# 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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### I'm looking for resources that involve concretely taking Lebesgue integral of functions (non-axiomatic and computation focused)

I want to practice finding the Lebesgue integrals of certain functions. My source of inspiration is integrating Dirac delta functions and anything relating to differential equations like Green's ...
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### Calculus of variations with linear Lagrangian and two optimisations

I am working on a two-part system, that jointly determines two functions $F$ and $G$. $F'(x) = f(x)$ and $G'(x) = g(x)$. $F$ and $G$ are both defined on $[0, 1]$, $F(0) = G(0) = 0$, they are both non-...
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### Limit behaviour of a distribution

Consider the distribution, with $a>0,p>0$ $$F(x)=\left\{\begin{matrix}\frac{J_2(a\sqrt{p^2-x^2})}{x- p} & -p<x<p \\ 0 & \text{otherwise}\end{matrix}\right.$$ I claim that $F$ ...
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### Derivative of a sign-like function

Assume a function $f: \mathbb{R}^2 \to \mathbb{R}$ such that $$f(x_1,x_2)=\begin{cases} 1 & x_2\le h(x_1,t) \\ 0 & x_2 > h(x_1,t) \end{cases}$$. I am trying to find the derivative of f in ...
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### How to show that the Gaussian sequence satisfies the sifting property of the delta function?

I am trying to show that the Gaussian sequence of functions, defined by $g_a (x) = (\frac{1}{a\sqrt{\pi}}) e^\frac{-(x-x_0)^2}{a^2}$ satisfies the sifting property of the $\delta$ - function, namely ...
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### Interpreting Green's function in Evans' Partial Differential Equations

Pictures below is from Evans' Partial Differential Equations. $U\subset \mathbb R^n$ is open, bounded. And $\nu$ is the normal vector of $\partial U$. I want to get the last red box from the first ...
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### Dirac Delta Function Integral Over Finite Interval

I am taking a course on Partial Differential Equations at the moment. We were introduced to the Dirac Delta function, and were told it satisfies $$\int^b_a \delta(x)=1$$ for $a<0<b$. Having ...
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### Modified Bessel Function and Dirac Delta function

Is there a representation of the Dirac delta function of the form $$\int_0^\infty \frac{d z}{z} K_{i \nu} (z) K_{i \nu'} (z) = f(\nu) \delta ( \nu - \nu' ) , \qquad \nu,\nu'>0.$$ for some ...
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### Problems with understanding Dirac Delta Properties [closed]

I need to check equation ${\bf 6.15}$ in an $\tt arxiv$ paper for my work, but I don't get why the Dirac Distributions change and why there is that exponential ...
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### Why is the autocorrelation of an uncorrelated random noise process the dirac delta distribution?

I am reading Stochastic Methods by Gardiner and in the beginning of chapter 4 he motivates the rigorous interpretation of a Stochastic Differential equation by describing the properties of a "...
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### is $\int_0^t \int_0^s \delta(u - v) \, du \, dv = \min(t, s)$? [duplicate]

Is the following integral equation true? $\int_0^t \int_0^s \delta(u - v) \, du \, dv = \min(t, s)$ where $\delta$ is the Dirac delta function.
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### Which Method Correctly Computes the Fourier Transform of the Ramp Function?

While calculating the Fourier transform of the ramp function, I found two different methods that yield different results. Here is how each method works before I share my confusion. Define the ramp ...
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### Proof of space white-noise limit for Milstein algorithm

I was trying to proof Milstein Algorithm so I read the book "Stochastic Numerical Methods_ An Introduction for Students and Scientists" of Toral. Here, they show all the proof in one ...
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### Fundamental solution of Poisson equation on the torus

The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. this MathSE post): \begin{align} &\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y) \quad \Rightarrow ...
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### Evaluating the integral $\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$

I want to evaluate the following integral $$\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$$ According to the book where I found the exercise, the answer is $$C\Theta(1-v^2)\sqrt{1-v^2}$$ ...
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### Inverse Mellin transform of the Mellin transform of the binomial PMF

The probability mass function of the Binomial distribution is given by $$$$f(x)=\binom{n}{x} p^x (1-p)^{n-x},$$$$ where $p \in [0,1]$ and $x=\{0,1,\dots,n\}$ (finite ...
I know that a representation of the Dirac delta function is $\sum_{n=-\infty}^\infty e^{inx}=2 \pi \delta(x)$. I am trying to figure out if the series with positive $n$ only $\sum_{n=0}^\infty e^{inx}$...
### Find the solution to the given derivative of the product: $\big(\cos{x}\delta{x}\big)^{(k)}$
Find the solution to the given derivative of the product $$\big(\cos{x}\delta{x}\big)^{(k)}$$ We need to use the differentiation formula for generalized functions: \big(D^\alpha f, \phi)=(-1)^{|\...