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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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3
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0answers
176 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 ...
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430 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})\...
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76 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
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3answers
153 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?
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391 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
469 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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185 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
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2answers
748 views

Discontinuous solution to first order ODE with delta function coefficients

Consider the following first-order ODE: $$y'(x) = f'(x) y(x);$$ this has solution $y(x) = C e^{f(x)}$. Now, consider taking $$f_\lambda(x) = \frac{\alpha}{2}\left(1+\tanh(\lambda x)\right);$$ in ...
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1answer
391 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(...
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2answers
62 views

Disintegration of a of delta function

A delta function has been written as $$\delta(x^2-a^2)$$ Do you think I can write the function as $$\delta (x+a)+ \delta (x-a)$$ ? My plan is to locate the delta function potential coordinates in ...
2
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2answers
6k views

Use the Inverse Fourier transform to show the Dirac-Delta function as a limit of the sinc function [duplicate]

This question is not the same as Dirac delta function as a limit of sinc function because I am asking about the inverse Fourier transform and more specifically the relation of equation $(1)$ below to ...
2
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1answer
110 views

Computing the integral $\int\limits_{-\infty}^{\infty}(t^2-1)\delta(t)\:dt$

Have I solved this problem correctly? \begin{align} \int\limits_{-\infty}^{\infty}\left(t^2-1\right)\delta\left(t\right)\:dt&=\int\limits_{-\infty}^{\infty}t^2\delta\left(t\right)\:dt-\int\...
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2answers
1k views

What's the point of Dirac delta function?

I have heard that The main useful property of Dirac delta function is it's fundamental property that $$ \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) $$ I don't understanding why this equation is ...
2
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3answers
137 views

Where does the $2\pi$ in Fourier Transform Dirac delta identity come from?

$$ \chi( \omega - \omega ')= \int_{-\infty} ^ {\infty} dt e^{j( \omega - \omega ')t} = 2 \pi \delta ( \omega - \omega ') $$ That is the identity to proof. I have seen different ways to proof ...
2
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3answers
347 views

Integral of delta dirac function

I try to calculate the following integral: \begin{equation} \int^{+\infty}_{-\infty} \frac{x^4 \exp{(ixa)}}{1+x^2b^2} \mathrm{d}x \end{equation} where $a,b$ are real positive numbers. This integral ...
2
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2answers
31 views

How to prove scaled delta function relation mathematically?

I am working through Shankar's Introduction to Quantum Mechanics. I have come across exercise 1.10.1, which asks the reader to show that: $$\delta(ax)=\frac{\delta(x)}{|a|}.$$ I can understand it ...
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2answers
106 views

Integral of a generalized function

Let $\mathcal{S}$ be a Schwartz space and $\delta_{a}$ the following distribution: $$\delta_{a}: \phi \rightarrow \phi(a) \ \ \ \ \text{ for each } \phi\in\mathcal{S}$$ Now, we routinely see ...
2
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1answer
118 views

Delta Dirac and duality property

I know that the property of duality says: $$x(t) \iff X(f)$$ $$X(t) \iff x(-f) "="x(t=-f)$$ and I know that: $$\delta(t-t_0) \iff exp(-j2\pi ft_0)$$ If I apply the duality property, I get: $$exp(-...
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2answers
17k views

Laplace Transform of Dirac Delta function

I've seen everywhere that that the Laplace Transform of Dirac Delta function is: $$L[\delta(t-a)] = e^{-sa} \text{ when } a > 0$$ But they never explain what happens when $a < 0$. Can I assume ...
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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 &...
2
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1answer
74 views

Delta derivative distribution identity?

It is easy to show that the Dirac $\delta(x)$ distribution satisfies the distributional identity $$\delta(x) = - x \delta'(x).$$ Can we conclude that the following also holds $$\delta'(x) = - \frac{\...
2
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1answer
53 views

Fourier representation of complex Dirac function

I have confusion on the Fourier representation of complex Dirac function, recently. As $t$ is real value, we have \begin{align} \delta(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{iwt}\text{d}w \end{...
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2answers
157 views

Identifying $\int_{-\infty}^\infty e^{i k x} dx$ as Dirac delta distribution

The expression $\int_{-\infty}^\infty e^{i k x} dx$ is sometimes identified as the Dirac delta function. This identification is said "formal" or "symbolic", and some physics texts say that the theory ...
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2answers
47 views

Interchanging differentiation and expectation

I have a nonnegative random variable $X$ with $E[X] < \infty$, that admits a density wrt to the Lebesgue measure. For arbitrary $K > 0$, I write $$P(K) = E[\max(X-K,0)]$$ I am interested in the ...
2
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2answers
287 views

Fourier transform of Singular Function $f(x)$ with $\frac{1}{x^2}$ as $x\to 0^{+}$ and $\frac{1}{x^2} +\frac{1}{x}$ as $x\to 0^{-}$

In Lighthill's `An Introduction to Fourier Analysis and Generalised Functions', the Fourier transform of a function $f(x)$ is defined as: $$ \mathscr{F}[f](y) \ = \ \int_{-\infty}^{\infty}dx\ f(x) e^{-...
2
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3answers
87 views

Why is $\int_0^t ds \int_0^{t'} ds' \delta(s-s')= \min(t,t')$

I don't understand this equation $\int_0^t ds \int_0^{t'} ds' \delta(s-s')= \min(t,t')$. I tried to work with the property of the dirac delta function that $\int_a^b \delta(x-c)dx = 1$ if $c \in [a,b]$...
2
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1answer
52 views

Mellin transform of rescaled delta distributions

There's something about the Mellin transform I don't get, so hopefully someone can tell me what it is that I'm doing wrong. Let's define the Mellin transform of $f(t)$ as $\mathcal{M}\{f(t)\}(s) = \...
2
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1answer
123 views

Verifying that $\lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x)$

I'd like to show that: $$ \lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x). $$
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2answers
1k views

Dirac delta function divided by Dirac delta function

Is the following defined: (Dirac delta function divided by Dirac delta function) $$f = \frac{\delta}{\delta} = ?$$
2
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1answer
63 views

Difficult Integral Involving the Dirac Delta Function

Hello fellow Stack heads, I am stuck on a difficult integral that almost looks like it can be accomplished with a one-sided Laplace transform but more than likely can be solved using Dirac Delta ...
2
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2answers
128 views

Find Laplace transform of a function multiplied by the Dirac delta function

I know that the Laplace transform of the delta function $\delta(t-a)$ is $e^{-as}$. Now my question is to find the Laplace transform of $te^{-3(t-2)}\delta(t-2) $. Is there any theorem or easy ...
2
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2answers
422 views

Convolution of Dirac comb with an exponential

I have to solve the following convolution: $$y(t)=e^{-t}u(t)*\sum_{k=-\infty}^{\infty}\delta(t-2k)$$ Here $\delta(.)$ is Dirac delta function. The summation is: $$\sum_{k=-\infty}^{\infty}\delta(...
2
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2answers
55 views

Simplify an integration

Can the following integral be reduced to simpler terms? $$\int_{-1}^0\mathrm{d}x_1 \int_{0}^1\mathrm{d}x_2 \int_{-1}^1\mathrm{d}x_3\, \delta(x_1+x_2+x_3) \exp(a_1x_1 + a_2x_2 + a_3x_3 + cx_3^2)$$ ...
2
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1answer
30 views

Problem defining a function via Step function and Dirac's Delta

First: I know Dirac's Delta isn't a function and hence shouldn't be treated like one. But this arose in a physics textbook so I'm looking for an answer that oversees that. Consider the following ...
2
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1answer
116 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 ...
2
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3answers
433 views

Fourier transform of $t^2$ discrepancy

I encountered a discrepancy when taking the fourier transform of $t^2$ that I don't understand. I would expect $\mathcal{F}[t^2]$, $$\mathcal{F}[t^2]=\int_{-\infty}^\infty t^2 e^{-i \omega t}dt$$ to ...
2
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2answers
54 views

What is the result of $x(at) * δ(t-k)$

It seems rational that $x(at) * δ(t-k) = x(at - ak)$. I tried to prove this: Let $x_1(t) = x(at)$ then $x_1(t) * δ(t-k) = x_1(t-k) = x(at - ak)$. But I thought this as well: $$x(at) * δ(t-k) = x(...
2
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2answers
255 views

Dirac delta derivatives and discontinuities

I have an integral of the form $$ I = \int_{-\infty}^{x} f(x) \delta'(x) dx $$ where $$ f(x) = \begin{cases} a & x<0 \\ b & x>0 \end{cases} $$ and $\delta'$ is the derivative of the ...
2
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2answers
75 views

How do you write $\int_0^\infty d\phi\ \cos(x\phi)\sin(y\phi)$ as a sum of Dirac deltas?

The following result is well-known (ie; I read it in a book) $$ \int_0^\infty d\phi\ \cos(x\phi)\cos(y\phi) = \frac{\pi}{2} \delta( x + y ) + \frac{\pi}{2} \delta( x - y ) $$ I read it in the ...
2
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1answer
220 views

2-D Dirac Delta in Polar Coordinate

In Section 7.3.2 of Fourier Methods in Imaging (by Dr. Roger Easton Jr), it is stated that for 2-D Dirac delta located on the x-axis at a distance $\alpha > 0$, $x = r$ and $y = r_0 \theta$ Thus ...
2
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2answers
186 views

Dirac delta definition with $i \epsilon$.

I am asked to prove that: $$\delta(x) = \frac{1}{\pi} \Im \bigg(\frac{1}{x-i\epsilon}\bigg)$$ defines a Dirac delta function. For one, $\Im\big(\frac{1}{x-i \epsilon}\big) = \frac{\epsilon}{x^2+\...
2
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1answer
378 views

$\nabla \cdot \big(\frac{\hat{r}}{r^{2}}\big)$ and Dirac Delta Function

So my textbook says that $\nabla \cdot \big(\frac{\hat{r}}{r^{2}}\big) = 4\pi\delta^{3}(r)$ and I was wondering if anyone could help me. I understand how the delta function is used and that it is ...
2
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2answers
116 views

Simple property of Dirac's $\delta$-function.

I'm on Page 63 of R. Shankar's "Principles of Quantum Mechanics". I'm trying to do Exercise 1.10.1 by proving that $\displaystyle{\delta(ax) = \frac{\delta(x)}{|a|}}$, where $a \in \mathbb R \...
2
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1answer
341 views

Can all linear operators on functions be represented as a convolution of the input function with the operator's impulse response?

Where $f, g$ are functions, $\delta$ is the Dirac Delta distribution ('aka Delta function'), and L is a linear (LTI) operator, if $g = L(f)$, then can we say in all cases $g = Li * f$ , where $ Li $...
2
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1answer
208 views

A problem about Dirac delta function

If $f(x)=g(x)$, then $f'(x)=g'(x)$, is it right? Assume there is a function $f(x)=m(x)\delta(x)$,where $m(x)$ is any function and $\delta(x)$ is Dirac-δ function. We know that $m(x)\delta(x)=m(0)\...
2
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1answer
493 views

Solve a linear ODE with dirac delta

I have to solve this linear differential equation $$ rf''(r) + f'(r) + k^2rf(r) = \delta(r) $$ on $\mathbb{R}^+$. I know the solutions to the homogeneous problem are $cJ_0(kr)+dY_0(kr)$, where $J_0,...
2
votes
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 ...
2
votes
1answer
83 views

Is it possible to have a $f(\vec{r})$ satisfy this relation?

It is known that: $$ (\nabla^2+k^2)(-\frac{e^{ikr}}{4\pi r})=\delta(\vec{r}) $$ where $k>0$ and $\delta(\vec{r})$ is the three dimensional Dirac delta function. My question is, is it possible to ...
2
votes
1answer
161 views

How can I prove $\lim_{\epsilon \to 0} \space \text{Im}\frac{1}{x+i \epsilon}=-\pi\delta(x)$?

I want to show that: $$\lim_{\epsilon \to 0} \space \text{Im}\frac{1}{x+i \epsilon}=-\pi\delta(x)$$ This is my attempt: I assumed that $\text{Im}$ stands for the imaginary part. Therefore, $$\...