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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17 views

The fundamental causal solution $g=\delta(t-r/c)/4\pi r$ satisfies the wave equation $\nabla^2 g - (1/c^2) g_{tt}=-\delta^3(r)\delta(t)$

I am trying to show that the fundamental causal solution g=δ(t-r/c)/4πr satisfies the 3d wave equation (del^2)g - (1/c^2)*gtt=-δ^3(r)δ(t) If I substitute the solution to the wave equation in ...
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21 views

Multiplying the dirac delta distribution by a function

In the theory of distributions, if $H(x)$ is the Heaviside function, we have seen that the distributional derivative of $H$ can be found as follows, for a test-function $\varphi$: \begin{align*} \left\...
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28 views

Dirac delta function composition rule

after reviewing the properties of the Dirac delta function, I have a hard time figuring out one property: the composition rule. Composition rule I understand the integral form comes from a change of ...
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121 views

Fourier transform of non-radial function in spherical using delta function

I'm trying to take the integral of non-radial function in spherical coordinates. $$\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty} \frac{x^2}{r^3} e^{2\pi i \vec{r}\vec{p}}dx dy dz= \int_{0}^{...
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23 views

Replacement of the delta function, Free-space Green Function

The following is how to find $G_{i j}$ in the free-space infinite unbounded flow from C. POZRIKIDIS's Boundary Integral and Singularity Methods for Linearized Viscous Flow: Replace the delta ...
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52 views

How can I compute the integral of the product of two Dirac delta and a polynomial?

$$\int_{0}^{\infty}dk~k^{d-2}\delta(k-a)\delta(k-b).$$ I tried substituting $k^{d-2}\delta(k-a)$ with other espressions such as $\frac{d}{dk}\biggl[k^{d-2}\Theta(k-a)\biggr]-(d-2)k^{d-3}\Theta(k-a)$ ...
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1k views

Integral of delta distribution in spherical coordinates

We know that, if $\mathcal D$ is a domain containing the origin $(0,0,0)$, then $$\int_{\mathcal D} \delta(\vec r) d \vec r= \int_{\mathcal D} \delta(x) \delta(y) \delta(z) dx dy dz=1$$ However, we ...
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32 views

Legendre Transformation of the functional $F=(\phi(x))^n$

I've been asked as a Homework to find Legendre Transform of the functional $F[\phi(x)]=\phi^n(x)$ $(i.e. (\phi(x))^n)$. If Legendre transform control function be $J(x)$, then I know that the L.T. ...
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22 views

Derivative of Heaviside step function multiplied by an exponential

Let $f(y)=e^y\mathscr{H}(y)$ where $\mathscr{H}$ is the Heaviside step function. We know that the derivative of $\mathscr{H}$ is given by $$\frac{d}{dy}\mathscr{H}(y)=\delta(y),$$ Then how come, in ...
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62 views

Derivative of Dirac delta distribution

How is the derivative of $\delta(x-y)$ with respect to $x$ related to the derivative with respect to $y$? I suspect they differ by a minus sign but I'm not sure. Both $x$ and $y$ are real variables.
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36 views

Solving Dirac delta function for $ \int_{-\infty }^{\infty}e^{x}\delta (x^{2}-2x)dx $

I have the next equation:$ \int_{-\infty }^{\infty}e^{x}\delta (x^{2}-2x)dx $ The solution for this equation is: $\int_{-\infty }^{\infty}e^{x}\delta (x^{2}-2x)dx= \int_{-\infty }^{\infty}e^{x}[\...
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1answer
50 views

Problem with an integral in statistical mechanics

I have problems with this integral: There are N identical particles contained in a circle with radius R with Hamiltonian $H=\frac{p^2}{2m} -Aq^2$, A is constant. Now the integral i want to calculate ...
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162 views

Are mixed derivatives of $1/r$ a distribution?

So, in three-dimensions we famously have the result that the Laplacian acting on $1/r$ is a distribution: $$\vec{\nabla}^2\frac{1}{4\pi r}=-\delta^3(\vec{r})$$ where $\delta^3(\vec{r})$ is the Dirac-...
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27 views

Integral exp(ax+H(x))

How I can calculate $$\int e^{(ax+bH(x))} dx $$ Where H is the Heaviside function.
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27 views

$X$ r.v of a uniform law, find cdf of $Y=\min(X,a)$ $(a \in [0,1])$ prove the law of Y is linear combination of a density and a Dirac measure

Problem : Let $X$ be r.v following a uniform law on $[0,1]$, find the cdf of $Y=\min(X,a)$ $(a \in [0,1])$ and prove that the law of Y is linear combination of of a law with a density and a Dirac ...
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230 views

Laplacian of $1/r$ in a tensor

As we know the $$\nabla^2(1/r) =- 4 \pi \delta^3(r).$$ However, I recently was readling an hydrodynamic book (An introduction to dynamics of colloids By J.K.G Dhont). The Oseen tensor is defined as: ...
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1answer
44 views

Dirac delta multiple integral bounds

I'm struggling to understand the bounds on integration region after performing integral over Delta function. Correct result from book: $$ \int_0^1dz \int_0^1 dy \int_0^1 dx \delta(x+y+z-1) = \int_0^...
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136 views

Can we integrate the dirac delta of an a.e. discontinuous function?

Let's say I have a real valued function $f$ and a subset $S \subset \mathbb{R}$. Let's also define the dirac measure to be $ \delta_{x}(S) = \begin{cases} 1 & x \in S \\ 0 & x ...
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34 views

Proof that $\int\delta(t)\cos(t)dt = 1$ using integration by parts

According to the properties of the Dirac Delta function $\int\delta(t)\cos(t)dt = \cos(0) = 1$. I tried proving this using integration by parts but got stumped: $ \require{cancel} \int_{-\infty}^{\...
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Why does integrating a complex exponential give the delta function?

How come, when we integrate a complex exponential from $ -\infty $ to $ \infty $, we get a scaled delta function? $$ \begin{align} \int_{-\infty}^{\infty} e^{i k x} \; dk & = 2 \pi \delta \left ( ...
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66 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?
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29 views

Fourier transform of delta distribution satisfies $ \hat\delta_{x-x_0} = \hat\delta_{x_0} $

Let $ \delta_{x_0} $ be the distribution defined by $ \delta_{x_0} = \varphi (x_0) $, $ x_0 \in \mathbb{R^n} $. One can show that it has compact support, so in particular $ \delta \in \mathcal{S}'$, ...
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3answers
66 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 ...
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1answer
20 views

Three dimensional delta function property

We know that delta function has the amazing property of $\int_{a}^{b} δ(x-X) f(x) $ = $f(X)$ So, can it be carried over to three dimensions? $$\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} δ(x-X)δ(y-Y)δ(...
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33 views

Application of the Lebesgue integral with the Dirac measure

Say that $(X,\mathcal{F}, μ)$ is a $\sigma$-finite measure space and $f:X→ℝ^+$ is $\mathcal{F}$-measurable and nonnegative. Given the Lebesgue measure $\lambda$ on $(ℝ,\mathcal{B}(ℝ))$, I want to show ...
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38 views

Green's Function for the Laplacian in 3D

Does anyone know where to find a good resource for solving for the Green's Function of the Laplacian in 3D or tips on where to start? $$\nabla^2G(\boldsymbol{x,x_0})=\delta (\boldsymbol{x,x_0})$$
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31 views

Does my proof show that the sequence of measures defined using mollifiers with shrinking support converges weakly to the Dirac measure?

Following Evan's PDE book, Appendix C4, PP. 629, let's define the function: $\eta(x):=C\exp\left(\frac{1}{\|x\|^2 - 1} \right) \forall x \in \mathbb{R}^d$ when $\|x\|\leq 1$ and $0$ otherwise, where $...
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24 views

what's the solution to the following Kepler-like problem

CONTEXT I'm working with a problem tangentially like the Kepler problem [1]. I'm trying to make sense of eccentricities $e$, where $e = -1$. There are some thoughts about such eccentricities [2]. ...
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15 views

Multivariate Integration with respect to Dirac Measure

Let $n\geq 2$, $A = \{x\in [0,1]^n|\exists t\in[0,1]: x = t\cdot (1,...,1)\}$, $B$ a Borel-measureable set and $f:\mathbb{R}^n\to\mathbb{R}$ a measureable function. I can not make sense of the ...
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53 views

Can we apply the Dirac Measure to itself in the Lebesgue Integral?

Say we have some function $f$ on the reals and a Borel set $\sigma$ from the $\sigma$-algebra of $\mathbb{R}$. My understanding from https://en.wikipedia.org/wiki/Dirac_measure is that the Dirac ...
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1answer
24 views

What property of $\delta(x)$ is used here?

In the solution of an exercise, there is the following equality: $$ \sum_{-\infty}^{\infty} \delta(\theta - \frac{1}{4} - \frac{k}{2}) = \sum_{-\infty}^{\infty} \delta(\theta - \frac{1}{4} - k) + \...
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58 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.
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62 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 $...
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32 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 \...
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1answer
34 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 ...
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1answer
112 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 (...
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281 views

A Differential Equation with Dirac Delta as the Non-homogeneity Term

I'm currently studying Computational Mechanics, and an important step to solve problems is knowing how to express physical interactions in terms of mathematical equations. Consider for example, the ...
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37 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)...
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3answers
54 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 ...
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1answer
42 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 ...
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38 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} \...
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217 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 "...
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2answers
529 views

First order ODE with Dirac delta funtcion

I am looking for a direct method to solve this first order ODE with Dirac delta funtcion $$\frac{dU(t)}{dt}+k^2U(t)=\frac{1}{\sqrt{2\pi}}\delta(t)$$ with the initial condition $U(0)=\frac{1}{\sqrt{2\...
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441 views

Solution to ODE with Dirac Delta satisfies ODE

I am working on a problem where I have the following ODE. $$m\dot{v}+bv=\delta_I(t)$$ where $$\delta_I(t)=\begin{cases}0, & \text{for}&t\ne0\\ I, & \text{for} &t=0\end{cases}.$$ The ...
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2k views

Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

I have a question about this integral with a Dirac delta $$ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $$ by integration by parts I get; $$ \displaystyle \delta(x-3)e^{x^2}\biggr\...
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100 views

Can we formulate a probability density distribution (not function!) for every probability measure?

I know that not every probability measure admits a probability density function, where a function is a mapping $f: X \to \mathbb R$. However, there is something, such as the Dirac delta distribution,...
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26 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)=\...
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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) (\...
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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 $$g(x)=\frac1{f(x)}-1$$ ...
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1answer
58 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 ...

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