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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3
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1answer
28 views

How to prove $\delta(kx) = \frac{\delta(x)}{|k|}$ by using properties of a test function.

So, I have a Fourier analisys course right now and got a problem to solve. Here's how it sounds Given that $\delta(x)$ is a Dirac delta function, $\phi(x)$ is a test function, by using properties of ...
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21 views

Confusion about empirical distributions notation that uses a dirac delta function in paper

I've been reading this deep learning paper https://arxiv.org/pdf/2110.12567.pdf in section 2.3.1 the authors define $pQ$ and $pK$ as two empirical distributions as: $pQ = \sum_{i=1}^{w}\frac{1}{w}\...
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0answers
31 views

Solving Dirac delta double integrals

I am solving a stochastic ODE with additive noise: $\frac{dx(t)}{dt} = -\alpha x(t) + L(t)$ where L(t) is Gaussian noise meaning $<L(t)> = 0, <L(s)L(t')> = 2D\delta(s -t').$ I am trying to ...
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20 views

Integrating $ \int_{\mathbb{C}}dze^{(1-b)|z|^{2}}\frac{\partial^{2}}{\partial{z}\partial{\bar{z}}}\delta(Re(z))\delta(Im(z))$ by parts

I was reading a paper where I came across the following integral $$ \int_{\mathbb{C}}dze^{(1-b)|z|^{2}}\frac{\partial^{2}}{\partial{z}\partial{\bar{z}}}\delta(Re(z))\delta(Im(z))$$ where it is ...
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1answer
40 views

Fourier transform definition

There are the different definition of Fourier transform (i.e scaling constant or sign on the kernel). How can I show if the following is a correct Fourier transform pair: $\hat{f}(\omega)=\int_{-\...
2
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1answer
40 views

Pendulum with Dirac Comb excitation

There is a pendulum that is excited by a Dirac Comb. $l \ddot\theta+b\dot \theta+g\theta=G\,\sum_{-\infty}^\infty\delta(t-nT)$ where $l, b, g, G$ are constants and $T=\dfrac{2\pi}{\omega}$. Show that ...
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33 views

Double integral with the Dirac's delta

Problem For a given $T\in\mathbb{R}$ and $K\in\mathbb{N}$, consider the following sequence of points \begin{equation*}t_k\triangleq kT \qquad k=0,1,\dots,K\end{equation*} I need to compute the ...
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1answer
47 views

Proving the sifting property of the Dirac delta

How to prove the following property of the Dirac delta? $$ f(x)= \int_0^1 f(a) \delta (x-a)da $$ for $ 0 < x < 1 $
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1answer
100 views

Deriving physics result for definite integral involving Dirac delta function

I'm reading electrodynamics physics notes that describe a cavity of length $L$. The cavity is said to lie along the $z$-axis from $-L$ to $0$. There is a reflective mirror at $z = 0$, represented by ...
3
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1answer
68 views

How to prove that the delta function is not actually a function

Show that there is no continuous bounded function $\delta : [−1/2, 1/2) → R$ with the following property: for all continuous bounded functions $f : [−1/2, 1/2) → R$, $$\int_{-1/2}^{1/2}f(x)\delta(x)=f(...
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0answers
99 views

Dirac Measure on a Group

In certain works https://arxiv.org/pdf/1803.11173.pdf (page 6, equation 8), the authors use the Dirac measure on the unitary group $\delta(V-U)dU$ to mean that "U must be of form V", where $...
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1answer
39 views

$f(0)=0 \neq f'(0) \Rightarrow \frac{\delta(x)}{f(x)} = \frac{\delta(x)}{x f'(0)}$?

In this reference (Eq. 15 and sentence below) the following property of Dirac's delta is used: $$ k_n \frac{ \delta(x)}{\sin(k_n x)} \rightarrow k_n \frac{ \delta(x)}{x \, k_n} = \frac{ \delta(x)}{x} $...
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2answers
84 views

The Dirac Delta Function: How do you prove the $f(0)$ property using rigorous mathematics?

Dirac, Reference 1, pg 59, says The most important property of $\delta(x)$ is exemplified by the following equation \begin{equation*} \int_{-\infty}^\infty f(x) \delta(x) dx =f(0),~~~~~~~~~~~~~~~...
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12 views

Is the $\delta(ix)$ sequence based on the function $\delta_{n}(ix) = ne^{-inx}$ a delta sequence?

I know that this sequence $\delta_{n}(x)$ based on the following function: $\delta_{n}(x)=\left\{\begin{array}{ll} 0, & x<0 \\ n e^{-n x} & x>0 \end{array}\right.$ satisfies: $\lim_{n\...
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1answer
38 views

How to solve $X_{xx}=(1+\delta{(x)})X$

I am trying to find the vibration modes of a string that has a uniform mass density, plus some point mass somewhere attached to it, modelled by an additional Dirac delta function in the mass density. ...
5
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1answer
85 views

Can you use a random variable as intput to the Dirac delta, i.e. $\delta(X)$?

Question: If $X$ is a random vairable and $\delta$ is the Dirac delta, is $\delta(X)$ meaningful? Useful? Motivation: In this other question, we have $X\sim Exp(p)$, and it is asked whether there ...
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1answer
47 views

How can the derivative of the dirac delta function of (-x) equal the negative derivative of the dirac delta function of x?

On the wikipedia page for the Dirac delta function derivative ( https://en.wikipedia.org/wiki/Dirac_delta_function#Distributional_derivatives currently by citation 50), It says that $$\delta'(-x) = -\...
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0answers
58 views

Derivative after applying residues theorem, that removes original poles

I have a question related to a typical integration in particle physics. Suppose one has a function $f(t) = \dfrac{i}{t^2-\omega^2 }$ with $t, \omega \in \mathbb{R}$ and $\omega>0$. I am interested ...
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40 views

How to prove $\delta [g(x)]= \sum_i \frac{\delta (x-x_i)}{|g'(x_i)|} $ [duplicate]

How to prove $\delta [g(x)]= \sum_i \frac{\delta (x-x_i)}{|g'(x_i)|} $ This is given as a property I came upon on MathWorld - Delta Function. This is very helpful in proving other properties of the ...
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0answers
53 views

Multiplication of 2 singular distributions(Dirac Delta times Shifted Dirac Delta)

I already know that $\delta^{2}(t)$ is not defined. However consider $A(t) = \delta(t) \cdot \delta(t-t_0)\ $ with $ t_0 \neq 0 $. Can I claim that $A(t) = 0$ ? If that is the case, is there a way to ...
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0answers
38 views

Orthogonality relation of $e^{i\vec k \cdot \vec x} $: how to pass from cartesian to spherical coordinates

Disclaimer: this is my first question here, english is not my first language and i'm a physicist, so sorry in advance for any mistakes! I'm dealing with the orthogonality relations of 3-dimensional ...
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0answers
41 views

Relating Dirac mass with surface integral

I know that for a smooth function $g:\mathbb{R}^n\to\mathbb{R}$ with $\nabla g(x)\neq 0$, there holds the formula $$\int_{\mathbb{R}^n}\delta(g(x))f(x)\,dx=\int_{g=0}\frac{1}{|\nabla g(x)|}f(x)\,d\...
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1answer
39 views

An apparent counterexample of the derivative proprety of Fourier transform

I learned that one of the properties of the Fourier transform is that $\mathcal{F}[\frac{df}{dx}] = ik\mathcal{F}[f(x)]$. However it seems to me that the function $f(x) = 1$ is a counterexample of ...
2
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1answer
89 views

What are the correct steps of integrating with a Dirac delta function of two variables that you're integrating over?

I have the following function where $F(x)$ is a normal distribution and in this context it represents a 'weight': $$R(t) = \int_\tau \int_x T(\tau)\cdot F(x) \cdot \delta(t -(mx+b) - \tau) \: dx \: d\...
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1answer
42 views

Using Inverse Laplace to find the frequency response of a transfer function - Help needed!

The frequency response is the inverse Laplace transform of a transfer function. I am tasked to apply the inverse Laplace on the transfer function below in order to convert it into the time domain. $$...
2
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2answers
64 views

Not sure how to finish this integral $\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$

Dirac delta is a symmetric function defined as $$\int_{-\infty}^\infty f(t)\delta(t-A)dt = f(A)$$ Find the value of $$\int_{-\infty}^\infty x^3 \delta(x^2-2)dx$$ SOLUTION: Let $t=x^2$ then $dt=2xdx \...
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1answer
50 views

Dirac delta in ODE (inhomogenous), where is the discontinuity?

Trying to understand where the discontinuity comes from in this equation coming from an ODE course (link at the bottom): $$ \ddot{x}+3\dot{x}+2x = \delta(t), $$ Initial conditions: $\dot{x}(0)=x(0)=0$...
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2answers
37 views

Is this expression identical to just a Dirac delta?

I have come across an expression like this, $$ \frac{f(x) + f(a)}{2\sqrt{f(x)f(a)}}\,\delta(x-a), $$ where I expected to find just $\delta(x-a)$. When I thought about it, though, I realised maybe... ...
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1answer
48 views

Derivative of a multiple of Heaviside step function

First of all, thanks for your time, I have a question. Let's assume that we have: f(x)=(Heaviside's step function)*e^(3x) Now let's assume to calculate the derivate ...
2
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1answer
38 views

Sum of Dirac measures is not regular

Prove that: $$\mu(A)=\sum_{n\in\mathbb{N}}\delta_{\frac{1}{n}}(A)$$ is not a regular measure. A measure is regular if: $$ \mu(A) = \inf \{ \mu (G)|A\subset G, G\text{ is open}\} = \sup \{ \mu (F)| F\...
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0answers
24 views

Double line integral of a feynman propagator

I'm having trouble in finding the result of the following double integral of a Feynman propagator in Euclidean signature: \begin{equation} I = -\frac{i}{(2\pi)^2}\oint_\gamma dx \oint_\gamma dy\,\frac{...
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0answers
35 views

Evaluate integral of dirac delta

I was solving some problems with Dirac delta. I don't know if my method is correct. But some answers matched coincidently, so I was thinking if my method is correct so I am going to write a question ...
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0answers
14 views

Question on Applying a Transfer Function to an Integrand

I am reading a book on Linear Time-Invariant systems and came across with the following: $$y(t) = T\{x(t)\}$$ where input $x(t) = \int_{-\infty}^\infty x(\tau)\delta(t-\tau)\text{d}\tau$, $\delta$ is ...
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1answer
42 views

A Loop problem-Inverse Fourier Transform and delta function definition

Suppose the fourier transform is defined as $$\hat{f}(k)=\frac{1}{2}\int_{\infty}^{\infty}f(x)e^{-i2\pi kx}dx$$ and the inverse: $$f(x)=B\int_{-\infty}^{\infty}\hat{f}(k)e^{+i2\pi kx}dk$$ I want to ...
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1answer
58 views

Delta function centered on the boundary of an integral

Suppose I have a function $f : \mathbb{R} \rightarrow \mathbb{R}$ evaluated at some $k\in\mathbb{R}$. Often (espacially in physics contexts), one must represent $f(k)$ in the following way: $$f(k) = \...
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1answer
38 views

Integral of the delta distribution

What is the integral of the Dirac delta in this case? \begin{equation} \int_{0}^{a} dx \delta (x-a) \end{equation} I was working out the length of a circumference first in Euclidean space and it's ...
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1answer
49 views

PDF for a random variable with a step in density at one point (Dirac delta?)

Dirac background and problem statement: The Dirac delta function $\delta(x)$ defined as $$ \delta(x) = \lim_{c \to 0} \delta_c(x) \\ \delta_c(x) = \begin{cases} 1/c && |x| \leq \frac c2 \\ 0 &...
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0answers
14 views

Periodicity in function and its fourier transform

I have a periodic function $f(x)$ in $x$ with period $x$, i.e. $$f(x+T)=f(x)$$ The Fourier transform of this function, $$\hat f(\omega)=\mathcal F[f](\omega)$$ is also periodic in $\omega$ with some ...
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1answer
37 views

delta function with uniform distribution

I am trying to evaluate the integral, that comes from a physics problem, where i need to calculate the probability of hitting a crocodile 100 meters away from the launching point and the crocodile is ...
2
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2answers
32 views

Can I exchange these operations: Absolute square, limit and time derivative in my specific example involving a Dirac sequence. If yes: why?

I am rederiving some physics stuff (related to Fermi's Golden rule) so I know what the result should be. ( I am a physicist, so I lack some math training.) However to get this result I have to ...
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1answer
46 views

Equality of Integrands Involving Dirac Delta Functions

Say, I have the following integral: $$\int_{-\infty}^{\infty} f(x) \delta(x-y) \, dx = f(y) \int_{-\infty}^{\infty} \delta(x-y) \, dx = f(y).$$ Again consider the following integral: $$\int_{-\infty}^...
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1answer
95 views

A controversy regarding the generalization of the Sign function to dual numbers

Here is a link to a long discussion regarding generalization of $\operatorname{sign}z$ function to dual numbers. There are basically two proposed versions: $\operatorname{sign}(a+\varepsilon b) = \...
2
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1answer
69 views

Normalization of One-Particle States using Dirac Delta Function

I am attempting to understand the normalization of one-particle state $|\textbf{p}\rangle \propto a_{\textbf{p}}^\dagger$ in the context of Klein-Gordon field quantization from Peskin & Schroder's ...
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0answers
25 views

Fourier transform the mass density having delta functions?

If we denote the density (or even the pmf in some sense?) $\rho(x)$ of a $N$ point masses of mass $m_j \in \mathbb{R}^+$ at locations $x_j \in \mathbb{R}^n$ as the sum of delta functions: $$\rho(x) = \...
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1answer
88 views

Delta (Dirac) Sequence - $\delta [(x-a)(x-b)] = \frac{[\delta (x-a) + \delta (x-b)]}{|a-b|}$

Im trying to prove the identity above, what I got until now is: for $y = (x-a)(x-b) = x^2 -(a+b)x + ab $ we have $ dy = [2x - (a+b)] dx $ so $dx = \frac{dy}{2x - (a+b)}$ from here I used: $0 = x^2 -(a+...
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1answer
126 views

Problems with deriving FT from FS using impulse train

There are two of the same equation given for the Inverse Fourier Transform: $$ f(t) = {1 \over {2\pi}} \int_{-\infty}^{\infty} X(\omega) \, e^{j \omega t} \operatorname{d\omega} \;\;\;\;\;\text{or}\;\;...
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2answers
55 views

Fourier Integral Representation of Delta Function and Equity of Complex Integrands

In Peskin & Schroeder's QFT textbook, the following Fourier integral representation of Dirac delta function is given on page no. xxi. $$\int d^4x \, \exp(ik\cdot x) = (2\pi)^4 \, \delta^{(4)}(k). $...
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0answers
21 views

Int Jdv= int Idl how would i find a suitable J

Given that: the triple integral of $J(x,y,z)dv$ = Line integral of $I(t)dr$ (scalar $dr$) where J,I are vector fields (but "I" is obviously confined to a line path set by r(t)) r(t) ...
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1answer
36 views

Fourier Transform of complex exponential $ \omega $ vs. $ 2 \pi f $

The Fourier Transform of a complex exponential is an impulse (or Dirac) function. Depending on how frequency is written ($ \omega $ or $ f $) we have two identical definitions of the same FT: $$ \...
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0answers
65 views

Proof that an integral of an exponential is a Dirac delta

The following formula is often used: $$\int_{-\infty}^{+\infty}e^{ikx}dk = \delta(x).$$ This is equivalent to $$\int_{-\infty}^{+\infty}\left( \int_{-\infty}^{+\infty}e^{ikx}f(x)dx\right)dk = 2 \pi f(...

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