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. J

    Charge Densities & Dirac's Delta Function

    Homework Statement What is the (volume) charge density of a ring of radius r_0 and uniform charge density \lambda? Homework Equations The Dirac Delta Function The Attempt at a Solution I've done a few line charge densities of straight wires along an axis (usually z, but on x as...
  2. I

    Square root of Dirac Delta function

    Homework Statement I wonder how to deal with the square root of Dirac Delta function, \sqrt{\delta(x)}. Actually, this comes from a problem which asking readers to calculate the wave function of a free particle and of a harmonic oscillator at time t, provided that the wave function at time...
  3. R

    How to apply the definition of the derivative of a delta function

    I am supposed to prove that δ'(ax) = (1/a)*(1/|a|)*δ'(x) but I cannot figure out where the (1/a) term comes from. Using the scaling theorem I know that δ(ax) = (1/|a|)*δ(x), but how does this apply to the first derivative and does it explain where the (1/a) comes from?
  4. R

    Proof of the derivative of delta function

    The problem is to prove that δ'(ax) = (1/a)*(1/a)*δ'(x), where a is a constant. I tried applying the scaling theorem with the formal definition of δ'(x) but I can not get the second (1/a) term. Does anyone have some insight on this problem? Thank you...
  5. Z

    Can a distribution or delta function solve a NONlinear ODE or PDE

    the question is , can a delta function /distribution \delta (x-a) solve a NOnlinear problem of the form F(y,y',y'',x) the question is that in many cases you can NOT multiply a distribution by itself so you could not deal with Nonlinear terms such as (y)^{3} or yy'
  6. A

    Dirac delta function is continuous and differential

    since dirac delta function is not a literally a function but a limit of function,does it mean that dirac delta function is continuous and differentiable through out the infinity? is there any example of dirac delta function if yes then give meeeeeeee?
  7. T

    Integral over a sphere with the dirac delta function

    Homework Statement \[ \underset{\left|\underline{\xi}\right|=1}{\int}\delta_{0}\left(\underline{\xi}\cdot\underline{z}\right)dS_{\xi}=\intop_{0}^{2\pi}d\varphi\intop_{-r}^{+r}\delta_{0}\left(\varsigma\right)\frac{d\varsigma}{r}=\frac{2\pi}{r}\] The \delta_{0} is the dirac delta function.the...
  8. T

    Integral Over a Sphere with dirac delta function

    Hi, I am not really sure whether its over the surface of the sphere or the Volume, the problem and the solution are given below, I want to know how it has been solved. The \delta_{0} is the dirac delta function. \[...
  9. J

    An identity involving a Dirac delta function.

    I have been reading papers for my research and I came across this equation twice: \lim_{\eta\to 0+}\frac{1}{x+i \eta} = P\left(\frac{1}{x}\right) - i \pi \delta(x) Where P is the pricipal part. It has been quite a while since I have had complex variables, but might it come from the...
  10. D

    Proving 3D Delta Function with Laplacian

    Hi. How do we argue that \nabla^2\frac{1}{r} is a three dimensional delta function? I have seen some people do it using the divergence theorem, i.e. saying that \int_V \nabla\cdot\nabla\frac{1}{r}dv=-\oint_S \nabla\frac{1}{r}\cdot ds=-4\pi if S is a surface containing the origin, but I...
  11. M

    How to handle the Dirac delta function as a boundary condition

    Using perturbation theory, I'm trying to solve the following problem \frac{\partial P}{\partial \tau} = \frac{1}{2}\varepsilon^2 \alpha^2 \frac{\partial^2 P}{\partial f^2} + \rho \varepsilon^2 \nu \alpha^2 \frac{\partial^2 P}{\partial f \partial \alpha} + \frac{1}{2}\varepsilon^2 \nu^2...
  12. C

    Integrating the delta function

    Homework Statement By using the substitution u=cosx obtain the value of the integral \int\delta(cosx-1/2)dx between 0 and pi Homework Equations I have no idea how to go any further with this apart from substituting in for u!? The Attempt at a Solution
  13. O

    How to manually write the code for a matlab delta function

    that is 0 everywhere and 1 at 0. the code I wrote was this: n = -20:1:20; if n==0 imp = 1 else imp = 0; end >> stem (n, imp) ? Error using ==> stem at 40 X must be same length as Y. but i got that error. Using vectors and matrices is useless cause the delta...
  14. J

    Property of the Dirac Delta Function

    Homework Statement How do you show that int[delta(t)]dt from negative infinity to infinity is 1? Homework Equations Dirac delta function defined as infinity if t = 0, 0 otherwise The Attempt at a Solution My teacher said that it has to do with m->infinity for the following...
  15. Z

    How can integration by parts be used to prove the Dirac delta function?

    1. The problem statement Show that: \int_{-\infty}^{\infty} f(x) \delta^{(n)}(x-a) dx = (-1)^n f^{(n)}(a) The Attempt at a Solution I am trying to understand how to prove: \int_{-\infty}^{\infty} f(x) \delta '(x) dx =- f'(x) I know that we need to use integration by parts, but I'm...
  16. E

    Understanding the Dirac Delta Function in Spherical Coordinates

    Homework Statement Justify the following expretion, in spherical coordinates; delta (vector r) = (1 / r^2 * sin (theta) ) * delta(r) * delta(theta) * delta(phi) Homework Equations The Attempt at a Solution I don't know what it means... please help?
  17. C

    Can someone explain the 3D Dirac Delta Function in Griffiths' Section 1.5.3?

    Griffiths' section 1.5.3 states that the divergence of the vector function r/r^2 = 4*Pi*δ^3(r). Can someone show me how this is derived and what it means physically? Thanks in advance.
  18. S

    Starting with the definition of the Dirac delta function,

    Homework Statement Starting with the definition of the Dirac delta function, show that \delta( \sqrt{x}) um... i have looked in my book and looked online for a problem like this and i really have no clue where to start. the only time i have used the dirac delta function is in an integral...
  19. J

    Dirac delta function - its confusing

    Hi I have been trying to learn dirac delta function. but its kind of confusing. I come across 2 contrasting definitions for it. The first one states that the function delta(x-xo) is infinite at x=x0 while the other states that delta(x-x0) tends to infinite as x tends to x0. Now both of them...
  20. S

    Analysis of A-Module Endomorphism \phi: Understanding Kronecker Delta Function

    Let A, M be a commutative ring and a finitely generated A-module respectively. Let \phi be an A-module endomorphism of M such that \phi (M)\subseteq \alpha\ M where \alpha is an ideal of A. Let x_1,\dots,x_n be the generators of M. Then we know that \displaystyle{\phi(x_i)=\sum_{j=1}^{n}...
  21. 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...
  22. K

    How to find the second derivitive of delta function?

    how to find the second derivitive of delta function?
  23. 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'...
  24. E

    Evaluating integral of delta function

    Homework Statement Evaluate the integral: Homework Equations To integrate this, should one use a dummy variable to get the delta function only of t, then integrate, then substitute back in after integration?
  25. A

    Why Does the Delta Function Cause Different Charge Densities in Electrodynamics?

    Homework Statement This is problem 2.46 from Griffith's Electrodynamics. I've already solved the problem but there is one aspect of the solution which bothers me and I can't think of where it is originating. I have found that the potential given in the problem produces an electric field...
  26. 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
  27. Peeter

    What is the Interpretation of the Inverse Property of a Delta Function in QM?

    In a book on QM are listed a few properties of the delta function, one of which is: x \delta^{-1}(x) = - \delta(x) I can't figure out how to interpret that? Putting the statement in integral form isn't particularily enlightening looking: f(x) = \int f(x-x') \delta(x') dx' = \int...
  28. 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?
  29. R

    Does the Jacobian in delta function change in non-Cartesian coordinates?

    when using delta function \delta(r) in cordinates othe then cartesian when does it needs to be divided by the jacobian for example in spherical coordinates \delta(x)=\frac{1}{r^2sin(\vartheta)}\delta(r-r_{0})\delta(\vartheta-\vartheta_{0})\delta(\phi-\phi_{0}) but if you want a delta...
  30. Fredrik

    Question about the delta function

    No answer in the linear algebra section, so I'll try here. ("Calculus & analysis" would probably have been more appropriate than "linear algebra"). I have a question about the delta function. Link.
  31. J

    How do I convolve delta functions in Fourier transform calculations?

    Hi everyone, I need help finding the Fourier transform of Cos(10t)sin(t) i know that i need to find the transform of cos and sin and then convolve them, but i m not sure how to convolve delta function. I would really appreciate any helps.
  32. P

    Getting a delta function from an indefinite integral

    Hey everybody, One question that I've had for a week or so now is how the following integral can equal a Dirac delta function: \frac{1}{2\pi} \int_{-\infty}^{\infty}{dt} \:e^{i(\omega - \omega^{'})t}\: = \: \delta(\omega - \omega^{'}) A text that I was reading discusses Fourier transforms...
  33. F

    Delta function antisymmetric potential problem

    Homework Statement particle of mass m is subjected to antisymmetric delta-function potential V(x) =V'Delta(x+a)-V'Delta(x-a) where V'>0 Show that there is only one bound state, and find its energy Homework Equations Assuming free particle eqn for x<-a for particle incident from -ve...
  34. D

    Explicitly Deriving the Delta Function

    When working with Fourier transforms in Quantum mechanics you get the result that \int_{-\infty}^{\infty}e^{-ikx}e^{ik'x} = \delta(k-k') I understand conceptually why this must be true, since you are taking the Fourier transform of a plane wave with a single frequency element. I have also...
  35. 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...
  36. 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
  37. M

    How Does the Delta Function Affect Integral Values?

    Homework Statement I want to measure this integral ( Sd(x)δ(χ-c) , where c is constant Thanks in advancE Homework Equations The Attempt at a Solution
  38. J

    Distributions and delta function

    where can I read about distributions and the delta function. esp. to solve singular integrals. I have seen that you could write 1/x = \delta (x) + P.V (1/x) and all that stuff.. where can i read about it ...
  39. maverick280857

    Integral of a delta function from -infinity to 0 or 0 to +infinity

    Hello everyone Today in my QM class, a discussion arose on the definition of the delta function using the Heaviside step function \Theta(x) (= 0 for x < 0 and 1 for x > 0). Specifically, \Theta(x) = \int_{-\infty}^{x}\delta(t) dt which of course gives \frac{d\Theta(x)}{dx} =...
  40. maverick280857

    Delta function of a function with multiple zeros

    Hi everyone, I was wondering how to deal with delta functions of functions that have double zeros. For instance, how does one compute an integral of the form \int_{-\infty}^{\infty}dx g(x)\delta(x^2) where g(x) is a well behaved continuous everywhere function? In general how does one find...
  41. J

    Normalization of a delta function in curved spacetime

    Which of the following are true in curved spacetime? \int d^4 x \delta^4(x - x_0) = 1 (1) \int d^4 x \sqrt{-g} \delta^4(x - x_0) = 1 (2) I think the first one is incorrect in curved spacetime, or in general when the metric is non-constant. I would argue this by saying that the delta...
  42. D

    Is the Reflection Coefficient for a Delta Function Potential Always Close to 1?

    So let's say we have a particle in the delta function potential, V = - \alpha \delta(x). I calculated that the reflection coefficient (scattering state) is R = \frac{1}{1 + (2 \hbar^2 E/m\alpha^2)} Now, clearly, the term 2 \hbar^2 E/m\alpha^2 is very small, as \hbar^2 has an order of magnitude...
  43. 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...
  44. D

    Fourier Transform, Delta Function

    Hey everybody. I was studying Fourier transforms today, and I thought, what if you took the transform of an ordinary sine or cosine? Well, since they only have one frequency, shouldn't the transform have only one value? That is, a delta function centered at the angular frequency of the wave...
  45. 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 >...
  46. M

    Contour integral with delta function

    Using Cauchy's integral theorem how could we compute \oint _{C}dz D^{r} \delta (z) z^{-m} since delta (z) is not strictly an analytic function and we have a pole of order 'm' here C is a closed contour in complex plane
  47. M

    Strange representation of Heaviside and Delta function

    in the .pdf article http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6027/1/jfs080104.pdf i have found the strange representation \delta (x) = -\frac{1}{2i \pi} [z^{-1}]_{z=x} and a similar formula for Heaviside function replacing 1/z by log(-z) , what is the meaning ...
  48. V

    When can we ignore the delta function in th Feynman rules?

    in peskin-schroeder and http://www.hep.phy.cam.ac.uk/batley/particles/handout_04.pdf" the amplitude for e^-e^+\rightarrow \mu^- \mu^+ is written using feynman rules as follows -iM=[\bar{v}(p_2)(-ie\gamma^\mu )u(p_1)] \frac{-ig_{\mu\nu}}{q^2}[\bar{u}(k_1)(-ie\gamma^\nu )v(k_2)] but what...
  49. M

    How Does a Delta Function Potential Affect a Quantum Particle's Radial Equation?

    Homework Statement write the radial equation for a particle with mass m and angular momentum l=0 which is under the influence of the following potential: V(r)=-a*delta(r-R) a,R>0 write all the conditions for the solution of the problem.Homework Equations Schroedinger's equation: Hu=Eu...
  50. R

    Double delta function potential

    Homework Statement Consider the double delts-function potential V(x)=-\alpha[\delta(x+a)+\delta(x-a)] How many bound states does this possess? Find the allowed energies for \alpha=\frac{\hbar^{2}}{ma^{2}}and\alpha=\frac{\hbar^{2}}{4ma^{2}}Homework Equations The Attempt at a Solution I divided...
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