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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4
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581 views

How can we prove the scaling property of the Dirac delta function rigorously?

Let $(\Omega,\mathcal A)$ be a measurable space $\omega\in\Omega$ $\delta_\omega$ denote the Dirac measureat $\omega$ on $(\Omega,\mathcal A)$ $E$ be a $\mathbb R$-Banach space $\mathcal M$ denote ...
4
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1answer
751 views

Surface area of sphere using Dirac delta

This question is related to this one. Suppose I want to calculate the surface area $S(R)$ of a sphere of radius $R$. I can express $S(R)$ as $$S(R)=\int_{\mathbb{R}^3} \delta (\| \vec x \|-R) \ d \...
4
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1answer
3k views

Proof of an identity of the dirac delta

How can I prove this property for delta function? ($a$ is a root of $f$) $$\delta(f(x)-f(a)) = \frac{\delta(x-a)}{|f ' (a)|}$$ I tried to prove it by expanding $f$ using Taylor series, and inserting ...
4
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1answer
243 views

Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 +\...
4
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2answers
1k views

Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where $G(...
4
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1answer
119 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 ...
4
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1answer
170 views

Integration with Dirac delta function of two-argument function

I would like to solve that problem: $$ \int d^2 \mathbf{k} \, \delta(f(k,\phi)) = \int_{k_1}^{k_2} dk \, k \int_0^{2\pi} d\phi \, \delta(f(k,\phi)) \,, $$ where $f(k,\phi) = a - bk^2 - ck^3 |\sin(2\...
4
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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)...
4
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2answers
81 views

What is the inverse Laplace transform of my equation?

What is the inverse Laplace transform of L(s)=$exp⁡[-(1/2)sI]$*$I_0[(1/2)sI]$, where $I_0$ is the modified bessel function of first kind. I am told that the answer is $\frac{1}{π}$ $\left(\frac{1}{(...
4
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1answer
119 views

Integral involving delta functions and vector quantities

This integral comes from equation (3.15) in an older paper I've been reading: $$ \int \mathrm{d} \Omega_k \, \delta\left(|\vec{k}|^2 - |\vec{k}+\vec{q}_1|^2\right) \delta\left(|\vec{k}|^2 - |\vec{k}-\...
4
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1answer
278 views

Prove that the Green´s Function satisfy the Differential Equation

Prove that the Green´s Function: $$G(t,t´)=\theta(t-t´) \dfrac{\sin{[(t-t´)(\omega_0^2-\gamma^2)^{1/2}}]}{(\omega_0^2-\gamma^2)^{1/2}}\cdot e^{-\gamma(t-t´)} \tag{1}$$ satisfy the Differential ...
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0answers
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Fourier series expansion of Dirac delta function

My question is from Arfken & Weber (Ed. 7) 19.2.2: In the first part, the question asks for Fourier series expansion of $\delta(x)$. I have found $$\delta(x)=1/2\pi + 1/\pi\sum^{\infty}_{n=1} ...
4
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1answer
126 views

Find $\sum_{k \geq 1} e^{itk}$ in the sense of distribution - $\delta(x-a)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i(x-a)t}dt$

I have to solve $Z(t)=\sum_{k \geq 1} e^{itk}$ in the sense of distribution (generalized function), i.e., $<\sum_{k \geq 1} e^{itk}, \varphi>$, where $\varphi$ is a test function. So far, by the ...
4
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1answer
332 views

Generalised derivatives of discontinuous real functions

Does the generalised derivative of every discontinuous real-valued function always yield a Dirac-delta 'function' at the point(s) of discontinuity? My limited experience with generalised distributions ...
4
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1answer
116 views

Radon measure determined by the intersection of half lines in the plane

Consider a vector $r$ in the euclidian plane $\mathbb R^2$ and two unit vectors $u,v\in\mathbb U$ ($\mathbb U$ is the unit circle). Let $s>0$ be a real number. I am looking for an expression of the ...
4
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0answers
272 views

Proving an identity of the composition of the delta distribution with a differentiable function

Given a differentiable function $f$, some $x_j$ ($j \in \{1, ..., n\}$) such that $f(x_j) = 0$ $\forall j$ and $f'(x_j) \ne 0$ $\forall j$, and the following definition of the composition of a ...
4
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0answers
75 views

Computing the limit of this integral,

This is Part 6 (last part) of a problem statement of an old comprehensive exam question that I am working on. It asks to evaluate $$\lim_{r_0 \to 0} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\...
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0answers
142 views

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
3
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3answers
29k views

How to solve integration with Dirac Delta function?

I am struggling to solve several problems in my 'Signals and Systems' textbook. However, I just met a confused problem. Q) $\displaystyle\int_{-4}^{4}\left(t-2\right)^2\delta'\left(-\frac13t+\frac12\...
3
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2answers
457 views

Is dirac comb a tempered distribution?

Prove or disprove the statement $$ III:=\sum_{k=-\infty}^\infty\delta_k\in \mathscr{S}'(\mathbb{R}^n)$$ where $\delta_k\varphi:=\varphi(k),\,\,\forall\varphi\in\mathscr{S}(\mathbb{R}^n)$ I tried to ...
3
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2answers
946 views

How to solve this integral with dirac delta functions?

$$F(t)=\displaystyle\int_{-\infty}^{\infty}\left[e^{-2t^2}\left\{\dot\delta(t-2)\right\}+\delta(t^2-16)\right]\,dt$$ How to get rid of the derivative ? and for the second function i wrote it as $\...
3
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4answers
673 views

Dirac delta property: $f(x)\delta(x-x_0) = f(x_0)\delta(x-x_0)$

Suppose you want to prove that $$ f(x)\delta(x-x_0) = f(x_0)\delta(x-x_0) $$ In my homework, I was instructed to show that the integral of both sides of the equations will lead to the fact that the ...
3
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3answers
156 views

Why $0 \delta(0)=0$?

I defined the "function": $$f(t)=t \delta(t)$$ I know that Dirac "function" is undefined at $t=0$ (see http://web.mit.edu/2.14/www/Handouts/Convolution.pdf). In Wolfram I get $0 \delta(0)=0$ (http:/...
3
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2answers
3k views

evaluating a Dirac delta function in an integral

I have a problem evaluating this integral $$ \int\limits_{-\infty}^\infty \left[ e^{-at^3} \delta(t-10) + \sin (5 \pi t) \delta(t) \right] \mathrm{d}t $$ Can you please help me evaluate the ...
3
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5answers
243 views

Delta function is not in $ L^1(\mathbb{R})$

Let $f_n$ be a sequence in $L^1(\mathcal{R})$ such that $$ \lim_{n\to \infty} \int_{\mathbb{R}}f_n(x)g(x)dx = g(0) $$ for each $g\in C_0(\mathbb{R})$. Show $f_n$ is not Cauchy. My approach was to ...
3
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1answer
2k views

Derivative of Heaviside step Function

Is this mathematically correct for a basic proof that the derivative of the Heaviside function is equal to the delta function? I don't know much about distributions so I kept everything integrated. $$...
3
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3answers
245 views

What is $\int_0^1 \delta(0)dx$?

What is $\int_0^1 \delta(0)dx$? Should not it be equal to the number of reals in that interval? My attempt: $$\delta(x)=\frac2\pi\int_0^\infty \cos(xt)dt$$ (http://functions.wolfram.com/...
3
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4answers
1k views

Dirac delta function integral

How should i integrate this? $$\int_{0}^{t} \int_{0}^{t} \delta(x-y)dxdy$$ where $\delta$ represents Dirac delta function My try: $\int_{0}^{t} \int_{0}^{t} \delta(x-y)dxdy = t$ is it right?
3
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2answers
140 views

From $\displaystyle\int_{-\infty}^{\infty}f(x)\delta(x)\mathrm{d}x=f(0)$ derive $\displaystyle\int_{-\infty}^{\infty}f(x)\delta(x-a)\mathrm{d}x=f(a)$

Firstly, apologies for abusing the notation by placing the Dirac measure inside an integral for which I was told that this should not be done from a previous question asked by me. But given the ...
3
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2answers
104 views

$\int_{-1}^{1}x^{2}\delta(x^3)dx$

How to solve $$\int_{-1}^{1}x^{2}\delta(x^3)dx$$ where $\delta(x)$ is dirac delta. I tried a substitution $$y=x^3$$$$\frac{1}{3x^2}dy=dx$$ $$\int_{-1}^{1} \frac{1}{3}\delta(y)dy=\frac{1}{3}$$ ...
3
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2answers
499 views

Representation of dirac delta function

Can $ \delta (x-x') \cdot e^{k(x-x') } $ equivalent to $ \delta(x-x') $ in the sense of generalised function identity? As $ \int_{i=-\infty}^\infty \delta (x-x') e^{k(x-x')} G(x) ~dx= \int_{i=-\...
3
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2answers
135 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: $$\frac{-\hslash^2}{2m}\psi''(x)-V_0\delta(x)\psi(x)=E\psi(x)$$ ...
3
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3answers
157 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)?...
3
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1answer
251 views

Is the delta function in $L^2$ (even though it's not a function)?

I'm studying for a qualifying exam and in our study group someone asked the question whether the delta function is in $L^2$ spaces. My argument is that it is; since the delta function function can be ...
3
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2answers
106 views

Integrating over a shell in $d$-dimensional space.

In M. E. Tuckermann's book Statistical Mechanics: Theory and Molecular Simulation, par. 3.2, the following approximation is made: $$\int_{c<f(\vec x)< c+\epsilon} d^D x \simeq \epsilon \int_{\...
3
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1answer
730 views

How to prove this Dirac delta identity involving $\int_{-\infty}^\infty \delta(f(x)) \,s(x) \,dx$?

How to prove the identity $$\int_{-\infty}^\infty \delta(f(x)) \,s(x) \,dx = \sum_i \frac{s(x_i)}{|f'(x_i)|},$$ where $x_i$ are the zeros of $f$? I am supposed to use an identity that I've already ...
3
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1answer
2k views

Dirac delta function of x squared

Sorry to add to all the delta function questions already here but I couldn't find a related one. Can anyone explain why $ \int_{-\infty}^{\infty} f(x) \delta(x^2-9)=\frac 16(f(-3)+f(3)) $? I can ...
3
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2answers
96 views

Double integral involving the derivative of a delta 'function': contradicting results

I'm trying to evaluate an integral of the form: $$I\equiv \int_{-\infty}^{\infty}dx\space f(x)\int_{-\infty}^{\infty}dy\space h(y)\int_{-\infty}^{\infty}d\omega\space\omega e^{i\omega(x-y)}$$ with $...
3
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1answer
205 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}\...
3
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2answers
182 views

Dirac Delta of a Function When the Derivative of the Function is Discontinuous at its Roots

We know that the Dirac delta function of a function can be written as $$ \delta(f(x))=\sum_i\frac{\delta(x-x_i)}{|f'(x_i)|}, $$ where $x_i$ are roots of function $f(x_i)$. Now my question is the ...
3
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1answer
332 views

How to prove this Dirac delta limit representation is correct?

According to the 7th representation in this site: $$\lim_{M\to \infty} \frac{1}{2\pi\sin(\omega/2)}\sin\left(\omega\left(M+\frac12\right)\right) = \delta(\omega)$$ I'm trying to understand why this ...
3
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2answers
266 views

A Differential Equation with Dirac Delta as the Non-homogeneity Term

I'm currently studying Computational Mechanics, and an important step to solve problems is knowing how to express physical interactions in terms of mathematical equations. Consider for example, the ...
3
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5answers
601 views

Why does the Dirac delta function satisfy $f(x)\delta(x-a) = f(a)\delta(x-a)$?

Why does the Dirac delta function have the property that $$ f(x)\delta(x-a)= f(a)\delta(x-a) , $$ where $\delta(x-a)$ is the Dirac delta function? Won't the Dirac delta function just stay the same ...
3
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2answers
1k views

Why is the delta function the continuous generalization of the kronecker delta and not the identity function?

In a discrete $n$ dimensional vector space the Kronecker delta $\delta_{ij}$ is basically the $n \times n$ identity matrix. When generalizing from a discrete $n$ dimensional vector space to an ...
3
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2answers
64 views

Why is $\frac{d^2}{dx^2} e^{-|x|} = e^{-|x|} - 2 \delta(x)$

How can you show that $$\frac{d^2}{dx^2} e^{-|x|} = e^{-|x|} - 2 \delta(x) ? $$ I found this result using Wolfram Alpha and it seems strage to me, how the delta function appears here ...
3
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1answer
204 views

Can the Dominated Convergence Theorem be applied here?

$$\lim_{\epsilon \to 0} \frac{1}{2\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y) \frac{y}{\epsilon^2} \exp\left(-\frac{x^2+y^2}{2\epsilon}\right) \, \mathrm{d}x \, \mathrm{d}y$$ here $f$ and ...
3
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1answer
5k views

Delta function in spherical coordinates. Does my professor have a mistake?

This is a homework question and it goes like this: "In spherical coordinates the Delta function is written in the form $\frac{1}{r^2}\delta(r-r_o)\delta(\cos\theta-\cos\theta_o)\delta(\phi-\phi_o)$ ...
3
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2answers
68 views

What is the meaning of $1^\lor=\delta$

I tried to look up the definition of a $\lor$ and it does not seem to explain this particular usage $$1^\lor=\delta$$ This is used in a proof that inverse fourier transform of $1$ is $\delta$, but ...
3
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1answer
1k views

Composition of a Dirac delta and a function in higher dimensions

Coming from a physics background, I was taught the formula for the composition of a Dirac delta and a function. Indeed, if we consider a nice function $ f : \mathbb{R} \to \mathbb{R} $, one can write $...
3
votes
1answer
91 views

Numerical solution of ODE with Delta function

I want to model a dynamical system of the form $\frac{\text{d}x}{\text{d}t} = f(x)+nx\delta(\pi(t-0.2)). $ The problem is that I have a point source which is reoccurring at fixed time steps (say at ...