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. S

    1D wave equation with dirac delta function as an external force.

    Hey there! I'm faced with this problem: http://img7.imageshack.us/img7/4381/25686658nz9.png It's a 1D nonhomogeneous wave equation with a "right hand side" equaling to the dirac delta function in x * a sinusoidal function in t. I have to find its general solution with the constraints...
  2. L

    Dirac delta function definition

    By definition of the Dirac delta function, we have: \int f(x) \delta(x-a) dx=f(a) This is fair enough. But in ym notes there is a step that goes like the following: \mathbf{\nabla} \wedge \mathbf{B}(\mathbf{r})=-\frac{\mu_0}{4 \pi} \int_V dV'...
  3. Y

    Is the Dirac Delta Function Defined at Zero or Infinity?

    I cannot get the answer as from the solution manuel. Please tell me what am I assuming wrong. Thanks
  4. S

    The math of the Dirac delta function?

    I'm posting this here because I'm asking about the mathematical properties of the Dirac delta function, delta(x) which is zero for all non-zero real values of x and infinite when x is zero. The integral (-inf to +inf) of this function is said to be 1. How is this derived?
  5. G

    [Q]Some confusing about Dirac Delta Function

    Hi. Recently day, I tried to solve quantum mechanics problem in liboff fourth version to prepare graduate school. But what make me be confused a lot is Dirac Delta Function. One of my confusing on Dirac Delta is what i wrote below. -One of the formula describing Dira Delta...
  6. M

    Dirac Delta Function Explained: Simplified for M.S Students

    hello every body i am a new M.S student and i can't understand the Dirac delta function can anyone simply describe it to me in order to simplify it. thank you
  7. O

    Dirac Delta Function - unfamiliar definition

    Given: f(x)=\delta(x-a) Other than the standard definitions where f(x) equals zero everywhere except at a, where it's infinity, and that: \int_{-\infty}^{\infty} g(x)\delta(x-a)\,dx=g(a) Is there some kind of other definition involving exponentials, like: \int...
  8. B

    Bound state for a Dirac delta function potential

    Homework Statement Find the bound state energy for a particle in a Dirac delta function potential. Homework Equations \newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } - \frac{\hbar^2}{2 m} \ \pd{\psi}{x}{2} - \alpha \delta (x) \psi (x) = E\psi (x) where \alpha >...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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 *...
  14. 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...
  15. 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)...
  16. 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?
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  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. 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...
  23. 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)...
  24. 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.
  25. 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).
  26. 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...
  27. 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...
  28. 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...
  29. 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?
  30. 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)
  31. P

    Dirac Delta Function vs probability distribution

    Hello, What is the dirac delta function and how is it different from a probability distribution?
  32. 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.
  33. M

    Dirac Delta Function: Scaling and Shifting

    Just have a question about the dirac delta function. I understand how you would write it if you want to shift it but how would you scale it assuming we are using discrete time. Would you write 2*diracdelta[n] or diracdelta[2n]. Also, would that increase it or reduce it by 2 meaning that...
  34. Cincinnatus

    Exploring the Mystery of the Dirac Delta Function

    I've recently come across this function in one of my science classes and am wondering were this identity comes from: \displaystyle{\int{\delta(t-\tau)f(\tau)d\tau}=f(t)} Where \delta(t) is the dirac delta function and f(t) is any (continuous?) function.
  35. A

    Proving Dirac Delta Function Does Not Exist

    How can I prove that no continuous function exists that satisfies the property of the dirac delta function? I thought it should be pretty easy, but it's actually giving me quite a hard time! I know that the integral of such a function must be 1, and that it must also be even (symmetric about the...
  36. P

    Properties of the dirac delta function

    I'm trying to show that \int \delta \prime(x-x')f(x') dx = f\prime(x) can I differentiate delta with respect to x' instead (giving me a minus sign), and then integrate by parts and note that the delta function is zero at the boundaries? this will give me an integral involving f' and delta...
  37. E

    Integrals and dirac delta function

    hello again, i have an integral to solve and not sure how to approach this: \int f(q+T)\delta (t-q)dq and the boundaries of integral are -inf +inf couldn't figure it out with latex. what I know about this is that if delta function is integrated like this, it would be just the value of...
  38. C

    Understanding Transition btwn Steps of Dirac Delta Function

    Can someone help me understand the transition between these two steps? <x> = \iint \Phi^* (p',t) \delta (p - p') \left( - \frac{\hbar}{i} \frac{\partial}{\partial p} \Phi (p,t) \right) dp' dp = <x> = \int \Phi^* (p,t) \left( - \frac{\hbar}{i} \frac{\partial}{\partial p} \Phi (p,t)...
  39. D

    Help with Dirac Delta Function Problem

    i can't solve there problem, please help me 1) delta(y^2-a^2) = 1/absolute 2a[delta(y-a)+delta(y+a)] 2) f(y)delta(y-a) = f(a)delta(y-a)
  40. E

    Dirac delta function (DE problem) solved

    NOTE: I actually found the correct answer while I was typing this :rolleyes: and since I already had it typed, I figured i would post anyway. mods you can do with it as you please or leave it for reference. thanks Here's the problem: A uniform beam of length L carries a concentrated...
  41. A

    What is the simplest way to understand the Dirac Delta function?

    1. INTRODUCTION Many students become frustrated when they first meet the Dirac Delta function, typically in a course involving electrostatics, or Laplace transforms. As it is commonly presented, the Dirac function seems totally meaningless: Either, it is "defined" as...
  42. S

    Dirac Delta Function: Understanding Laplace & Inverse Laplace Properties

    I have a test in Diff Eq. tommorow and part of the test is inovling the Dirac Delta function. I have no clue as to what it is at all. More specifically its Laplace and Inverse Laplace. If anyone could explain to me what the delta function is and how to use in in diff eq and what are its...
  43. Reshma

    Explaining Dirac Delta Function: \vec A

    Can someone explain me the Dirac Delta function for the function: \vec A = \frac{\hat r}{r^2}
  44. K

    Laplace transform of dirac delta function

    let S be the Unit Step function for a function with a finite jump at t0 we have: (*) L{F'(t)}=s f(s)-F(0)-[F(t0+0)-F(t0-0)]*exp(-s t0)] so: L{S'(t-k)}=s exp(-s k)/s-0-[1-0]*exp(-s k) = 0 & k>0 but S'(t-k)=deltadirac(t-k) and we know that L{deltadirac(t-k)}=exp(-s k) so...
  45. E

    Dirac Delta Function Properties

    Okay...so here's the thing. I have been researching the dirac Delta properties. The sights I've visited, thus far, are moderately helpful. I'm looking to tackle this question I'm about to propose, so for you Brains out there (the truly remarkable :rolleyes:) please don't post a solution...
  46. MathematicalPhysicist

    The Dirac delta function question

    in the attatch file there is the dd function. what i want to know is: when x doesn't equal 0 the function equals 0 and the inegral is the integral of the number 0 which is any constant therefore i think the integral should be equal 0. can someone show me how this integral equals 1? for...
  47. pellman

    Dirac delta function on the complex plane?

    Supposedly, &int; ez*(z - z0)f(z) dz*dz is proportional to f(z0) much in the same way that (1/2&pi;)&int; eiy(x - x0)f(x) dxdy = &int; &delta;(x - x0)f(x) dx = f(x0) Is this true? Could someone help convince me of it, or point me to a text? I would say that even if true, it...
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