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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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I'm looking for resources that involve concretely taking Lebesgue integral of functions (non-axiomatic and computation focused)

I want to practice finding the Lebesgue integrals of certain functions. My source of inspiration is integrating Dirac delta functions and anything relating to differential equations like Green's ...
Nate's user avatar
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Calculus of variations with linear Lagrangian and two optimisations

I am working on a two-part system, that jointly determines two functions $F$ and $G$. $F'(x) = f(x)$ and $G'(x) = g(x)$. $F$ and $G$ are both defined on $[0, 1]$, $F(0) = G(0) = 0$, they are both non-...
Ishan Kashyap Hazarika's user avatar
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The Helmholtz equation for the spherical harmonics with delta functions

In three dimensions, the Green’s function for the Helmholtz equation with a radiating point source $$ (\nabla^{2}+k_{0}^{2})g(\textbf{r},\textbf{r}')=\delta(\textbf{r}-\textbf{r}') $$ is $$ g(\textbf{...
Chris's user avatar
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Integral representation of Dirac delta? [closed]

Is there any integral representation of the Dirac delta distribution as follows: $$\delta(x-x_{0}) = \lim_{n\to \infty}\int dx f_{n}(x-x_{0})\frac{1}{\sqrt{|x|^{2}+m^{2}}}$$ for a fixed parameter $m &...
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Special case of sifting property of dirac-delta at singularity.

Note: This is not a duplicate. You can see my linked old question below in the text and realize that my question is different. I have an integral of the following form: $$ I = \lim_{a\ \to\ \infty} \...
Srini's user avatar
  • 862
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Laplacian of arbitrary power of arbitrary norm

So I have the function $f : \mathbb R^d \to \mathbb R$ given by $f(x) = \lVert x \rVert_p^q$, where $\lVert \cdot \rVert_p$ denotes the $p$-norm on $\mathbb R^d$, given by $$ \lVert x \rVert_p = \left(...
markusas's user avatar
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On a particular uniform convergence

Let's consider a function $\phi(x)$ that we call "fairly good", meaning that $\phi(x)$ and all its derivatives of all orders are $O(|x|^N)$, for $|x|\to\infty$, with $N\in\mathbb{N}$ known. ...
Nameless's user avatar
1 vote
2 answers
22 views

Mismatch between impulse response gotten from differentiation of unit step response vs solving directly with input $\delta(t)$ for operator $ 2D + 1 $

To find the unit step for the operator $ 2D + 1 $, where $ D $ is the differential operator, we proceed as follows: Unit Step Response The unit step function $ u(t) $ is defined as: $$ u(t) = \begin{...
Rishav Dhariwal's user avatar
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Limit behaviour of a distribution

Consider the distribution, with $a>0,p>0$ $$ F(x)=\left\{\begin{matrix}\frac{J_2(a\sqrt{p^2-x^2})}{x- p} & -p<x<p \\ 0 & \text{otherwise}\end{matrix}\right. $$ I claim that $F$ ...
Wouter's user avatar
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Derivative of a sign-like function

Assume a function $f: \mathbb{R}^2 \to \mathbb{R} $ such that $$f(x_1,x_2)=\begin{cases} 1 & x_2\le h(x_1,t) \\ 0 & x_2 > h(x_1,t) \end{cases}$$. I am trying to find the derivative of f in ...
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How to show that the Gaussian sequence satisfies the sifting property of the delta function?

I am trying to show that the Gaussian sequence of functions, defined by $g_a (x) = (\frac{1}{a\sqrt{\pi}}) e^\frac{-(x-x_0)^2}{a^2}$ satisfies the sifting property of the $\delta$ - function, namely ...
Falgun Sukhija's user avatar
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1 answer
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Interpreting Green's function in Evans' Partial Differential Equations

Pictures below is from Evans' Partial Differential Equations. $U\subset \mathbb R^n$ is open, bounded. And $\nu$ is the normal vector of $\partial U$. I want to get the last red box from the first ...
Enhao Lan's user avatar
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Dirac Delta Function Integral Over Finite Interval

I am taking a course on Partial Differential Equations at the moment. We were introduced to the Dirac Delta function, and were told it satisfies $$\int^b_a \delta(x)=1$$ for $a<0<b$. Having ...
raynerk's user avatar
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Modified Bessel Function and Dirac Delta function

Is there a representation of the Dirac delta function of the form $$ \int_0^\infty \frac{d z}{z} K_{i \nu} (z) K_{i \nu'} (z) = f(\nu) \delta ( \nu - \nu' ) , \qquad \nu,\nu'>0. $$ for some ...
Prahar's user avatar
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Problems with understanding Dirac Delta Properties [closed]

I need to check equation ${\bf 6.15}$ in an $\tt arxiv$ paper for my work, but I don't get why the Dirac Distributions change and why there is that exponential ...
Tscheburaschka's user avatar
1 vote
1 answer
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Differential Entropy of the Dirac Delta Function

We are working on finding differential entropy of Dirac delta function $\delta(x)$, that is $$H(X)= - \int_{-\infty}^{\infty} \delta(x) \ln \delta(x) \, \mathrm{d}x$$ We found answer that $H(X)= -\...
Jay's user avatar
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2 votes
2 answers
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Intuition for why $\int_{-\infty}^{\infty} f(t) \delta(t) dt = f(0)$

What is a good, intuitive, 1st-year Calculus student, layman's-terms reason that the Dirac delta function $\delta(t)$ is defined to have the characteristic $\int_{-\infty}^{\infty} f(t) \delta(t) dt = ...
The Math Potato's user avatar
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Why is the autocorrelation of an uncorrelated random noise process the dirac delta distribution?

I am reading Stochastic Methods by Gardiner and in the beginning of chapter 4 he motivates the rigorous interpretation of a Stochastic Differential equation by describing the properties of a "...
Mashe Burnedead's user avatar
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is $\int_0^t \int_0^s \delta(u - v) \, du \, dv = \min(t, s) $? [duplicate]

Is the following integral equation true? $ \int_0^t \int_0^s \delta(u - v) \, du \, dv = \min(t, s) $ where $\delta$ is the Dirac delta function.
user1326164's user avatar
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Fourier transform of $f(x)=x\mathcal{X}_{[a,b]}$, for $a<b$ reals.

As stated, this problem is pretty straigth forward. Using the normal definition of the Fourier transform I get, $$\widehat{x\mathcal{X}_{[a,b]}}=\int_{a}^{b}xe^{-2\pi i x \xi}dx=\frac{e^{-2\pi i a \xi}...
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Proof review Tempered Distributions: $(T*\varphi)^\wedge = (2\pi)^{n/2} \hat{\varphi}\hat{T}$ and computing $(x^\alpha)^\wedge$

I am learning about tempered distributions on schwartz space. But there are 2 questions in this text where Im unsatisfied with my solution... I ask kindly for your review of my solutions below, and if ...
NazimJ's user avatar
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My proof of $\delta_0\left(t^2-|x|^2\right)=\frac{1}{2 \sqrt{2} t} \mathrm{~d} \sigma_{C_0^{+}}(t, x)$

Here is the proof of Lemma 4.10. On $\mathbb{R}^{1+3} \backslash\{0\}$, we have the identity $$ \delta_0\left(t^2-|x|^2\right)=\frac{1}{2 \sqrt{2} t} \mathrm{~d} \sigma_{C_0^{+}}(t, x) $$ where $C_0^{...
YuerCauchy's user avatar
3 votes
2 answers
74 views

Differentiating Dirac delta with product rule

I have here an equation. $$ h'(t_2) \delta(t_1 - t_2) = [h(t_2) - h(t_1)] \delta'(t_1 - t_2) $$ I checked the equality by integrating both sides with a test function. $$ \int d t_1 \phi(t_1) \ldots \...
Bio's user avatar
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Derivative of Lebesgue integral with indicator functions

Suppose we have a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$. I want to take the derivative of $$ W\left(x\right)\equiv\int_{\Omega}\mathbf{1}\left\{ \omega\in R\left(x\right)\...
Thomas's user avatar
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DiracDelta/x = -DiracDelta'? - Use and Correctness of Statement [duplicate]

One property of the Dirac Delta Distribution is $x \delta'(x) = -\delta(x)$ because of $\int x \delta'(x) f(x) dx = -\int \delta(x) (xf(x))' dx = -\int \delta(x) (f(x)+xf'(x)) dx = -\int \delta(x) f(x)...
theta_phi's user avatar
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Derivation of sifting property of Dirac Delta Distribution from Dirac sequence, one property isn't used

I want to start from the properties of the dirac sequence to derive its defining characteristic. To show: $\int_{-\infty}^\infty \delta(x) f(x) dx =f(0)$ Dirac sequence $\delta(x)_{n \in \mathbb{N}}$: ...
theta_phi's user avatar
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Laplace Transform of Derivative by Limit and Dirac Delta Distribution

The Laplace Transformation of the Derivative is given by $\mathcal{L}[f'(x)](s) = \int_0^\infty f'(x) e^{-s x} dx = s \mathcal{L}[f(x)](s) - f(0)$, which can be easily shown by partial integration. I ...
theta_phi's user avatar
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1 vote
3 answers
49 views

3D Inverse Fourier Transform

I am trying to prove the following: $$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \frac{\sin(kt)}{k} e^{i \vec k . \vec x} d^3 k = 2\pi^2 \times \frac{\delta (t- |x|)}{|...
user354595's user avatar
1 vote
1 answer
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Which Method Correctly Computes the Fourier Transform of the Ramp Function?

While calculating the Fourier transform of the ramp function, I found two different methods that yield different results. Here is how each method works before I share my confusion. Define the ramp ...
John Title's user avatar
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2 answers
59 views

Divergence of $\hat r$

I have read the other posts such as divergence of $\,\hat{r}$ divided by $r^{2}$: But my question is very basic - where does the extra $r^{2}$ come from in the below proof $?$. I realize why $\frac{1}...
Rasputin's user avatar
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Why $d\mu (q)\delta (k,q)$ is $G$-invariant?

Let $G$ be a Lie group acting transitively on a smooth manifold $M$ endowed with a quasi-invariant measure $\mu$ (then there exists Radon-Nikodym derivative $\rho_f$ for every $f\in G$). For $k\in M$,...
Mahtab's user avatar
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2 votes
1 answer
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How to integrate $\delta(\omega - E(\boldsymbol{k})$? [duplicate]

When I was reading section 3.3, Density of States, in the theses ‘Electronic Phenomena in 2D Dirac-like Systems: Silicene and Topological Insulator Surface States’, I encountered a problem. I don’t ...
HRTCWQ's user avatar
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0 answers
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How has this limit of the sequence been achieved?

I am reading a section from a book where some basic/introductory material on Delta functions has been presented. The author has explained the Delta sequences through the usage of Fourier series ...
ishan_ae's user avatar
3 votes
1 answer
143 views

Rate of convergence of Fourier modes of a mollifier on the torus in a negative regilarity Besov norm

Consider a standard mollifier $\rho_\delta$ on $\mathbb{T}^2$ (the 2 dimensional torus) and let $$ \hat{\rho_\delta}(m): = \int_{\mathbb{T}^2} \rho_\delta(z) e^{2 \pi i m \cdot z} dz $$ for any $m \...
Marco's user avatar
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2 votes
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Is there another way of deriving the Fourier transform of 1?

Reasoning with generalized functions we can derive that the Fourier transform of the Dirac delta function is: $$\langle \mathcal{F}[\delta(x)](p),\phi(p) \rangle=\langle \delta(x),\mathcal{F}[\phi(p)](...
Krum Kutsarov's user avatar
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1 answer
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Applying Integration by parts to the Dirac Delta Sifting Property

I've seen explanations/proofs of the dirac delta sifting property which involve treating the given function as a constant close to the impulse point, but I wanted to try and prove it using regular ...
John Smith's user avatar
3 votes
0 answers
49 views

Help with "delta dirac" integral

Let $f_0(x)$ be a positive, symmetric and bounded function and $c > 0$ some constant. I came across the following integral: $$ I = \int_{\mathbb{R}} \frac{d}{dx} \left[\text{sign}( |x| -c )\right]...
lohe's user avatar
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1 answer
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Calculate the integration about Dirac Delta function

How to calculate the integration $$\int_{-1/2}^{+\infty}\cos\left(\frac {\pi x}2\right)\delta\left(e^x\sin\left(\frac {\pi x}2\right)\right)\,\mathrm dx$$ My ideal is that let $$e^x\sin\left(\frac {\...
Shaoran Sun's user avatar
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0 answers
37 views

The product of a Heaviside function and Dirac function centered around different points.

Let $a,b$ be two real-numbers such that $a \neq b$. Let $\iota(x \leq a)$ be the Heaviside step function in variable $x$ and $\delta_b(x)$ be the Dirac Delta function centered around $b$. I have two ...
sp1122's user avatar
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1 vote
1 answer
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Proving a useful identity with the Dirac Delta function

I am studying QFT and renormalization of QED, and in a passage is exploited the following identity: $$\frac{1}{ABC}=2∫_{0}^{1} dxdydzδ(x+y+z-1)\frac{1}{(Ax+By+Cz)^{3}}$$ for every $A,B,C\in\mathbb{C}$....
Filippo's user avatar
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Strong & weak solution of $xy' + y = 0$ [Friedlander ex. 2.3]

The ODE $xy' + y = 0$ has no strong solution over $\mathbb R$ but has solution $y(x) = \begin{cases} c_1/x & x<0 \\ c_2/x & x>0 \end{cases}$ over $\mathbb R^*$, which may equivalently be ...
phaedo's user avatar
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A reference for the definition of Dirac delta function $\delta (\cdot ,\cdot)$

Assume that $\mu$ a quasi invariant measure on a homogenous space $M$. Pick $k\in M$. The Dirac distribution $\delta_k$ at $k$ associated with $\mu$ is defined as the distribution such that $\langle \...
Mahtab's user avatar
  • 763
1 vote
1 answer
232 views

Derivative of expected value

Let $X\geq 0$ be a real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the derivative of $$F(z)=E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$ $X$ and $Y(X)$ have no ...
marc's user avatar
  • 242
3 votes
2 answers
130 views

Second derivatives of 1/r

Let $r = \sqrt{x^2 + y^2 + z^2}$. From the fact that $\nabla^2 r^{-1} = -4\pi \delta^{(3)}(\vec{r})$, is it correct to say that $$ \frac{\partial^2}{\partial x^2}(r^{-1}) = \frac{3x^2 - r^2}{r^5} - \...
Bio's user avatar
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1 vote
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Proof of space white-noise limit for Milstein algorithm

I was trying to proof Milstein Algorithm so I read the book "Stochastic Numerical Methods_ An Introduction for Students and Scientists" of Toral. Here, they show all the proof in one ...
Daniel's user avatar
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2 votes
1 answer
202 views

Fundamental solution of Poisson equation on the torus

The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. this MathSE post): \begin{align} &\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y) \quad \Rightarrow ...
Quillo's user avatar
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2 votes
1 answer
109 views

Evaluating the integral $\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$

I want to evaluate the following integral $$\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$$ According to the book where I found the exercise, the answer is $$C\Theta(1-v^2)\sqrt{1-v^2}$$ ...
William's user avatar
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0 votes
2 answers
117 views

Inverse Mellin transform of the Mellin transform of the binomial PMF

The probability mass function of the Binomial distribution is given by $$\begin{equation} f(x)=\binom{n}{x} p^x (1-p)^{n-x}, \end{equation}$$ where $p \in [0,1]$ and $x=\{0,1,\dots,n\}$ (finite ...
Efficiency's user avatar
4 votes
2 answers
79 views

What is the limit (as a distribution) of this Fourier series, similar to that of a Dirac Delta?

I know that a representation of the Dirac delta function is $\sum_{n=-\infty}^\infty e^{inx}=2 \pi \delta(x)$. I am trying to figure out if the series with positive $n$ only $\sum_{n=0}^\infty e^{inx}$...
Fabio Caceffo's user avatar
-1 votes
2 answers
90 views

Find the solution to the given derivative of the product: $\big(\cos{x}\delta{x}\big)^{(k)}$

Find the solution to the given derivative of the product $$\big(\cos{x}\delta{x}\big)^{(k)}$$ We need to use the differentiation formula for generalized functions: $$\big(D^\alpha f, \phi)=(-1)^{|\...
Superunknown's user avatar
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