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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17
votes
1answer
575 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
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3answers
21k 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 ( ...
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
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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
309 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
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8answers
3k 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
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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
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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
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0answers
523 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
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4answers
22k 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
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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\...
9
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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
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5answers
284 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
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1answer
247 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
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2answers
40k 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
votes
1answer
11k 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
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2answers
918 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 ...
8
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1answer
3k 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
884 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
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5answers
2k 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
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3answers
525 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
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
1answer
91 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
3answers
182 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
2answers
228 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: ...
7
votes
1answer
109 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
141 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
241 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
16k 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
607 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 ...
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
254 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
1answer
1k views

Dirac Delta Function as a Measure

I was always told in my college physics classes to not worry too much about the dirac delta function because it can be made rigorous using distributions or measure theory. I've just started learning ...
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
448 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
101 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
638 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
410 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
521 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
4answers
215 views

Teaching Dirac delta “function” $\delta(t)$

I am about to teach applied mathematics for engineering. So I will teach how to use Laplace Transform to solve differential equations. Some of these differential equations involve the Dirac delta "...
5
votes
3answers
1k 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}(\...
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
872 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
308 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 ...

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