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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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10
votes
0answers
497 views

zeros/poles of Laplace transforms of Dirac combs (Riemann zeta function)

let's define $p_\alpha(n) = \displaystyle\int_1^n x^\alpha dx$ so that $\left\{\begin{array}{lll} p_0(n) &=& n-1 \\ p_{-1}(n) &=& \ln n \\ p_\alpha(n) &=& \frac{\textstyle n^...
6
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0answers
94 views

What is $\int_{-\infty}^{\infty}\exp(\mathrm{i} n \cosh{x}) \, \mathrm{d}x$?

I'm hoping to determine the value of the following integral: $$\int_{-\infty}^{\infty}\exp(\mathrm{i} n \cosh{x}) \, \mathrm{d}x$$ Here is a plot of the integrand as a function of $x$ with parameter ...
6
votes
1answer
568 views

Delta function at the origin in polar coordinates

I have some problems understanding what the best way of dealing with the delta functions in polar coordinates (I know there are many questions on the subjects on this website but they are all not ...
5
votes
0answers
267 views

Poisson summation formula for positive integers

I am trying to evaluate the following expression for $\lambda \in \mathbb{R}$ : $$f(\lambda)=\sum_{n=1}^{+\infty}e^{-i\lambda n}$$ My idea is to introduce an epsilon prescription, so I choose $\...
5
votes
2answers
400 views

Singular differential forms and $\nabla^2\left(\frac{1}{r}\right)$

The delta function identity $$\nabla^2\left(\frac{1}{\lvert\mathbf{x-x'}\rvert}\right)=-4\pi\delta^{(3)}(\mathbf{x-x'})$$ is often casually derived using the divergence theorem, since the divergence ...
5
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0answers
535 views

Heaviside unit step- and delta function

The following question is right from the book: Show that $$ H(x-x_i) = \int_{-\infty}^x \delta(x_0-x_i)dx_0\, $$ satisfies $$ H(x-x_i) \equiv \begin{cases} 0 & x < x_i \\ 1 &...
4
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1answer
82 views

Strong definition of the radial Dirac delta function and its derivative

The G. Barton textbook “Elements of Green’s functions and Propagation,”, Oxford University Press, 1991 has a very nice introduction on the Dirac delta function. When the 3-D delta function is ...
4
votes
1answer
136 views

Integration with Dirac delta function of two-argument function

I would like to solve that problem: $$ \int d^2 \mathbf{k} \, \delta(f(k,\phi)) = \int_{k_1}^{k_2} dk \, k \int_0^{2\pi} d\phi \, \delta(f(k,\phi)) \,, $$ where $f(k,\phi) = a - bk^2 - ck^3 |\sin(2\...
4
votes
1answer
119 views

Traffic flow with Dirac-$\delta$ source (on ramp)

I have been trying to solve the traffic flow equation with a singular source ($D>0$ large): $$ \rho_t + f(\rho)_x = D\delta(x) $$ with the flux $f(\rho)=\rho(1-\rho)$ and the initial data $\rho(x,0)...
4
votes
2answers
81 views

What is the inverse Laplace transform of my equation?

What is the inverse Laplace transform of L(s)=$exp⁡[-(1/2)sI]$*$I_0[(1/2)sI]$, where $I_0$ is the modified bessel function of first kind. I am told that the answer is $\frac{1}{π}$ $\left(\frac{1}{(...
4
votes
1answer
114 views

Integral involving delta functions and vector quantities

This integral comes from equation (3.15) in an older paper I've been reading: $$ \int \mathrm{d} \Omega_k \, \delta\left(|\vec{k}|^2 - |\vec{k}+\vec{q}_1|^2\right) \delta\left(|\vec{k}|^2 - |\vec{k}-\...
4
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0answers
10k views

Fourier series expansion of Dirac delta function

My question is from Arfken & Weber (Ed. 7) 19.2.2: In the first part, the question asks for Fourier series expansion of $\delta(x)$. I have found $$\delta(x)=1/2\pi + 1/\pi\sum^{\infty}_{n=1} ...
4
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0answers
269 views

Proving an identity of the composition of the delta distribution with a differentiable function

Given a differentiable function $f$, some $x_j$ ($j \in \{1, ..., n\}$) such that $f(x_j) = 0$ $\forall j$ and $f'(x_j) \ne 0$ $\forall j$, and the following definition of the composition of a ...
4
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0answers
75 views

Computing the limit of this integral,

This is Part 6 (last part) of a problem statement of an old comprehensive exam question that I am working on. It asks to evaluate $$\lim_{r_0 \to 0} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\...
4
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0answers
136 views

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
3
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0answers
56 views

Limit distribution equal to Dirac delta

This is the problem 6.19 from the book Distributions and Operators, Gerd Grubb. I already have done parts (a) and (b). The part (a) of this problem is proving that for $r\in(0,1]$, the sequence $$\{\...
3
votes
1answer
93 views

Dirac delta uder coordinate transformation

I am having some issues understanding Dirac's delta function/distribution behaviour under change of coordinates. There is a statement, if $(x_1,\ldots,x_n)$ are cartesian coordinates and $y_1,\ldots,...
3
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0answers
69 views

Dirac delta from poles of a function

Suppose we are given the simple expression $$ F(k) = \frac{1}{E^2-E(k)^2} $$ which has a pole when $E^2 = E(k)^2$ and where $E, E(k)$ are real numbers. When working with this expression (e.g. inside ...
3
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0answers
41 views

General solutions to first-order differential equations with disitrubutional coefficients

Consider the first-order differential equation $$\dot{x}+p(t)x=q(t).$$ This can be generally solved using an integrating factor $$a(t)=\exp\left(\int p(t)dt\right)$$ and the solution is $$x(t)=\...
3
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0answers
477 views

Using Green's function to solve 2d laplace equation

Consider a domain $D : {(x,y) : x>0 , y>0}$ Let $\mathbf{x}= (x,y)$ and $\mathbf{\xi}= (\xi_x, \xi_y)$. Then the Green's function satisfying $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
3
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0answers
100 views

Delta “function” and calculus

In a comment on Math.SE, I came today across the remark that the expression $\delta(f(x))$ has no meaning. Being a physicist, I have been surprised but I then remembered Schwartz distribution theory ...
3
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0answers
118 views

Integrating derivatives of dirac delta from zero

I wish to compute the following integral. $$ I_1 = \underbrace{\int_{-\infty}^{\infty} \dots\int_{-\infty}^\infty}_{n} \frac{\partial^n}{\partial x_1 \dots \partial x_n}\Bigg(\max \Big( \sum_i^n ...
3
votes
1answer
369 views

Is it true that the integral of $\delta(x)/x$ between symmetrical limits is zero?

My professor is claiming that the following is true: $$\int_{-\infty}^{\infty}\frac{\delta(x)}{x}dx=0,$$ where $\delta(x)$ is the Dirac delta "function", as he calls it. I think the integral ...
3
votes
0answers
175 views

How to show this property of the delta function?

Let $\mathcal{D}(\mathbb{R})$ be the space of test-functions in $\mathbb{R}$ and let $f$ be a $C^\infty$ function. I want to show that if $f$ has $n$ zeroes $x_1,\dots,x_n$ in the interval where it is ...
3
votes
1answer
324 views

Generalised derivatives of discontinuous real functions

Does the generalised derivative of every discontinuous real-valued function always yield a Dirac-delta 'function' at the point(s) of discontinuity? My limited experience with generalised distributions ...
3
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0answers
419 views

Green's function for the 3D wave equation

Suppose we have the 3D wave equation which, in terms of Green's functions, can be written as $$ \left( \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla \right)G(\bar{x},t) = \delta(\bar{x})\...
3
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0answers
75 views

pdf of transformed random variable $g(X)$ as integral over $X$?

I am not a mathematician, so I am sorry if this question is too easy or some notational detail is not correct. I am trying my best! I have got a random Variable $X$ in $\mathbb{R}^N$ with pdf $p(X)$ ...
3
votes
3answers
151 views

Dirac delta - sifting

We know $\int_{-\infty}^\infty \delta(x-a)f(x) \, dx=f(a) $ Is this still true for: $\int_{-\infty}^\infty \delta(a-x)f(x) \, dx=f(a) $ In general, can we call dirac delta even function?
3
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0answers
367 views

Represent Dirac Delta function in Finite Difference method

I recently solving $-\Delta u=\delta$ where $\delta$ is dirac delta function using FDM on 2 dimensional space. Since dirac delta function is undefined at origin, and 0 elsewhere, I will use $\delta(...
3
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0answers
460 views

Cauchy principal value for solving the integral of complex exponential

I need to solve the following integral (if it is possible): $$\int_0^{\infty}dx\,f(x) \left\{ \lim_{t \rightarrow \infty}\int_0^{t}e^{i(x-x_0) \tau}d\, \tau \right \}$$ I found an expression in an ...
3
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0answers
183 views

Dirac Delta Properties

From Mathworld, for example, we have the following properties of the Dirac delta: $x^n\delta^{(n)}(x)=(-1)^n\, n! \, \delta(x)$ $x^2 \, \delta'(x)=0$ So, if $f(x)$ is $C^\infty(R)$, is it correct ...
3
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0answers
94 views

impossible ODE using delta functions?

I'm working on the problems in the book "Asymptotic Methods of Differential Equations", by Roscoe White. It's a pretty legit book, and all the problems are quite non-trivial and very rich. However, ...
3
votes
1answer
381 views

How should the Calculus of Variations deal with $\delta(t-t_0)$ variations?

I'm familiar with using the Calculus of variations to find the condition for which first order variations of a functional wrt a function are zero: We start with a functional $J[x]= \int_{t_f}^{t_i}L(...
2
votes
0answers
85 views

Is the formula $\;\oint H(x) \, \delta(y) \, dy = \frac{1}{2\pi} \oint d\phi\;$ well known?

Explanation - everything real-valued: $$ (x,y) = \mbox{cartesian coordinates} \\ \phi = \mbox{angle, in polar coordinates} \\ H(x) = \begin{cases} 0 & \mbox{for} & x < 0 \\ 1 & \mbox{...
2
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0answers
61 views

Integral limit that behaves like a delta function

I am having difficulty seeing why this integral reduces to $1$ or $0$, just like the delta function. In the following statement, $\Delta$ is small, whereas $R$ goes to infinity: $$\lim_{R\rightarrow \...
2
votes
1answer
63 views

Fourier transform rules for Dirac-deltas, to avoid integration

I'm following through a worked example but have a different solution to the given solution. Could someone tell me what I'm missing please, if anything? I have taken the 2nd derivative of a piece-...
2
votes
2answers
105 views

How can we evaluate the following integral using the tricks of delta functions?

I am trying to teach myself the statistical field theory formulation of statistical mechanics. Not part of a class, just self study in my free time. I appreciate any help here. I am starting with ...
2
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0answers
76 views

How to deal with a Dirac delta function numerically?

I need to solve some differential equations with a Dirac delta function. e.g. the source terms are like, $\delta(x)$ and $\partial_x \delta(x)$. Could I just use the Gaussian type function? e.g. $$...
2
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0answers
47 views

show that $\delta''(\frac13x^3 +x) = \delta''(x) - 2\delta(x)$

I tried to go about about this using the definition : let $\phi$ be a test function $$\begin{align}\langle \delta''(\frac13x^3 +x) ,\phi\rangle &= \langle \delta''(\frac13x^3 +x) ,\frac{x^2+1}{x^...
2
votes
1answer
48 views

Equality between two equations

at the moment I am reading the following paper Benno, Steven A., and José MF Moura. "On translation invariant subspaces and critically sampled wavelet transforms." Multidimensional Systems and ...
2
votes
1answer
69 views

Higher order derivatives of Composition of Dirac delta distributions

There are two equations showed in Gel'fand and Shilov's book (Generalized Functions I Properties and Operations) on page 183 and 185: $${\delta}^{(k-1)}(1-x^2)=\frac{(-1)^{k-1}}{2^kx^{k-1}}[{\delta}^{...
2
votes
0answers
52 views

Hidden Fourier multiplier in integral expression?

After some (formal!) manipulations I stumbled upon the following expression: $$ \hat{f}\left(\xi,\eta\right)=\iint_{\mathbb{R}^{2n}}e^{2\pi i\left\langle x,t-\xi\right\rangle }e^{2\pi i\left\langle \...
2
votes
0answers
64 views

Issues after taking the integral by using Dirac delta: a jump of the integrand

Introduction Consider the integral $$ \tag 1 I= \int f(x,t)\ g(y)\ \delta(m(x,y,t,a,b)) \ dx\ dy\ dt , $$ where $a,b$ are positive parameters, and $b>a$, The integration ranges for $x,y,t$ are ...
2
votes
1answer
76 views

How do we need to read the Dirac-comb identity?

Let $(\Omega,\mathcal A)$ be a measurable space $\omega\in\Omega$ $\delta_\omega$ denote the Dirac measureat $\omega$ on $(\Omega,\mathcal A)$ $E$ be a $\mathbb R$-Banach space $\mathcal M$ denote ...
2
votes
2answers
96 views

Fourier Transform and Dirac Delta Function

A Fourier transform of function R into function Q is defined as: $$Q(\underline{k}) = \int_{}^{}R(\underline{x}) e^{-i\underline{k}·\underline{x}} \mathrm{d}\underline{x}.$$ where I've underlined $\...
2
votes
0answers
57 views

Dirac delta of multi-variant function with infinite zeros

I am trying to evaluate the following integral $$\iint \limits_{[0,1]^2} {\rm d}x{\rm d}y\;f(x,y) \delta(g(x,y))$$ where $$g(x,y) = a_0 + a_1 x + a_2 y + a_3 xy $$ and $\delta(\cdot)$ is the Dirac ...
2
votes
0answers
32 views

Solving general linear ODE with Dirac delta as input, regarding continuity

I am interested in solving the following form of ODE, of arbitrary order $n$ (Equation 1): $$a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1 \frac{dy}{dx} + a_0 y = \delta(...
2
votes
0answers
68 views

Question about differential forms and delta functions

EDIT: Related to "Derivative of a logarithm and Dirac delta function", "2-dimensional delta function (complex plane)" and "How to define a delta function on complex plane?". In a physics paper by ...
2
votes
0answers
84 views

If $f$ is a Sobolev $H^1$-function and vanishes everywhere except for a point, is it constantly zero?

Let $H^1(\mathbb{R}^3)$ be the usual $L^2$-based Sobolev space. If $f\in H^1(\mathbb{R}^3)$ satisfies $f=0$ in $\mathbb{R}^3\setminus\{0\}$, can one conclude that $f=0$ in $H^1(\mathbb{R}^3)$? I ...
2
votes
1answer
262 views

Dirac delta in Dirichlet Distribution explanation

I am familiar with the following definition of the Dirac delta function: $$ \delta = \begin{cases} \infty, & \text{if } x=0 \\ 0 & \text{if } x \neq 0 \end{cases} $$ Now, I am reading this ...