Linked Questions

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 $0$ ...
1
vote
6answers
162 views

Proving $\int_\limits{\infty}^{\infty}\delta(x)dx=1$

From Wikipedia I found this about the delta-function $$\delta(x)=\begin{cases}\infty,\:\:x=0\\0,\:\:x\neq 0\end{cases}$$ $$\int_\limits{-\infty}^{\infty}\delta(x)dx=1$$ I tried to prove that $...
2
votes
2answers
16k 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 ...
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 $$\delta(x) = \begin{cases} 0 & \space \mathrm{for} \space x \ne 0 \\\infty&\ \mathrm{for} \space x = 0 \end{cases} $$ ...
4
votes
3answers
903 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}(\...
0
votes
2answers
3k views

Integrating a dirac delta 'function' on a definite domain

Came across a question that requires evaluation of $\int_{-3}^{+1} \left(x^{3}-3x^{2}+2x-1\right)\delta\left(x+2\right)dx$ Here's my attempt: Recall that: $\int_{-\infty}^{\infty} f\left(x\right)\...
1
vote
2answers
2k views

Differential of Dirac Delta Function

By considering $\int_{-\infty}^{\infty}f(x)\frac{d(\delta(x))}{dx}dx$ and $\int_{-\infty}^{\infty}f(x)\frac{\delta(x)}{x}dx$ show that $\frac{d(\delta(x))}{dx}=-\frac{1}{x}\delta(x)$ The hint that I'...
1
vote
2answers
2k views

What is the laplace transform of $\delta(t-\pi /6)\sin (t)$

What is the laplace transform of $\delta(t-\pi /6)\sin (t)$ I know that $L\{\delta(t-\pi/6) \}=e^{-s\pi/6}$ I also know that $L\{\sin (t) \}=1/(s^2+1)$ I also know that $L\{(u(t-\pi/6)f(t-\pi/6)\}=e^...
2
votes
3answers
2k views

Integrating on open vs. closed intervals

What is one difference in the values of $$\int\limits_{\left[0,1\right]}y\, dx$$$$\int\limits_{\left(0,1\right)}y\, dx$$ and how would you calculate the values? For the sake of simplicity, let $y=x$....
4
votes
1answer
638 views

Integral of delta function and derivative of delta function

Can anyone rigorously prove this? $$\int dx \, \delta(x-\alpha)\delta^{\prime} (x-\beta) = \delta^{\prime} (\alpha-\beta).$$
4
votes
1answer
680 views

Solving Poisson's equation for a point charge in 1-D

I Apologize that this is a continuation of a question that I just asked. Anyway here is where I am: Ok so I was trying to solve the Poisson's equation for a point charge with a Fourier transform to ...
2
votes
1answer
795 views

Using the Dirac delta function to find the density of point masses/charges

Here is an example from a textbook: Suppose there is a unit charge or unit mass at the point $(x,y,z)=(-1,\sqrt{3},-2)$; then in rectangular coordinates, the ...
0
votes
1answer
770 views

Impulse function and exponential, integral

I'm having trouble understanding this integral: $$ y =\int\limits_{-\infty}^{\infty}\delta(t+2)e^{-t}dt = e^2$$ I'm having trouble visualizing what this function looks like. Is $\delta(t+2)$ an ...
1
vote
3answers
667 views

How to change the sign of the Dirac delta function argument?

How to proof the latter equality of, $$ f(a) = \int f(x)\,\delta(x-a)\,dx =\int f(x)\,\delta(a-x)\,dx. $$
1
vote
3answers
511 views

Integral Unit Impulse Function

I am having trouble integrating the following unit impulse function. $$\int_{0}^{4} \delta(t - \tau) \, d\tau$$ I got the answer $$u(t-4) - u(t)$$ but my professor says its $$u(t) - u(t-4)$$ ...

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