What is Dirac delta: Definition and 333 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.
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).
For my statistical mechanics class I need to find the cumulants of a special distribution of which all moments are constant and equal to a. I followed two different approaches and obtained imcompatible results, something is wrong but I couldn't figure it out.
Here's what I did:
Since all...
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...
Suppose that we have a compact manifold \mathcal{M} with a positive definite metric g_{ij}. The volume of the manifold is then
V = \int_\mathcal{M} d^3x \sqrt{g(x)},
where x^i are coordinates on \mathcal{M} and \sqrt{g(x)} is the square root of the determinant of the metric. Suppose now that...
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...
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...
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?
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.
Question:
Consider the motion of a particle of mass m in a 1D potential V(x) = \lambda \delta (x). For \lambda > 0 (repulsive potential), obtain the reflection R and transmission T coefficients.
[Hint] Integrate the Schordinger equation from -\eta to \eta i.e.
\Psi^{'}(x=\epsilon...
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.
Let be the exponential:
e^{inx}=cos(nx)+isin(nx) n\rightarrow \infty
Using the definition (approximate ) for the delta function when n-->oo
\delta (x) \sim \frac{sin(nx)}{\pi x} then differentiating..
\delta ' (x) \sim \frac{ncos(nx)}{\pi x}- \frac{\delta (x)}{\pi x}...
Hi,
After reading DJGriffiths sections on the DDF I have a question about evaluating it in regards Prob. 1.46 (a), to wit:
"Write an expression for the electric charge density \rho(\vec{r}) of a point charge q at \vec{r}'. Make sure that the volume integral of \rho equals q."
This is easily...
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...
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.
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...
Hey everyone, a quick question: what is the Fourier space representation of the dirac delta function in minkowski space? It should be some integral over e^{ikx} (with some normalization with 2*pi's). I'm curious if the "kx" is a dot product in the minkowski or euclidean sense, and how one...
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...
If I had a function g(x) defined by
g(x) = \int_{-\infty}^{\infty} f(x) \delta(x) dx
where \delta(x) is the dirac delta function, what would dg(x)/dx be? The fundamental theorem of calculus requires that f(x) \delta(x) needs to be a continuous and differentiable function before I can...
I want to prove coulomb's low for a single charge point from the general form of coulomb's low:
E→=1/(4∏€) ∫∫∫ ρv(ŕ) * (r→- ŕ→)/│(r→- ŕ→)│^3 dŕ
using Dirac Delta function
where r→ is the field point vector
ŕ→ is the source point vector
ρv(ŕ) is the volume charge density
I really don't...
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...
just curiosity, if you integrate dirac-delta from exactly zero to infinity, will you get one or a-half?
since it is symmetrical about zero, i think it is a half. is it?
i mean:
\int_{0}^{\infty} \delta(x) dx=\frac{1}{2}
or is it 1?
thanks.
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...
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...
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...
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...
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...
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...
Supposedly,
∫ ez*(z - z0)f(z) dz*dz
is proportional to f(z0) much in the same way that
(1/2π)∫ eiy(x - x0)f(x) dxdy
= ∫ δ(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...