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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1answer
66 views

Dirac delta function integral over given limits

I know that $\int$$λδ(x-a)=1.λ$ with the limits $$-inf$$ to $$+inf$$ but what if the limits are $$a-ε$$ to $$a+ε$$?? I am guessing it's the same as the former limits but still not sure Can someone ...
2
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1answer
51 views

Convolution and second derivatives of Dirac Delta function

In some class notes I have found the following statement: Let $f(x)$ be a continuous funtion, $\delta(x)$ the Dirac delta function and $\ast$ the convolution operation given by $(f \ast g)(x) = \int_{...
4
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1answer
104 views

Strong definition of the radial Dirac delta function and its derivative

The G. Barton textbook “Elements of Green’s functions and Propagation,”, Oxford University Press, 1991 has a very nice introduction on the Dirac delta function. When the 3-D delta function is ...
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0answers
24 views

Integral of a delta sequence

Let us define the function $g_{n,p}(\eta) := n^{-1/p} g_0(\frac{\eta }{n})$ where $g_0$ is even, non-negative and smooth. In the limit $n \to 0^+$, $g_{n,1}(\eta) \to \delta(\eta)$. With such a ...
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1answer
137 views

Need help understanding distributions and dirac delta function

I recently came across dirac delta function and trying to learn about it has led me to learn that it is a distribution/generalized function and is not a ordinary function. But most of the explanations ...
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0answers
28 views

Dirac delta and Minkowski content measure?

In Wikipedia page of Dirac delta distribution (here), there is the generalization for a property of Dirac delta distribution given by I know this is a generalization of the case when $g(\mathbf{x})$ ...
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0answers
40 views

Solving Equations of operators with the delta function (Example)

I currently read a paper about solving Fredholm integral equations. There is a theorem given to solve them, but I cant seem to solve the given example. Theorem: The solution can be calculated by: $$...
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1answer
33 views

Solution of ODE with point sounce

I have a two-point boundary value problem $$\left(\tau u_{x}\right)_{x}=f(x), \quad 0<x<1, \quad u(0)=0, \quad u(1)=0$$ where $\tau$ is constant, is the surface tension coefficient of a ...
2
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2answers
81 views

Distributional “Antiderivative”

Suppose that $\mathbb{R}^3$, fix $f \in L^1(\mathbb{R}^3;\mathbb{R})$ and let $g \in L^1(\mathbb{R}^3;\mathbb{R}^3)$ satisfies $$ div(g)=\delta_a -f. $$ Then what is $g$? It what is the ...
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1answer
37 views

Setting up mixing problem involving Dirac delta

A salt tank initially contains $50$ gallons of pure water. A brine solution containing $\frac15$ lb/gal of salt flows into the tank at a rate of $5$ gal/min. Brine flows out of the tank at the same ...
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0answers
44 views

Does the use of $\delta(\omega-\omega_1)$ for retrieving amplitude of a single Fourier-component of a periodic function violate Fubuni's requirements?

I've got a non-negative function $f(x)=a+b\cos(\omega_1 x)$ for $a,b,\omega_1 \in \mathbb R$ and I need to represent $b^2$ as a square of the Fourier-transform of a whatever function. As a physicist I ...
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1answer
51 views

Delta function of a product

Manoukian in QFT I page 236 writes an equality of the form $$\delta\left((k^0)^2-|\mathbf{k}|^2\right)=\frac{\delta(k^0-|\mathbf{k}|)+\delta(k^0+|\mathbf{k}|)}{2|\mathbf{k}|}$$ (with $k$ the ...
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0answers
31 views

Fourier Transform of the product of two Dirac functions [closed]

How to calculate the Fourier transform of the product of two Dirac functions like $\delta (x-x_s) \delta (y-y_s)$, where $x_s$ and $y_s$ are both constant ? Thank you in advance
2
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2answers
108 views

Find Laplace transform of a function multiplied by the Dirac delta function

I know that the Laplace transform of the delta function $\delta(t-a)$ is $e^{-as}$. Now my question is to find the Laplace transform of $te^{-3(t-2)}\delta(t-2) $. Is there any theorem or easy ...
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0answers
19 views

derivative of sum of indicator functions which are functions of time

I have this function, and I want to take the derivative $S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\ I(R_j) = \begin{array}{cc} \{ & \begin{array}{cc} 0 & R_j < 7 \\ 1 & R_j ...
2
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1answer
57 views

Dirac Delta distribution (function) confusion

Given this identity $\delta(kx)=\frac{1}{|k|}\delta(x)$ is the following correct? $$\delta(ct-x)=\delta(c(t-x/c))=\frac{1}{|c|}\delta(t-x/c)$$ It seems like the impulses would be located at the same ...
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1answer
55 views

Derivative of Dirac delta of f(x)

I'm trying to prove this relation: $(\delta (f(x)))' = f'(x) \delta' (f(x)) $, where $f(x)$ is a monotone function. I just end up tangled in different derivatives of Dirac delta function or ...
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2answers
33 views

What is the best way to find the Inverse Laplace with Convolution Theorem? [closed]

I have to calculate the inverse laplace using Convolution, but I get stuck when I have to integrate with the delta dirac. $$F(s)=\frac{s}{s^{2}+2s+2}$$
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1answer
53 views

Why is the covariance of white noise not finite and not a Kronecker delta?

Suppose we have a Gaussian white noise $f(t)$. Then the covariance $\langle f(t)f(t')\rangle = c\times\delta(t - t')$ for some $c > 0$. However, since $f(t)$ is a Gaussian random variable (with ...
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2answers
134 views

Two dimensional integral involving Dirac delta

It seems to me that $$ \int_{-\infty}^\infty \int_{-\infty}^{\infty} \delta(x^2 + y^2 - R^2) dx dy $$ should evaluate to $2\pi R$, the perimeter of a circle of radius $R$, but I'm having trouble ...
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1answer
46 views

Writing a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$?

How to write a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$? For example; I read something in a book. You can find the following picture. But ...
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2answers
78 views

Integrals with Dirac delta functions in 3-D

I've been trying to make sense of these two integrals, somehow the result seems intuitive, yet very hard to compute. We define $$ f(x)=\frac{1}{4\pi}\delta(|x|-R)$$ and then note that $$ -\frac{1}{2}\...
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3answers
102 views

Where does the $2\pi$ in Fourier Transform Dirac delta identity come from?

$$ \chi( \omega - \omega ')= \int_{-\infty} ^ {\infty} dt e^{j( \omega - \omega ')t} = 2 \pi \delta ( \omega - \omega ') $$ That is the identity to proof. I have seen different ways to proof ...
0
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1answer
35 views

Integral of a function with dirac on one lower bound

Let's consider the function: $$ \int_{K}^{+\infty}f(x)dx$$ with $f(x) = \delta(x-K)$ if $x=K$ where $\delta$ is the Dirac delta function (cf https://en.wikipedia.org/wiki/Dirac_delta_function#...
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2answers
70 views

How can I numerically implement $\delta(f(x,y))$

I would like to numerically implement a Dirac Delta function whose argument is another 2 variable function. I know that I can model a Dirac Delta numerically using a Gaussian. What can I do if I want ...
3
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2answers
113 views

Schrödinger equation involving the Dirac-Delta

I am taking a course on quantum mechanics and I try to understand the time-independent Schrödinger-equation with the Delta-potential: $$\frac{-\hslash^2}{2m}\psi''(x)-V_0\delta(x)\psi(x)=E\psi(x)$$ ...
2
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2answers
108 views

How can we evaluate the following integral using the tricks of delta functions?

I am trying to teach myself the statistical field theory formulation of statistical mechanics. Not part of a class, just self study in my free time. I appreciate any help here. I am starting with ...
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0answers
60 views

Integration of delta function with non-isolated zeros

I have a problem of the form $$\int_0^{2\pi}dx_1\int_0^{\pi}dx_2\delta(-a+\cos(x_1)+\cos(x_2))$$ where $a\in (-2,0)$. I have reduced it to the following $$\int_0^{2\pi}dx_1\sum_{j\in I}\frac{1}{\...
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1answer
37 views

How to calculate the Integral $\int_0^t\delta'(t-t')d t'$ (where $\delta'$ is a derivative w.r.t $t$, and $t'\in(0,t)$)?

Without choosing a specific representation for this distribution (by using something like a limit law on a Gaussian function): How to calculate the Integral $\int_0^t\delta'(t-t')d t'$ (where $\...
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1answer
31 views

Fundamental property of Green's function is violated

$$\langle x| D|x'\rangle=D_x\langle x|x'\rangle=D_x\,\delta(x-x')$$ $DD^{-1} = I$ , Where '$I$ ' represents the Identity position representation of this equation is $\langle x|D|x'\rangle \langle ...
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0answers
17 views

Solve the differential equation $R*C*y'(t)+y(t)=x(t)$ using Convolution

I have the next differential equation: $R*C*y'(t)+y(t)=x(t)$ Where $x(t)$ is a piecewise funtion defined by: \begin{equation} \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} x(t)...
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0answers
38 views

How boundary conditions for a green function is same a original function

I am trying to study green's function.suppose we have a differential equation, $D_x f(x) =g(x)$ , We also provided with some boundary conditions for $f(x)$, But we supply this boundary conditions in ...
0
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1answer
60 views

Distributional derivative of a function with jumps

Let $f(x)$ be a real function of a single variable x that is continuously differentiable on the real line except at $x=a$ where $f$ has a jump discontinuity. Let's suppose also that left- and right-...
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0answers
38 views

Continuity of solutions to ordinary differential equations under a limit

Consider $\eta(x;\delta) \in C^\infty$ a family of functions parameterized by $\delta$ with domain $x \in [0,1]$ such that, \begin{align} \lim_{\delta \to 0} \eta(x;\delta) &= 0 \qquad x < R \\...
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1answer
53 views

Understanding Empirical Data Distribution

I've been trying to understand this paper and am having trouble understanding this part: "We can approximate $p(x,y)=p(x)p(y|x)$ using the empirical data distribution $$p(x,y) =\frac{1}{N} \...
0
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1answer
57 views

Position representation of an operator

$$\langle x| M|x'\rangle=M(x)\langle x|x'\rangle=M(x)\,\delta(x-x')$$ I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general operator $M $? $\...
0
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1answer
51 views

Simplifying unit step functions

I'm having quite a hard time trying to simplify the following equations: $sinc(t)\delta(t) \\ u(t)u(t) $ (by u(t) I mean the unit step function) also, the integral $ \int^{+\infty}_{-\infty} cos(...
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1answer
40 views

Solving Differential Equations with the delta function

so I am revising the Delta Dirac function, and how to solve differential equations with it. I don't want to be employing the Laplace transform at this point, because the question I am practicing ...
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1answer
61 views

Find all distributions $T$ belonging to $\mathcal{D}(\mathbb{R})$ such that $(x^2)T=0$

My exercise is to find all distributions $T$ on $\mathbb{R}$, such that $(x^2)T=0$. On the lecture we solved the equation $xT=0$, where the solution where all distributions $T$, such that $T=C\delta.$ ...
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0answers
40 views

How can I evaluate the following involving dirac delta functions?

\begin{align} \delta(x)\delta(x) = ? \end{align} \begin{align} \delta(x)\delta(x-a) = ? \end{align} \begin{align} \delta(x)/x = ? \end{align} In the above problems, a is a real number and a $\ne$...
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0answers
35 views

series of the Dirac function

How to prove that the series $$\sum_{k=-\infty}^{\infty}a_k\delta^{(k)}(x-k)$$ converges in $D'$ for all values of $a_k$? I Understand that the partial sum $$ s_n=\sum_{k=-n}^{n}a_k\delta^{(k)}(x-k)$$...
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3answers
57 views

two dimensional integral of delta function

For $x,y \in \mathbb{R}$, function $f(x,y)$ is defined as $$f(x,y) = 1 \quad\textrm{if}\quad x=y$$ $$f(x,y) = 0 \quad\textrm{if}\quad x\neq y$$ It seems to me that the integral $I = \int_0^1 \int_0^1 ...
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0answers
22 views

Is The Dirac Delta Communitive Over a Cross Product?

Is the statement, $\delta(\vec{x}) \vec{A}\times \vec{B} \equiv \vec{A}\times \vec{B}\ \delta(\vec{x}) $ True? If so, how would I go about proving this?
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0answers
44 views

Proof of Dirac Delta Sifting Property With Volume Integral

The Dirac delta function possess the sifting property which states, $ \int _{a}^{b} f'( x) \delta ( x-x') dx'=\begin{cases} f( x) & a< x< b\\ 0 & otherwise \end{cases} $ I suspect by ...
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1answer
54 views

Integrating a Dirac delta with multidimensional periodic argument

I am trying to solve the following $$\int_0^{2\pi}dx_1\int_0^{\frac{2\pi}{B}}dx_2\delta\left(\lambda+\cos(x_1)+\cos(x_2)\right)$$ where $B,\lambda$ are constants. Since this is not a vector function, ...
0
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3answers
144 views

Dirac delta function properties

Good afternoon! I can't prove $$x \cdot \delta^m(x)=-m\delta^{(m-1)}(x), m=1,2,3....$$ I have found that $\int x \cdot \delta'(x)dx =x \cdot \delta(x)-\int \delta(x)dx$, as a result $ x\cdot\delta'(x)...
2
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1answer
79 views

Non-Homogeneous Differential Equation with Dirac Delta

I am trying to solve the following differential equation: $$ u^{\prime \prime} = -\frac{F}{EA}\delta(x-L) $$ subjected to the boundary conditions: $$ u(0) = 0 \qquad \frac{du}{dx} \biggr\vert_{(x=L)...
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1answer
85 views

Question about $\delta\big(f(x)\big) = \frac{\delta(x)}{\left|f'(x) \right|}$ (that is-non discrete zeros in f(x))

The following holds for Dirac delta function where $f(x)$ has discrete zeros at $a_i$ $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$ See: Dirac Delta ...
0
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1answer
41 views

Properties of integration a delta function

I have great difficulties understanding why the following relation holds: $$\int^{t}_{0} \int^{t’}_{0}\delta(t_1-t_2)\mathrm{d}t_1\mathrm{d}t_2=\min\{t,t’\}.$$ Our teacher gave us an explanation ...
0
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2answers
71 views

Prove Dirac delta function equation

Show that $\displaystyle x\frac{d}{dx}(\delta(x))=-\delta(x)$. My attempt: Using the product rule of differentiation, $\displaystyle\frac{d}{dx}(x\delta(x))=x\frac{d}{dx}(\delta(x))+\delta(x)\cdot ...