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.

Filter by
Sorted by
Tagged with
-3
votes
0answers
23 views

Why we cannot say that integration of $\delta(x)$ is Lebesgue integration over zero-dimentional manifold?

Lebesgue integral is defined as $$\int 1_{S}\,d\mu =\mu (S)$$ For instance, integral over n-dimentional unit cube is always $1$. But a point is a $0$-dimentional unit cube and its Lebesgue measure ...
-1
votes
1answer
41 views

Integral of the square of the Dirac delta

What is the following integral? $$\int_{-\infty}^{\infty} \delta^2(x) {\,\rm d} x$$ I think it should be one, by Parseval's Theorem.
3
votes
1answer
49 views

Gaussian integral over a surface

I have to solve the following integral: $$ I(s)=\int_{S(s)} \frac{1}{2\pi^{3/2}}e^{-\frac{x^2+y^2+2z^2}{2}}dxdydz $$ where $S(s)$ is the surface defined by $s=\sqrt{(x-y)^2+4z^2}$. I parametrised $...
0
votes
1answer
30 views

Significance of measure valued solution to a PDE

I know the motivation to define weak solution to a PDE, for example if we consider transport equation $$ u_{t}+(f(u))_{x}=0, ~~ (x,t) \in \mathbb{R} \times (0, \infty) $$ $$ u(x,0)=u_{0}(x), ~~ x \...
2
votes
1answer
32 views

Why is the DFT of $1 = \sum_{k=-\infty}^{\infty}\delta(\theta-k)$?

I struggle with a part of the solution of an exercise and would be grateful for your help. "Given a time discrete LTI (Linear time-invariant) system H, and an input signal $x[n]$, we have the ...
1
vote
3answers
52 views

A sufficient condition for a Borel probability measure to be a Dirac measure

Let $\mu$ be a Borel probability measure on $\mathbb{R}$ and $\mu^k=\mu$ for some natural number $k>1$. Then I have been asked to prove that $\mu=\delta_0$. I sense that I have to somehow exploit ...
0
votes
0answers
36 views

A Dirac delta integral with parameter

Let $u:[0,\infty)\times\mathbb{R}\to\mathbb{R}$ be continuous in $t$ and consider $$\lim_{\varepsilon\to 0}\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi \varepsilon}}e^{-y^2/2\varepsilon}u^2(t-\varepsilon,y)...
0
votes
1answer
38 views

Laplace transform: $\mathcal{L}(f')=s\mathcal{L}(f)-f(0)$ for weak derivatives

Let $f:[0,\infty[\to\mathbb {R}$ be the periodic function with period $T=1$ defined on $[0,\infty[$ by $$f(t)=t,\,\,\,\,\,\text{pour}\,\,0\leq t<1.$$ The book I am reading used the following ...
2
votes
3answers
47 views

Integrating Dirac delta function over two variables

I am trying to evaluate one equation like this $$ \int_{-\infty}^\infty \int_{-\infty}^\infty xy\delta \left[ (x-1)^2 + y^2-2 \right] \, dx\, dy $$ but found my result is not the same compared with ...
1
vote
1answer
57 views

How can I prove $\delta_{\lambda} \in \mathcal{E}'(\mathbb{R})$?

How can I prove, formally, by its definition, that $\delta_{\lambda}=\delta(x-\lambda)$ is a distribution in $\mathcal{E}'(\mathbb{R})$ for every $\lambda$ where I choose to center my distribution?
0
votes
2answers
36 views

Probability density of random variable with one value

Let X be the real-valued number of sides on a six-sided dice. So $\Omega(X) = ${$6$}. Therefore, the probability density function is: \begin{equation} f_{X}(x) = \begin{cases} \...
0
votes
1answer
25 views

Are there examples of $f(x)$ for which $\mathcal{F}_x[f(x)](s)=\mathcal{F}_x^{-1}[f(x)](s)$ other than $f(x)=\cos(x)$ and $f(x)=\text{sinc}(x)$?

I've noticed the Fourier and inverse Fourier transforms of $f(x)=\text{sinc}(x)$ (as well as $f(x)=\cos(x)$) are identical: $$\mathcal{F}_x[\text{sinc}(x)](s)=\mathcal{F}_x^{-1}[\text{sinc}(x)](s)=\...
5
votes
4answers
194 views

Teaching Dirac delta “function” $\delta(t)$

I am about to teach applied mathematics for engineering. So I will teach how to use Laplace Transform to solve differential equations. Some of these differential equations involve the Dirac delta "...
0
votes
0answers
33 views

Derivative of a function defined by parts involving dirac delta

Given a function $$\rho (x_1,x_2,x_3,t)=\begin{cases} \rho_1,\text{ if } x_3 \geq f(x_1,x_2,t) \\ \rho_2,\text{ otherwise}\end{cases}$$ I found its gradient as $$\nabla \rho = (\rho_2 - \rho_1) (\...
0
votes
0answers
30 views

Is this a Dirac comb?

While playing around, I came across the following function $$f(x)=\lceil\{|x|\}\rceil$$ Its $0$ at integers, $1$ otherwise.(The mod is to extend it to $x<0$) So I made $$g(x)=\frac1{f(x)}-1$$ ...
0
votes
1answer
51 views

Convergence to Delta Dirac Distribution

This question derived from my previous question. When I took a course on the theory of distributions, I was first introduced to the Dirac delta as an usual distribution, that is, as a linear ...
0
votes
0answers
40 views

Dirac delta distribution as the limit of a succession

In a class on distribution theory, the professor stated that a succession of function $\phi_n(x)$, in order to converge to a Dirac distribution needs the following property: $\lim_{n\rightarrow\infty}...
3
votes
1answer
109 views

What is the right way to calculate $ \partial_z\frac{1}{\bar{z}} $?

In the appendix B of a physics paper arXiv: 1902.01434, it says $$ \partial_z\frac{1}{\bar{z}}=\partial_{\bar{z}}\frac{1}{z}=2\pi\delta(z)\delta(\bar{z}), $$ same as 2-dimensional delta function (...
0
votes
1answer
39 views

Is this passage correct?

How to justify the following passage $$\int_{j-1}^j \int_{k-1}^{k} \delta (t-\tau) \, dt \, d\tau=\delta_{jk}$$ $j, k\in \mathbb{Z}\; t, \tau \in \mathbb{R} $
0
votes
0answers
26 views

Show that $2a\delta(at+x)\delta(at-x) = \delta(x)\delta(t)$.

Firstly we should examine the action of $2a\delta(at+x)\delta(at-x)$ and then can we use transformation of coordinate system $s=at-x$, $y=at+x$. so, can we show $2a\delta(at+x)\delta(at-x) = \delta(x)...
0
votes
0answers
20 views

Integral of variance of uncorrelated random number

Given an uncorrelated, Guassian-distributed random number $\xi$ is unitary variance and zero mean, i.e. $\langle \xi\rangle = 0$ and $\langle \xi(t)\xi(0)\rangle = \delta(t)$. My question is: If $\...
0
votes
1answer
20 views

How to solve this problem involving convolution of negative input to delta function

How to solve this problem involving convolution of negative input to delta function $$ x(-t) \ast \delta(-t-t_0) $$ $\ast$ - stands for convolution and $\delta$ stand for - dirac - delta function
0
votes
1answer
39 views

What does δt mean in this equation?

I am trying to understand this page which comes to the following equation: $$f_{\mathrm{ext}}(t) = mv_0\delta(t).$$ They are attempting to describe the force being applied to a mass as a momentary ...
0
votes
0answers
38 views

Infinite sum of delta function when calculating Fourier Transform

I am trying to find the Fourier Transform of the periodic function $$ f (\theta) = \sum_{n=- \infty}^{\infty} a_n \space \exp(i n \theta)$$. Using the formula, $$ f(k) = \int_{- \infty}^{\infty} d \...
0
votes
0answers
9 views

Boundary value in a convolution integral

I am trying to understand the following equation: $$\int^1_0dz\int^1_0dy f(y)g(z)\delta(x-yz)=\int^1_x\frac{dy}{y}f(y)g(\frac{x}{y}).$$ My result is instead $\int^1_0\frac{dy}{y}f(y)g(\frac{x}{y})$. ...
0
votes
0answers
23 views

Relating the Convolution to a Bank Account

In this video, the professor references an assignment that relates the convolution to a bank account. However, the link in the description redirects to a later version of the course which omits this ...
0
votes
1answer
20 views

How to find particular solution for Green's function with $g(x)$ as Dirac delta function?

So I have done the first part of this question i.e. finding a homogeneous solution and going through the usual steps to get $G(x,s)$ in terms of $x<s$ and $x>s$. My question is for the last part,...
0
votes
2answers
32 views

Dirac Delta Integration by Parts

$$\int_{-\infty}^{\infty}\left(\sin ^{2} x+2 \tan x\right) \delta^{\prime}(x)$$ $1.$ Is this integral missing a $dx$ at the end? $2.$ Assuming that it is then performing integration by parts leaves ...
0
votes
1answer
25 views

Second Derivative of the Dirac Delta function

I want to simplify: $\int dp dp' f(p,p') \delta''(p-p')$ where f(p,p') is an unknown function. How do I deal with the second derivative of the delta function?
1
vote
1answer
38 views

Table of integrals for complex exponentials involving delta functions

I am interested in finding a list of integrals of the form: $$ \int a^n e^{iax} da$$ For $n = 0$, I found it to be $\delta(x)$. For $n = -1$, I found it to be sgn$(x)$. For $n = -2$, I found it to ...
0
votes
2answers
56 views

The Heaviside step function at zero and the integral of the Dirac delta function

Just for fun I was reading about the Heaviside step function on Wikipedia. The definition in terms of the Dirac delta function makes sense: $$ H(x) = \int_{-\infty}^x \delta(s)\ ds $$ $\delta(s) = 0$...
1
vote
1answer
41 views

Can I rewrite this expression with a Dirac $\delta$-function?

I am looking at the following scalar product of divergences in $4$ dimensions (in momentum space): $$\partial_{p_1} \cdot \partial_{p_2} \frac{1}{p_1^2p_2^2}. \tag{1}$$ There is a further ...
0
votes
1answer
29 views

Dirac delta function - why the requirement on compact support?

I am reading the wikipedia articla on Dirac delta, and as far as I understand it, it is saying that only for functions with compact support $f$: $$\int_\mathbb R \delta_t(s)f(s)ds=f(t)$$ Why the ...
1
vote
1answer
49 views

Solving ODEs with Delta Functions

I was given a homework question Suppose a bone described by the Kelvin model is deformed by a force $$F(t)=F_0(\Theta(t)-\Theta(t-\tau)+\Theta(t-2\tau)),$$ where $\Theta$ is the ...
0
votes
1answer
17 views

Why is integral of differential delta function equal to delta function?

Why is integral of differential delta function equal to delta function? I.e. as I've read elsewhere: $$\int_A 1 d \delta_A=\int_A d \delta_A=\delta_A=\delta_x(A)$$ https://planetmath.org/...
2
votes
1answer
49 views

Derivative of Contour Integral Representation of Step Function

The following is Problem 11.9 in "Mathematical Physics: A Modern Introduction to Its Foundations, Second Edition" by Sadri Hassani. Given the following representation of the step function: $$\theta(x)...
0
votes
0answers
14 views

PDE with a dirac source

i have to solve this PDE (lossy Wave equation) : $$ \partial_{x x}^{2} p^{\prime}-\frac{\alpha}{c_{0}^{2}} \partial_{t} p^{\prime}-\frac{1}{c_{0}^{2}} \partial_{t t}^{2} p^{\prime}=-\rho_{0} \Gamma \...
0
votes
0answers
49 views

Integration of two dimensional dirac delta

I want to integrate the two dimensional dirac delta function: $ \delta^2(x,y)=\delta(x)\delta(y) $ I know that for the one dimensional case the integral is $\int_{-\infty}^{+\infty}\delta(x)dx = 1$ ...
-1
votes
1answer
43 views

How to solve this integrals (delta Dirac)

i have the following problem : $$\int_0^L\sum_{i=1}^{n=3} \delta\left(x-x_{i}\right) \cos(2\pi x/L ) \left\{ A_{1}'(t) \cos \left(\frac{2 \pi x}{L}\right) + B_{1}'(t) \sin \left(\frac{2 \pi x}{L}\...
0
votes
1answer
19 views

Integral of dirac delta function times another another function

Dirac delta function $\delta(x)$ is defined with two properties: 1) At $x=0$ its value is $\infty$ and everywhere else it is $0$ 2) Area under the curve is $1$ How does above definition result in $$\...
0
votes
1answer
44 views

Dirac delta questions

I'd like to ask for help in solving 3 I suppose quite easy questions about Dirac delta. $$\delta (-x) = \delta (x)$$ $$\delta(ax) = \frac{1}{|a|}\delta (x)$$ $$\delta ' (-x) = -\delta ' (x)$$ ...
0
votes
0answers
28 views

Dirac Delta Evaluation of Function

I am struggling to understand what does the following expression evaluate to. "I am looking for general answer, not actual evaluation - i.e is dirac dealta making the integral center around h(x)? ...
1
vote
0answers
71 views

Why does the method using the Green's function for $\nabla^2 \Phi(\mathbf{x}) = \delta(x)\delta(y)$ not work?

I have the Poisson Equation (with $\mathbf{x}\in\mathbb{R}^3$) with the following form: $$\nabla^2 \Phi (\mathbf{x}) = \delta(x)\delta(y)$$ I used 2 methods for the resolution of this PDE. I am ...
0
votes
0answers
24 views

Laplacian and Dirac function gives contradictory result.

The following equation is correct for all non-negative real numbers: $$4\pi\delta^{(3)}(\mathbf{r})=\nabla\cdot\frac{\mathbf{r}}{r^{3}},$$ $$r\in[0,+\infty)$$ especially, when r=0, both sides give ...
2
votes
0answers
42 views

Show that $\sum\limits_{n=-\infty}^{\infty}\delta^{(|n|)}(x-n)$ diverges in S'

by Schwartz’s theorem, any generalized function from $S'$ has a finite singularity order. In this example, it is infinite and I want to show that the series $\notin S'$. ($g^{(l)}$ means $l$th ...
0
votes
1answer
29 views

How to solve second order ODE with Dirac Delta?

I'm trying to solve a non-homogeneous second order ODE. I've read similar other questions, but all use the method of Laplace transformations, which I've not seen/used before. The ODE is: $$y''(x) - ...
0
votes
1answer
30 views

Dirac delta integral for evolving networks

I'm reading Dynamical Processes on Complex Networks (link), which makes frequent use of dirac delta integrals to examine evolving networks. I'm trying to get a good sense of how to evalute them and ...
0
votes
0answers
41 views

Dirac delta and convolution

Is there a way to simplify the following equation? $$\int_{-\infty}^{\infty} f(q) \delta(q-k) * g(q) dq =\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(q) \delta(q-k) g(q-\tau) d\tau dq $$ The $...
1
vote
0answers
29 views

Fundamental solution of the derivative operator

Let $x \in \mathbb{R}$ and define $x_+:=xH(x)$, where $H$ is the Heaviside function. By induction we can show that $E(x)=\dfrac{x_+^{k-1}}{(k-1)!}$ is a fundamental solution of $$P=\frac{d^{k}}{dx^{k}}...
0
votes
2answers
35 views

How to prove the two formulas are equal in the sense of distribution

$1+2\sum_{n=1}^\infty \cos2n\pi x=\sum_{k=-\infty}^\infty \delta(x-k)$. I couldn't have an idea to prove it, maybe we can discuss how to get it clearly.