulse response.(* is the convolution) The definition of convolution is y(t)=integration from -infinity to +infinity (h(tau)x(t-tau)d(tau) p/s: i don't know how to use...
  28. E

    How Do You Correctly Integrate the Delta Function in This Equation?

    Homework Statement Evaluate: \int_{-3}^{5} e^{-2t} sin(t-3) \delta(t-5) dt Homework Equations \int_{-\infty}^{\infty} f(t) \delta(at-t_0) dt = \frac{1}{|a|}f(\frac{t_0}{a}) The Attempt at a Solution e^{-2(5)} sin (5-3) = e^{-10} sin (2) The solution given by the professor...
  29. nicksauce

    Does the delta function integral still hold true for non-continuous functions?

    Evaluate: \int^{\infty}_{-\infty} f(x)\delta(x-x_0)dx Where f(x)=ln(x+3), x_0=-2 Ordinarily, you would just evaluate f(x_0), so it would be 0, but in this case, since f(x) is -\infty at x=-3, does that make a difference?
  30. C

    Understanding the Kronecker Delta function

    Homework Statement I'm having some trouble understanding the Kronecker Delta function and how it is used. I understand the basics of it, if i=j, delta=1, if not, delta=0. However, I don't understand why: \delta_{ii}=3 and \delta_{ij}\delta_{ij}=3Homework Equations \delta_{ij}=...
  31. G

    2 variable delta function integration

    Homework Statement \int^{A}_{-A}\int^{Bx}_{-Bx}c\delta(xcos\varphi+ysin\varphi-d)dydx where A, B, c, d are constant Homework Equations The Attempt at a Solution I have tried a few different ways to integrate this, but am completely confused with what happens to this kind of delta...
  32. radou

    Dirac delta function confusion

    OK, I'm currently reading Hughes' Finite Element Method book, and I'm stuck on a chapter the goal of which is to prove that the Galerkin solution to a boundary value problem is exact at the nodes. So, the author first speaks about the Dirac delta function: "Let \delta_{y}(x) = \delta(x-y)...
  33. M

    Most general form of nascent delta function

    Dear all I'm wondering if you can help me find the most general formula of all nascent delta functions. all i have found a somewhat random forms . I'm looking for a general elegant formula that all the forms can be derived from . thanks in advance .
  34. S

    Delta function from divergence

    We know that div \; (\hat{r} / r ) = 4 \pi \delta (r) Why is there no generalized function (distribution) for div \; (\hat{r} / r^2) = ??
  35. M

    Understanding the Equivalence of Dirac Delta Functions in Quantum Mechanics

    Dirac developed his delta function in the context of QM. But there are various functions under the integral that give the delta function. My question is does one Dirac delta function equal any other? Are all ways of getting the Dirac delta function equivalent? Thanks.
  36. P

    Dirac Delta Function: Integral at x=a

    Homework Statement int[d(x-a)f(x)dx]=f(a) is the dirac delta fn but is int[d(a-x)f(x)]=f(a) as well? If so why?The Attempt at a Solution Is it because at x=a, d(0)=infinite and integrate dirac delta over a region including x=0 when d(0) is in the value in the integral will produce 1 hence f(a).
  37. A

    Can delta function equality hold for m not equal to n?

    Hi I am not a mathematician so my question might be silly. I really came across it in physics but I think it is purely mathematical: I came across an equation of the form: delta(m-n)*A= delta(m-n)*B my question is now for what cases can I conclude A=B? Does this only hold for m=n, or can I...
  38. G

    Dirac Delta function as a limit

    Dear all, I need a simple proof of the following: Let [tex]u \in C(\mathbb{R}^3)[\tex] and [tex]\|u\|_{L^1(\mathbb{R}^3)} = 1[\tex]. For [tex]\lambda \geq 1[\tex], let us define the transformation [tex]u\mapsto u_{\lambda}[\tex], where [tex] u_{\lambda}(x)={\lambda}^3 u(\lambda...
  39. C

    Convultion with Delta Function

    I can remember from Differential Equations that any function convolved with a delta function results in a copy of the function located at the impulese. That is, x(t) * \delta(t-5) = x(t-5) However, I can't remember why. This is really irritating me since I need to use this concept for my...
  40. G01

    Double Delta Function Potential

    Homework Statement How many stationary states exist for this potential? What are the allowed energies if the strength of the well, \alpha= \hbar^2/ma and \hbar^2/4ma where a= the position of the well(one at a, one at -a) Homework Equations V(x) = -\alpha(\delta(x+a) +\delta(x-a)) E_{one...
  41. B

    Is this a graph of a delta function?

    Homework Statement My question asks me to sketch the following: g(x) = \delta (y+a) + \delta (y) + \delta (y-a) Homework Equations The Attempt at a Solution I think this is it, but am I correct? I don't recall actually seeing a delta function other than a Kronicker(sp?) delta...
  42. P

    A question of Dirac Delta function

    A vector function V(\vec{r}) = \frac{ \hat r}{r^2} If we calculate it's divergence directly: \nabla \cdot \vec{V} = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{1}{r^2} \right) = 0 However, by divergence theorem, the surface integral is 4\pi . This paradox can be solved by...
  43. T

    How Do You Compute the Integral of a Delta Function with a Quadratic Argument?

    let f(y)=\int_0^2 \delta(y-x(2-x))dx. Find f(y) and plot it from -2 to 2. I know how to calculate \delta (g(x)) but i am not sure how to treat it with the y. I thought possibly to solve the quadratic in the delta function to find what x will equal for the roots in terms of y and got...
  44. S

    Is There a Simpler Definition for the Dirac Delta Function?

    https://www.physicsforums.com/showthread.php?t=73447 I saw the above tutorial by arildno and looked at how he defined the Dirac Delta "function" as a functional. But isn't there a more easier way to do this. I have seen the following definition in a lot of textbooks. \delta(t) \triangleq...
  45. Q

    What is a Dirac Delta function?

    I often see this in electrodynamics in the form of a point charge density function. There are some rules on how to manipulate the thing in integrals. But what is it mathematically?
  46. T

    Deriving the Dirac Delta Function Equation in Field Theory

    I found this equation in a field theory book, which I can't figure how it was derived: \delta(x-a) \delta(x-a) = \delta(0) \delta(x-a)
  47. S

    Wave Function for Delta Function Barrier with E<0

    Given a delta function barrier located at x=0: V(x) = +a * delta(x) If you have a particle incident from the left with E<0, what does the wave function look like?? I have trouble with this because I thought the particle energy needed to be greater than the minimum potential (E > Vmin) for...
  48. S

    Delta function potential and Schrodinger Equation

    I have a time dependent wavefunction for inside a delta function potential well: V(x) = -a delta(x). It reads Psi(x,t) = (sqrt(m*a)/hbar) * exp(-m*a*abs(x)/hbar^2) * exp(-iEt/hbar) I'm supposed to stick this back into the time dependent Schrodinger Equation and solve for E. Taking my...
  49. P

    Dirac Delta Function vs probability distribution

    Hello, What is the dirac delta function and how is it different from a probability distribution?
  50. S

    Dirac delta function homework help

    Suppose that we take the delta function \delta(x) and a function f(x). We know that \int_{-\infty}^{\infty} f(x)\delta(x-a)\,dx = f(a). However, does the following have any meaning? \int_{-\infty}^{\infty} f(x)\delta(x-a)\delta(x-b)dx, for some constants -\infty<a,b<\infty.
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