1
vote
1answer
64 views

Cauchy Schwarz in an integral with distributions.

I am working with energy methods for PDE's and I have a expression of the following form: \begin{equation} \int f \phi\phi_{j} \end{equation} under the conditions that $f\in L^{\infty}, \phi\in ...
1
vote
1answer
37 views

Integrating a Dirac delta function with the argument dependent of a parameter

How can I handle the integral $$ \int_{t_1}^{t_2} \delta(D - x(t)) dt, $$ with $D$ a constant. I want to do a change of variables to perform the integral over $x$ but I am not sure how to proceed.
4
votes
0answers
24 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
0
votes
2answers
67 views

How to treat Dirac delta function of two variable?

We can treat one variable delta function as $$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$ Then how do we treat two variable delta function, such as $\delta(f(x,y))$? ...
4
votes
1answer
79 views

Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 ...
0
votes
1answer
41 views

Line integral along an implicit curve and dirac distibution

Let $\varphi : \Bbb{R}^2 \rightarrow \Bbb{R}$ defining an implicite curve $C = \{ (x,y), \varphi(x,y) = 0 \}$, and $u : \Bbb{R}^2 : \rightarrow \Bbb{R}$ Does the line integral $\int_C u(x,y)\ dC$ ...
1
vote
0answers
42 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
0
votes
0answers
127 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 ...
1
vote
1answer
37 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in ...
2
votes
1answer
58 views

The $\frac{1}{x+i\varepsilon}$ distribution.

I read that the distribution defined as: $$ \lim_{\varepsilon \rightarrow 0}\frac{1}{x+i\varepsilon}$$ is equal to $$p.v. \frac{1}{x} -i\pi \delta(x)$$ So that for any test function $f$, ...
1
vote
2answers
110 views

Strange Dirac delta distribution contradiction

Consider the following integrals in variables $x,y$ over the whole $\mathbb{R}$, where $a,b\in\mathbb{R}/0$ are constants: $$\int dx \int dy ~\delta(x-a)\delta(y-b\,x)=\int dy ~\delta(y-b\,a)=1$$ In ...
1
vote
0answers
53 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 ...
3
votes
1answer
65 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
0
votes
1answer
53 views

How to prove that limit is equal to zero

How to prove that: $$\lim_{\epsilon\rightarrow 0}(-\log(-x) \phi(x)|_{-\infty}^{-\epsilon} -\log(x) \phi(x)|_{\epsilon}^{+\infty})$$ where $\phi(x) $ is any test function is equal to $0$. It seems ...
2
votes
2answers
91 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 $t$ ...
6
votes
4answers
266 views

Dirac delta of nonlinear 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) ...
0
votes
1answer
402 views

integral of Dirac delta function with sine

It i well known that the Dirac Delta Function has the following property $\int_{-\infty}^{\infty}\delta(t-a)f(t)dt=f(a)$ if $g(t)=\int_{0}^{t}\sin(t-\tau)\delta(\tau-\pi)d\tau$ then $g(t) = ...
1
vote
2answers
83 views

Integral transform with Dirac delta

Let $f,g: \mathbb{R}^n \to \mathbb{R}$. Let $\delta$ denote the Dirac delta function. How can I write the integral over $\mathbb{R}^n$ (on the left hand side) as an integral over $g^{-1}(0)$ $$ ...
2
votes
1answer
58 views

How to finish some complex integration

How to finish some integration as following below: $$\int_x^{\infty} \frac{\mathrm \beta^{\alpha+\gamma} X^{\alpha-1}(y-x)^{\gamma-1}\exp^{-\beta y}}{\Gamma(\alpha) \Gamma(\gamma)}dy\;$$ and ...
2
votes
1answer
70 views

Correct notation when integrating Dirac distribution

I have a question regarding the correct notation when integrating the Dirac distribution $\mu$. When treating it as a measure, I can write the Lebesgue inetgral $\int_{\mathbb{R}}\mu(dx)=1.$ What if I ...
1
vote
1answer
723 views

Proving that the delta function is symmetric

to prove that the delta function is symmetric, I need to show that $\delta(x) = \delta(-x)$ by employing a change in variables. $$\delta(x) = {1\over 2\pi}\int_{-\infty}^\infty\exp(ikx)dk\tag{1}$$ ...
3
votes
1answer
119 views

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
1
vote
0answers
64 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Is the following operations OK (this is related to the Feynman parameter trick)? $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
2
votes
0answers
92 views

Dirac $\delta\left( \left[\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right]^2 -k^2-m^2 \right)$

This question is related to $f(k) = 0$, but we now we consider $\delta(f(k))$, i.e. $\delta\left( \left[\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right]^2 -k^2-m^2 \right)$ We ...
1
vote
2answers
221 views

Dirac $\delta(g(x))$ with complex roots $x_i$

Hey we all know about this infamous Wikipedia page Dirac-delta composition related to the Dirac-$\delta$ in composition with a function $g(x)$. But I wonder if the roots of $g(x) = 0$ happens to be ...
5
votes
0answers
143 views
+50

Delta function and integrating over level sets?

Consider the three-dimensional integral $$ \int_{\mathbb R^3} d^3x\,f(x)\delta(g(x)) $$ where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there ...
2
votes
1answer
87 views

Dirac function and integration by parts

I have some problems to show the following relation, apparently using integration by parts and knowing that $\phi$ denotes the density of the standard one dimensional normal distribution. $$\int ...
4
votes
1answer
846 views

Fourier transform of Cauchy principal value

I try to understand the direct computation of the Fourier transform of the distribution `Cauchy principal value' $v.p \frac{1}{x}$. I don't understand the following change of order of integration: $$ ...
5
votes
1answer
431 views

Proving the integral of the Dirac delta function is 1

Was wondering if my solution is mathematically accurate enough: The question in the book yields: Derive $$ 1=\int_{-\infty}^{\infty} \delta(x-x_i)\ dx_i $$ From $$ ...
2
votes
1answer
137 views

generalized functions (Distributions) elementary question

I am working with Strichartz's "A Guide to Distribution Theory and Fourier Transforms" (self-study -> not a homework question). He says none of the distributions that correspond to $1/|x|$ are ...
3
votes
1answer
575 views

Normalization parameter, properties of Dirac delta functions

Suppose $\psi_E (x)=N(E)\exp (ikx)$ where $\psi_E (x)$ is a momentum eigenfunction, $N(E)$ is the normalization constant on the energy scale such that $\langle E'|E\rangle=\int_{-\infty}^\infty ...
1
vote
0answers
169 views

Integration methods for functions with Delta distributions

Which Monte-Carlo methods are available for computing a multidimensional integral with Delta distributions (in case one cannot sample them explicitly)? PS: I also asked a similar question at ...
4
votes
2answers
461 views

limit of an integral with a Lorentzian function

We want to calculate the $\lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \frac{f(x)}{x^2 + \epsilon^2} dx $ for a function $f(x)$ such that $f(0)=0$. We are physicist, so the function $f(x)$ is smooth ...
4
votes
2answers
4k views

Dirac Delta function

I know that $\int_{-\epsilon}^\infty f(x)\delta(x)dx=f(0)$ but what about $\int_0^\infty f(x)\delta(x)dx$? I suppose we have to do this by definition since the lower limit is bang on $0$?
3
votes
2answers
447 views

Verifying the 2-dimensional fundamental solution of the wave equation

I'm trying to verify that $$u(t,x)=H(t-|x|)(t^2-|x|^2)^{-1/2}$$ is the fundamental solution of the 2-dimensional wave equation; that is, $\Box u = u_{tt}-\Delta u = \delta_{0}$. I know there are ...
0
votes
1answer
154 views

Verify this distribution convolution: $E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$

In our class notes we are asked to verify the following equality: $$E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$$ where ...
2
votes
2answers
200 views

Simplifying the generalized function $x^{\lambda}_+$ in the strip $-n - 1 < \mbox{Re}\lambda < -n$

Note: this post is a follow up to an earlier question. The (divergent) integral of $x^{\lambda}_+$ can be analytically continued into the region Re $\lambda > -n - 1$, $\lambda \ne -1, -2 , \ldots ...
6
votes
1answer
185 views

Showing that $\int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx$ is convergent for $\lambda > -2$

Id' appreciate help understanding why the integral $$ \int_0^1 x^{\lambda} [ \: \phi(x) - \phi(0)\: ] dx $$ is convergent provided $\lambda > -2$, where $\phi \in \mathcal{D}(\mathbb{R})$. To ...
9
votes
1answer
173 views

Oscillatory integral giving me the willies

So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification: ...
2
votes
0answers
305 views

Seeking rationale for Hadamard's finite part of a divergent integral

I have a problem justifying the throwing away the divergent term in order to obtain Hadamard's finite part. I find this step to be highly unusual and it is not obvious to me how the resulting ...