# Tagged Questions

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### 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 ...
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### 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 $t$ ...
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### 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 ...
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### 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 ...
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### 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. $$\delta(x) = {1\over 2\pi}\int_{-\infty}^\infty\exp(ikx)dk\tag{1}$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...