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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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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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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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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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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33 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{...
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53 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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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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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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53 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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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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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 ...
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36 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 ...
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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 ...
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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 ...
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38 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. ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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49 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 $a_1=a_2=\ldots=a_n=0$. ...
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30 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{...
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Use Laplace transforms to find the solutions of $y''(t) +9y(t) = 3\delta_{2\pi/3} +9H_{\pi}(t)$ [closed]

Use Laplace transforms to find the solutions of the following IVP $y''(t) +9y(t) = 3\delta_{2\pi/3} +9H_{\pi}(t), $ (where $H_{\pi}(t)$ is the Heaviside function of $\pi$) with initial values: $y(...
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Equivalence of two formulations for an elliptic problem with Dirac source

Let $\Omega \subset \mathbb{R}^3$ be a Lipschitz bounded domain and $x_0 \in \Omega$. Recall the definitions of some weighted Sobolev spaces: \begin{align} &H^1(\omega; \Omega) := \{ v \in L^2(\...
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1answer
59 views

Difficult Integral Involving the Dirac Delta Function

Hello fellow Stack heads, I am stuck on a difficult integral that almost looks like it can be accomplished with a one-sided Laplace transform but more than likely can be solved using Dirac Delta ...
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1answer
58 views

Question on Heaviside step function (distribution) identities

I am trying to show the equality between two Heaviside step functions using their derivatives.For $c<0$, and in the sense of distributions; First: $$\frac {d}{dt} \theta (ct-z)=c\delta(ct-z)=\...
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1answer
44 views

Sum of all derivatives of Dirac Delta function

I wonder whether this expression has any meaning? $$\pi \sum _{n=0}^{\infty } i^n \delta ^{(n)}(x)$$ I encountered it while trying to make it into a Maclaurin series but unfortunately it has no ...
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27 views

Calculating the integral of a function multiplied by two delta functions

I would like to calculate $\int_{a}^{d}\delta(x-b)\delta(x-c)f(x)dx$, where $a \le b \le d$ and $a \le c \le d$. My imediate thought is to integrate by parts to obtain $$\int_{a}^{d}\delta(x-b)\...
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48 views

Expectation with Dirac measure

Let $\mathbf{P}_X [A] := \frac{1}{2} \delta_0 (A) + \frac{1}{2} \int_{A \cap (0, \infty)} e^{-t} dt$ for $A \in \mathcal{B} (\mathbb{R})$. What is $\mathbf{E} [X]$? I tried finding the density and ...
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20 views

Mutual Information of a Multivariate Sinus Distribution

I want to calculate the mutual information of the trajectory of two cosines $$ g(t) = \cos(t) $$ and $$ h(t) = \cos(t+\omega) $$ I wanted to do this via the definition of the mutual information $$ I(X;...
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1answer
119 views

Proof for the following nascent delta function

Let $f(x)$ be a (nonzero) rapidly decreasing function in $\mathbb{R}$ (i.e. $f(x)$ and all its derivatives go to zero as $x\to\pm\infty$ faster than any negative power of $x$). The sequence of ...
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44 views

Fourier Transform of (Dirac delta*f(x)) [closed]

I am looking for Fourier transform of $f(x)\delta(x)$. Thanks
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1answer
101 views

Replacing with a Dirac function in a differential equation

I was reading ISBN: 9782868836373 which is dealing with a differential equation at page 139 with a source term in 3D but with three different types of source of neutrons (source term): A point A wire ...
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58 views

verification: find a function $f$ such that $\frac{df}{d\phi} = \sin(\theta+\beta)\, \delta(\phi-\theta-\beta)$

Comments I initially posed the problem statement incorrectly. The problem statement has been altered. Problem Statement: By $\delta$ I denote the Dirac delta function. By $\frac{df}{d\phi} : \...
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Derivatives of delta function as a basis for distributions

Is there some sense in which one could write any distribution as a sum of this sort? $$A(x,y)=\sum_{n=0}^{\infty}a_n(x)i^n\frac{\partial^n}{\partial x^n}\delta (x-y)$$ Provided that the rhs acting ...
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1answer
157 views

Dirac delta function, sinx

How to solve integration of del(sin x)dx The limits are from minus infinity to plus infinity. Please explain.
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Symmetry of delta functions

Proving that the delta function is symmetric The above link has answers for the symmetry of the delta function as: $V(x)=δ(x)$ If suppose I have a $V(x)=λ(δ(x-ap)+δ(x-aq))$ And $V(x)=λ(δ(x-a)+δ(...
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36 views

Contour integration with delta functions

One wants to calculate the time dependent voltage, $V(t)$, of an electric circuit with impedance $$Z(\omega) = \frac{R + i \omega L}{(1 - \omega^2 CL) + i \omega CR}$$ and a current $$I(t) = I_0 \cos(\...
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2answers
133 views

Dirac's Delta function

On Wikipedia, the definition of the dirac delta function is given as: Suppose I have a function where at two points, the function goes to infinity. Given that the distance between the two points is $...
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How to compute $\partial \frac{1}{z^*}$?

I have trouble understanding some basic concepts in Complex Analysis: For $z=x+\mathrm{i}y$, we define: $$\partial \equiv \frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i \...
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50 views

Fourier Series for a Dirac Train

I'm trying to find out by myself the Fourier Series of a Dirac Train, but I'm getting after Integration by Parts that Dn equals to 0 and not to 1 as needed to be. Could you please help me find my ...
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14 views

Singularity conditions

$f(x)=\cot(k(x-c))$ for $[0,a]$ Now I have singularities at 0 and π So I choose the end points to be points of singularities since it is allowed and have the whole f(x) continuous in the interval (0,...
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62 views

Integral limit that behaves like a delta function

I am having difficulty seeing why this integral reduces to $1$ or $0$, just like the delta function. In the following statement, $\Delta$ is small, whereas $R$ goes to infinity: $$\lim_{R\rightarrow \...
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1answer
63 views

Fourier transform rules for Dirac-deltas, to avoid integration

I'm following through a worked example but have a different solution to the given solution. Could someone tell me what I'm missing please, if anything? I have taken the 2nd derivative of a piece-...
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58 views

Can I split Dirac functions? [closed]

I’m doing some calculations with Dirac delta function and I don’t know if I can split te function. I’ll show in a picture Edit1: The integral I was talking about is: $$\int \mathrm{d^4}{p}\,\mathrm{d^...
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53 views

Is the Laplace distribution a nascent delta function?

Define the normalized Laplace distribution as $$ L(x;a) = \frac{1}{2a} \exp\Big(-\frac{|x|}{a}\Big) \quad (a>0). $$ With the normalization out of the way, what we then need to prove is that for ...