# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...