# 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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### Let X be a continuous random variable and u a measurable function. Show that u(X) is not necessarily a continuous random variable:

I would like to know if my proof is valid, because I am new to probability theory and not sure if my reasoning is valid for this proof: Thrm: Let $X:Ω → \mathbb{R}$ be a continuous random variable and ...
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### Integrate DiracDelta[l^2 - a] from -inf to inf

I would like to know why the integral of the delta function: \begin{equation} \int_{-\infty}^\infty \delta[l^2 - a] dl \end{equation} is equal to $1/\sqrt{-a}$ if a<0 and $1\sqrt{a}$ otherwise. ...
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### Showing that the Dirac comb is a tempered distribution

Context: I am currently working through Chapter $8$ of Anders Vretblad's Fourier Analysis and Its Applications. This particular chapter focuses on distributions, and builds up to the Fourier transform ...
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### A partition function problem under conserved total momentum, involving Gaussian integration under a delta constraint

Recently I learned about the following expression showing the kinetic part of the partition function in an N-atom ideal gas under conserved total momentum: \begin{aligned} Q_{\mathrm{Kin}}^{\mathrm{CM}...
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### dirac delta confusion

From the Wikipedia page, dirac delta function is described such as: Description 1: "whose value is zero everywhere except at zero" or if you scroll down a little bit more, you will find: ...
1 vote
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### Derivatives for step function

This question might be related a little bit to physics, but wanted to hear the answer from mathematics perspective and mathematician's intuition. Imagine we have an object that moves in the time ...
1 vote
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### Which subsets of $\mathbb{R}^{N}$ are measurable with respect to the dirac measure?

Let $a \in \mathbb{R}^{N}$ and $\delta_{a}$ be the corresponding Dirac measure. Which subsets of $\mathbb{R}^{N}$ are measurable with respect to $\delta_{a}$? Is $\delta_{a}$ Borel-measure, ...
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### Is it possible to Expand the Dirac Delta function into the summation of a serie of certain Bessel functions?

I am trying to solve an axisymmetric special plate problem, but stuck at the expansion of the Dirac Delta function $\delta(r)$ in the interval $[0,c]$, consisting of $1$ and orthonormal Bessel ...
1 vote
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### Green's third identity - unit inconsistency

I have been trying to understand Green's functions and using them to solve differential equations. I have hit a road block in terms of dimensionality and units. My understanding is that the Green's ...
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### Sampling property of the delta distribution

Let's take for granted the following sampling property of Dirac's delta $\delta_w(x)\triangleq \delta(x-w)$ centered in $w$ \begin{equation*} \int_S \delta_w(x)\,f(w)\text{ d}w= \begin{cases} \quad \...
1 vote
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### Derivative of delta function integrated from 0 to infinity

I am aware this is not a perfectly rigorous question, but bear in mind I am a physicist (though I would be open to a rigorous discussion). Suppose I want to integrate a function $f$ together with the ...
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### Sketch the graph $x(t)[\delta(t+\frac{3}{2}) - \delta (t + \frac{3}{2})]$

Sketch and label the following. When I solve this question, I get two delta functions with the value of $-0.5$, at $t = -3/2$ and $3/2$. But the solution says it should be positive $0.5$ (aka ...
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### Dirac delta function as a limit of a function

For some nice function $f(x)$, how to show the following using contour integration? $$\lim_{\epsilon\to 0^+} \int_{-\infty}^\infty f(x) \frac{\epsilon}{\epsilon^2+x^2} dx = \pi f(0)$$ The way I ...
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### Evaluating $\int_{p} f(p) \delta(p+a) \Theta(p)= f(-a) \Theta(-a)$ for Heaviside theta function

I want to evaluate the following integral For given arbitrary function $f$ and Dirac Delta function $\delta$ with Heaviside Theta function $\Theta$, what is the form of \begin{align} \int_{p} f(p) \...
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### Why is it sufficient when proving dirac function identities to show that the action inside an integral is the same on both sides?

I am just starting with the Dirac function $\delta(x)$ (for physics) and proving some identities such as $$\delta(g(x))= \Sigma \frac{\delta(x-x_n)}{|g'(x_n)|}$$. I am told that it is sufficient to ...