# 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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### 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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### Find $\sum_{k \geq 1} e^{itk}$ in the sense of distribution - $\delta(x-a)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i(x-a)t}dt$

I have to solve $Z(t)=\sum_{k \geq 1} e^{itk}$ in the sense of distribution (generalized function), i.e., $<\sum_{k \geq 1} e^{itk}, \varphi>$, where $\varphi$ is a test function. So far, by the ...
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### Generalised derivatives of discontinuous real functions

Does the generalised derivative of every discontinuous real-valued function always yield a Dirac-delta 'function' at the point(s) of discontinuity? My limited experience with generalised distributions ...
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### Radon measure determined by the intersection of half lines in the plane

Consider a vector $r$ in the euclidian plane $\mathbb R^2$ and two unit vectors $u,v\in\mathbb U$ ($\mathbb U$ is the unit circle). Let $s>0$ be a real number. I am looking for an expression of the ...
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### Proving an identity of the composition of the delta distribution with a differentiable function

Given a differentiable function $f$, some $x_j$ ($j \in \{1, ..., n\}$) such that $f(x_j) = 0$ $\forall j$ and $f'(x_j) \ne 0$ $\forall j$, and the following definition of the composition of a ...
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### What is $\int_0^1 \delta(0)dx$?

What is $\int_0^1 \delta(0)dx$? Should not it be equal to the number of reals in that interval? My attempt: $$\delta(x)=\frac2\pi\int_0^\infty \cos(xt)dt$$ (http://functions.wolfram.com/...
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### Dirac delta function integral

How should i integrate this? $$\int_{0}^{t} \int_{0}^{t} \delta(x-y)dxdy$$ where $\delta$ represents Dirac delta function My try: $\int_{0}^{t} \int_{0}^{t} \delta(x-y)dxdy = t$ is it right?
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### From $\displaystyle\int_{-\infty}^{\infty}f(x)\delta(x)\mathrm{d}x=f(0)$ derive $\displaystyle\int_{-\infty}^{\infty}f(x)\delta(x-a)\mathrm{d}x=f(a)$

Firstly, apologies for abusing the notation by placing the Dirac measure inside an integral for which I was told that this should not be done from a previous question asked by me. But given the ...
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### $\int_{-1}^{1}x^{2}\delta(x^3)dx$

How to solve $$\int_{-1}^{1}x^{2}\delta(x^3)dx$$ where $\delta(x)$ is dirac delta. I tried a substitution $$y=x^3$$$$\frac{1}{3x^2}dy=dx$$ $$\int_{-1}^{1} \frac{1}{3}\delta(y)dy=\frac{1}{3}$$ ...
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