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.
2,022
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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 ...
- 1
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0
answers
33
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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.
...
3
votes
1
answer
57
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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 ...
- 42.1k
3
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2
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38
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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}...
- 33
1
vote
1
answer
80
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Clarifications about dirac delta
Imagine you got a Heaviside step function $H(x-x_0)$ and the problem requires one to solve the integral $\displaystyle\int_{-\infty}^{\infty} H'(x-x_0) f(x) dx$.
Way 1: One immediately thinks that $H'(...
- 45
-1
votes
1
answer
105
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Dirac delta confusions [closed]
I might be hitting the wall and would appreciate following me on this.
Some explain dirac delta such as it's 0 everywhere except at a single point where it's infinite and some explain it as a ...
- 45
-2
votes
0
answers
32
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Is DiracDelta function $\delta(t)$ equals zero in the neighborhood of $0$?
I know that DiracDelta function equals zero everywhere except at $0$, but what about $\delta(0^-)$ and $\delta(0^+)$?
Should they be evaluated as $\delta(0)$ or to be equal $0$ ?
I ask this question ...
- 1,134
0
votes
1
answer
28
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Proving Dirac Delta property if the approximating functions aren't necessarily non-negative
Here is my problem, as written:
A Dirac approximation sequence is a family of smooth functions $\{ \delta_1, \delta_2, \dots\}$ defined over $\mathbb R$ such that $\delta_n(x) = 0$ whenever $|x| >...
- 123
0
votes
1
answer
80
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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:
...
- 45
1
vote
1
answer
52
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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 ...
- 45
1
vote
2
answers
75
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Dirac-Delta distribution lies in $H^{-1}(B_1(0))$ only for $n = 1$
Show that the delta distribution $\delta_0$ is in $H^{-1}(B_1(0))$ ($B_1$ denotes the unit ball in $\mathbb{R}^n$) if and only if $n = 1$.
I've successfully proven that for $n = 1$ we have $\delta_0 \...
- 39
1
vote
1
answer
52
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Hankel transform of arbitrary order of $f(r)=1$
The Hankel transform of order $\nu$ of a function $f(r)$ is defined by
\begin{equation*}
F_\nu(k) = \int_0 ^\infty r dr f(r) J_\nu(kr ),
\end{equation*}
where $J_v$ is a Bessel function of the first ...
- 113
1
vote
1
answer
50
views
differential equation with delta and heaviside functions
Suppose we have a differential equation given by:
$$ \frac{d}{dx}f(x) = g\big(c.H(x)\big)\delta(x) $$
where $H(x)=1_{x\ge0}$ is the Heaviside step function, $c$ is a constant and $g$ is a function (...
- 243
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0
answers
41
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Help make sense of these integral manipulations
In the context of statistical mechanics (so functions of real output are often defined on a 2n dimensional vector space called the phase space, of whose measure I will indicate with $d\Gamma$ ), the ...
- 1,401
0
votes
0
answers
45
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Relationship Between Green's Function and Dirac Delta
In chapter 9 (Green's Functions for Time-Independent Problems) of Haberman's text 'Applied Partial Differential Equations', the author presents the following fundamental result
$$ L[G(x,x_s)] = \delta(...
0
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0
answers
14
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Calculating the complex coefficient for the dirac comb
Im trying to calculate the complex coefficient $c_n$ for the dirac comb but im stuck in this step which leads to 2/T instead of 1/T: so the sum vanished because $t \in [0, T]$ so k must be either $ 0; ...
- 227
1
vote
1
answer
89
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Interpretation of Dirac Delta
In Haberman's 'Applied Partial Differential Equations with Fourier Series and Boundary Value Problems', he says the following,
Dirac delta function. Our source $f(x)$ represents a forcing of our ...
1
vote
0
answers
19
views
Proof of the completeness property of Hermite polynomial related solutions of the harmonic oscillator
I am searching for tips towards a proof of the completeness property for the solutions of the harmonic oscillator based on real Hermite polynomials $H_n(x)$, i.e.:
$$e^{-(x^2+a^2)/2}\sum_{n}^\infty \...
- 111
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votes
0
answers
11
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How to express the concentrated Force at the center of a circular domain into a distributed pressure with the help of Dirac Delta?
Background: the beam case (1D)
For a beam bearing concentrated froce $P$ at the middle, the concentrated force $P$ can be expressed in a distributed manner, i.e.,
$$
q=P\delta(r)
$$
such that the ...
- 65
0
votes
0
answers
18
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Weak derivative, constants infront of Dirac-delta
I'm currently taking a PDE course, and would like some help with understanding weak derivatives.
My question is: when computing weak derivative and getting a result like $f'=-2\delta_1 + 4\delta_2 + g$...
- 583
1
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0
answers
28
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Integral representation of Dirac Delta in cylindrical coordinates
Dirac delta have the representation
$$
(2\pi)^4\delta^4(x) := \int e^{ik.x} d^4 k
$$
I would like to know how such integral representation realized in cylindrical coordinates. I tried the following ...
- 221
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0
answers
25
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Signal and Dirac delta functions
I have been having a rough time understanding how to solve this. If anyone can help, thank you.
P3: Consider the system $\displaystyle y(t) = \int\limits^t_{-\infty} e^{-(\tau-t)} x(\tau+2) \,\Bbb d\...
0
votes
1
answer
56
views
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,
...
- 121
3
votes
1
answer
142
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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 ...
- 65
1
vote
1
answer
34
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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 ...
- 131
0
votes
0
answers
28
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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 \...
- 501
1
vote
1
answer
59
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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 ...
0
votes
0
answers
25
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Is there a well-defined notion of "maximally correlated" probability distribution on $[a,b]^2$?
For some interval $[a, b]$ of the real line, consider the family of probability distributions $p: [a,b]\times [a,b] \rightarrow \mathbb{R}^+$ such that $p\geq 0$ and $\int dx dy p(x, y) = 1$. So $p(x, ...
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votes
0
answers
43
views
Is this integral operator involving Dirac delta diagonalizable?
For a Hilbert space $\mathcal{H} = L^2([0, 1])$ and a linear operator $L: \mathcal{H} \rightarrow \mathcal{H}$, then if $L$ is Hilbert-Schmidt (e.g. $\sum_{i} \lVert L e_i \rVert^2 < \infty$ for ...
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0
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59
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Alternate definition of the Dirac Delta function
I was reading the wikipedia article on the Dirac Delta function, and what I saw was that the function needed to have a integral over the complete number line to be $1$, and have a value of zero ...
- 317
1
vote
1
answer
56
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Fourier Transform of $x \delta(x)$
I'm looking to understand how to compute this distributional Fourier transform:
$$\mathcal{F}(x \delta(x)) = \int x \delta(x) e^{-ikx} \, dx$$
Were it $x f(x)$ I would use the common rule that $\...
- 478
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votes
0
answers
41
views
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 ...
0
votes
0
answers
96
views
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 ...
- 727
2
votes
1
answer
52
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The derivative of the Heaviside step function in $\mathbb{R}^n$
Here is a problem I have:
Consider the Heaviside function
$$H(x)=H(x_1)H(x_2)\cdots H(x_n), ~~~x\in\mathbb{R}^n$$
and prove that
$$\dfrac{\partial^n H(x)}{\partial x_1\partial x_2\cdots\partial x_n}=\...
- 29
0
votes
1
answer
59
views
How do we derive the composed Dirac Delta function?
Suppose we have to compute $∂_{x}\theta(y-x)$, where $θ$ is the Heaviside step function, how do I find the result?
Intuitively I would just use the chain rule
$\partial_{x}\theta(y-x)=-\delta(y-x)$
...
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votes
1
answer
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Heaviside step function properties $\theta(t-t_1) \theta(t_1-t_2) + \theta(t-t_2) \theta(t_2-t_1) = \theta(t-t_1) \theta(t-t_2)$
I want to prove the following Heaviside theta function property :
\begin{align}
\theta(t-t_1) \theta(t_1-t_2) + \theta(t-t_2) \theta(t_2-t_1) = \theta(t-t_1) \theta(t-t_2)
\end{align}
Recalling the ...
- 6,358
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votes
1
answer
45
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Dirac delta doublet function in simple harmonic oscillation. Conditions imposed?
I'm considering an at-rest simple harmonic oscillator (m,k) and want to model the force by a doublet (derivative of dirac delta) at t=0.
$$f = \delta'(t)$$
I've already considered the case for a dirac ...
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0
votes
1
answer
102
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What is the limit for $x$ approaching zero of the dirac delta distribution $\delta(x)$?
I am trying to prove that
$$
\lim_{y\rightarrow 0} \delta(y-x) = \delta(x)
.$$
To justify this problem, this comes from the orthogonality of the position eigenstates in quantum mechanics. Indeed, we ...
- 3
0
votes
0
answers
29
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Trouble visualizing a graph of the sum of infinite Dirac functions
I'm currently trying to solve a problem on my "Signals and Systems" class and I'm stumped.The question has 2 parts:
1-) The function T.I can be defined as:
$$
T\cdot I(t) = \sum_{n=-\infty}^{...
1
vote
1
answer
42
views
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) \...
- 6,358
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0
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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 ...
- 1
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votes
1
answer
41
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Derivative of the Dirac delta function. The Fourier transform of this function is the a-th derivative of the Dirac delta function. How to interprete?
Evaluating this integral
$$\int_{-\infty}^{\infty} x^{a} e^{-i(k-k')x}dx=\sqrt{2\pi}(-i)^a\delta^{(a)}(k-k')$$
I got this result that lead to the definition of $$\delta^{(a)}(k-k')=\sqrt{2\pi}(i)^a\...
- 11
1
vote
2
answers
64
views
What technique is to use for differentiation regarding $\delta$-functions?
Interesting differentiation regarding $\delta$-functions. Let we define a function $h(x)=e^{-ax}H(x)+e^{ax}H(-x)$ and we want to find the $n$-th derivative.
\begin{align}
h(x)&=e^{-ax}H(x)+e^{ax}H(...
- 793
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votes
0
answers
65
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Derivative of Dirac delta
To represent formally the part of an algorithm
if $r \in [0,p (\mathbf{v})]$, set $y = A(\mathbf{v})$; else set $y = B(\mathbf{v})$,
where $\mathbf{v}$ is a vector of parameters and $r$ is a uniform ...
- 668
0
votes
1
answer
58
views
Divergence of a function involving delta distribution
Given the vector field $\vec{b}$ defined in this way: $$ \vec{b}=-4\pi g\int_0^{\infty} dt \frac{d\vec{s}}{dt}\delta^3(\vec{r}-\vec{s(t)}),$$ where $\vec{s(t)}: t\in[0,+\infty)\rightarrow \mathbb{R^3}$...
1
vote
0
answers
61
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What is the best way to numerically integrate this function?
I am trying to numerically evaluate an integral that came up in my work. The integral takes the form
$$
\int_0^{\infty}dx\tanh(\alpha(x-x^*))\int_0^{\infty}dzf(x,z)\delta(g(x,z)-\beta t(z))
$$
with $\...
- 11
0
votes
0
answers
48
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Is there any method to solve the integral including Kronecker delta, not Dirac delta?
It's a weird question to me, too. Nevertheless, please refer to below formula.
$V(r_1,r_2)=\langle\int\varepsilon_\nu(R_1)\varepsilon_\nu(R_2)\frac{e^{i\omega|R_1-r_1|}}{|R_1-r_1|}\frac{e^{i\omega|R_2-...
- 5
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votes
1
answer
70
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Solving two coupled ODEs with delta function sources
I am interested in solving two coupled ODEs for two functions $f(r)$ and $g(r)$ of the following form:
$$r(1-g(r))+(r^2+1)\frac{f'(r)}{2f(r)}=E\delta(r-r_0)$$
$$rf(r)[2r(g(r)-1)g(r)+(r^2+1)g'(r)]=E\...
- 25
1
vote
0
answers
45
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Multiple change of variables in integral to difference between variables
When changing variables in multiple integrals one needs to find the Jacobian matrix of the transformation
$$\int dx_1\cdots dx_n f(x_1,...,x_n)=\int dy_1\cdots dy_n f(y_1,...,y_n)\left|\frac{\partial(...
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1
answer
65
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Dirac delta - how to get this identity?
I have a question concerning this paper, page 8.
On page 8, it is said that
$$
\int_{K_l}1\delta(x-\hat{x})\, dx=\begin{cases}1, & \hat{x}\in K_l\\
0, & \hat{x}\notin K_l\end{cases}.\tag{1}
$$
...
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