# Tagged Questions

0answers
33 views

0answers
126 views

### Delta function?

I came across this integral: \begin{align} \int_{-\infty}^{\infty} \mathrm{d}k_1 |f(k_1)| \int_{-\infty}^{\infty} \mathrm{d} k_2 |g(k_2)|& \nonumber \times\bigg\{\lim _{V \to ...
0answers
112 views

### Derivative of Dirac delta behavior at 0

I calculated the curl of the vector field $\vec j = \delta(z) f(\rho) \vec e_\phi$ in cylindrical coordinates (ρ,φ,z). For the $\vec e_ρ$ unti vector I got: $$-\delta'(z)f(\rho)$$ The main question ...
1answer
19 views

### Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0answers
52 views

### I need to integrate with $\delta$ against something that isn't a test function!

In relation to How does integration over $\delta^{(n)}(x)$ work?,'' I need to evaluate $\int_{-a}^{a}f(x)\delta^{(n)}(x)\, dx$. However, while my $f$ is smooth on its domain, it can't be a test ...
2answers
74 views

### 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?
2answers
90 views

### Computation of integral involving Heaviside function

Let $H : \mathbb{R} \to \mathbb{R}$ denote the Heaviside function: $$H(y) = \begin{cases} 0 & y < 0, \\ 1 & y \ge 0. \end{cases}$$ Suppose that $c > 1$ is fixed with $t$ ...
1answer
394 views

1answer
64 views

### Non-regularity of non-elliptic operator

Let $\Omega\subset\mathbb{R}^d$ be open,and $P(D)=\operatorname{\sum_{|\alpha|\le N}}f_{\alpha}D^{\alpha}$ be an elliptic differential operator. Rudin proves in Functional Analysis Part II the ...
1answer
106 views

### Show that a functional is a distribution

Consider the following functional $$\langle u , \phi \rangle = \int_0^{\infty} \phi(t) \frac{\mathrm{d}t}{t^{\alpha}}$$ I want to show that it is a functional in $\mathcal{D'}{(\mathbb{R})}$. ...
2answers
36 views

### what could be said about the series of test-functions?

in another thread an offtopic question came to my mind. Consider a test/bump function $\phi$, then consider all its derivatives $\phi^{(k)}$, of course they are bounded by constants $M_k$, next form ...
2answers
4k 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 ...
3answers
1k views

### Delta function integrated from zero

I am trying to understand the motivation behind the following identity stated in Bracewell's book on Fourier transforms: $$\delta^{(2)}(x,y)=\frac{\delta(r)}{\pi r},$$ where $\delta^{(2)}$ is a ...
3answers
2k views

### How to prove $\frac{d\theta}{dx} = \delta(x)$?

Here is a problem from Griffith's book Introduction to E&M. Let $\theta(x)$ be the step function $$\theta = \begin{cases} 0, & x \le 0, \\ 1, & x \gt 0. \end{cases}$$ The ...