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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236 views

Question about identity of Dirac delta function [duplicate]

I am trying to understand an identity of the $\delta$-function written on this Wikipedia page: \begin{equation} \int \mathrm{d} x \; f(x) \delta[g(x)] = \sum\limits_i \frac{f(x_i)}{\left| \frac{dg(x_i)...
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Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...
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1answer
308 views

Derivative of Delta dirac integral

I have a problem with this integral: $I=\int\limits_{ - 5}^5 ( x^2 - 4 )^9 \delta^{(9)} (x + 2) \, dx $ I did it as this: Use Newton expansion: ${\left( {{x^2} - 4} \right)^9} = {x^{18}} - 9.4{x^{16}}...
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1answer
583 views

Fourier decomposition of Dirac-delta function under finite limit.

Dirac delta function is said to be Fourier transformation of 1, $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\ e^{i bx} . 1\ =\ \delta(b) $$ This is usually shown by considering the definition ...
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2answers
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Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$ f'(x)=\frac{|x|}{x} $$ and $$ f''(x)=2\delta(x). $$ Can you help me?
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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, ...
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2answers
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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} ...
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3answers
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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 ...
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Convolution of two delta distributions

Show ${\int}_0^{\infty}\delta(x+z)\delta(y-z)dz =\delta(y+x)$ It seems obvious, and I don't think we need a rigorous proof for this (statistical mechanics homework) but I want a rigorous proof of ...
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2answers
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Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where $G(...
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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 ...
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1answer
238 views

Weighted Dirac comb as a tempered distribution?

I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists N\in\mathbb{...
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a question about a Green function but with $ \delta (\frac{x}{s}-1) $

for a second order differential equation the Green function is defined as $$ a(x)y''(x)+b(x)y'(x)+c(x)y(x)G(x-s)= \delta (x-s) $$ however could exist examples so we have a Green function of the form ...
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1answer
661 views

Write this expression with series & sum in terms of a single variable

I know this is very specific, but is there a way to represent the expression $$\frac{3}{5} \sum_{n = 0}^\infty \left(\frac{2}{5}\right)^n \sum_{j = 0}^n {n \choose j} \delta_{2j - n, k}$$ in terms ...
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550 views

Mathematical derivation of the Chan Vese algorithm

This is my first question. I am working on an algorithm in Image processing which is popularly known as the Chan Vese segmentation algorithm. I have a problem with it. Let me first try to explain the ...
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1answer
588 views

Integral of a piecewise continuous function

Consider a piecewise-continuous function at the origin $f(x)$, such that $\lim_{x\to0^\pm} f(x) = f_\pm$. Let us say that the domain of the function is ${\mathbb R} - \{0\}$. We are interested in the ...
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2answers
202 views

Integral in 3 dimensions

I am trying to integrate $$ \iiint \delta(|\mathbf r| -R)\:\mathrm{d}^{3}\mathbf{r} $$ I know that $ \int f(r) \delta(r-R) d^3 \mathbf r =f(R) $, but when I try to apply this here I end up ...
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1answer
2k views

How to calculate this integral in 3 dimensions involving the Dirac delta function?

How would I go about calculating the integral $ \int d^3 \mathbf r {1\over 1+ \mathbf r \cdot \mathbf r} \delta(\mathbf r - \mathbf r_0) $ where $\mathbf r_0 = (2,-1,3)$ My attempt so far: I have ...
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3answers
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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 ...
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4answers
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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[...
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2answers
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Dirac delta function divided by Dirac delta function

Is the following defined: (Dirac delta function divided by Dirac delta function) $$f = \frac{\delta}{\delta} = ?$$
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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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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}^\...
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0answers
537 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 &...
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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 ...
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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?