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Questions tagged [dirac-delta]

Questions involving the Dirac delta function, either in the informal sense, or in the distribution sense.

2
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1answer
35 views

A puzzle with derivative of delta-functions

I will assume as a given the fact that in terms of complex variables $z,\bar{z}$ the following formula holds (normalization is not essential) $$\partial_{\bar{z}}\frac{1}{z}=\delta(z)$$ Then, by the ...
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0answers
17 views

Green’s function of an ODE [on hold]

How to compute Green’s functions for $$-u^{\prime\prime}+k^2u=0, -\infty<x<\infty$$
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0answers
18 views

How to handle a differential equation whose RHS contains the derivative of the Dirac delta function?

I have a differential equation of the following form. \begin{align} \begin{split} \frac{\mathrm{d}^4\psi(\eta)}{\mathrm{d}\eta^4}-\beta^4\psi(\eta)=\psi'(\zeta)\,\delta'(\eta-\zeta) \...
2
votes
1answer
52 views

question on integration of $f(x)\delta(g(x))$

I am learning integration with dirac delta. I do not understand this result by Maple Dirac When I work it by hand, using known relation (here is an image of the page) Therefore $$ \int_{0}^{\pi}f\...
3
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2answers
87 views

Double integral involving the derivative of a delta 'function': contradicting results

I'm trying to evaluate an integral of the form: $$I\equiv \int_{-\infty}^{\infty}dx\space f(x)\int_{-\infty}^{\infty}dy\space h(y)\int_{-\infty}^{\infty}d\omega\space\omega e^{i\omega(x-y)}$$ with $...
3
votes
1answer
45 views

Integration with Dirac delta function of two-argument function

I would like to solve that problem: $$ \int d^2 \mathbf{k} \, \delta(f(k,\phi)) = \int_{k_1}^{k_2} dk \, k \int_0^{2\pi} d\phi \, \delta(f(k,\phi)) \,, $$ where $f(k,\phi) = a - bk^2 - ck^3 |\sin(2\...
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0answers
28 views

Bessel function and delta of dirac

I'm trying to make sense of the next integral, I don't know how to proceed when I take into account the value of $m$ specially when it is equal to 0, and$ n =0 $ as well. $$J_{m}(\rho)=\sum^{\infty}_{...
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0answers
33 views

How to deal with a Dirac delta function numerically?

I need to solve some differential equations with a Dirac delta function. e.g. the source terms are like, $\delta(x)$ and $\partial_x \delta(x)$. Could I just use the Gaussian type function? e.g. $$...
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0answers
10 views

Derivative of Probability Transform formula

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a Borel measurable function. (One may further assume that $|\frac{\partial f}{\partial z_i}|>0$. $\overrightarrow{Z}=(Z_1, Z_2, \dots, Z_n$ is a ...
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1answer
24 views

Integral involving an impulse function (no Laplace) [closed]

Integrals Problems Please, help me solve these problems without using Laplace. So on http://mathworld.wolfram.com/DeltaFunction.html I found these fundamenetal properties that I thought I could use ...
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4answers
57 views

Solving $y' + ay = \delta$ in $S'$ space

I am solving ordinary differential equation in $S'$ (dual to Schwartz space) given as: $y' + ay = \delta$, where $\delta$ is a Dirac delta function. The general solution of homogenous equation is $...
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1answer
27 views

Need help in finding integral value of $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} dx \int_{-1}^{1} dy\ \delta(\sin2x)\ \delta(x-y)$?

I am trying to find value of the following integral: $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} dx \int_{-1}^{1} dy\ \delta(\sin2x)\ \delta(x-y)$ I have some experience in solving integrals that contain ...
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0answers
37 views

Initial condition Impulse Answer of a first order system

Given the following first order system $\tau y'(t) + y(t) = ku(t)$ The LaPlace transform yields $\tau [s y(s) - y(t=0)] + y(s) = ku(s)$ Given $y(t=0) = 0$, this simplifies to $y(s)[\tau s + 1] = ...
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0answers
22 views

Optimal Control: Proof of Adjoint Lagrange method?

Consider an optimal control problem: Let $x_0\in\mathbb{R}^n$, $f\in C^{0,1}([0,T]\times (\mathbb{R}^n\times \mathbb{R}^m),E)$ be bounded, let the state equation be $$\dot{x}(t) = f(t,x(t),u(t)),\, t\...
0
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1answer
56 views

Explanation of solution: integral calculation with Dirac $\delta$ ( Fourier transform)

Can someone explain to me this solution : $$\int_{-\infty}^{\infty}\delta (\tau -\frac{1}{4}t)e^{-j2\pi f_dt}dt=e^{-j2\pi f_d4\tau }$$ When I have solved, I got $e^{-j2\pi f_d4t }$ But the answer ...
2
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1answer
48 views

Numerical solution of ODE with Delta function

I want to model a dynamical system of the form $\frac{\text{d}x}{\text{d}t} = f(x)+nx\delta(\pi(t-0.2)). $ The problem is that I have a point source which is reoccurring at fixed time steps (say at ...
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0answers
22 views

Dirac measure on periodic orbits for Hamiltonian Dynamcis

An iterative map $T: x_{n+1} = T(x_n)$, given an initial state $x$, defines a periodic orbit: $O_x=\lbrace x, S(x), ...,S^{n-1}(x) \rbrace$ , with $S^n(x) = x$. The periodic orbit supports a measure ...
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1answer
89 views

Filtering property of Dirac $\delta$ function

Evaluate $$\int_0^\infty f(x)\delta(x-1)dx$$where$$f(x)=\begin{cases}x^2,&0\le x<1\\\sin 2,&x=1\\x,&x>1\end{cases}$$ Attempt Since the function is discontinuous at $1$, I couldn't ...
6
votes
2answers
94 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: ...
0
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0answers
38 views

Integral of dirac-delta times the log of dirac-detla

$$ \int_x \delta(x)~\ln(\delta(d))~dx = 0 ? $$ Where $\delta(x)$ denotes the Dirac-Delta function, $ln(\cdot)$ is a logarithm, and $dx$ is simply the diferential of $x$ for the integral. I'm ...
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1answer
36 views

How to solve the differential equation with delta function on right hand side

I have a differential equation of the form below. How to solve this. It is actually a fourth order beam equation with a derivative of the delta function. I have solved for just delta function in the ...
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2answers
56 views

Can someone explain Dirac Distribution?

I have been reading book on Deep Learning and in the chapter of probability and information theory I found this topic "Dirac Distribution and Empirical Distribution". It says: In some cases, we wish ...
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2answers
58 views

Lebesgue-integrability of the Dirac delta function?

I stumbled upon this question, which I think was answered incorrectly. Considering the Dirac delta function $$ \delta\colon\mathbb R\to[0,\infty],x\mapsto\begin{cases}\infty,&\text{if }x=0,\\0,&...
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0answers
21 views

For $\delta(x) = \lim_{e \to 0} \mu(x,e)$ definition, express $\mu(x,e)$ in terms of Bessel functions

Based on the definition of Dirac Delta as: $\delta(x) = \lim_{e\to 0} \mu(x,e)$ Is it possible to obtain an expression of $\mu$ as a series of Bessel's functions $J_n$ or in which satisfy the ...
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3answers
76 views

What's the deal with this $\frac1\pi$?

I recently learned about the very interesting Dirac Delta function, defined as $$\delta(x)=\frac1\pi\lim_{\epsilon\to 0}\frac{\epsilon}{x^2+\epsilon^2}$$ Which is a very majestic definition, as the ...
2
votes
2answers
39 views

Function of a dirac delta

I would like to know if it is possible to compute something like $$\int_{-\infty}^{\infty}f\left(\delta(x-a)\right)dx,$$ where $f(x)$ is a function, or if it is even defined. Thanks in advance!
2
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1answer
45 views

non-constant coefficient Differential Equation with Dirac delta (unsure how to properly write the solution)

$\zeta$ is a constant, $g(M) < M$, $g^{-1}(\zeta) > 0$, and the equation is: $$ a'(M) - a(M)\frac{g'(M)}{M-g(M)} + \frac{\delta_{g^{-1}(\zeta)}(M)}{M-g(M)} = 0 $$ Things can be left in terms of ...
0
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1answer
42 views

Heat equation PDE (nonhomogeneous); Green's function; Dirac delta

(Sorry for the messy title, trying to include the keypoints of the problem.) I am new to the theory on how to solve this kind of PDE problem which is presented below; I am unsure on which method to ...
0
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1answer
31 views

Example of a compactly supported $\delta$-sequence.

Recall that a $\delta$-sequence can be defined as a sequence, $(\phi_n)_{n\in\mathbb N}$, of continuously differentiable, non-negative, real-valued functions for which $\int_\mathbb R\phi_ndx=1$ for ...
2
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0answers
42 views

show that $\delta''(\frac13x^3 +x) = \delta''(x) - 2\delta(x)$

I tried to go about about this using the definition : let $\phi$ be a test function $$\begin{align}\langle \delta''(\frac13x^3 +x) ,\phi\rangle &= \langle \delta''(\frac13x^3 +x) ,\frac{x^2+1}{x^...
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1answer
44 views

Impulse response of integrator

I want to get the impulse response of an LTI system where $$y(t) = \int_{t-2T}^{t-T} x(\alpha) d\alpha $$ To solve this I did: $$h(t) = \int_{t-2T}^{t-T} \delta(\alpha) d\alpha $$ Then you see that ...
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1answer
20 views

Integrating the composition of a Heaviside function with a smooth function

I am trying to find how to compute an integral of the form: $\int_{R^n}{\Theta(g(x))f(x)\,dx}$, where $\Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is ...
2
votes
1answer
40 views

Integral involving Dirac delta $\delta(ax-b)$

I am trying to evaluate the integral $$\int_{-\infty}^{\infty} f(x) \delta(ax-b) \, dx$$ for $a\neq0$. From what I was taught, I would expect the answer to be $f\bigl(\frac{b}{a}\bigr)$ since $a\cdot\...
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1answer
32 views

Need help about references for 2D delta “function”

I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true: $$\left( \frac{\partial^2}{\partial x^2} + \frac{\...
1
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1answer
61 views

Computing (distributional) gradient of a singular function

This question could well belong better to the physics stackexchange, but I'm hoping that posting it here could give me a more mathematical perspective. I am trying to find the expression for the ...
1
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1answer
25 views

Are these distributions the same?

Consider the following distribution, where $\delta$ is the Dirac delta: $$f(x,y)=\delta(x)+\delta(y).\tag1$$ This can be viewed as a limit of the following sequence of smooth functions: $$\...
1
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1answer
52 views

Integral evaluation with delta Dirac

I am having doubts about the following integral:$$\int \limits _{0}^{10} \int \limits _{0}^{10} \frac{x^2y^2}{(x^2+y^2)^{5/2}}\ \delta(x)\ \mathrm{d}x\mathrm{d}y$$ If we apply the definition of the ...
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votes
1answer
67 views

Dirac Delta and its evaluation in a complicated integral

I would like to better understand how to use and manipulate the Dirac Delta function. It seems to me that whenever the delta function appears in an integral, it reduces the dimension of the domain of ...
7
votes
3answers
147 views

How to solve a second order partial differential equation involving a delta Dirac function?

In a mathematical physical problem, I came across the following partial differential equation involving a delta Dirac function: $$ a \, \frac{\partial^2 w}{\partial x^2} + b \, \frac{\partial^2 w}{\...
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0answers
102 views

Fourier transform of the convolution of a Dirac comb with the product of a complex exponential function and a rect function

Straight to the math question: How can I calculate the following 1-dimensional spatial Fourier transform? $\frac{1}{2\pi}\int_{-\infty}^{\infty}\left(e^{i(n-1)k_0\frac{x^2}{2R}}\mathrm{rect}(x/w)*\...
2
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2answers
72 views

Integrating over Dirac delta

TL;DR: How does one show that $(\clubsuit)$ holds. Some context and how I arrived at my problem. In a Thermodynamics problem set I was asked to calculate the partition function of the ideal gas in ...
3
votes
2answers
64 views

Difficult Fourier integral giving a distribution

I would like to understand the distribution defined by $$ b(x)=\int_{-\infty}^{\infty}\lvert y\rvert e^{-ixy} dy $$ What I've understood so far is that $$ b(x)=\lim_{\alpha\to0^+}\int_{-\infty}^{\...
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0answers
49 views

I am evaluating the fourier transform of a function + a constant: $\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx$ equals what?

I am evaluating the fourier transform of a function plus a constant $c$: $$\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx.$$ As a result, I should get the fourier transform of the function ...
1
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1answer
58 views

Double integral of Dirac delta distribution with more than one root

I found a double integral involving a Dirac distribution of a sine function, $\int_{-1}^{1} \Big( \int_{0}^{2\pi} g(\theta,\epsilon)\delta(\epsilon-\frac{1}{3}\sin\theta)d\theta\Big)f(\epsilon)d\...
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1answer
65 views

Dirac delta function + constant [closed]

If we sum the Dirac delta function with a constant, what is the result? I.e., $k+\delta(x)$, where $k$ is a constant.
2
votes
1answer
46 views

Equality between two equations

at the moment I am reading the following paper Benno, Steven A., and José MF Moura. "On translation invariant subspaces and critically sampled wavelet transforms." Multidimensional Systems and ...
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vote
2answers
61 views

Delta distribution and the Schrödinger equation

While studying the lecture notes of my quantum mechanics course I came across something that seemed a bit odd. There we want to solve the Schrödinger equation for the potential $V(x)=V_0 \delta(x)$, ...
2
votes
1answer
59 views

Higher order derivatives of Composition of Dirac delta distributions

There are two equations showed in Gel'fand and Shilov's book (Generalized Functions I Properties and Operations) on page 183 and 185: $${\delta}^{(k-1)}(1-x^2)=\frac{(-1)^{k-1}}{2^kx^{k-1}}[{\delta}^{...
4
votes
0answers
67 views

Traffic flow with Dirac-$\delta$ source (on ramp)

I have been trying to solve the traffic flow equation with a singular source ($D>0$ large): $$ \rho_t + f(\rho)_x = D\delta(x) $$ with the flux $f(\rho)=\rho(1-\rho)$ and the initial data $\rho(x,0)...
0
votes
1answer
21 views

An exemple of integral of distributions

Need to solve this integral: $$I=\int_{-1}^{1}dx(\lim_{\varepsilon\to 0^+}\frac{\varepsilon}{\varepsilon^2+x^2}f(x)+\pi\vartheta(x)\frac{df(x)}{dx}(x)) $$ I think I should recognize the limit as a ...