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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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29 views

Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation

$y'' + y =\mu_\pi \big(t\big)$ $y''+y= \delta (x- \pi )$ wih initial conditions: $y \big(0\big) =0$ $y' \big(0\big) =0$ It is obvious to me that the first equation is a Heaviside distribution ...
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1answer
81 views

Is $\int^{\infty}_{-\infty}\delta(x-x_0)f(x) \, dx = f(x_0)$ sufficient to define delta distributions?

Most of the sources start introductory section of the delta distributions by defining \begin{eqnarray} \delta(x-x_0)&=&\begin{cases} \infty, & \text{if $x=x_0$}.\\ 0, & \text{...
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401 views

Proof with Dirac delta function

I have to prove that Dirac delta function $\delta$ is equal to this limit: $$ \delta(x) = \lim_{a\to \infty}\frac{sin(ax)}{\pi x}.$$ We can say, show that the result of the right side of equality is ...
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25 views

Any way to rewrite/simplify $\int dp ~f(p)~\delta[c-g(p)-E(p)]$?

Im thinking of using $$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$ but in my case $$\int dp ~f(p)~\delta[c-g(p)-E(p)]$$ the functions $f$ and $E$ are basically unknown, while $g$ is a ...
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143 views

Simple proof for a continuous-time linear system and impulse $\delta$?

From Schaum's Outlines of Signals & Systems: Let's work with continuous-time signals. Let $T$ be a linear time-invariant system (LTI). Input $x(t)$ can be expressed as $x(t) = \int_{-\infty}^{\...
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1answer
41 views

What is the average of two stochastic processes multiplied?

Consider two random processes $X(t)$ and $Y(t)$ for which $$\langle X(t) X(t') \rangle = \mu_X^2 + \sigma_X^2 \delta(t-t')$$ $$\langle Y(t) Y(t') \rangle = \mu_Y^2 + \sigma_Y^2 \delta(t-t')$$ ie. the ...
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1answer
39 views

Transforming a function in the dirac delta function

I need help finding $k(\sigma)$ such that the family of functions $$ \delta_\sigma (x,y) = k(\sigma) e^{-\frac{1}{2}\frac{x^2 + y^2}{\sigma^2}} $$ defines the unit impulse $\delta(x,y)$ as $\sigma \...
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37 views

polar co ordinates integration

integrate the polar co ordinates $$ \int^{r=\infty}_{r=0} \int^{z=\infty}_{z=-\infty} \delta(r) \delta(z-z_s) dz dr$$ => I want to integrate the above equation. integral of $ \int^ {z=\infty} _ {z= ...
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2answers
358 views

Obtaining PDF of continuous random variable from CDF

I'm trying to study for an exam and I'm having a bit of difficulty understanding the given solution to a particular problem. Given $F_X(x)$ it asks the reader to find $f_X(x)$ $$ F_X(x) = \begin{...
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1answer
83 views

Differential Question about Laplace/Delta/Convolution

I need help understanding a part of this question. Let $a.) y''+4y = \delta (x)$, $y(0)=y'(0)=0$. and $b.) y'' + 4y = f(x)$, $y(0)=y'(0)=0$ where $f(x)$ is some continuous function of finite ...
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2answers
179 views

Integrability of sums of Dirac deltas

this is my first post in the forum and I am an engineer, so I apologize in advance if my question is not clearly stated. Consider the function $f(x)=\sum_{i=1}^N a_i\delta(x-x_i)$ where $a_i\in\...
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1answer
661 views

Write this expression with series & sum in terms of a single variable

I know this is very specific, but is there a way to represent the expression $$\frac{3}{5} \sum_{n = 0}^\infty \left(\frac{2}{5}\right)^n \sum_{j = 0}^n {n \choose j} \delta_{2j - n, k}$$ in terms ...
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1answer
38 views

What should I do if limits of integration are ones where Dirac Delta gets Infinite?

I am working on an integral involving Dirac delta function. I am from Physics background. I know that if I integrate from a to b and in between there is a point where argument of Dirac delta function ...
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1answer
766 views

Expected value of Dirac measure: $E[\delta_{\tau}(t)]$

Thx for the efforts! I tried to clean up the question and remove the ambiguous notations. For the dirac measure, I use the one found in the Wikipedia article. I try to formally understand a statement ...
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2answers
585 views

solve $y''-4y=\delta(t-1)$ with initial conditions $y(0)=0, \; y'(0)=1$ using Laplace transforms

I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ ...
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71 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 ...
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3answers
1k views

Integration involving dirac delta function

I know that $\delta(0)=1$, and $\delta(x)=0$ otherwise. So for the integral $\int_{-\infty}^{\infty}\delta(6-2x)x^2$, why can't you say that $\delta(6-2x)= \delta(0)$ at $x=3$, and therefore ...
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1answer
776 views

Understanding Dirac Delta

I found this: here: http://www.nada.kth.se/~annak/diracdelta.pdf on page 2 Can anyone explain how and why all the terms are cancelled in the second step?
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2answers
44 views

Use Laplace transforms to find the solutions of $y''(t) +9y(t) = 3\delta_{2\pi/3} +9H_{\pi}(t)$ [closed]

Use Laplace transforms to find the solutions of the following IVP $y''(t) +9y(t) = 3\delta_{2\pi/3} +9H_{\pi}(t), $ (where $H_{\pi}(t)$ is the Heaviside function of $\pi$) with initial values: $y(...
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2answers
42 views

$\int_{-\epsilon}^\epsilon\delta(f(x))g(x)dx=\frac{g(0)}{f'(0)}$?

$\delta(x)$ is the Dirac delta function and $\epsilon>0$ $\int_{-\epsilon}^\epsilon\delta(ax)dx$=$\int_{-\epsilon}^\epsilon\delta(u)d(\frac ua)=\frac1a$ My question is does this hold more ...
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1answer
242 views

Probability measures and dirac measures [closed]

Let $X$ be a compact set. Consider the set $H$ of non-negative probability measure $\eta$ defined on $X$ satisfying $$ \eta(dx) = \delta_\xi(dx),\quad \xi \in X. $$ How to understand, how to ...
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1answer
67 views

Can I split Dirac functions? [closed]

I’m doing some calculations with Dirac delta function and I don’t know if I can split te function. I’ll show in a picture Edit1: The integral I was talking about is: $$\int \mathrm{d^4}{p}\,\mathrm{d^...
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1answer
107 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
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
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1answer
44 views

Citation on heuristic definition of the Dirac delta function

The wikipedia page on the Dirac delta function offers the following heuristic definition of the Dirac delta function: $$\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$$...
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1answer
36 views

Derivation involving delta function

In some of my notes on physics I am currently reading, I stumbled across something I don't understand... How do I derive$\frac{1}{\hbar^2}|\int_0^tdt'e^{i(E_n-E_m)t'/\hbar}|^2\space\space$$\rightarrow$...
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1answer
2k views

The area under the impulse (Dirac delta) function

Why is the area under the Dirac delta function equal to one and not zero? Shouldn't it be zero since the function is symmetric and the area under of each side cancels out the other?
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1answer
57 views

operator method? non-autonomous differential equation. [closed]

Professor told me that solving: $\ddot{\theta}+\dot{\theta} +f(\theta) = c_0 \times \delta(t)$ can be done by taking a limit $\lim_{e \to 0} \int_{-e}^{e} \left ( \ddot{\theta}+\dot{\theta} +f(\...
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1answer
58 views

What is the meaning of $\delta(ax + by - t)$

How one deals with 2D delta function $\delta(x+y)$? Is $\delta (x+y)$ same as $\delta(x,y)=\delta(x) \delta(y)$ ? It appears in radon transforms. Below is special case that I am interested in. $$\...
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1answer
46 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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1answer
44 views

How to solve this integrals (delta Dirac)

i have the following problem : $$\int_0^L\sum_{i=1}^{n=3} \delta\left(x-x_{i}\right) \cos(2\pi x/L ) \left\{ A_{1}'(t) \cos \left(\frac{2 \pi x}{L}\right) + B_{1}'(t) \sin \left(\frac{2 \pi x}{L}\...
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1answer
210 views

Dirac delta function, sinx

How to solve integration of del(sin x)dx The limits are from minus infinity to plus infinity. Please explain.
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1answer
41 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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2answers
103 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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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: $$ \lim_{T\to0} f(t,T) $$ ?
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2answers
411 views

What's Laplace Transform of $u(t-1)\delta(t)$?

If I took LT : $L(u(t-1)\delta(t)) = u((0)-1)$ as Laplace of $\delta(t)=1$, and we substitute $t$ with zero don't we get a zero answer as $u(t-1) = 0$ when $t<1$? but what if I did it the other ...
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1answer
73 views

Help with a step in the Parseval Theorem

I (think I) understand everything up until the step that integrates $(f(x))^2$ at $x \in [-\pi, \pi]$. I understand why the zero occurred, my understanding is that $\cos$ and $\sin$ cancel out after ...
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1answer
70 views

Integration in $\Bbb{R}$ involving Complex Transformation

$$\int^{\infty}_{0}dx\int^{\infty}_{0}dy \; \delta(\sqrt{y^2-x^2}-a)$$ Here, $$ a>0 $$ and using the Jacobian $$x=x , \quad y=\sqrt{z+x^2}$$ Then, $$\int^{\infty}_{0}dx\int^{\infty}_{0}dy \; \delta(...
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2answers
74 views

Solve this integration $\int_{-\infty}^{+\infty}(x^2+1)\delta{(x^2-x-6)}\,dx$?

I know that $\delta(g(x))=\sum_i\dfrac{\delta(x-x_i)}{g'(x)}$, but I want it's proof, am not getting how to proof this.
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1answer
151 views

Dirac Delta Function definition with ksi (ξ)

The dirac delta function has a definition $$f(0)=\int_∞^∞f(x)δ(x)dx$$ and $$ f(x)=\int_∞^∞f(x-ξ)δ(ξ)dξ $$ (the lower bound is minus infinity but I couldn't add a minus :/) I do understand the ...
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3answers
88 views

Prove that $\int f(\theta(x))\delta(x)dx = \int f(y)dy$ ?? [closed]

Show that the above is true. where $\delta(x)$ is the delta and $\theta(x)$ is the Heaviside Step function. We know that $$\frac{d\theta(x)}{dx} = \delta(x)$$ I am stuck on this. Please help me? ...
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1answer
102 views

Taking the Laplace Transform of $\sum_{n=0}^{\infty} f(nT)\delta(t-nT)$.

I am trying to show that $$\mathcal{L}\left(\sum_{n=0}^{\infty} f(nT)\delta(t-nT)\right)=\sum_{n=0}^{\infty} f(n)z^{-n},$$ where $n\in\mathbb{Z^+}$, $z\in\mathbb{C}$ and $\delta$ is the Dirac delta ...
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1answer
625 views

What is the expectation of the Dirac delta function of a random variable

I have a Dirac delta function as follows: $\delta_{\epsilon = y}$ where $\epsilon \sim N(0, \sigma^2)$, and $y \in \mathbb{R}$. I want to take the expectation of this function, but it appears that ...
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1answer
36 views

addition of unit step functions

Recently, I received an answer key for a test and I am trying to understand this line: $$u[n-3]-u[n-4]=δ[n-3]$$ How is this possible? I cant seem to find any resource to this. Here is the original ...
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1answer
98 views

Approximate delta distribution with indicator function

Is it true that $$\int_{[0,1]}\int_{[0,1]} f(t)g(s)2^m\sum_{\substack{i=0}}^{2^{m}-1}\mathbf{1}_{\left[\frac{i}{2^m},\frac{i+1}{2^m} \right]}(t)\mathbf{1}_{\left[\frac{i}{2^m},\frac{i+1}{2^m} \right]}...
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0answers
25 views

Why we cannot say that integration of $\delta(x)$ is Lebesgue integration over zero-dimentional manifold?

Lebesgue integral is defined as $$\int 1_{S}\,d\mu =\mu (S)$$ For instance, integral over n-dimentional unit cube is always $1$. But a point is a $0$-dimentional unit cube and its Lebesgue measure ...