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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26 views

Infinite sum of delta function when calculating Fourier Transform

I am trying to find the Fourier Transform of the periodic function $$ f (\theta) = \sum_{n=- \infty}^{\infty} a_n \space \exp(i n \theta)$$. Using the formula, $$ f(k) = \int_{- \infty}^{\infty} d \...
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9 views

Boundary value in a convolution integral

I am trying to understand the following equation: $$\int^1_0dz\int^1_0dy f(y)g(z)\delta(x-yz)=\int^1_x\frac{dy}{y}f(y)g(\frac{x}{y}).$$ My result is instead $\int^1_0\frac{dy}{y}f(y)g(\frac{x}{y})$. ...
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19 views

Relating the Convolution to a Bank Account

In this video, the professor references an assignment that relates the convolution to a bank account. However, the link in the description redirects to a later version of the course which omits this ...
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1answer
15 views

How to find particular solution for Green's function with $g(x)$ as Dirac delta function?

So I have done the first part of this question i.e. finding a homogeneous solution and going through the usual steps to get $G(x,s)$ in terms of $x<s$ and $x>s$. My question is for the last part,...
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29 views

Dirac Delta Integration by Parts

$$\int_{-\infty}^{\infty}\left(\sin ^{2} x+2 \tan x\right) \delta^{\prime}(x)$$ $1.$ Is this integral missing a $dx$ at the end? $2.$ Assuming that it is then performing integration by parts leaves ...
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1answer
23 views

Second Derivative of the Dirac Delta function

I want to simplify: $\int dp dp' f(p,p') \delta''(p-p')$ where f(p,p') is an unknown function. How do I deal with the second derivative of the delta function?
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44 views

Integral - product of delta functions

How to compute this integral please? $$\int_{-1}^1 \delta (\mu - \mu_0) \delta (\varphi - \varphi_0) d\mu $$ I compute the coefficient of a scatter: I don't understand to this: I think that it ...
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1answer
36 views

Table of integrals for complex exponentials involving delta functions

I am interested in finding a list of integrals of the form: $$ \int a^n e^{iax} da$$ For $n = 0$, I found it to be $\delta(x)$. For $n = -1$, I found it to be sgn$(x)$. For $n = -2$, I found it to ...
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2answers
50 views

The Heaviside step function at zero and the integral of the Dirac delta function

Just for fun I was reading about the Heaviside step function on Wikipedia. The definition in terms of the Dirac delta function makes sense: $$ H(x) = \int_{-\infty}^x \delta(s)\ ds $$ $\delta(s) = 0$...
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37 views

Can I rewrite this expression with a Dirac $\delta$-function?

I am looking at the following scalar product of divergences in $4$ dimensions (in momentum space): $$\partial_{p_1} \cdot \partial_{p_2} \frac{1}{p_1^2p_2^2}. \tag{1}$$ There is a further ...
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21 views

Dirac delta function - why the requirement on compact support?

I am reading the wikipedia articla on Dirac delta, and as far as I understand it, it is saying that only for functions with compact support $f$: $$\int_\mathbb R \delta_t(s)f(s)ds=f(t)$$ Why the ...
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47 views

Solving ODEs with Delta Functions

I was given a homework question Suppose a bone described by the Kelvin model is deformed by a force $$F(t)=F_0(\Theta(t)-\Theta(t-\tau)+\Theta(t-2\tau)),$$ where $\Theta$ is the ...
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17 views

Why is integral of differential delta function equal to delta function?

Why is integral of differential delta function equal to delta function? I.e. as I've read elsewhere: $$\int_A 1 d \delta_A=\int_A d \delta_A=\delta_A=\delta_x(A)$$ https://planetmath.org/...
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31 views

Derivative of Contour Integral Representation of Step Function

The following is Problem 11.9 in "Mathematical Physics: A Modern Introduction to Its Foundations, Second Edition" by Sadri Hassani. Given the following representation of the step function: $$\theta(x)...
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13 views

PDE with a dirac source

i have to solve this PDE (lossy Wave equation) : $$ \partial_{x x}^{2} p^{\prime}-\frac{\alpha}{c_{0}^{2}} \partial_{t} p^{\prime}-\frac{1}{c_{0}^{2}} \partial_{t t}^{2} p^{\prime}=-\rho_{0} \Gamma \...
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35 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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41 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 ) \left\{ A_{1}'(t) \cos \left(\frac{2 \pi x}{L}\right) + B_{1}'(t) \sin \left(\frac{2 \pi x}{L}\...
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1answer
18 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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1answer
40 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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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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67 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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41 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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1answer
29 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
29 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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39 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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27 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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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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40 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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19 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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33 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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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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20 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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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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31 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
25 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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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

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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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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153 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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2answers
40 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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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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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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1answer
34 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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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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378 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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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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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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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 ...
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93 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{...