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.

Filter by
Sorted by
Tagged with
17
votes
1answer
542 views

Real Analysis question that affects how to think about the Dirac delta function.

Okay, here are the ingredients to this question. Me: 60 years old. 39 years ago I took two semesters of Real Analysis using the Royden textbook. Rusty is an understatement. But I am still quite ...
16
votes
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[...
14
votes
3answers
19k views

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 ( ...
14
votes
6answers
2k views

Where is the wild use of the Dirac delta function in physics justfied?

Wikipedia has a wild article about the Dirac delta function. Are the things listed correct? Or is there no proof that they are correct? For my master thesis I want to refer to rigorous proofs of these ...
13
votes
1answer
268 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 ...
12
votes
8answers
2k views

Two Dirac delta functions in an integral?

For context, this is from a quantum mechanics lecture in which we were considering continuous eigenvalues of the position operator. Starting with the position eigenvalue equation, $$\hat{x}\,\phi(x_m, ...
12
votes
4answers
2k views

When are definite integrals undefined?

We have $$\int_{-1}^{1} \dfrac{1}{x} \, dx$$ as undefined and then we have $$\int^1_{-1} f(x)\delta(x) = f(0)$$ assuming $f(x)$ is continuous everywhere and $$\delta(x) = \begin{cases} 0 & x\ne ...
10
votes
2answers
3k views

Intuition behind the derivative of dirac delta function

Let me first begin what I mean by saying the intuition behind the " $\delta'(x)$ ". For example the smooth approximations of the delta function looks like the following: (Left:the smooth ...
10
votes
0answers
506 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^...
9
votes
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?
9
votes
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\...
9
votes
2answers
1k views

Confusion with derivative of the Dirac Delta function.

So I was told by my instructor that $$L(\delta(t)) = 1 $$ And that $$\delta * f(t) = f(t)$$ For any $f(t)$ So $$\delta *1 = 1.$$ But this is $$\int_0^{t}\delta(z)dz.$$ So $$(1)' = 0 = (\int_0^{t}\...
9
votes
5answers
278 views

“Least Squares” of Dirac Delta?

It is well known that the first $N$ terms of a Fourier series of an even function $f$ corresponds to the least squares approximation of $f$ on $[-\pi,\pi]$ using the functions $S = \{1,\cos(x), \cos(...
9
votes
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 ...
8
votes
1answer
10k views

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. ...
8
votes
1answer
2k views

Change of variables for a Dirac delta function

I have often seen the following equality in Physics textbooks. $$\int_{\mathbb{R}}\delta\left(\alpha x\right)f\left(\alpha x\right)|\alpha|dx=\int_{\mathbb{R}}\delta(u)f(u)du$$ or $$\int_{-\infty}^\...
8
votes
2answers
803 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_{-\...
7
votes
5answers
1k views

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
votes
3answers
500 views

Computing integral involving Dirac Delta Function

Compute $$ \int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|} \mathrm dt $$ In closed form, where $\delta(t)$ is the Dirac Delta function . My attempt: $$ \int_{-\infty}^{\infty} t^2 \delta(\...
7
votes
3answers
2k views

Dirac delta of a function with zero derivative

It is known that: $$\int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}$$ Where $x_i$ are the roots of $g(x)$. My question is, what happens when $g'(x_i)$ is ...
7
votes
2answers
38k 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-...
7
votes
2answers
852 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 ...
7
votes
2answers
1k views

Vector Delta Function Identity

I'm trying to prove the the vector extension of the identity \begin{equation} 1 = \int \left|\sum_i\frac{ \partial g }{ \partial a }\big| _{a =a _i} \right| \delta ( g ( a ) ) da \end{equation} ...
7
votes
2answers
2k views

Dirac delta sequences

Is it true that any sequence of real functions $(\delta_n)_n$, such that $$\lim_{n\to\infty} \delta_n(x) = 0 \qquad \forall\,x\ne 0$$ and $$\int_{-\infty}^\infty \delta_n(x)\,dx = 1 \ ,$$ tends to a ...
7
votes
3answers
174 views

How to solve a second order partial differential equation involving a delta Dirac function?

In a mathematical physical problem, I came across the following partial differential equation involving a delta Dirac function: $$ a \, \frac{\partial^2 w}{\partial x^2} + b \, \frac{\partial^2 w}{\...
7
votes
1answer
88 views

How to compute $\partial \frac{1}{z^*}$?

I have trouble understanding some basic concepts in Complex Analysis: For $z=x+\mathrm{i}y$, we define: $$\partial \equiv \frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i \...
7
votes
1answer
105 views

Curious ODE with Dirac comb

I got stuck in my calculations trying to solve the following problem: Given the ODE $$\dot{x} = -\alpha x + a\sum_{n=0}^\infty\delta(t-n\tau)$$ where $\alpha \gt 0$, define $$x_k = x(k\tau +0 )$$ ...
7
votes
1answer
2k views

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 ...
7
votes
2answers
132 views

Distributional limit of a sequence of Dirac delta

I have to evaluate the following distributional limit: $$ \lim_{n \to \infty} T_n = \lim_{n \to \infty} \frac 1n \sum_{k=-2n}^{5n} \delta_{\frac kn}$$ We have that $$\lim_{n \to \infty}\langle T_n, \...
6
votes
3answers
158 views

integral of delta function of x^2

The name says what I need to calculate. When trying to integrate I stumble upon interpretation problem $$ \int\limits_{-\infty}^{+\infty} \delta(x^2) dx = \{y=x^2\} = 2\int\limits_{0}^{+\infty} \delta(...
6
votes
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
votes
1answer
259 views

Where am I making a mistake in following delta function integral?

If the given integral is $$\int_{-\infty}^{+\infty} dx \delta (x-x^{'})f(x)$$ The answer is $f(x')$. However if we make a transformation$$x\rightarrow\alpha x=y$$and $$x^{'}\rightarrow\alpha x^{'}=y^{'...
6
votes
2answers
2k views

Proving that the delta function is the derivative of the step function.

I want to prove $\frac{\mathrm{d} }{\mathrm{d} x}\Theta =\delta (x)$ using this representation of the delta function: $\delta(x)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk $ This should be ...
6
votes
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} = \...
6
votes
1answer
247 views

Derivative of the Logarithm - Dirac

So I stumbled across P.Dirac's book Principles of Quantum Mechanics and I found something really peculiar on page 61 of the Fourth Edition. He states that usually we accept that $$\frac{d}{dx}\log(x)=...
6
votes
2answers
1k views

Solution of differential equation with Dirac Delta

Is it possible to solve a differential equation of the following form? $\partial_x^2y + \delta(x) \partial_x y = 0$ where $\delta(x)$ is the dirac delta function. I need the solution for periodic ...
6
votes
2answers
182 views

Laplacian of $1/r$ in a tensor

As we know the $$\nabla^2(1/r) =- 4 \pi \delta^3(r).$$ However, I recently was readling an hydrodynamic book (An introduction to dynamics of colloids By J.K.G Dhont). The Oseen tensor is defined as: ...
6
votes
2answers
437 views

Dirac Delta definition in non-standard analysis?

What is the definition of Dirac Delta in non-standard analysis? I would define it either as a standard distribution with $\sigma=\epsilon$ or maximum equal to $\omega$. Which is the correct answer?
6
votes
1answer
2k views

Dirac delta integral form proof

While reading the book "Modern Electrodynamics" by Andrew Zangwill, on page 13 I've encountered a Dirac function integral representation. The proof the book provides is the following: \begin{...
6
votes
0answers
96 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
603 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 ...
6
votes
1answer
408 views

Correlation Theory for stationary Random process

I have read the following derivation in a book about correlation theory (Correlation theory of stationary and related random functions) and I need help understanding how the correlation function is ...
5
votes
3answers
513 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=\...
5
votes
3answers
1k views

Why is $f(x) \delta(x) = f(0)\delta(x)$ only true when $x=0$?

This is a follow up from a previous question asked by me. I know that $$\delta(x) = \begin{cases} 0 & \space \mathrm{for} \space x \ne 0 \\\infty&\ \mathrm{for} \space x = 0 \end{cases} $$ ...
5
votes
2answers
585 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 ...
5
votes
2answers
850 views

Dirac delta distribution and sin(x) - what can be a test function?

I read about the Dirac delta distribution some days ago to better understand distributions (or generalized functions), but I've become a bit confused. I used $\delta$ as a "function" ($\delta(x)$) ...
5
votes
2answers
298 views

Formally derive $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$

I have been searching for a derivation of the defining property for the Dirac-delta function: $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$ and found this derivation ...
5
votes
1answer
330 views

Integral Representation of Dirac Delta on a Manifold

In Cartesian coordinates on an $\Bbb{R}^d$ plane the Dirac delta can be represented as a Fourier transform: $$\delta^d(\vec r-\vec r')=\int \frac{d^d \vec k}{(2\pi)^d}\;\;\exp\left({i\vec k\cdot (\vec ...
5
votes
1answer
3k views

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
votes
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 ...