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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

Dirac delta questions

I'd like to ask for help in solving 3 I suppose quite easy questions about Dirac delta. $$\delta (-x) = \delta (x)$$ $$\delta(ax) = \frac{1}{|a|}\delta (x)$$ $$\delta ' (-x) = -\delta ' (x)$$ ...
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19 views

Integration of two dimensional dirac delta

I want to integrate the two dimensional dirac delta function: $ \delta^2(x,y)=\delta(x)\delta(y) $ I know that for the one dimensional case the integral is $\int_{-\infty}^{+\infty}\delta(x)dx = 1$ ...
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1answer
22 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 )( A_{1}'(t) \cos \left(\frac{2 \pi x}{L}\right)+B_{1}'(t) \sin \left(\frac{2 \pi x}{L}\right))dx $ ...
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1answer
17 views

Integral of dirac delta function times another another function

Dirac delta function $\delta(x)$ is defined with two properties: 1) At $x=0$ its value is $\infty$ and everywhere else it is $0$ 2) Area under the curve is $1$ How does above definition result in $$\...
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25 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)? ...
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35 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 ...
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227 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 ...
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0answers
63 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 ...
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23 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 ...
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1answer
28 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) - ...
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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 ...
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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 $...
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25 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}}...
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2answers
32 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.
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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 ...
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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\...
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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 ...
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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 ...
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1answer
30 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 \...
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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} \...
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1answer
724 views

Inverse Laplace by convolution with Dirac Delta function

I'm trying to find the inverse laplace of the following function by using convolution. $$\mathcal{L^{-1}}(\frac{s}{(s+1)^2})$$ What I did was to separate into: $$\mathcal{L^{-1}}(\frac{s}{s+1}\,\...
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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 $...
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3answers
144 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(...
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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)\...
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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? $$\nabla^2G(\boldsymbol{x,x_0})=\delta (\boldsymbol{x,x_0})$$
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1answer
523 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 ...
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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:- $$\...
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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 $\pi\sum_{k}\delta(E-E_k)$.
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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(\...
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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 $\sin(\theta)$...
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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}^...
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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 ...
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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 ...
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4answers
21k views

What is the product of a Dirac delta function with itself? [closed]

What is the product of a Dirac delta function with itself? What is the dot product with itself?
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2answers
933 views

Integral of delta distribution in spherical coordinates

We know that, if $\mathcal D$ is a domain containing the origin $(0,0,0)$, then $$\int_{\mathcal D} \delta(\vec r) d \vec r= \int_{\mathcal D} \delta(x) \delta(y) \delta(z) dx dy dz=1$$ However, we ...
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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 ...
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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 ...
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0answers
90 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{...
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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 ...
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1answer
2k views

Formal derivation of the Fourier transform of Dirac delta using a distribution

The Fourier transform of Dirac delta is often naively calculated by considering Delta function as a function that makes sense within an integral and by using its fundamental property: $$ \int_{-\...
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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 ...
3
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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 $\delta$ ...
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1answer
2k views

How to calculate this integral in 3 dimensions involving the Dirac delta function?

How would I go about calculating the integral $ \int d^3 \mathbf r {1\over 1+ \mathbf r \cdot \mathbf r} \delta(\mathbf r - \mathbf r_0) $ where $\mathbf r_0 = (2,-1,3)$ My attempt so far: I have ...
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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 \...
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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 ...
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2answers
80 views

Distributional “Antiderivative”

Suppose that $\mathbb{R}^3$, fix $f \in L^1(\mathbb{R}^3;\mathbb{R})$ and let $g \in L^1(\mathbb{R}^3;\mathbb{R}^3)$ satisfies $$ div(g)=\delta_a -f. $$ Then what is $g$? It what is the ...
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1answer
55 views

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

Particularly, what gets broken if $$\pi^2\delta^2(x)=2i\pi\delta'(x)-\frac1{12}$$?
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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. ...
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2answers
38 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 ...
3
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5answers
548 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 ...