Tagged Questions
1
vote
3answers
58 views
How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?
Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.
I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
1
vote
2answers
96 views
Dirac delta function
1)Prove that the dirac delta function property:
$$ x\delta'(x)=-\delta(x)$$
2)and :
$$\int_{-\infty}^\infty \delta'(x)f(x)dx=-f'(0) \ $$
3
votes
4answers
983 views
Which of these two ways to take the derivative of a delta function times another function is correct?
A well known identity of the Dirac delta function is that for any function $f(x)$:
$$
\delta(x) f(x) = \delta(x) f(0).
$$
If we take the derivative of the right hand side we get:
$$
...
4
votes
1answer
243 views
Laplace transform and differentiation
Let $F(s)$ be the Laplace transform of $f(t)$:
$$F\left(s\right)=\int_{0}^{\infty}e^{-st}f\left(t\right)dt$$
It then follows that $f(t)$ can be recovered from $F(s)$ by the inverse Laplace ...
3
votes
2answers
2k views
Proof of Dirac Delta's sifting property
A common way to characterize the dirac delta function $\delta$ is by the following two properties:
$$1)\ \delta(x) = 0\ \ \text{for}\ \ x \neq 0$$
$$2)\ \int_{-\infty}^{\infty}\delta(x)\ dx = 1$$
I ...
6
votes
3answers
783 views
Property of Dirac delta function in $\mathbb{R}^n$
How does one prove the following identity?
$$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$
where $S$ is the surface inside $V$ where ...
2
votes
2answers
248 views
Delta function in curvilinear coordinates
I have been looking everywhere but I am unable to prove $$\delta(\vec{x}-\vec{a}) = \frac{1}{fgh}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$$
Where $f,g,h$ are scale factors for an orthogonal ...
