Questions tagged [dirac-delta]

This tag is for questions involving the Dirac delta function, either in the informal sense, or in the distribution sense. The Dirac delta function is a mathematical construct which is called a generalized function or a distribution and was originally introduced by the British theoretical physicist Paul Dirac.

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How to write $\delta (f(x))$ in terms of $\delta (x)$?

I've seen this identity in my electrodynamics book: $$\delta (f(x))=\sum_i{ \frac{1}{|{df\over dx}(x_i)|}\delta (x-x_i)}$$ Where $x_i$ shows the $i$th zero of $f(x)$. How can I prove it? I've tried ...
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delta function on a complex number

for a real number we know that $$ f(a)= \int_{-\infty}^{\infty}dx \delta (x-a)f(x) $$ but what happens for $$ \int_{-\infty}^{\infty}dx \delta (x-2i)f(x) $$ ? is this equal to $ f(2i) $ or equal ...
5
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2answers
209 views

Integral in 3 dimensions

I am trying to integrate $$ \iiint \delta(|\mathbf r| -R)\:\mathrm{d}^{3}\mathbf{r} $$ I know that $ \int f(r) \delta(r-R) d^3 \mathbf r =f(R) $, but when I try to apply this here I end up ...
5
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1answer
307 views

Laplace transform of the derivative of the dirac delta function times another function

I'm trying to solve a DE involving terms of the form $\dot{\delta}(t-k)f(t)$ and $\ddot{\delta}(t-k)f(t)$, where $k>0$. I would therefore like to find the Laplace transform of these terms. My ...
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1answer
104 views

Prove that $\frac{1}{\epsilon}\int_{\mathbb{R}}f(t).\exp\left(\frac{-\pi(x-t)^2}{\epsilon^2}\right)dt \xrightarrow{\epsilon \to 0}f(x) $

Prove that, for any $x \in \mathbb{R}$ and $f \in (L^1\cap C)(\mathbb{R})$, $$\frac{1}{\epsilon}\int_{\mathbb{R}}f(t).\exp\left(\frac{-\pi(x-t)^2}{\epsilon^2}\right)dt \xrightarrow{\epsilon \to 0}f(...
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2answers
181 views

When does one consider the laplacian as a dirac delta function?

If we consider that $$\nabla^2\left(\frac{1}{r}\right) = -4\pi\delta(\vec{r})$$ we can explain the dirac-delta function here via the results of Gauss' law $$\int_V \nabla^2\left(\frac{1}{r}\right) =...
5
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1answer
314 views

Why delta function is a tempered distribution?

Tempered distribution $T\in \mathcal{S}'$ is a continuous linear functional defined on the space $\mathcal{S}$ of infinitely differentiable test functions $f(x)$ on $\mathbf{R}$ with finite norm $$ ||...
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1answer
166 views

How to prove that a delta function belongs to the Besov space $B^{-1}$?

$\def\R{\mathbb{R}}$ $\DeclareMathOperator{\supp}{supp}$ I am trying to understand the definition of the Besov space and to prove that the delta function in $\R^1$. However I got stuck. First, let ...
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3answers
283 views

Trivial or not: Dirac delta function is the unit of convolution.

My task is to prove that the Dirac delta function is the unit of convolution and all I find always is this formula but no further explanation: $$[f*\delta](t)=\int_{-\infty}^{\infty}f(t-\sigma)\delta(...
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1answer
773 views

Convergence to the Dirac Delta Function

Let $h\colon[0,1]\to \mathbb{R}^+$ be any bounded measurable non-negative function with a unique maximum at $a$ and $h$ is continuous at $a$. For $\lambda>0$ define $h_\lambda(x)=C_\lambda h(x)^\...
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1answer
440 views

Cauchy's Integral Formula and Delta Functions

As we know from complex analysis, Cauchy's integral formula states: $f(z_o)=\frac{1}{2\pi i}\int_\gamma{\frac{f(z)}{z-z_o}dz}$ for a closed contour $\gamma$. However there is also the result from ...
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2answers
519 views

What is the total variation of a dirac delta function $\delta(x)$?

What is the total variation of a dirac delta function $\delta(x)$? My guess is that it is something like $\infty$. If not defined, what would be the best way to define?
5
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1answer
140 views

Schrödinger operator with delta (zero range) interaction.

I am reading the book of Albeverio named Solvable models in quantum mechanics. In the first chapter it is explained how to realize the operator $"-\Delta+\delta_0"$ as a self adjoint operator on $L^2(\...
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1answer
287 views

A representation of Dirac-$\delta$

Prove that $$g_\epsilon (x)=\lim_{\epsilon \to 0} \frac1 \epsilon \frac1 \pi e^{-x^2/\epsilon^2}$$ is a Dirac-$\delta$ function. This is a homework question I'm stuck with. I'm probably missing a ...
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0answers
268 views

Poisson summation formula for positive integers

I am trying to evaluate the following expression for $\lambda \in \mathbb{R}$ : $$f(\lambda)=\sum_{n=1}^{+\infty}e^{-i\lambda n}$$ My idea is to introduce an epsilon prescription, so I choose $\...
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2answers
421 views

Singular differential forms and $\nabla^2\left(\frac{1}{r}\right)$

The delta function identity $$\nabla^2\left(\frac{1}{\lvert\mathbf{x-x'}\rvert}\right)=-4\pi\delta^{(3)}(\mathbf{x-x'})$$ is often casually derived using the divergence theorem, since the divergence ...
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0answers
540 views

Heaviside unit step- and delta function

The following question is right from the book: Show that $$ H(x-x_i) = \int_{-\infty}^x \delta(x_0-x_i)dx_0\, $$ satisfies $$ H(x-x_i) \equiv \begin{cases} 0 & x < x_i \\ 1 &...
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669 views

Derivation of the Dirac-Delta function property: $\delta(bt)=\frac{\delta(t)}{\mid b \mid}$

Considering the case such that $b \gt 0$ and $b \in \mathbb{R^+}$ and making the substitution $t'=bt$, it follows that $$\int_{t=-\infty}^{t=\infty}f(t)\delta(t)\mathrm{d}t =\color{red}{\int_{t'=-\...
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3answers
309 views

Double Dirac delta integral

Let $I=\int_{-1}^0\int_{0}^1 \delta(x-y)dxdy$ where $\delta(t)$ is defined as the limit of a symmetric Gaussian pdf. The ranges overlap only on the zero-length range $x=y=0$. Is the result here $0$ ...
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3answers
958 views

Divergence of $\vec{F} = \frac{\hat{\mathrm{r}}}{r^{2}}$

Consider the vector field $$\vec{\mathrm{F}} = \frac{\hat{\mathrm{r}}}{r^{2}},$$ then the divergence of this field is: $$\vec{\nabla}\cdot\left(\frac{\hat{\mathrm{r}}}{r^{2}}\right) = 4\pi\delta^{3}(\...
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3answers
792 views

How to prove that $\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x)$

It is well-known that: $$\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x).$$ This can also be written as $$ 2\pi\delta(x)=\int^{+\infty}_{-\infty}e^{ikx}\,\mathrm dk.$$ However, I don't know how to ...
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2answers
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Integral of Dirac delta function from zero to infinity

I know that: $$\int_{-\infty}^{+\infty} \mathrm{d}t \, f(t) \delta(t) = f(0)$$ However, I cannot figure out the result of the integral below: $$\int_{0}^{+\infty} \mathrm{d}t \, f(t) \delta(t) = ?$$ ...
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2answers
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Legendre polynomial expansion of the unit step function.

The problem is to determine the expansion of the unit step function in terms of Legendre polynomials on the interval $[-1,1]$. Here the Legendre polynomials are the family of orthogonal polynomials ...
4
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3answers
106 views

If $f(x) = e^{-|x|}$, show that $f''(x) - f(x) = -2\delta(x)$ (in the sense of distributions)

Exercise : If $f(x) = e^{-|x|}$, show that for its derivatives, it is : $f''(x)-f(x) = -2\delta(x)$, in the sense of distributions. Attempt : I am completely at a loss on how to handle such an ...
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2answers
52 views

How to expalin the pathologie of this zero distribution(or measure).

Assume $\delta$ is the zero Dirac distribution (measure) on $\Bbb R$. namely, $$(\delta,f)= \int_\Bbb R f\delta(dx) =f(0)$$ We know if $T\in \mathcal {D'}(\Bbb R)$ is a distribution then for every $ ...
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4answers
798 views

Integration of $\int_{0}^{\infty} e^{-itw} dt$

We know that $\int_{-\infty}^{\infty}e^{-itw}dt=2\pi\delta(w)$, but how to calculate the half integration $\int_{0}^{\infty}e^{-itw}dt$?
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1answer
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How to prove that the Dirac delta is not a function?

I am currently taking a course on test functions and distributions and my task is to prove that the Dirac delta is not a function. Furthermore, I would also like to prove that it is continuous as a ...
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3answers
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Approximating dirac delta function with sinc functions

I am trying to prove the following $$\lim_{a\rightarrow\infty}~\int_0^\infty\frac{f(x)\sin(ax)}{x}dx=(\pi/2)f(0)$$ for all square integrable functions $f(x)$ continuous at $0$. I tried to do the ...
4
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2answers
395 views

By substituting $z = h(t)$ show that $\delta(h(t))=\sum\limits_{i}\frac{\delta(t−t_i)}{\mid h^{\prime}(t_i)\mid}$

Firstly, apologies in advance for using an abuse of notation by placing the Dirac-delta inside an integral. But for my level of understanding, I have no choice. This question involves one of the ...
4
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4answers
79 views

Regarding Dirac Delta

I'm having a little bit of trouble understanding Dirac Delta, or rather, finding a proper definition. I understand the way it is "found" by using Fourier transforms on a function, and that it isn't ...
4
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2answers
534 views

Using the Dirac Delta function in PDE's

Solve the diffusion equation on the positive half-line $\frac{∂u}{∂t}−a^2\frac{∂^2u}{∂x^2}=0,0≤x<∞$ subject to the initial and the boundary condition $u(x,0)=Qδ(x−x_0),u_x(0,t)=0.$ Where $Q≠0$ ...
4
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1answer
73 views

Integration of dirac, why this is the result?

I don't understand why this example: $$\int_{-\infty}^{\infty} \left(\ \delta(x)+ \frac{\delta(x-1)}{2} + \frac{\delta(x+1)}{2}\right)\ e^{-x} \ dx $$ Gives the following anwser: $$ 1 \ + \ \frac{...
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2answers
397 views

Solving Second Order PDE with Dirac Delta

I want to find the functional form of the Green function G(x,t) for a parabolic differential equation: $$ \frac{\partial{}G(x,t)}{\partial{}t}=a\frac{\partial{}^2G(x,t)}{\partial{}x^2}+\delta(t)\...
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2answers
514 views

How to prove this Dirac Delta Function property?

How to prove the equation below, using Dirac Delta function properties? $$ \delta(x^2-m^2)=\frac{1}{2|w|}(\delta(x-w)+\delta(x+w)) $$ where $$ w^2=|x|^2+m^2 $$ I tried to show it using $$ \delta(f(x)...
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2answers
171 views

An identity on $\delta(x+y)$?

I am dealing with some derivatives on Dirac delta function, $(x\partial_x+y\partial_y)\delta(x+y)$. Consider the following integral ($f(x,y)$ is bounded) and perform integration by parts, \begin{...
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2answers
599 views

Derivation of Dirac delta function

Is there anyone could give me a hint how to find the distributional derivative of the delta function $\delta$? I don't know how to deal with the infinite point.
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2answers
605 views

Dirac Delta function at a point

From my understanding of the Dirac Delta function, it is infinitely thin and has a value of infinity at only a particular point. I also learned that $$\int_{-\infty}^{\infty} \delta(x-a) dx = 1$$ What ...
4
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1answer
691 views

Derivative of Heaviside Function and Equivalence

The derivative of the Heaviside function $\theta(x - a)$ is normally taken to be the delta function $\delta(x - a)$. This question has two parts, the first is whether a constant coefficient is ...
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2answers
434 views

Expanding Dirac delta function with Hermite polynomial

My question is related to a formula in this paper In that paper, they try to expand Dirac delta function $\delta(x)$, which has the property $$ \int \delta(x)f(x) \, dx = f(0), $$ using Hermite ...
4
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1answer
238 views

Integrating the dirac delta function with a periodic argument

For example $$\int_0^\infty \frac{\delta \big (\cos(x^2) \big)}{x^2} dx$$ I'm just not sure how to handle there being multiple solutions for $\delta (cos(x^2))=0$
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2answers
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Integral of Dirac delta function/distribution $\delta(x)$ with upper boundary equal to zero

I would like to find the value of $$\int_{a<0}^0 \delta(x) dx$$ In particular, I would like to know if I can break down the integral $$\int_a^b \delta(x)f(x) dx=\int_a^0 \delta(x)f(x) dx + \...
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1answer
222 views

Integral involving Dirac delta: two different results?

I am evaluating the integral over all space $$\int \delta \left(r^2 - R^2\right) d \vec r$$ At first, I did this: $$\int \delta \left(r^2 - R^2\right) d \vec r = 4 \pi \int_0^\infty \delta \left(r^...
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2answers
533 views

Prove the Dirac Delta Function satisfies $ x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = -\delta(x) $

$ x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = -\delta(x)$ I've been told that this answer involves integration by parts. I began like this: $\int x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = x\...
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1answer
169 views

Closed form of multi-variable integral with Dirac delta function

I want to give you this exercise to see if there are simpler solutions to the one that I've chosen! I found it a very interesting exercise. Question Find the closed form for all N of following ...
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1answer
3k views

Proof of Convolution Theorem for three functions, using Dirac delta

I am stuck on proving the convolution theorem for the product of three functions using the Dirac delta function. Please excuse any nonstandard notation--I am a physics major who has not been formally ...
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1answer
481 views

A Property of the Dirac Delta Function

I can't seem to prove the following property of te $\delta$- function. Please help.
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1answer
549 views

Traveling delta function $\delta(x - ct)$ as a distributional solution of the wave equation

I'm trying to show that a delta function $\delta(x - ct)$ is a distributional solution of the PDE $$ D_{(0,2)}u(x, t) = c^2 D_{(2,0)}u(x,t). $$ Here $D_{(i,j)}$ means $i$-th partial differentiation on ...
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1answer
632 views

Computing an explicit Radon-Nikodym derivative

Q/ let $\lambda$ be the Lebesgue measure and $\delta_0$ be the Dirac measure at 0. Show that $\lambda$ is abs cts wrt $\lambda+\delta_0$ (have done this part) and find the R-N derivative $\frac{d\...
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1answer
222 views

'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
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2answers
136 views

$I = \int_{- \infty}^{\infty} \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $ should not diverge

I'm having trouble evaluating this integral: $$I = \int_{-\infty}^\infty \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $$ where $\delta(x)$ is Dirac delta function, $\mathbf x$ is the real $n$-...