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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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3
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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}^{\...
3
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1answer
246 views

Dirac Delta functions and Sobolev Embeddings

This is a question about the action of the Dirac delta on Sobolev spaces $H^s(\mathbb{R}^d) = W^{s,2}(\mathbb{R}^d)$. We know that $\delta(\underline{x})\in H^s(\mathbb{R}^d)$ for $s<-d/2$. In 2D,...
3
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2answers
146 views

How to simplify $\delta(x^2)$

How to simplify the dirac delta of squared x? How should i solve $\int f(x)\delta(x^2)dx$ for an enough smooth function $f$? Is this related to $\int f'(x) \delta(x) dx$ ? I can understand symbols ...
3
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1answer
194 views

Approximating derivative of Dirac delta function using mollifiers

Given $\rho_\epsilon (x)$ a delta approximating sequence, what is the limit in distribution of the function $$\frac{\rho_\epsilon (x) - \rho_\epsilon (x-\epsilon)}{\epsilon} \text{?}$$ Intuitively it ...
3
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2answers
463 views

Differential Equation with Delta Dirac

This is my first question, and it was my last solution, since no article could help me solve this differential equation. The equation is in the following form: $$\dfrac{d^2 f(x)}{dx^2}-Af(x)+B\...
3
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1answer
974 views

What does Dirac delta function of a constant mean?

I have seen a formula that unit step function is the integration of Dirac delta function. $$H(x) = \int_{-\infty}^{x} \delta(t)~\mathrm dt $$ In evaluating the integral if we take the integral as ...
3
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1answer
406 views

Convergence to Delta-distribution

I was trying to prove that function $\frac{\sin{nx}}{\pi x}$ sutisfies these conditions: 1) $ \forall D \; \exists c : \forall a,b \;\; |a|<D, \; |b|<D \quad \left|\int_a^b \frac{\sin{nx}}{\pi ...
3
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3answers
78 views

Does Gaussian convolution respects order?

Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$ Now let us define the ...
3
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1answer
783 views

How does the posterior of a dirac prior look like?

Edit for the Moderators: Should this question migrate to stats.stackexchange? I have a very basic question concerning updating from a prior to a posterior in bayesian statistics. Setting: I enter ...
3
votes
1answer
913 views

Greens function for 2d laplace equation with neumann boundary conditions

I have a domain, $ D : {(x,y) : x>0 , y>0}$ Let $ \mathbf{x}= (x,y) $ and $\mathbf{\xi}= (\xi_x, \xi_y)$, The Greens function satisfying: $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
3
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1answer
555 views

2-dimensional delta function (complex plane)

I have a task to show that $$\partial_{\bar{z}} \frac{1}{z - \zeta} = \pi \delta^{(2)}(z - \zeta) $$ But I thought, that delta-function is determined by $\int f(\zeta) \delta(z-\zeta) d\zeta = f(z)$,...
3
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1answer
6k views

How does one find the Laplace transform for the product of the Dirac delta function and a continuous function?

As an example, what is the Laplace transform for the following: $$g(t)=\delta(t-2\pi) cos t$$ I've worked through a few examples that required finding $\mathcal{L}\{\delta(t-t_0)\}=e^{-st_0}$, but I'...
3
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1answer
116 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
3
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1answer
56 views

Gaussian integral over a surface

I have to solve the following integral: $$ I(s)=\int_{S(s)} \frac{1}{2\pi^{3/2}}e^{-\frac{x^2+y^2+2z^2}{2}}dxdydz $$ where $S(s)$ is the surface defined by $s=\sqrt{(x-y)^2+4z^2}$. I parametrised $...
3
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1answer
56 views

Move integral inside logarithm

I want to simplify the integral $$I=\int_y \log \left( \int_x f(y) \delta(x-y) dx \right)dy,$$ where $x$, $y$ are real numbers, $f$ is a "nice" real fuction of real argument (eg. exp) and $\delta$ ...
3
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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{...
3
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1answer
151 views

Integral with delta-function in denominator

I have the integral $$ I=\int\limits_{-\infty}^{+\infty} dx\,\frac{f(x)}{a(x)+[b(x)+\delta(x)]^2}, $$ where $f(x)$, $a(x)$, and $b(x)$ are some smooth functions, and $\delta(x)$ is the Dirac delta-...
3
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1answer
132 views

Dirac Delta's Ill-defineness Property

This problem arises from the following property of Dirac $\delta-$function: $$\delta(f(x))=\sum_{a_i\in Z(f)}\frac{\delta(x-a_i)}{|\frac{df}{dx}(a_i)|} $$ where $Z(f):=\{x\in dom(f)|\,f(x)=0\}$, the ...
3
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1answer
503 views

Verifying the delta function satisfies Poisson's Equation

Suppose we have the equation $\nabla^2 G= \delta(\mathbf{r-r_0})$ where $\delta$ is the Dirac Delta function in $\mathbb R^3$ and I want to verify that the solution to this equation is given by Green'...
3
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2answers
79 views

Can a non Lebesgue integrable function tend to a $\delta$ distribution?

Consider this definition of $\delta (x)$ $$\int_\mathbb{R} \delta(x)f(x)\mathrm{d}x=f(0) $$ I saw a question asking to prove that $$ \frac{1}{\pi x}\sin\left(\frac{\pi x}{\varepsilon}\right)$$ ...
3
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2answers
476 views

Phase of Dirac delta

How can I get the phase of the following Fourier transform: $$X(f)=\frac 1 2 \delta\left(f-\frac B 2\right)+\frac 1 2 \delta\left(f+\frac B 2\right)+\frac 1 {2j} \delta\left(f-\frac {3B} 2\right)-\...
3
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3answers
127 views

How to solve a differential equation with a distributional free term?

I tried to solve this type of differential equation $$y'' + y = \delta + \delta' .$$ I tried using the Laplace Transform, but I'm stuck at that $\delta$ (Dirac function). The only thing I know is ...
3
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1answer
222 views

$\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$ with distributions defined on Schwartz space

I know, from a recent enlightening answers received here, that, if we define the distribution represented by Dirac's $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is ...
3
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1answer
3k views

Double Integral with a Delta Function

Consider the integral $$\int_0^b\int_0^a\delta(x-y)f(x,y)dxdy$$ where $b>a$. I know that we need to integrate over the larger range first (i.e do the $y$ integral) and then do the remaining ...
3
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1answer
413 views

Surface integral of a partially constant Dirac delta

I am trying to integrate the product of a function and a partially constant delta function over a sphere of constant radius $r$. The integral is of the form $$\int^{2\pi}_0 \int^{\pi}_0 f(\mathbf{r}...
3
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1answer
126 views

Integral question of Dirac's Delta function

I came across this exercise in a book that I was reading, it says: What value would you assign to $$\int_{-\infty}^{\infty} (\delta(x))^2dx$$ ? My attempt at a solution was to think of as so: This ...
3
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1answer
157 views

Dirac-$\delta$ integral evaluation. Infinite series value?

Evaluate $$\int_0^\infty \delta(cos(x))e^{-x}dx$$ This is a homework problem I'm dealing with. Here's my solution: $\delta$ function of a function is given by $$\delta (g(t))=\sum_n \frac{\delta (t-...
3
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1answer
134 views

Derivative of the composition of delta distribution with a differentiable function

Is there an explicit representation of what $\frac{\partial }{ \partial x} \delta(f(x,t))$ is? Here $\delta $ is the delta distribution and $f$ is an arbitrary differentiable function which is ...
3
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1answer
34 views

Why is the DFT of $1 = \sum_{k=-\infty}^{\infty}\delta(\theta-k)$?

I struggle with a part of the solution of an exercise and would be grateful for your help. "Given a time discrete LTI (Linear time-invariant) system H, and an input signal $x[n]$, we have the ...
3
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1answer
171 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 ...
3
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1answer
216 views

Spherical harmonics and Dirac delta integrals

I have an equation given in the following form: $$D_{l,m} = \int_{\mathbf{\Omega}}^{} \mathrm{d}\Omega \ Y_{l,m}(\mathbf{\hat{s}})\int_{\mathbf{\Omega'}}^{} \mathrm{d}\Omega' \ K(\mathbf{\hat{s}} \...
3
votes
1answer
500 views

Lighthill explanation of delta function/generalized functions

I am reading Jaynes' "probability theory: the logic of science" which refers at one point to Dirac's delta function an example of a so called "generalized function". Jaynes then says that "...
3
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1answer
144 views

Evaluating an integral containing a delta function

When calculating the Green's Function to the Wave Equation in two spatial dimensions, I came across the integral $$\int_0^{\infty}e^{ickt}J_0(kr)dk$$ The only idea I had was to put in the integral ...
3
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1answer
1k views

Proving delta dirac properties

I'm currently studying electromagnetism from Reitz's Foundations on electromagnetic theory, in that book delta dirac is presented as a "function" $\delta$ satisfying: $\delta(x)=0$ for any $x \not=0$...
3
votes
1answer
142 views

General equation for the change of variable (random variables' functions)

I have learned that for monotonic functions, one can obtain the formula for the pdf of a random variable, by using the following: $$f_Y(y) = \left| \frac{dx}{dy}\right|f_X(x)$$ where $x$ is $g^{-1}(y)...
3
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1answer
84 views

Where do the step function integral boundaries come from?

EDIT: I have a confusion about Heavyside step function. Suppose I have integral like $$ \int_{0}^{\infty}dE_1\int_{0}^{\infty}dE_2\int_{0}^{\infty}dE_3 \delta(2- \gamma-E_1-E_2-E_3) $$ my first ...
3
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1answer
85 views

$\int_{a-\epsilon}^{a+\epsilon} \delta(x - a)dx = 1$

$$\int_{a-\epsilon}^{a+\epsilon} \delta(x - a)dx = 1$$ I can see intuitively why this is so as $a$ is inside the the domain of integration and all other values in the domain contribute $0$ to the ...
3
votes
1answer
369 views

Fourier transform of exponent?

Mathematica fails to find a Fourier transform of exponent. Yet according to this page $$\mathcal{F}[e^{2\pi iat}]=\delta(t-a)$$ and via substitution, $$\mathcal{F}[e^{at}]=\delta\left(t-\frac a{2\...
3
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1answer
109 views

Evaluate the following Dirac delta integrals:

a) $ \int^{+\infty}_{-\infty} \delta'(t-\pi)e^{-t^2} \; dt$ b) $ \int^{+\infty}_{-\infty} \delta(-3t)(\frac{e^{-t^2}}{\ln(t^2 + 3)}) \; dt $ c) $ \int^{+\infty}_{-\infty} \delta(4t)\sinh{t^2} \;...
3
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1answer
243 views

Weighted Dirac comb as a tempered distribution?

I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists N\in\mathbb{...
3
votes
1answer
110 views

What is the right way to calculate $ \partial_z\frac{1}{\bar{z}} $?

In the appendix B of a physics paper arXiv: 1902.01434, it says $$ \partial_z\frac{1}{\bar{z}}=\partial_{\bar{z}}\frac{1}{z}=2\pi\delta(z)\delta(\bar{z}), $$ same as 2-dimensional delta function (...
3
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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 $$\{\...
3
votes
1answer
110 views

Dirac delta uder coordinate transformation

I am having some issues understanding Dirac's delta function/distribution behaviour under change of coordinates. There is a statement, if $(x_1,\ldots,x_n)$ are cartesian coordinates and $y_1,\ldots,...
3
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0answers
88 views

Dirac delta from poles of a function

Suppose we are given the simple expression $$ F(k) = \frac{1}{E^2-E(k)^2} $$ which has a pole when $E^2 = E(k)^2$ and where $E, E(k)$ are real numbers. When working with this expression (e.g. inside ...
3
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0answers
43 views

General solutions to first-order differential equations with disitrubutional coefficients

Consider the first-order differential equation $$\dot{x}+p(t)x=q(t).$$ This can be generally solved using an integrating factor $$a(t)=\exp\left(\int p(t)dt\right)$$ and the solution is $$x(t)=\...
3
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0answers
571 views

Using Green's function to solve 2d laplace equation

Consider a domain $D : {(x,y) : x>0 , y>0}$ Let $\mathbf{x}= (x,y)$ and $\mathbf{\xi}= (\xi_x, \xi_y)$. Then the Green's function satisfying $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
3
votes
2answers
383 views

Solution to ODE with Dirac Delta satisfies ODE

I am working on a problem where I have the following ODE. $$m\dot{v}+bv=\delta_I(t)$$ where $$\delta_I(t)=\begin{cases}0, & \text{for}&t\ne0\\ I, & \text{for} &t=0\end{cases}.$$ The ...
3
votes
0answers
101 views

Delta “function” and calculus

In a comment on Math.SE, I came today across the remark that the expression $\delta(f(x))$ has no meaning. Being a physicist, I have been surprised but I then remembered Schwartz distribution theory ...
3
votes
0answers
118 views

Integrating derivatives of dirac delta from zero

I wish to compute the following integral. $$ I_1 = \underbrace{\int_{-\infty}^{\infty} \dots\int_{-\infty}^\infty}_{n} \frac{\partial^n}{\partial x_1 \dots \partial x_n}\Bigg(\max \Big( \sum_i^n ...
3
votes
1answer
384 views

Is it true that the integral of $\delta(x)/x$ between symmetrical limits is zero?

My professor is claiming that the following is true: $$\int_{-\infty}^{\infty}\frac{\delta(x)}{x}dx=0,$$ where $\delta(x)$ is the Dirac delta "function", as he calls it. I think the integral ...