Delta function Definition and 13 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. eradan

    Double delta potential

    Summary:: I have a problem with a particle, which gets scatterd at a double delta-potential Hello, I am really stuck with the floowing problem: A particle moves from the left along the x-axis and gets scatterd at a one-dimensional potential V(x)=a[dirac delta of x) +b [dirac delta of x-c]...
  2. E

    S-wave phase shift for quantum mechanical scattering

    a.) The potential is a delta function, so ##V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma \delta \left(r-a \right)##, therefore ##V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma ## at ##r=a##, and ##V \left( r \right) = 0## otherwise. I've tried a few different approaches: 1.) In...
  3. Alan Lins Alves

    Problem with the Finite Element Method applied to Electrostatics

    Hi! I have a code that solve the poisson equation for FEM in temperature problems. I tested the code for temperature problems and it works! Now i have to solve an Electrostatic problem. There is the mesh of my problem (img 01). At the left side of the mesh we have V=0 (potencial). There is a...
  4. P

    I Is the derivative of a discontinuity a delta function?

    In these notes,, at the end of page 4, it is mentioned: (3) V(x) contains delta functions. In this case ψ'' also contains delta functions: it is proportional to the product of a...
  5. Tbonewillsone

    Recovering the delta function with sin⁡(nx)/x

    Homework Statement Ultimately, I would like a expression that is the result of an integral with the sin(nx)/x function, with extra terms from the expansion. This expression would then reconstruct the delta function behaviour as n goes to infty, with the extra terms decaying to zero. I...
  6. J

    2D Integrating With Quadratic Arg. of Delta Function

    Homework Statement I have a 2D integral that contains a delta function: ##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##, where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real...
  7. cg78ithaca

    A Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)

    This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...
  8. J

    I Square root of the delta function

    Is square root of delta function a delta function again? $$\int_{-\infty}^\infty f(x) \sqrt{\delta(x-a)} dx$$ How is this integral evaluated?
  9. 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...
  10. debajyoti datta

    A Divergence of 1/r^n....n>0

    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...
  11. 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%...
  12. blue_leaf77

    Integral of sin and cos

    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.
  13. 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 Equations The Attempt at a Solution This integral actually came up in a homework problem for my E&M class and...