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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9
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6answers
2k views

Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

I have a question about this integral with a Dirac delta $$ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $$ by integration by parts I get; $$ \displaystyle \delta(x-3)e^{x^2}\biggr\...
7
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5answers
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Question about the dirac $\delta$-function

I have a basic question about the dirac $\delta$-function based on the beginning of Chapter 1 of these notes. The dirac $\delta$-function can be defined heuristically as the function that is $0$ ...
7
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2answers
848 views

Dirac's delta in 3 dimensions: proof of $\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$

If $T_f$ is a distribution, i.e. a linear functional, continuous according to the convergence defined here, defined on the space $K$ of the functions of class $C^\infty$ that are null outside a ...
4
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3answers
896 views

Divergence of $\vec{F} = \frac{\hat{\mathrm{r}}}{r^{2}}$

Consider the vector field $$\vec{\mathrm{F}} = \frac{\hat{\mathrm{r}}}{r^{2}},$$ then the divergence of this field is: $$\vec{\nabla}\cdot\left(\frac{\hat{\mathrm{r}}}{r^{2}}\right) = 4\pi\delta^{3}(\...
9
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4answers
21k views

What is the product of a Dirac delta function with itself? [closed]

What is the product of a Dirac delta function with itself? What is the dot product with itself?
16
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4answers
5k views

Dirac delta function of non-linear multivariable arguments

How does one compute a dirac delta function with a multivariable argument? For example, compute: $$ \int^{\infty}_{-\infty}{\rm d}x\,{\rm d}y\, \delta\left(x^{2} + y^{2} - 4\right) \delta\left(\left[...
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 ...
1
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1answer
651 views

composition of the derivative of Dirac delta with a function

I found this question where a nice formula is given for the composition $\delta(f(x))$. Is there a similar general formula for $\delta'(f(x))$? In other words, is there a nice way to express the ...
4
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4answers
768 views

Integration of $\int_{0}^{\infty} e^{-itw} dt$

We know that $\int_{-\infty}^{\infty}e^{-itw}dt=2\pi\delta(w)$, but how to calculate the half integration $\int_{0}^{\infty}e^{-itw}dt$?
5
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1answer
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How to write $\delta (f(x))$ in terms of $\delta (x)$?

I've seen this identity in my electrodynamics book: $$\delta (f(x))=\sum_i{ \frac{1}{|{df\over dx}(x_i)|}\delta (x-x_i)}$$ Where $x_i$ shows the $i$th zero of $f(x)$. How can I prove it? I've tried ...
5
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3answers
512 views

Why is $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$?

I understand that $\delta(x)=0$ whenever $x \ne 0$ and that $\displaystyle\int_{x=-a}^{x=b} \delta(x) \, \mathrm{d}x = 1 \space$ $\forall\, a,b \gt 0$ and also that $\displaystyle\int_{x=-\infty}^{x=\...
8
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2answers
792 views

Dirac delta function $\delta(f(x))$ of function $f$ with a higher-order zero

Dirac delta function have this property: \begin{equation} \delta(f(x))=\textstyle \sum_i\frac{\delta(x-a_i)}{\lvert f^\prime(a_i)\rvert}. \end{equation} And its derivation is: \begin{eqnarray} \int_{-\...
3
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1answer
220 views

$\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$ with distributions defined on Schwartz space

I know, from a recent enlightening answers received here, that, if we define the distribution represented by Dirac's $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is ...
3
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1answer
722 views

How to prove this Dirac delta identity involving $\int_{-\infty}^\infty \delta(f(x)) \,s(x) \,dx$?

How to prove the identity $$\int_{-\infty}^\infty \delta(f(x)) \,s(x) \,dx = \sum_i \frac{s(x_i)}{|f'(x_i)|},$$ where $x_i$ are the zeros of $f$? I am supposed to use an identity that I've already ...
2
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1answer
110 views

What is the integral of $\int_{-\infty}^{+\infty} H(t)\delta(t)dt$ ($H(t)$ Heaviside step, $\delta(t)$ Dirac delta)?

I was trying to figure out what is the integral of $$\int_{-\infty}^{+\infty} H(t)\delta(t)dt,$$ where $H(t)$ is the Heaviside step and $\delta(t)$ is the Dirac delta. A first approach: We ...
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1answer
2k views

Product of 2 dirac delta functions

Solve for a parabolic diff. eq: $$ \frac{\partial{}G(x,t)}{\partial{}t}=a\frac{\partial{}^2G(x,t)}{\partial{}x^2}+\delta(t)\delta(x)$$ Using the result, write the general solution of: $$\frac{\...
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2answers
2k views

Dirac Delta and Exponential integral

I am able to derive the following equation by substituting the definition of a Fourier transform into it's inverse. $$2\pi\delta(x-x') = \int_{-\infty}^{\infty} e^{ik(x-x')} dk$$ How do you prove ...
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2answers
272 views

Is $\delta$ in $L^\infty$?

I think the question title says is all. I am wondering, is the Dirac delta in the Lebesgue space $L^\infty$?
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3answers
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Why does integrating a complex exponential give the delta function?

How come, when we integrate a complex exponential from $ -\infty $ to $ \infty $, we get a scaled delta function? $$ \begin{align} \int_{-\infty}^{\infty} e^{i k x} \; dk & = 2 \pi \delta \left ( ...
7
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2answers
37k views

Convolution with delta function

I am merely looking for the result of the convolution of a function and a delta function. I know there is some sort of identity but I can't seem to find it. $\int_{-\infty}^{\infty} f(u-x)\delta(u-...
8
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1answer
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Dirac delta function as a limit of sinc function

I'm looking for a rigorous proof of the statement: $\delta(x) = \lim_{\epsilon->0} \frac{\sin(x/\epsilon)}{\pi x}$ (see (37)). For any non-zero value of x, LHS of the above is by definition zero. ...
4
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2answers
643 views

Derivation of the Dirac-Delta function property: $\delta(bt)=\frac{\delta(t)}{\mid b \mid}$

Considering the case such that $b \gt 0$ and $b \in \mathbb{R^+}$ and making the substitution $t'=bt$, it follows that $$\int_{t=-\infty}^{t=\infty}f(t)\delta(t)\mathrm{d}t =\color{red}{\int_{t'=-\...
2
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1answer
2k views

How to define a delta function on complex plane?

I understand that it makes perfect sense to define a 2-dimensional delta function on the complex plane by $$\int dz\wedge d\bar{z}\delta(z)\delta(\bar{z})=1.$$ However, is there any chance to define a ...
5
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3answers
3k views

delta function on a complex number

for a real number we know that $$ f(a)= \int_{-\infty}^{\infty}dx \delta (x-a)f(x) $$ but what happens for $$ \int_{-\infty}^{\infty}dx \delta (x-2i)f(x) $$ ? is this equal to $ f(2i) $ or equal ...
0
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1answer
757 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta (x-x');...
5
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1answer
280 views

A representation of Dirac-$\delta$

Prove that $$g_\epsilon (x)=\lim_{\epsilon \to 0} \frac1 \epsilon \frac1 \pi e^{-x^2/\epsilon^2}$$ is a Dirac-$\delta$ function. This is a homework question I'm stuck with. I'm probably missing a ...
4
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2answers
384 views

By substituting $z = h(t)$ show that $\delta(h(t))=\sum\limits_{i}\frac{\delta(t−t_i)}{\mid h^{\prime}(t_i)\mid}$

Firstly, apologies in advance for using an abuse of notation by placing the Dirac-delta inside an integral. But for my level of understanding, I have no choice. This question involves one of the ...
3
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1answer
1k views

Composition of a Dirac delta and a function in higher dimensions

Coming from a physics background, I was taught the formula for the composition of a Dirac delta and a function. Indeed, if we consider a nice function $ f : \mathbb{R} \to \mathbb{R} $, one can write $...
1
vote
2answers
968 views

Dirac Delta function and Lebesgue-measurability

Baaquie, in "Quantum Finance", states that the Dirac Delta function is unmeasurable, since it "has support on a set that has zero measure" What is a "support"? What kind of mathematical object is it (...
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2answers
273 views

Fresnel function converging to delta distribution

i need to show that the function known from the Fresnel integral (wikipedia) converges to the Dirac delta-distribution. This function is defined as $f_{\epsilon}(x) = \sqrt{\frac{a}{i \pi}\frac{1}{\...
0
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0answers
685 views

How does the Dirac delta function operate when its peak is at the boundary of an integral?

As far as I can tell the Dirac delta function in an integral picks the value of the multiplying function at the peak provided the peak is within the boundary, i.e. $$\int^{a+e}_{a-e} \delta (x-a) f(x)...
2
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2answers
2k views

Second derivative of absolute value function proportional to Dirac delta function?

I have recently discovered the relation \begin{equation} \frac{\mathrm d^2}{\mathrm dx^2} \big| x \big| = 2\delta (x). \end{equation} I was very intrigued when I found this expression, and as it ...
1
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2answers
419 views

Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
6
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4answers
15k views

Dirac Delta function inverse Fourier transform

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-i\omega t} dt = 1,$$ and if I were to reconstruct the function back in time domain,...
6
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1answer
2k views

Representation of Dirac Delta Function in complex plane

I am stuck on a representation of the Dirac Delta function that is used in several books I am reading. They state: $\begin{equation} \delta^{(2)} = \frac{1}{2\pi} \partial_{\bar{z}} \frac{1}{z} = \...
13
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1answer
267 views

Why do physicists get away with thinking of the Dirac Delta functional as a function?

For instance they use it for finding solutions to things like Poisson's Equation, i.e. the method of Green's functions. Moreover in Quantum Mechanics, it's common practise to think of the delta ...
4
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2answers
5k views

Integral of Dirac delta function from zero to infinity

I know that: $$\int_{-\infty}^{+\infty} \mathrm{d}t \, f(t) \delta(t) = f(0)$$ However, I cannot figure out the result of the integral below: $$\int_{0}^{+\infty} \mathrm{d}t \, f(t) \delta(t) = ?$$ ...
6
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1answer
596 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 ...
9
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1answer
244 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to0^+.$$ Obviously, this sequence of functions ...
5
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2answers
580 views

Dirac delta and non-test functions

Normalization of the delta function (distribution) is often informally written as an integral $$\int_{-\infty}^{+\infty} \delta(x) \, dx = 1$$ An attempt to write this formally would be expression ...
4
votes
3answers
720 views

How to prove that $\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x)$

It is well-known that: $$\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x).$$ This can also be written as $$ 2\pi\delta(x)=\int^{+\infty}_{-\infty}e^{ikx}\,\mathrm dk.$$ However, I don't know how to ...
4
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2answers
168 views

An identity on $\delta(x+y)$?

I am dealing with some derivatives on Dirac delta function, $(x\partial_x+y\partial_y)\delta(x+y)$. Consider the following integral ($f(x,y)$ is bounded) and perform integration by parts, \begin{...
3
votes
0answers
426 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})\...
2
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2answers
5k views

Laplace transform of the derivative of the Dirac delta function

If asked to find the Laplace transform of the derivative of the Dirac delta function, I would naively integrate by parts and conclude that $$ \begin{align}\int_{0}^{\infty} \delta'(t) e^{-st} \, dt &...
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2answers
1k views

Numerical way to deal with Dirac delta.

I have been wondering about this: I have a differential equation $y'(t) = y(t) + n \delta(t) y(t)$ with $y(-1) :=y_0$ Thus I want to apply a short delta pulse at some particular point $0$ to my ...
7
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1answer
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ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
4
votes
2answers
581 views

Dirac Delta function at a point

From my understanding of the Dirac Delta function, it is infinitely thin and has a value of infinity at only a particular point. I also learned that $$\int_{-\infty}^{\infty} \delta(x-a) dx = 1$$ What ...
4
votes
1answer
726 views

Surface area of sphere using Dirac delta

This question is related to this one. Suppose I want to calculate the surface area $S(R)$ of a sphere of radius $R$. I can express $S(R)$ as $$S(R)=\int_{\mathbb{R}^3} \delta (\| \vec x \|-R) \ d \...
4
votes
3answers
5k views

Approximating dirac delta function with sinc functions

I am trying to prove the following $$\lim_{a\rightarrow\infty}~\int_0^\infty\frac{f(x)\sin(ax)}{x}dx=(\pi/2)f(0)$$ for all square integrable functions $f(x)$ continuous at $0$. I tried to do the ...
3
votes
2answers
934 views

How to solve this integral with dirac delta functions?

$$F(t)=\displaystyle\int_{-\infty}^{\infty}\left[e^{-2t^2}\left\{\dot\delta(t-2)\right\}+\delta(t^2-16)\right]\,dt$$ How to get rid of the derivative ? and for the second function i wrote it as $\...