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Questions tagged [integral-transforms]

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53 views

On the necessitation of $(-1)^n$ within the series expansion of $f(x)$ concerning the usage of Ramanujan's Master Theorem

Ramanujan's well known Master Theorem states that the series expansion of the transformed function $f(x)$ has to be in form of $$f(x)~=~\sum_{n=0}^{\infty}(-1)^n\frac{\phi(n)}{n!}x^n\tag1$$ ...
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55 views

A complicated integral / a complicated Laplace transform involving the error function

For some reason I am interested in solving a complicated integral, which is $$\int_0^\infty (x+1)erf\left(\frac{-c_1+c_2x}{b}\right)e^{-c_3x^2-c_4x}dx,$$ where all $c_i$ are positive real numbers and $...
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0answers
30 views

Y-transforms of products of Struve functions and exponential functions?

In the Bateman Manuscript Project's Table of Integral Transforms, there are several identities for Y-transforming (or similarly but more familiarly Hankel-transforming) certain special functions with ...
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3answers
132 views

Reconstructing a curve in $S^2$ from intersections with great circles

Take $S^2$ with its standard metric. The space of great circles in $S^2$ can be identified with the real projective plane $\mathbb{R}P^2$. Let $X$ be an embedded circle in $S^2$; associate to it a ...
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41 views

Kernel of Radon transform in $\mathbb{R}^3$

Consider the Radon transform from the space of functions on the manifold of affine lines in $\mathbb{R}^3$ to functions on the manifold of affine 2-planes in $\mathbb{R}^3$: $$(Rf)(H):=\int_{l\subset ...
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48 views

On parametric integrals with elliptic functions

Can a simple transform like $$\int_0^az^2\wp(z+y)\,dz$$ be treated like an ordinary elliptic function in $y$? The integrals for lower powers of $z$ can be evaluated in terms of the Weierstrass $\sigma$...
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34 views

Name for a variant of the Mellin transform

The Mellin transform is usually expressed as \begin{eqnarray} \mathcal{M}[f](s) &=& \int_0^\infty x^{s-1} f(x)dx \\ \mathcal{M}^{-1}[F](x) &=& \frac{1}{2\pi i}\int_{\alpha -i\infty}^{\...
5
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2answers
229 views

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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1answer
310 views

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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45 views

Positivity of an integral with product of functions and their Fourier transforms

What condition two functions $f$ and $g$ defined on $\mathbb{R^+}$ must fulfilled to have: $$\int\limits_{0}^\infty \ln(x) (fg+ \hat{f} \hat{g} ) dx >0$$ Where we note $\hat{f}(x)= 2\int\...
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1answer
62 views

A System of generalized Abel's integral equation

Is there a method for solving the following system of generalized Abel's integral equation:? $(x^2 -1)\int_0^x \frac{u(t)}{(x-t)^{\frac{1}{2}}}\; dt + x\int_0^x \frac{v(t)}{(x-t)^{\frac{1}{3}}}\; dt ...
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173 views

Does $\int_0^\infty f(x+\theta)g(x) \, dx=0\, \forall \theta \in \mathbb{R}$ imply $f=0$ almost everywhere, if $g$ is smooth and strictly positive?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an integrable function, and $g:(0,\infty)\rightarrow(0,\infty)$ a smooth, strictly positive function. If $$\int_0^\infty f(x+\theta)g(x)\,dx=0\qquad\forall\...
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24 views

Calculating the Radon Transform from the Hartley Transform

It is wellknown, that the Radon Transform can be calculated from the Fourier Transform via the Central Slice Theorem. The Hartley Transform can be seen as a "purely real" version of the Fourier ...
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1answer
71 views

transformation of two measures on different space

Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$. Let $\Sigma_U$ be the smallest $\sigma-$ field ...
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1answer
120 views

Extended convolution theorem for Laplace transform

Let $(f*g)(t):=\int_0^t f(s) g(t-s)ds.$ Then the Laplace transform $L$ satisfies $L(f*g)(t)=L(f)(t)L(g)(t).$ This is known as the convolution theorem. I would like to know whether something similar ...
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64 views

Closed form expression of multi-dimensional integral across the multi-dimensional cube

Let $\mathbf{y}$ be a complex-valued $M\times 1$ vector and $\mathbf{B}$ a complex-valued $M\times N$ matrix. Let $\mathbf{b}_j$ denote the $j$:th column of $\mathbf{B}$. I am wondering if there is a ...
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67 views

Complex integral transforms

Is there an invertible (for some specified classes of functions) integral transform performed as a contour integral over $\mathbb{C}$, for example $F[f](w)=\oint_\Gamma K(w|z)f(z)\mathrm{d}z$ (let say,...
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1answer
118 views

CTRW: solve a renewal equation

Let X(t) be a continuous time random walk, with exponentially distributed waiting times of pdf $f_T(t)= k e^{-k t}\; t\geq 0$ and a jump sizes pdf $f_J(x)$. Suppose the initial distribution $\rho_{X(0)...
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2answers
309 views

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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2answers
427 views

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 ?
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212 views

Hankel transform

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 ...
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1answer
537 views

volume over a hypercube, over simplex: twist by Euler numbers

Let $\square_n=\{(x_1,\dots,x_n): 0\leq x_i\leq1,\, \forall i\}$ be an $n$-dimensional unit hypercube, and let $\Delta_n=\{(u_1,\dots,u_n):u_1+\cdots+u_n\leq\frac{\pi}2,\, u_i\geq0,\, \forall i\}$ be $...
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1answer
249 views

Fourier transform with cubic exponential

Please give references for the integral transform of the next kind: $$ F_3(f(x))(t)=\int_{-\infty}^{\infty} \exp(Q_3(x,t)) f(t)\,dt , $$ with $Q_3(x,t)$ - a cubic polynomial of its arguments. Special ...
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3answers
420 views

Asymptotic behavior of integral with gamma functions

Consider the following function defined for complex numbers $z\in\mathbb{C}$ with $\Re(z)\geq \frac{1}{2}$: $$F(z)=\frac{1}{5^{\Re(z)}}\int_0^\infty \left| \frac{\Gamma(z+ix)\Gamma(z-ix)}{\Gamma(z)^2}...
4
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2answers
206 views

An integral identity relate to the Gamma function or the Beta function

I encountered the following identity in a paper on number theory, $$\int_{-\infty}^{\infty}\frac{dW}{(W+i)^{\frac{3}{2}}(W^2+1)^s}=\frac{e^\frac{-3\pi i}{4}\sqrt{2}\pi \Gamma(2s+\frac{1}{2})}{2^{2s}\...
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0answers
52 views

Is there a name for the general type of operation that sweeps a kernel over a function (e.g. like convolution, morph. dilation, registration, etc)

There is a certain family of 'sweeping' operators / functions $S(y; f,k,g)$, where: $f$ is a function $f : x \mapsto \mathbb{R}^N$ $k$ is a 'kernel' function $k : x \mapsto \mathbb{R}^N$ $y$ ...
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86 views

Kontorovich Lebedev transform

By the title I mean [reference: ``Spectral methods of Automorphic forms" by Iwaniec (B.41)-(B.43)] for $f\in C^\infty_c(\mathbb{R^+})$, one has $$f(x)=\pi^{-2}\int_{-\infty}^\infty K_{it}(x)F_f(t)t\...
2
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0answers
184 views

Hilbert transform

Let $\mathscr H$ be the Hilbert transform, that is the Fourier multiplier by $\text{sign } \xi$: $$ (\mathscr H u)(x)=\int_{\mathbb R} e^{2iπ x\xi }(\text{sign } \xi) \hat u(\xi) d\xi. $$ The Hilbert ...
2
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1answer
226 views

Fourier sine and cosine transforms and Laguerre polynomials

Let $S$ and $C$ denote the Fourier sine transform and the Fourier cosine transform, respectively, i.e., \begin{align*} S f(k) &= \sqrt{\frac2\pi} \, \int_0^\infty f(x) \sin(kx)\,dx, \\ C f(k)...
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1answer
79 views

Representation of Hilbert transform by a singular integral

Hilbert transform defines as follow: $$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$ $$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$ Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \...
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1answer
322 views

how to reduce the integral into hypergeometric function?

The equation is $\int_{0}^{1}dy\left( \sqrt{1+\Pi ^{2}+2\Pi \sqrt{1-y^{2}}}-\sqrt{1+\Pi ^{2}-2\Pi \sqrt{1-y^{2}}}\right) =\frac{\pi \Pi }{2}\ _{2}F_{1}(-\frac{1}{2},% \frac{1}{2};2;\Pi ^{2})$, where $...
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265 views

The Hubbard Stratonovich transformation (for multiple variables)

I have found that $e^{\frac{1}{2}Ks^2} = \left(\frac{K}{2\pi}\right)^{1/2} \int^{\infty}_{-\infty}e^{-\frac{1}{2}K\phi^2 + Ks\phi} d\phi $ or $e^{\frac{1}{2}\sum_{ij}K_{ij}s_is_j} =\left(\frac{\...
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45 views

DIfference between transforms,

I have a question regarding the difference between Laplace transform and so called Carson-Laplace transform. I mean what's the motivation behind the second one, why was it invented? Is there some area ...
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0answers
88 views

Integral involving exponential and Marcum-Q function to a power

Do you have any suggestions to achieve a closed-form solution for the following integral: $$\int_{0}^{\infty}\exp\left[-ax\right]\times \left\{ 1-Q_{\mu}\left[\sqrt{2\kappa\mu},\sqrt{\frac{2\mu\left(\...
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1answer
261 views

Integral over Haar measure

$\mathcal{H}$ is a $d$-dimensional complex vector space. $\mathcal{E}$ maps matrix on $\mathcal{H}^{\otimes m+k}$ to matrix on $\mathcal{H}^{\otimes m}$ through $$\mathcal{E}(X)=EXE^{+},$$ where $m,k$ ...
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0answers
127 views

Convergence of the Mellin transform of $\zeta(s)\, \Gamma(s)$ for line integrals with real part $\le 1$

This question is inspired by this one, however I believe is quite different. The Mellin transform for: $$\displaystyle \Gamma(s)\,\zeta(s)= \int_0^\infty x^{s-1} \frac{1}{e^x-1}\,dx$$ equals: $$\...
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1answer
189 views

A convolution integral of airy functions

I wonder whether the following integral of Airy functions can be computed? \begin{equation} F(x,y):=\int_{-\infty}^\infty \int_{-\infty}^\infty Ai(x-u)Ai(y-v) e^{ituv}du dv,\quad t \in \mathbb R. \end{...
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187 views

Calculate an integral

For $x\in \mathbb R_+$, let us define $$ I_\lambda(x)=\frac2π\int_0^{+\infty}\frac{\sin t}{t\sqrt{1+t^2x^2}}\cos(\lambda \arctan (xt)) \,dt,\quad \lambda\in 1+2\mathbb N. $$ We see that $I_\lambda(0)=...
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2answers
328 views

Infinite dimensional version of a simple fact on certain singular matrices

We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds; Let $L$ be the collection of all $n \times n$ real matrices $A=(a_{ij})$ with ...
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1answer
124 views

Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
3
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1answer
99 views

Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...
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0answers
187 views

A question about multidimensional integral

Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$ Is there a necessary condition on the $\Omega_i$'...
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1answer
345 views

Integration of Bessel Function of the first kind

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows: $$\int_{0}^\infty F(x)[Bx^3J_0(xy)+x^4J_1(xy)]dx=G(y)$$ where $B$ is a constant, ...
1
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1answer
97 views

Definite intergal with two K-Bessel functions and x

I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers): $$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$ I could not find it in ...
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0answers
233 views

Evaluating an integral of a periodic function. It's positive?

My purpose is to show that this integral  \begin{equation} I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du\,\,,\,\,x,...
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1answer
48 views

Integral transforms involving square roots

I am considering the following integral equation $\frac{1}{y} = \int_a^{\infty} g(x,y) x^{-1/2} dx$, where $g(x,y)$ is to-be-determined and $a$ is a positive constant (if it is instructive, it can ...
1
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0answers
133 views

Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform, $$ f(s)=\int_0^\infty e^...
1
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1answer
121 views

Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$. If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$, $$ A_wf(y)=\int_y^1\frac{f(x)w(x,y)}{\sqrt{x^2-...
1
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0answers
216 views

Integration involving modified bessel function, exponential and power

I need to find the following integration. $$ \int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N $$ where $$ a>0, \quad N \geq 1 $$ Any help will be much appreciated. BR Frank
3
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1answer
556 views

Definite integral with modified Bessel functions, trigonometric function and a power

I require the following integral involving the modified Bessel functions of the first and second kinds of order one $$I(a, b, c) = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x, \...