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
2answers
67 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}^{\...
0
votes
0answers
57 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
vote
1answer
90 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\...
0
votes
1answer
121 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
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 ...
1
vote
2answers
93 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
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}^{...
4
votes
1answer
123 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
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 ...
3
votes
3answers
162 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)?...
0
votes
1answer
69 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 ...
1
vote
1answer
61 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 ...
0
votes
2answers
219 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 ...
1
vote
0answers
22 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 ...
0
votes
0answers
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 $$\tag{1}\rho(z)=\int\delta(z-\bar{z}(z_0))\rho_0 (z_0)dz_0$$ ...
6
votes
0answers
97 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 ...
0
votes
1answer
126 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, $\...
2
votes
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 ...
0
votes
0answers
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_{\...
1
vote
2answers
92 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 ...
4
votes
2answers
136 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 $n$-...
1
vote
1answer
63 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 $\...
2
votes
1answer
135 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 ...
1
vote
1answer
119 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 $\delta_x(E)=0$ ...
0
votes
2answers
189 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\...
-1
votes
1answer
81 views

Limit of The Dirac Comb

The Dirac comb function with period T is: $$ f(t,T):=\sum_{k=-\infty}^{k=\infty}\delta(t-kT) $$ What is the limit of: $$ \lim_{T\to0} f(t,T) $$ ?
0
votes
1answer
31 views

A function in $\textit{D}(\mathbb{R})$

I was solving an exercise and I faced these 2 parts 1) Prove that there exist a function $\gamma \in D(\mathbb{R})$ such that $\gamma(0)=0$ and $\gamma'(0)=1$. I tried for this part the function $$\...
0
votes
2answers
203 views

Continuity and Differentiability of Step function?

All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How ...
2
votes
1answer
74 views

Spherical Laplacians on an Exponential

I looked around a bit and couldn't find a resolution to this. I was curious about the scalar function $u(r) = e^{-r}$ with $r \in [0,\infty)$ and acting Spherical Laplacians on it. $$\Delta u(r) = \...
0
votes
1answer
67 views

Problem in limit involving Dirac Delta

I want to compute the following integral: $$ \lim_{\epsilon\rightarrow0}\int_{-2}^{\infty}f(x)\phi\left(\frac{x+\epsilon}{\sqrt{2\epsilon}}\right)dx\tag{1} $$ where $f(x)=\frac{1}{1+\exp\left\{ -x\...
2
votes
0answers
52 views

Hidden Fourier multiplier in integral expression?

After some (formal!) manipulations I stumbled upon the following expression: $$ \hat{f}\left(\xi,\eta\right)=\iint_{\mathbb{R}^{2n}}e^{2\pi i\left\langle x,t-\xi\right\rangle }e^{2\pi i\left\langle \...
2
votes
1answer
75 views

Delta derivative distribution identity?

It is easy to show that the Dirac $\delta(x)$ distribution satisfies the distributional identity $$\delta(x) = - x \delta'(x).$$ Can we conclude that the following also holds $$\delta'(x) = - \frac{\...
1
vote
1answer
108 views

Delta function with 2 variables

Suppose we have a double delta function: $$\delta(ax+by+c) \delta(Ax + By + C)$$ where $ax+by+c=0$ and $Ax + By + C=0$ has a solution $x=x_0, y=y_0$. Is it possible to simplify the double delta ...
1
vote
0answers
37 views

how to derive Laplace transform, delta function?

This is an awfully specific question that is extremely similar to previous question (linked below). This problem utilized a quality of the delta function that allows an indefinite integral to be ...
0
votes
1answer
74 views

Why $\lim_{\varepsilon \to 0}\int _{[0,1]}\delta _\varepsilon =1/2$ and not $1$ if $\delta (A)=\lim_{\varepsilon \to 0}\int_A \delta _\varepsilon $?

I asked here a question a Dirac $\delta $ function, and Surb in Hamza Boulahia answers says that $$\delta (A)=\lim_{\varepsilon\to 0 }\int_A \delta _\varepsilon (x)dx,$$ for $$\delta _\varepsilon (x)=\...
0
votes
2answers
53 views

Understand $\delta $ function, why do me write $\int \delta (x)dx$ instead of $\int d\delta $ ? Since $\int \delta (x)dx$ should be $0$

I have difficulties to understand delta function. We defined $\delta $ function as $$\delta (x)=\begin{cases}\infty&x=0\\ 0&\text{otherwise}\end{cases},$$ and $$\int_{\mathbb R}\delta (x)dx=1.$...
2
votes
1answer
66 views

Necessary and sufficient condition for $a_n\delta_n\to0$ in $\mathcal{S}'$; example of sequence convergent in $\mathcal{D}'$ but not in $\mathcal{S}'$

The sequence $(a_n\delta_n)_{n\in\mathbb{N}}$, where, for each $n\in\mathbb{N}$, $a_n$ is a complex number and $\delta_n$ the Dirac delta translated of $n$, i.e. $\langle \delta_n,\phi\rangle=\phi(n)$ ...
0
votes
0answers
74 views

How to plot the graph of this expression which involves dirac delta function?

I Was Doing a Problem on Electrostatics which required finding the charge density from the given electric field and then plot a graph of the charge density. I was able to find the charge density which ...
0
votes
1answer
49 views

Solving Differential equation with g(t) involving Dirac Delta function

$y"+2y'-15y=6\delta(t-9), y(0)=-5, y'(0)=7$ Solutions to this differential equation is $f(t)=\frac18e^{3t}-\frac18e^{-5t}, g(t)=\frac94e^{3t}+\frac{11}{4}e^{-5t},Y(s)=\frac{6e^{-9s}}{(s+5)(s-3)}-\...
0
votes
1answer
244 views

Confusion with the integral of Dirac delta function

Is the integral of delta function a scalar or a function u(t)? Imagine integral of Dirac delta between minus infinity to plus infinity; or from minus infinity to a particular time x. In some texts ...
3
votes
1answer
216 views

Finding the laplace transform of $\delta(t^2-3t+2)$

I have to find the laplace transform of $\delta(t^2-3t+2)$. The answer is : $e^{-s}+e^{-2s}$ I tried using the definition of laplace transform : $$\int_{-\infty}^{\infty}\delta(t^2-3t+2)e^{-st}\...
0
votes
1answer
51 views

How to integrate a delta distribution function by changing variable?

I'm considering the integral $$\int_0^\infty (\frac{1}{mc})^2dp \;\delta\left[\frac{p}{mc}-\sqrt{\frac{\nu}{B}}\right]$$ If I change the variable of the $\delta$ function, such that $$u=\frac{p}{...
0
votes
2answers
61 views

Integrating a delta function

I'm trying to express a flux $F$ in terms of frequency $\nu$ using: $$F=\frac{2 \pi}{D^2}\frac{a_0}{\sqrt{a_1\nu}}B^\frac{3}{2}Q\int p^2 dp \int r^2 (\frac{p}{mc})^2p^{-4}dr\;\delta\left[\frac{p}{mc}-...
-1
votes
1answer
52 views

proof of Delta dirac sifting [closed]

I need to proof this expression : $\int_{-\infty}^\infty \delta(x-a)f(x) \, dx=f(a) $ Starting with this one: $\int_{-\infty}^\infty \delta(x)f(x) \, dx=f(0) $ Thanks in advance
1
vote
0answers
241 views

Numerical integration with delta functions?

I would like to integrate the following numerically: $\int\limits_{-\infty}^{+\infty} \int\limits_{-\infty}^{+\infty} \delta(x^2+y^2-1) dx dy = \pi$ I could replace the dirac delta function with a ...
3
votes
0answers
61 views

Limit distribution equal to Dirac delta

This is the problem 6.19 from the book Distributions and Operators, Gerd Grubb. I already have done parts (a) and (b). The part (a) of this problem is proving that for $r\in(0,1]$, the sequence $$\{\...
1
vote
1answer
1k views

Fourier Transform for Cosine-Squared

I'm having trouble finding the Fourier transform of $g(t) = \cos^2{a x}$. I know the answer has to be a summation of $3$ dirac delta functions, but I'm having trouble showing this. I'll show you ...
3
votes
1answer
130 views

Solve $f'(t)=0$ and $f'(t)=1$ using Fourier transform

I'm trying to solve $f'(t)=0$ and $f'(t)=1$ using Fourier transform, but no luck: a) $f'(t)=0$ $$ f'(t)=0 \Rightarrow jwF(w)=0 \Rightarrow \begin{cases}F(w)=0 ~ \text{if} ~ w \ne 0\\ F(0) = \text{...
5
votes
1answer
312 views

Why delta function is a tempered distribution?

Tempered distribution $T\in \mathcal{S}'$ is a continuous linear functional defined on the space $\mathcal{S}$ of infinitely differentiable test functions $f(x)$ on $\mathbf{R}$ with finite norm $$ ||...
1
vote
0answers
58 views

Limit of integral involving Dirac Delta

I have to compute the following integral $$\lim_{\epsilon \rightarrow 0} \int_a^\infty \delta_\epsilon (x) f(x) dx$$ where $\delta_\epsilon$(x) is such that $\int_{-\infty}^\infty \delta_\epsilon (x)...