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.

Filter by
Sorted by
Tagged with
0
votes
0answers
21 views

Dirac Delta Evaluation of Function

I am struggling to understand what does the following expression evaluate to. "I am looking for general answer, not actual evaluation - i.e is dirac dealta making the integral center around h(x)? ...
1
vote
0answers
60 views

Why does the method using the Green's function for $\nabla^2 \Phi(\mathbf{x}) = \delta(x)\delta(y)$ not work?

I have the Poisson Equation (with $\mathbf{x}\in\mathbb{R}^3$) with the following form: $$\nabla^2 \Phi (\mathbf{x}) = \delta(x)\delta(y)$$ I used 2 methods for the resolution of this PDE. I am ...
0
votes
0answers
21 views

Laplacian and Dirac function gives contradictory result.

The following equation is correct for all non-negative real numbers: $$4\pi\delta^{(3)}(\mathbf{r})=\nabla\cdot\frac{\mathbf{r}}{r^{3}},$$ $$r\in[0,+\infty)$$ especially, when r=0, both sides give ...
0
votes
1answer
30 views

Show that $\sum\limits_{n=-\infty}^{\infty}\delta^{(|n|)}(x-n)$ diverges in S'

by Schwartz’s theorem, any generalized function from $S'$ has a finite singularity order. In this example, it is infinite and I want to show that the series $\notin S'$. ($g^{(l)}$ means $l$th ...
0
votes
1answer
27 views

How to solve second order ODE with Dirac Delta?

I'm trying to solve a non-homogeneous second order ODE. I've read similar other questions, but all use the method of Laplace transformations, which I've not seen/used before. The ODE is: $$y''(x) - ...
0
votes
1answer
26 views

Dirac delta integral for evolving networks

I'm reading Dynamical Processes on Complex Networks (link), which makes frequent use of dirac delta integrals to examine evolving networks. I'm trying to get a good sense of how to evalute them and ...
0
votes
0answers
35 views

Dirac delta and convolution

Is there a way to simplify the following equation? $$\int_{-\infty}^{\infty} f(q) \delta(q-k) * g(q) dq =\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(q) \delta(q-k) g(q-\tau) d\tau dq $$ The $...
1
vote
0answers
24 views

Fundamental solution of the derivative operator

Let $x \in \mathbb{R}$ and define $x_+:=xH(x)$, where $H$ is the Heaviside function. By induction we can show that $E(x)=\dfrac{x_+^{k-1}}{(k-1)!}$ is a fundamental solution of $$P=\frac{d^{k}}{dx^{k}}...
0
votes
2answers
31 views

How to prove the two formulas are equal in the sense of distribution

$1+2\sum_{n=1}^\infty \cos2n\pi x=\sum_{k=-\infty}^\infty \delta(x-k)$. I couldn't have an idea to prove it, maybe we can discuss how to get it clearly.
0
votes
1answer
39 views

2nd order non-homogenous ODE with delta function & B.C.S

I'm relatively new to the concept of the Dirac Delta function have come across a problem in dealing with ODE with delta Solve the ODE: $$A''(y) - λ^2 A(y) = δ(y - ξ)$$ Subject to B.C (Hint: Use ...
0
votes
0answers
18 views

Laplace Transform of Dirac Delta, issue with the limits

I am physicist so sorry for being not very formal. One normally finds the formulas for following Laplace Transforms: $\mathcal{L}(1)=\frac{1}{u}$ $\mathcal{L}(\delta(t))=1$, $\mathcal{L}(\int\...
0
votes
1answer
29 views

Writing a distribution as a tensor product

Definition about tensor product of distributions: Let $u_j \in \mathcal{D}'(\Omega_j)$, $j=1,2$. There is only $u \in \mathcal{D}'(\Omega_1 \times \Omega_2)$ such that $$\langle u, \varphi_1 \...
0
votes
0answers
19 views

Finding fourier coefficients of dirac comb

Let's consider a dirac comb $f(x) = \sum_{n=-\infty}^{\infty} \delta(x-nb)$. I know it has a period of $b$. So, using the definition of finding fourier coefficients, I can write: $C_n = \frac{1}{b} \...
0
votes
0answers
19 views

Is there an analog of integrating to $1$ property for a skyscraper sheaf?

Let $A$ be an abelian group. Let $X$ be a smooth manifold with a point $x \in X$. Then we can define a skyscraper sheaf $\delta_x$ valued in $A$ and supported at $x$, similarly to a delta function on $...
0
votes
0answers
41 views

Simple question on the antiderivative of a two variable function (or distribution)?

I am confused about antiderivatives of multivariable functions, specifically $\delta(ct-|x|)$ and $\delta(t-|x/c|)$. Here $\delta(.)$ is the Dirac delta function (distribution)and $x$ and $t$ are ...
0
votes
0answers
30 views

Integral of delta function along a curve

Let $\gamma\colon I\to\mathbb R^d$ be a closed curve with $\gamma\left(0\right)=\gamma\left(1\right)=p$ and $\gamma\left(\lambda\right)=\left(x_1\left(\lambda\right),\dots,x_d\left(\lambda\right)\...
0
votes
1answer
17 views

Green's Function for the Laplacian in 3D

Does anyone know where to find a good resource for solving for the Green's Function of the Laplacian in 3D or tips on where to start? $$\nabla^2G(\boldsymbol{x,x_0})=\delta (\boldsymbol{x,x_0})$$
0
votes
2answers
49 views

Using identity for the derivative of Dirac Delta function

I know that you can define the derivative of the delta function as:- $$\delta'(x)=-\frac{1}{x}\delta(x)$$ If i use this to calculate the integral with $f(x)$, I get 2 different results. Method 1:- $$\...
0
votes
1answer
40 views

Question regarding a non-rigourous proof that the Fourier transform of $1$ is the Dirac-delta function

I know this question has been asked a lot and I have already read these questions; Fourier Transform Dirac Delta, Fourier Transform of Dirac Delta Function, The inverse Fourier transform of $1$ is ...
1
vote
1answer
40 views

a fraction containing a very small value is equal to Dirac's delta function

When $\epsilon$ goes to infinite small value ($\epsilon\rightarrow 0$), how can I show $\sum_{k}\frac{\epsilon}{(E-E_k)^2+\epsilon^2}$ is equal to $\pi\sum_{k}\delta(E-E_k)$.
6
votes
3answers
143 views

integral of delta function of x^2

The name says what I need to calculate. When trying to integrate I stumble upon interpretation problem $$ \int\limits_{-\infty}^{+\infty} \delta(x^2) dx = \{y=x^2\} = 2\int\limits_{0}^{+\infty} \delta(...
1
vote
2answers
39 views

Homogeneity of dirac delta

I know that $\delta(ax) = \frac{1}{|a|}\delta(x)$ for $a\neq0$ in the sense of distributions. However I am confused about the inconsistency of the following computation. Consider $g\in C^\infty_c(\...
0
votes
0answers
8 views

Cosine function completeness relations

Does anyone how to prove the following completeness relations? $$\delta(\theta-\theta')=\frac{1}{\pi}\sum_{n\in\mathbb{Z}}\cos(n\theta)\cos(n\theta')$$ Is there an analogous relation for $\sin(\theta)$...
1
vote
0answers
20 views

When does $P_n \to\delta$ implies that $f *P_n \to f$ in $L^\infty(\mathbb T)$?

In the question $\mathbb T$ is a unit circle. For one example, even the continuity of $f$ does not suffice. If we let $P_n$ be the Dirichlet kernel $$P_n=\frac{1}{2\pi} \sum_{m=-n}^n e^{imx},$$ which ...
-2
votes
0answers
20 views

convolution between unit step function and impulse function

question In the question, it asks me to find the convolution between an impulse and a unit step function. I know that differentiating a unit step function results in an impulse. However, I don't ...
0
votes
1answer
32 views

Integral over product of Dirac delta functions

We can define the Dirac Delta function as a distribution satisfying $$\int_{-\infty}^{\infty} \text{dx}\; \delta(x-a)\; f(x) = f(a) .$$ What if I have a product of delta functions? $$\int_{-\infty}^...
1
vote
0answers
29 views

Question about a distribution definition in $D'(\Omega)$

So, let $\Omega :=B_1(0) \subset R^3$ and $\{x_n\} \subset \Omega$. If $x_n \rightarrow x \in \partial B_1(0)$, $T:= \sum_{n \in N}\delta_{x_n}$ define a distribution on $D'(\Omega)$? How can I check ...
1
vote
0answers
202 views

Approximation to the $n$-th derivative using reproducing kernels.

For integrable functions defined on the real line, the normalized gaussian function approximates the convolution identity, Dirac Delta, in the sense that if $$g(t):=N_0e^{-x²}$$ (denoting the ...
0
votes
0answers
3 views

replace input signal with $\delta $ will have the impulse response of the function?

Is my impulse response right? By definition,the impulse response is the output when the input is a impulse signal,so $y[n]=\sum\limits ^{n}_{k=-\infty}\frac{1}{2^{n-k}}\ x[k]$,the impulse response ...
1
vote
0answers
32 views

Apparent contradiction between the theory of hyperfunctions and the table of Fourier transforms

Graf's book on hyperfunction theory says (page $36$) that $$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$ while the table of Fourier transforms ...
0
votes
0answers
24 views

Proving Property of Dirac Delta

I need to prove the idendity of $$\delta(g(x)) = \Sigma \frac{\delta(x-x_i)} {|g'(x_i)|}$$ and we know that $$g(x_i) = 0$$ So I think we can write, $$g(x) = (x - x_1)(x-x_2)...(x-x_i)$$ so we are ...
2
votes
0answers
85 views

Is the formula $\;\oint H(x) \, \delta(y) \, dy = \frac{1}{2\pi} \oint d\phi\;$ well known?

Explanation - everything real-valued: $$ (x,y) = \mbox{cartesian coordinates} \\ \phi = \mbox{angle, in polar coordinates} \\ H(x) = \begin{cases} 0 & \mbox{for} & x < 0 \\ 1 & \mbox{...
3
votes
1answer
52 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 $\delta$ ...
0
votes
1answer
23 views

Difficult on distribution exercise with $\delta$

I'm facing this exercise; I'm new to distribution theory so I have lots of difficulties: let $f(x):R \rightarrow R, f(x) =4|x-5|$ for every $x \in R$. Are these statements TRUE or FALSE ? 1) $T'_f=4 \...
1
vote
1answer
36 views

Complicated Integral with a Delta Function

I am interested in the elastic theory of lipids and using geometric methods to model them so I've been reading Geometric Methods in Elastic Theory of Membranes in Liquid Crystal Phases. Within the ...
0
votes
0answers
52 views

Simplifying $\int f(t) \delta(\mathbf x - t\mathbf x_0) dt$

Consider two n-dimensional vectors $\mathbf{x, x_0}$ and the expression $$ \int_0^\infty f(t) \delta(\mathbf x - t\mathbf x_0) dt $$ where $\delta$ is the n-dimensional Dirac delta function. ...
0
votes
1answer
55 views

What gets broken if we assign a value to the square of Delta function?

Particularly, what gets broken if $$\pi^2\delta^2(x)=2i\pi\delta'(x)-\frac1{12}$$?
1
vote
2answers
36 views

Proving that the sum of delta functions is a measure on the Borel $\sigma$-algebra

I have the following problem and I also wrote my solution but I am not sure of its correctness, since I am new to this. Or if there is an easier solution. I would like if someone could check the ...
0
votes
0answers
32 views

Conjugate of Dirac delta

Does conjugation of a Dirac delta function makes sense: $\delta^*(x-y)$? Namely is $\delta^*_y = \delta_y$? I am asking specifically in the context of this identity (which is the same as Plancherel's ...
0
votes
0answers
37 views

Dirac delta inner product preserved under the Fourier transform

I have the following inner product: $$\langle \delta_y, f \rangle = \int_{\mathbb{R}^d} \delta(x-y)f^*(x)\,dx = f^*(y)$$ For it a property similar to Plancherel's theorem can be shown with respect to ...
0
votes
1answer
30 views

Find all distributions solving a differential equation

My problem is : Find all distributions $u \in D'(\mathbb{R}^{2})$ such that $(x_1+ix_2)u=0$. I know $c\delta$ is solution of $(x_1+ix_2)u=0$, but I am not sure if $c\delta$ is the general solution ...
0
votes
0answers
34 views

Extension of the Dirac delta function

For real number $a \in \mathbb{R}$, the heaviside step function $H_{a} : \mathbb{R} \to \{0,1\}$ is usually defined as \begin{equation} H_a(x) = \begin{cases} 1, & x \geq a; \\ 0, & x < a. ...
1
vote
1answer
28 views

Question about nomenclature of Dirac delta function as a measure

Introduction From [1], $$H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta\{dt\}.$$ From [2], $$\int_{X} f(y) \, \mathrm{d} \delta_x (y) = f(x).$$ Questions Question 1. Does the $\...
0
votes
1answer
23 views

How do I correctly report whether a unit step function is “increasing” or “decreasing”?

I have the (discontinuous) function that reports a $0$ if $X>(\frac{2y}{ln(y)})$ and a $1$ if $X \leq (\frac{2y}{ln(y)})$. I would simply like to describe something like the intuition that this ...
0
votes
0answers
15 views

Question on geometrically deriving the wave field from an impulsive planar source

The figure shows the geometry for deriving the wave field from an impulsive planar source. The impulse is approximated by a rectangle $c\epsilon$ wide and $\alpha=\frac{1}{c\epsilon}$ high so the ...
0
votes
1answer
34 views

Integral of counting measure

I am looking at a homework problem: Measure space ($\mathbb{N}, \mathcal{P}(\mathbb{N}),\mu)$) where $\mu$ is the counting measure. Let $\nu=\mu+\delta_2+\delta_5$ where $\delta$ is the Dirac ...
2
votes
2answers
30 views

How to prove scaled delta function relation mathematically?

I am working through Shankar's Introduction to Quantum Mechanics. I have come across exercise 1.10.1, which asks the reader to show that: $$\delta(ax)=\frac{\delta(x)}{|a|}.$$ I can understand it ...
0
votes
1answer
45 views

Delta function of two variables.

How can we transfer equation $$\iint \delta\left(f\left(x,y\right)-t\right)\, \mathrm{d}x\,\mathrm{d}y,$$ into line integral? Where $t$ is a parameter and a constant value of $t$ denotes a closed ...
1
vote
1answer
44 views

Linear combination of Dirac delta distribution and its derivatives

Let $f(x)=x$, $u=\sum_{j=1}^{n}a_j\delta^{(j)} \in \mathcal{D}'(\mathbb{R})$, where $a_j \in \mathbb{C}$ and $\delta$ is the Dirac delta distribution. Show that if $fu=0$ then $a_1=a_2=\ldots=a_n=0$. ...
0
votes
0answers
29 views

Expectation and Dirac delta function

consider following expression. where $\lambda$ is random variable with expectation $\mathbb{E}[\lambda] = a$ $= \mathbb{E}_{\lambda}\bigg[\delta \big(\tilde{x} = \lambda.x_1 + (1-\lambda)x_2 , \tilde{...