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 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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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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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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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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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 ...
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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. ...
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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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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 ...
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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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In Peskin & Schroeder's QFT textbook, the following Fourier integral representation of Dirac delta function is given on page no. xxi. $$\int d^4x \, \exp(ik\cdot x) = (2\pi)^4 \, \delta^{(4)}(k). ... 0answers 21 views Int Jdv= int Idl how would i find a suitable J Given that: the triple integral of J(x,y,z)dv = Line integral of I(t)dr (scalar dr) where J,I are vector fields (but "I" is obviously confined to a line path set by r(t)) r(t) ... 1answer 36 views Fourier Transform of complex exponential  \omega  vs.  2 \pi f  The Fourier Transform of a complex exponential is an impulse (or Dirac) function. Depending on how frequency is written ( \omega  or  f ) we have two identical definitions of the same FT:$$ \...
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(...