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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104 views

Surface Integral of Vector Field

Given the scalar field $$\phi(\vec{r})=\frac{1}{|\vec{r}-\vec{a}|},$$ where $\vec{a}=(-2,0,0)$, and the corresponding vector field $$\vec{F}(\vec{r})=\operatorname{grad}\phi,$$ as well as the surface $...
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1answer
227 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
63 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
305 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
145 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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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
83 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 ...
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2answers
55 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!
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1answer
58 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 ...
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1answer
107 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 ...
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38 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 ...
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50 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
224 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
35 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 ...
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1answer
125 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
39 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{\...
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1answer
93 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 ...
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1answer
31 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: $$\...
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1answer
75 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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2answers
97 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 ...
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3answers
174 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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217 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)*\...
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2answers
124 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 ...
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2answers
66 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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56 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 ...
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1answer
85 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
112 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.
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1answer
48 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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2answers
88 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 $V(x)=V_0 \delta(x)$, ...
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1answer
69 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}^{...
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1answer
119 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)...
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1answer
22 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 ...
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3answers
134 views

What is $\int\delta(x-y)\delta(y-z)f(y)\:{\rm d}y$?

Let $(\Omega,\mathcal A,\mu)$ be a measurable space and $\delta$ denote the Dirac delta function. If $f\in\mathcal L^1(\mu)$ and $x,z\in\Omega$, what is $$\int\delta(x-y)\delta(y-z)f(y)\:\mu({\rm d}y)?...
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1answer
66 views

Delta functions/Probability

Here in an answer they say Now note that its perfectly reasonable to have a prior that's say 2 delta functions at p=0.23 and p=0.88. Combining this prior with a likelihood coming from an ...
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1answer
59 views

Laplace Transformation of sin(t+1)*DiracDelta(t)

why is Laplace transformation of sin(t+1)*DiracDelta(t) = sin(1)? I thought: L[f(t)-u(t-a)] = e^(-as)*L[f(t+a)] and according ...
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2answers
194 views

Product of a Discrete Variable and a Continuous Variable

How would I go about obtaining the probability density function of a random variable that results in a product of a discrete variable and a continuous variable? I know that if $X$ and $Y$ are both ...
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39 views

Question on proving the inverse of the distribution function of the random variable

We can approximate the conditional distribution by a random variable with density $$ \\p\delta_0(x)+(1-p)\beta e^{-\beta{x}}1_{x>0}dx \ $$ that is to say a weighted mean between a Dirac mass at 0 ...
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21 views

Convergence of density estimates with parzen window

I am trying to understand why $lim_{||u|| \rightarrow+\infty}{\varphi(u)}\prod_{i=1}^{d}u_{i} = 0$ is necessary for convergence of Parzen density estimates. Similar question has been asked here ...
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28 views

Properties of delta function

Consider two arbitrary sets of coordinates $z_0$ and $z$ in some space, mapped via some $\bar{z} :$=$z_0 \rightarrow z$ I have a function $$\tag{1}\rho(z)=\int\delta(z-\bar{z}(z_0))\rho_0 (z_0)dz_0$$ ...
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96 views

What is $\int_{-\infty}^{\infty}\exp(\mathrm{i} n \cosh{x}) \, \mathrm{d}x$?

I'm hoping to determine the value of the following integral: $$\int_{-\infty}^{\infty}\exp(\mathrm{i} n \cosh{x}) \, \mathrm{d}x$$ Here is a plot of the integrand as a function of $x$ with parameter ...
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1answer
124 views

How to solve a Convolution Integral with one delta function.

I have a Convolution integral $$ \int_{t_0}^{t} \int_{t_0}^{\tau} C(t-t')C(\tau -t'') \delta(t''-t') dt'' dt'=\int_{t_0}^{t} C(t-t')C(\tau -t') dt'= ? $$ I do not know how to proceed any further, $\...
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34 views

Fourier transform of delta(dirac) function-multiplication with exp function

I am trying to find an integral of multiplication exponential function with a delta function. I know a property of delta function that if I would like to take the integral of the multiplication delta ...
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1answer
87 views

Calculation of some integrals

The next functions are defined: $$ f(y)=\frac{1}{1+e^{-2y}} \\ g_1(z)=\frac{1}{1+z^2},\quad g_2(z)=e^{-z^2},\quad g_3(z)=\frac{1}{cosh(z)},\quad g_4(z)=\frac{sin(z)}{z}$$ Is there a way to calculate ...
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37 views

Integral on Dirac delta with null argument

Making some calculus I end it up with the following expression: $$\int_{\mathbb{R^2}} dxdy\ \delta(x - y)\delta(x - y) = \int_{\mathbb{R}} dx\ \delta(0) \tag1$$ Taking into account that $$\int_{\...
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2answers
84 views

Limits of integration on a delta function of many arguments

I need to integrate the following expression involving a $\delta$-function $$\int_0^1 \mathrm{d}x \int_0^1 \mathrm{d}y \int_0^1 \mathrm{d}z \, \delta(x+y+z-1)$$ The textbook I'm using suggests this ...
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2answers
135 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 $n$-...
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1answer
61 views

Integral of delta function over a small interval around zero

I know that the dirac delta function, $\delta(x)$, satisfies the following properties $\int_{-\infty}^{\infty}\delta(x)=1$ and $\delta(0)=\infty$. However if I integrate the delta function over a $\...
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1answer
129 views

Is the mentioned method appropriate to solve $\int_{-\infty}^{\infty}\frac{\sin x}{x}\, dx$? [duplicate]

The integral is, $$I=\int_{-\infty}^{\infty}\frac{\sin x}{x}\, dx$$ I know the answer would be $\pi$ and I know how to solve this using Feynman's method and Fourier transform. However I was trying ...
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1answer
95 views

Finding a Hahn Decomposition involving a Dirac Measure

Let $(X,F,\mu)$ be a finite measure space, i.e. $\mu(X)<\infty$. And let $x\in X$, and let $\delta_x$ be the Dirac measure with respect to $x$, i.e. $\delta_x(E)=1$ if $x\in E$ and $\delta_x(E)=0$ ...
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2answers
158 views

Integral of Shifted Dirac Function

How to integrate this function: $$x[n] = \frac{1}{2\pi}\int_{-\pi}^{\pi}\delta(\omega-\omega_{0}) e^{jn\omega}d\omega$$ Where: $$\delta[\omega-\omega_0] = \begin{cases}1&n=\omega_o\\0&n\...