Articles of dirac delta

Is this interpretation of the Dirac-measure property correct?

First and foremost, apologies in advance for using an abuse of notation by placing the Dirac measure inside an integral for which I was told that this should not be done from a previous question asked by me. But given the circumstances, I have no choice. This is essentially a word by word copy of […]

How to solve integral of formula consisting of derivative of the delta function.

The question is $$ \int ^4 _{-4} (t-2)^2\delta’\left(-\frac13t+\frac12\right)dt $$ The solution of the text book is \begin{align} \int ^4 _{-4} (t-2)^2\delta’\left(-\frac13t+\frac12\right)dt &=\int ^4 _{-4} 3(t-2)^2\delta’\left(t-\frac32\right)dt \\ &=\int ^4 _{-4} \left[{\frac34\delta’\left(t-\frac32\right)+3\delta\left(t-\frac32\right)}\right]dt\\ &=3 \end{align} My solution is \begin{align*} \int ^4 _{-4} (t-2)^2\delta’\left(-\frac13t+\frac12\right)dt &=3\int ^4 _{-4} (t-2)^2\delta’\left(t-\frac32\right)dt \\ &=-3\int ^4 _{-4} \left[{\frac{d}{dt}(t-2)^2}\right]_\frac32 \delta\left(t-\frac32\right)dt\\ &=-3\cdot2\cdot\left(\frac32-2\right)\int ^4 _{-4}\delta\left(t-\frac32\right)dt\\ &=3\cdot1\\ &=3 […]

What is the relation between Fourier's Inversion theorem and the Dirac-Delta function?

This is a direct quote from page 472 of this book: From Fourier’s Inversion theorem $$f(t)= \int_{-\infty}^\infty f(u) \, \mathrm{d}{u} \left( \frac{1}{2\pi}\int_{-\infty}^\infty e^{-i\omega(t-{u})} \,\mathrm{d}\omega \right) \tag{1}$$ comparison of $(1)$ with the Dirac-Delta property: $$f(a)= \int f(x) \, \mathrm{d}x \, \delta(x-a)$$ shows we may write the $\delta$ function as $$\delta(t-u)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i\omega(t-{u})} \, \mathrm{d}\omega$$ My question is […]

Integral of Dirac delta function/distribution $\delta(x)$ with upper boundary equal to zero

I would like to find the value of $$\int_{a<0}^0 \delta(x) dx$$ In particular, I would like to know if I can break down the integral $$\int_a^b \delta(x)f(x) dx=\int_a^0 \delta(x)f(x) dx + \int_0^b \delta(x)f(x) dx $$ with $a<0$ and $b>0$ and $f(x)$ a well-behaved function. Is it wrong to break down the integral like this, doing […]

Integral involving Dirac delta: two different results?

I am evaluating the integral over all space $$\int \delta \left(r^2 – R^2\right) d \vec r$$ At first, I did this: $$\int \delta \left(r^2 – R^2\right) d \vec r = 4 \pi \int_0^\infty \delta \left(r^2 – R^2\right) r^2 dr = 4 \pi R^2 $$ But then someone made me notice that we can use the […]

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 very simple point, and can’t seem to figure it out. Any help to prompt me in the right direction would be much appreciated. What I’ve done so far […]

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 like $\delta[1] = 1$. However, the constant function $f(x) = 1$ has no compact support, therefore it is not a test function. Is there a precise way […]

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 prove this without using Fourier Transform. I have already searched google and looked for some books, but I just get nothing. In short, I want to know the proof of this equation: $$\lim_{k\to+\infty}\int^{+\infty}_{-\infty}\frac{\sin(kx)}{\pi x}f(x)dx=f(0).$$

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 rest again after one cycle. Determine the impulse $k\delta(t-t_{0})$ that should be applied to system in order to accomplish this objective (bring sys […]

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? $$ \begin{align} \int_{-\infty}^{\infty} e^{i k x} \; dk & = 2 \pi \delta \left ( x \right ) \end{align} $$ Specifically, why do we say that the integral converges for $ x \neq […]