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Questions tagged [dirac-delta]

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

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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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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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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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Differentiation and series of Dirac function

$$\sum_{k=-\infty}^{\infty}a_k\delta^{(k)}(x-k)$$ How to prove that the it converges in $D'$ for all values of $a_k$? What is the main idea? I think that I should use the properties of ...
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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, ...
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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)...
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42 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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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 ...
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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 ...
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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 ...
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What is the integral of the Dirac delta function between these limits?

I've learnt that $\displaystyle\int_{-k}^k\delta(x)dx=1$ where $k$ is any positive real number. But what is $\displaystyle\int_{-k}^0\delta(x)dx$?
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Integral of a generalized function

Let $\mathcal{S}$ be a Schwartz space and $\delta_{a}$ the following distribution: $$\delta_{a}: \phi \rightarrow \phi(a) \ \ \ \ \text{ for each } \phi\in\mathcal{S}$$ Now, we routinely see ...
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Delta Dirac integrals [closed]

Evaluate the following integrals: $$ \int_0^\infty e^x \sin \left(\frac{\pi x}{2} \right) \delta \left(x^2-1\right) dx\\ \int_0^\infty e^x \sin \left(\frac{\pi x}{2} \right) \delta'\left(x^2-1\right) ...
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Unilateral Laplace Transform Uniqueness and Dirac Delta “function”

Answering this question, I've tried to alert the OP about the misleading definition of $\delta(t)$ - used in one of the answers - as: $$ \delta(x) = \left\{\begin{array}{cc} \infty & x = 0 \\ 0 &...
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Question about a function in the argument of a Delta function (distribution)

referring to https://physics.stackexchange.com/questions/160503/what-is-the-wave-propagated-away-from-an-impulsively-excited-spherical-shell my question concerns the last part of the answer which I ...
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How to calculate the Laplace transform of $\sum_{k=0}^{+\infty}(-1)^{k}\delta (t-k)$?

On my midterm, I had the following question: Calculate the Laplace transform of $$\sum_{k=0}^{+\infty}(-1)^{k}\delta (t-k)$$ I was wondering how I should calculate it. I know that the transform ...
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Proof of $\sum_{i=0}^{N_s - 1} e^{-j2\pi n(k'-k)/N_s} = N_s\delta(k'-k)$

Currently, I'm faced with this problem: Prove that this property holds when $N_s$ is an even integer, for any integers $n$, $k'$, $k$: $\sum_{i=0}^{N_s - 1} e^{-j2\pi n(k'-k)/N_s} = N_s\delta(k'-...
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General Convergence of Exponential of Function to Dirac Delta

Let $f:\mathcal{X}\rightarrow \mathcal{F}$ be a function with a unique, positive maximum at $x^*=\arg\sup_{x}f(x)$ where $\mathcal{X}$ and $\mathcal{F}$ are both bounded. Let $f$ be locally smooth ...
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Application of the $\delta$ distribution to a function which is not a Schwartz-Function?

I come from the field of Physics, where lecturers roughly don't care that the $\delta$-distribution is not a function. In Physics, it is just used as if it were a function, in textbooks or lecture ...
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2nd order nonhomogenous ODE with delta function (Green's Fn)

I am trying to solve $\frac{u'}{(1+x^2)^2}-\frac{u''}{1+x^2}=\delta(x-\alpha), u(0)=u(1)=0$ Wolfram alpha gives $u(x)=-1/3(x^3+3x)(\sigma(x)-1)$ I have no idea how to set up an ansatz. Maybe some ...
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Verify Delta Dirac function

Verify that $\frac{1}{2π} \sum_{r=-∞}^{∞} e^{ir(x-x_0)} $ is a Dirac delta function $δ(x − x_0)$ by showing that it satisfies the definition, $$\int_{-π}^{π} f(x) \frac{1}{2π} \sum_{r=-∞}^{∞} e^{ir(x-...
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green's kernel goes to dirac delta? really?

Wikipedia claims that: In one variable, the Green's function is a solution of the initial value problem \begin{cases}u_{t}(x,t)-ku_{xx}(x,t)=0&(x,t)\in \mathbf {R} \times (0,\infty )\\u(x,0)=\...
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Evaluate the series $\sum_{n\in \mathbb{Z}} n^m e^{i n x }$

This is an exercise in the book 'integral transforms and their applications' by Davies. The problem is to evaluate the series as a generalized function. $$ \sum_{n\in \mathbb{Z}} n^m e^{i n x } ,$$...
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What does the symbol $\delta$ mean on this page?

On the Wikipedia page on Arithmetic Functions, the section Relations Among The Functions makes frequent references to a variable $\delta$ (or is it a function? Some other kind of value?). It's ...
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Dirac delta distribution [duplicate]

when I want to show that limit of the following function $$f_\epsilon(x)=\frac{1}{2\epsilon}e^{-\frac{|x|}{\epsilon}}~,$$ namely, $$\lim_{\epsilon\to 0} f_\epsilon(x)$$ is a presentation of $\delta$-...
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Integral over a finite Domain involving the Dirac-delta Function

In the formulation of the partial differential heat equation $$\frac{\partial \theta}{\partial t}=\frac{\partial^2 \theta}{\partial x^2}, \hspace{1 cm}0\le x \le D$$ there is an incompatibility ...
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Are $\delta(x-5)$ and $\delta(5-x)$ the same function?

As the title says, I'm confused whether $\delta(x-5)$ and $\delta(5-x)$ are the same function. Similarly, are $\delta(x-5,y-6)$ and $\delta(5-x,6-y)$ the same function? To, me it seems they are! But ...
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Proof of Dirac delta function identity involving exponential

Is the identity $$\delta(x-a)=\lim_{\epsilon \to 0}\frac{1}{\sqrt{2\pi \epsilon}}\exp{[-\frac{1}{2\epsilon}(x-a)^2]}$$ correct? If yes, what is the proof?
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Infinite sum of complex exponentials starting from $n=0$

I know that $$\sum_{k=-\infty}^\infty e^{i2\pi kt}=\sum_{k=-\infty}^\infty \delta(k-t)$$ which is known as the Dirac comb, but what is $$\sum_{k=0}^\infty e^{i2\pi kt}$$ Can it be expressed in terms ...
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Can the infinite convolution of the dirac delta give $1$ as the result?

I'm asking wether the dirac delta can be convoluted infinitely many times to give $1$ as the result, i.e. if $$\lim_{N\to\infty}\int_\Bbb R\prod_{i=1}^N\delta(x_i-x_{i+1})dx_{i+1}=1$$ I think this ...
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59 views

About derivative of delta function - chain rule for delta function containing a function

I have a problem that relates to derivative of a delta function. The problem originates from a paper I was reading https://aip.scitation.org/doi/full/10.1063/1.2938860 In the paper, it is said that ...
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1answer
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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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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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What is the derivative of Radon Transformation?

We have the Radon transformation, in the most common form: $g(\phi,r)=\int\int f(x,y)\delta(x\sin\phi-y\cos\phi-r)dxdy$ Question 1: So what is the closed form of the derivative of Radon transform w....
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Equivalence between Dirac Delta definition as a measure and as a distribution.

I always thought of Dirac Delta as the distribution $\delta_ {x_0}$ which performs $\phi\ \mapsto \phi(x_0)$. With respect to this definition we can think the Delta as the weak limit of some $L^1_{loc}...
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'Half-sifting' of Dirac Delta

Is the solution of the following integral $0$ or $1$ or something else? And why so? $$ \int_{0}^{t} f(\tau) \;\delta(\tau) \;d\tau$$ with $\;\;\;t \in \mathbb{R}\,,\; f: \mathbb{R}\rightarrow\...
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Analytic continuation of Dirac delta distribution

My question is a simple one. Is it possible to analytically continue the Fourier transform of the Dirac delta distribution, i.e., can we analytically continue \begin{equation} \delta(x)=\frac{1}{2\pi}\...
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Improper integral of a sinc function

I am trying to show that the sinc function $\frac{\sin(ax)}{x}$ behaves like a delta distribution when $\lim({a \to \infty})$. I can show that $$ \int_{-\infty}^{\infty} \frac{\sin(x)}{x}=\pi $$ ...
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Derivative of Dirac delta function

Is the relation of the Dirac delta function correct? $$ \frac{\partial}{\partial x''}\delta(x''-x') = -\frac{\partial}{\partial x'}\delta(x'-x'').\tag{1} $$ If it is, how to derive the above ...
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What is the Fourier transform of the identity function?

How to find the Fourier transform of $x \mapsto x$ using distribution $\delta$? Since $FT(1)=\sqrt{2\pi} \delta(k)$ then $FT(x \cdot 1)=\sqrt{2\pi} i \delta'(k)$ But also since $1=d/dx (x)$ then $...
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Convergence in distribution to delta

Let us assume $f_n\in S'(\mathbb{R}^k)$ (Schwartz space) such that $f_n\to\delta_t$ in distribution sense. That is, $$\int_{\mathbb{R}^k}f_n(s)F(s)\mathrm{d}s\to F(t)$$ for all $F\in S(\mathbb{R}^k)$. ...
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Integrate / Convolution of a dirac Delta function from 0 to t

I have a question about the convolution integral resulting due to an inverse Laplace transform. Considering one has a multiplication of 2 functions in the laplace Domain $G(s) F(s)= \exp(-a s) F(s)$ ...
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First and Second Quantization of the Hamiltonian - Quantum Mechanics

I am trying to show the relation between the first and second quantization in Quantum Mechanics. I have been told that the general relationship that holds is that in the first quantization, we can ...
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Why do we have $\int_{t_0}^t \delta(t-t') c(t')=\frac{1}{2} c(t)$?

I am reading a physics paper in which they claim that: $$\int_{t_0}^t \delta(t-t') c(t')=\frac{1}{2} c(t).$$ They just say about this equation: "The result will always hold when $\int_{-\infty}^...
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29 views

Proving Dirac delta identity

I'm having a hard proving an identity which includes the Dirac delta function and the Heaviside function. I want to show that for $m\in\mathbb{R},k\in\mathbb{R}^4$ and $\omega_{\bf k} \equiv \sqrt{{\...
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Understanding the process for plotting $x(t)=(1/2)*δ(t+2)$

I have stumbled upon an exercise where I have to plot/calculate the output of different variations of input functions in a LTI system similar to the one given as a example below (which is supposed to ...
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7 views

Defining the delta-function in multiple forms

I am trying to prove that the following is a valid definition of a delta-function: lim a→0 (1/π)(a/a^2+x^2) I am a bit unsure how to proceed, as I'm not sure what property I should be checking it ...
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1answer
42 views

$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

I know that function evaluation in $H^1(0,1)$ is continuous (see, e.g., Is the Delta distribution a continuous functional on $H^1(\mathbb R)$). So, $\delta_x : H^1(0,1)\to\mathbb C$ is a continuous ...
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Superposition of elementary Dirac measures: looking for reference or explanation

Reading a paper concerning transport of measures, I came accross the following sentence: "...the fundamental fact that a generic measure $\mu$ can be written as the superposition of elementary Dirac ...
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1answer
38 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 ...