What is Dirac delta functions: Definition and 17 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. Muthumanimaran

    Product of Dirac delta functions

    Homework Statement δ(z*-z0*)δ(z+z0)=? δ(z*+z0*)δ(z-z0)=? where 'z' is a complex variable 'z0' is a complex number Formula is just enough, derivation is not needed.
  2. Muthumanimaran

    What is the product of two Dirac delta functions

    Homework Statement What is the product of two Dirac delta functions δ(Real(z-c))δ(Img(z-c))=? 'z' and 'c' are complex numbers. This is not a problem, But I just need to use this formula in a derivation that I am currently doing. I just want the product of these two Dirac delta functions as a...
  3. ELB27

    Product of a delta function and functions of its arguments

    Homework Statement I am trying to determine whether $$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$ where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions. Homework Equations The defining equation of a delta function...
  4. K

    How do you integrate dirac delta functions?

    Homework Statement ∫δ(x3 - 4x2- 7x +10)dx. Between ±∞. Homework EquationsThe Attempt at a Solution Well I don't really know how to attempt this. In the case where inside the delta function there is simply 2x, or 5x, I know the answer would be 1/2 or 1/5. Or for say δ(x^2-5), the answer would...
  5. J

    Dirac delta function proof fourier space

    Homework Statement I am having trouble understanding this: I have a Dirac Delta function $$ \delta (t_1-t_2) $$ but I want to prove that in the frequency domain (Fourier Space), it is: $$\delta(\omega_1+\omega_2) $$ Would anyone have any ideas how to go about solving this problem? I know...
  6. genxium

    How is (d^3)r in Green's Function equivalent to volume element?

    Homework Statement This is part of the online tutorial I'm reading: http://farside.ph.utexas.edu/teaching/em/lectures/node49.html I'm so confused about the notation of Dirac Delta. It's said that 3-dimensional delta function is denoted as \delta^3(x, y, z)=\delta(x)\delta(y)\delta(z) in...
  7. S

    Dirac delta function identities

    hi deoes anyone know any online resource for proofs of Dirac delta function identities and confirming which representations are indeed DD functions Thanks a lot.
  8. 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?
  9. 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 -...
  10. F

    Algebraic structure of Dirac delta functions

    OK, the Dirac delta function has the following properties: \int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1 and \int_{ - \infty }^{ + \infty } {f({x_1})\delta ({x_1} - {x_0})d{x_1}} = f({x_0}) which is a convolution integral. Then if f({x_1}) = \delta (x - {x_1}) we get...
  11. E

    Solving an equation with Dirac delta functions

    Hello, I'm dealing with the following equation: A e^{jat} + B e^{jbt} = C e^{jct} \forall t \in \mathbb{R} My book says: given nonzero constants A,B,C, if the above equation yelds for any real t, then the a,b,c constants must be equal. The above statement is prooved by taking the Fourier...
  12. R

    Wavefunction collapse and dirac delta functions

    What is the experimental evidence that a wavefunction will collapse to a dirac delta function, and not something more 'smeared' out?
  13. Rasalhague

    Exploring Dirac Delta Functions in QM Theory

    I'm reading Daniel T. Gillespie's A QM Primer: An Elementary Introduction to the Formal Theory of QM. In the section on continuous eigenvalues, he admits to playing "fast and loose" with the laws of calculus, with respect to the Dirac delta function. I'd like to understand it better, or, if such...
  14. F

    Multivariable Dirac Delta Functions

    Hello all. So I am trying to integrate a function of this form: \int\intF(x,y)\delta[a(Cos[x]-1)+b(Cos[y]+1)]dxdy The limits of integration for x and y are both [0,2Pi). I know that this integral is only nonzero for x=0, y=Pi. So this should really only sample one point of F(x,y)...
  15. S

    Signals - Integration of Heavyside Step & Dirac Delta Functions

    Homework Statement \int_{-\infty}^{\infty}{u(t)e^{-t}(\delta(t+1)+\delta(t-1))dt Homework Equations \int_{-\infty}^{t}{u(t)dt = \left\{\begin{array}{cc}0,&\mbox{ if } t< 0\\t, & \mbox{ if } t>0\end{array}\right. \int_{-\infty}^{\infty}{f(t)\delta(t-a)dt} = f(a) The...
  16. S

    Simplifying the integral of dirac delta functions

    hello all, i am unaware of how to handle a delta function. from what i read online the integral will be 1 from one point to another since at zero the "function" is infinite. overall though i don't think i know the material well enough to trust my answer. and help on how to take the integral of...
  17. T

    Dirac delta functions integration

    I can't figure out how to integrate this: \int_{0}^{\infty} \frac{x}{\sqrt{m^2+x^2}}sin(kx)sin(t\sqrt{m^2+x^2}) dx m, k and t are constants. The book has for m = 0, the solution is some dirac delta functions.