# Questions tagged [integral-transforms]

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In dimension 2, the Radon transform range theorem states that a function $g(t,\theta)$ can be represented as a Radon transform of some function $f(x,y)$ (i.e. $g=R[f]$) if and only if for all integers ...
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### Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
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### A functional that occurs in Vlasov-Poisson equation

Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...
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### How to integrate an exponential function of a rational function?

Can anyone help me to calculate the following integral? \begin{align} \int\limits_0^t {{{(x - t)}^2}} x\,{e^{ - \left(x + \frac{a}{{bx + 1}}\right)}}\mathrm{d}x \end{align} where $a$ and $b$ are ...
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### “Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇”

This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
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### An integral involving three Bessel functions

I am looking for a closed form for the following integral $$I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx)$$ which can be thought of as a particular case of the more general integral ...
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### I want to disprove an equality involving a double integral

I want to show that the following equality does not hold: \begin{equation}\label{at3} \frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
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### Why Mellin transform is omitting infinite terms?

For instance, Mellin transform of function $f(x)=x$ $$\int_0^\infty f(x) x^{s-1} dx$$ returns the result $\delta(s)$ which is completely strange to me. Why only at $s=0$ the result is infinite? Why ...
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### Relationship between the Radon transform and Twistor spaces

I have often heard that the theory of Twistor spaces is a complex analogue" of the Radon transform. What is the precise connection ?
The Hankel transform of $f=h(a-r)$ and $g=1/\sqrt{r^2+b^2}$ are $a/s\,J_{1}(as)$ and $e^{-bs}/s$ respectively. Here, h is a Heaviside side function that is 0 if $r>a$ and $1$ otherwise. a and b are ...