The integral-transforms tag has no wiki summary.

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### Numerically computing $\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$

In the book, "Pi and the AGM" by Borwein and Borwein, it is mentioned that Gauss computed the following integral to the eleventh decimal palce.
$\int_0^1 \frac{1}{\sqrt{1-x^4}}dx$
How did he do it? ...

**16**

votes

**1**answer

484 views

### Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$.
(a) Some time ago, I convinced myself that
$f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...

**4**

votes

**4**answers

2k views

### Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral,
\begin{equation}
F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds
\end{equation}
...

**2**

votes

**0**answers

104 views

### A Convolution Integral Equation

Is there any close-form solution for a function $f(t)$ satisfied the below equation:
$f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known ...

**8**

votes

**3**answers

432 views

### Rate of growth of an explicit integral

Let $$J_1=\int_0^1\frac{1}{\sqrt{1-t_2}}dt_2,$$
$$J_2=\int_0^1 \int_0^{t_2}\frac{1}{\sqrt{1-t_2}}(\frac{1}{\sqrt{1-t_3}}+\frac{1}{\sqrt{t_2-t_3}})dt_3dt_2,$$
$J_3=\int_0^1 ...

**0**

votes

**1**answer

292 views

### Oscillatory Integral

Put $\eta = r+it, r>0$ and $K_{\eta}(z, w, \lambda)= \frac {\mid \lambda\mid }{2\sinh(\eta \mid \lambda \mid) } e^{-\frac{\mid\lambda\mid \coth(\eta | \lambda |) \mid z-w \mid ^{2}}{4}} e^{-i ...

**2**

votes

**0**answers

213 views

### Coutour Integral of Gamma Functions

How do I solve the Integral
$$ \frac{1}{2\pi j} \oint
\frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{
(2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$
This integral is an inverse ...

**1**

vote

**0**answers

124 views

### Fredholm Integral Equation of the first kind

I wish to solve the following integral equation, preferably analytically, and find $\int_{-l}^l f(x) dx$. If analytical solution is too complicated, any suggestion for the computational method?
...

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137 views

### Bounding an integral transform ouside a circle (or inside a strip)

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call
$$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$
the dispersion function of $g$.
Now, one can ...

**3**

votes

**1**answer

100 views

### Analogue of the integral Fourier operator with angle in some cone

Let $\Phi(x,y,\theta)$ be a phase function defined on $X \times X \times (\mathbb R^n-0)$ where $X$ is some domain in $\mathbb R^n$, let $A(x,y,\theta)$ be an amplitude function. As usually an ...

**4**

votes

**1**answer

169 views

### Continuity of integral

Assume that $f:[0,2\pi]\to [0,2\pi]$ is a continuous function such that $f(0)=f(2\pi)$ and define the function $$g(s)=\int_{-\pi}^\pi \frac{\sin f(t+s)-\sin f(s)}{\sin t/2} dt.$$ Is $g$ continuous or ...

**3**

votes

**1**answer

539 views

### Integral solving request

Dear all,
please help me solve the following integral.
I need to solve this integral for one of my problems.
$$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho R_0)J_0(\rho ...

**0**

votes

**1**answer

96 views

### General integral transform.

Hi, I was thinking about the next general integral transform, and would like to know what is already known on this integral transform.
The integral transform is:
$$F_{g(x,t)} (f(x)) = ...

**4**

votes

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234 views

### Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result:
Theorem. All straight lines are extremals of the variational problem
$$
...

**2**

votes

**1**answer

273 views

### Closed form for Fourier transform-like Integral on $S^{n}$

Hello!
It may be a stupid question, i'm trying to find a closed form for an integral similar to a Fourier transform on $S^{n}$ but i'm stuck...
Let $\alpha>0$, the integral i can't solve is
...

**2**

votes

**1**answer

298 views

### Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form.
Some background may be ...

**0**

votes

**2**answers

434 views

### Closed form for double integral?

I have the following double integral:
$\int\limits_0^x {\int\limits_0^y {{e^{ - {K_1}(u + v)}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)dudv} }$
where $K_1$ is a constant. Do you have any ideas of ...

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votes

**2**answers

410 views

### Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$

Hi,
I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$.
Fix $T>0$. ...

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votes

**0**answers

69 views

### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II

This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
...

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votes

**1**answer

196 views

### Forms satisfying the zero-energy condition on the projective plane

Theorem (Michel). A $1$-form on the projective plane is exact if and only if its integral over any projective line is equal to zero.
Is there a simple proof of this result due, I think, to R. Michel ...

**2**

votes

**1**answer

408 views

### Integral of Modified Bessel Function of the Second Type

Given the identity
$$ \int^\infty_0 K_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x = \frac{2^\mu \Gamma(\mu+1)}{\alpha^{\mu+1}z^{v-\mu-1}} ...

**3**

votes

**0**answers

204 views

### Using the Mellin transform to invert ill-posed problems of harmonic transforms

I will try to explain the problem using just words. Let's see how far I get!
I will use the harmonic transform of the Radon transform for 2 or more dimensions as an example, but there is a large ...

**0**

votes

**1**answer

157 views

### Transformation problem involving 2 random variables

Any help in this problem?
Suppose U and V are independent random variables with density f(u) and g(v) respectively.
The domain of U is the interval (0, 1) and the domain of V is v > 0. After the ...

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votes

**0**answers

283 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} ...

**7**

votes

**1**answer

360 views

### Inversion of Radon transform by incomplete data: specific case

Let $R[f](p,t)$ denote the Radon transform of smooth function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ with compact support in $\mathbb{R}^n_+$:
$$
R[f](p,t) = \int\limits_{x \cdot p = t} f(x) dx.
...

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votes

**2**answers

251 views

### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform

Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose
$$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$
Is $a(x,0)=0$, $\forall x\in[0,1]$?
...

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vote

**1**answer

175 views

### Expectation under a t-distribution

Given parameters $\lambda, \nu>0$, a covariance matrix $R$, a mean vector $\mu \in R^p$, the Arellano-Valle and Bolfarine's generalized $t$ distribution is given by (see, for example, the book by ...

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85 views

### Integral representation of an arbitrary polynomial

Consider a smooth function $q \colon \mathbb R^2_+ \to \mathbb R_+$ such that $q(\lambda x) = \lambda q(x)$ for any $\lambda > 0$. Denote $x \circ y$ the elementwise product of $x,y \in \mathbb ...

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votes

**2**answers

870 views

### Techniques to solve equations involving a definite integral [closed]

Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, ...

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276 views

### how to solve a singular integral equation involving the kernel $1/x$

Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

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**0**answers

346 views

### What kind of math is used to study this problem?

We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ :
$$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right ...

**2**

votes

**2**answers

473 views

### Resources for special functions, integral identities

In the past weeks, I have struggled with finding suitable tables for integral indentities for Beta functions, Chebyshev polynomials and their like.
I would like to ask for online/offline resources ...

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**0**answers

1k views

### An inverse Laplace transform involving Error function

Dear MOs,
I need to calculate the inverse Laplace transform of the following function
$$
g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0.
$$
I have checked, among many ...

**8**

votes

**1**answer

790 views

### Steinmetz, Laplace and Fourier Transforms

I am looking for references on Steinmetz Transform and its relation with Laplace and Fourier Transforms. There is an Italian Wikipedia page about this topic but with no references.

**0**

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**1**answer

205 views

### About an integral transform or generalized Laurent series

We start with a little of context. I needed that a function from $\mathbb{R}^+$ to $\mathbb{R}$ could be represented in the following form, not necessarily uniquely:
$$
...

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**2**answers

427 views

### One-sided Cauchy principal value

What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,
$ PV \int_a^b f(t) dt = ? $,
where the integral is convergent in the upper limit, but ...

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232 views

### Unitary Representations and Integral formulas

While reading the appendix to 4th chapter of Iwaniec and Kowalski's analytic number theory I came upon a remark relating unitary representations and some integral transforms involving J-Bessel ...

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143 views

### On the multidimensional Mellin transform of measures

Consider an integral transform of Borel measures supported on $\mathbb{R}^n_+$ given by
$$
f(z) =\int\limits_{\mathbb{R}^n_+} x^{z}\frac{\mu(dx)}{x}
$$
where $z = (z_1,...,z_n) \in \mathbb{C}^n$, ...

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195 views

### Solving a nonlinear integral equation

Consider the integral equation $$f=g^2+H[g]^2$$ where $f\colon\mathbf R\to \mathbf R$ is an even and integrable function, $g$ is the function to be solved for, and $H[g]$ is the Hilbert transform of ...

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150 views

### Delta function representation as an integral of Pfaffians over SO(2m)

Define $\mathcal S$ as the set of all $2^m$ skew-symmetric $2m \times 2m$ matrices with the form $\oplus_{j=1}^m\begin{pmatrix} 0&\pm 1 \\ \mp 1&0\end{pmatrix}$.
Let $S_i, S_j \in ...

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122 views

### Orthogonality of Pfaffian polynomials in $SO(2m)$

I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem.
Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = ...

**6**

votes

**1**answer

528 views

### Can elliptic integral singular values generate cubic polynomials with integer coefficients?

For the elliptic integral of first kind, $K(m)=\int_0^{\pi/2}\frac{d\theta}{\sqrt{1-m^2sin^2\theta}} $, it is well-known that $K(m)$ can be expressed in what Chowla and Selberg call "finite terms" ...

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**1**answer

382 views

### On the generalized Radon transform and currents

Given a family of hypersurfaces $H_{t,p} = $ {$x \in \mathbb{R}^n \mid g(x,p) = t $} one defines a generalized Radon transform $R$ of a function $u \colon \mathbb{R}^n \to \mathbb{R}$ as
$$
R[u] ...

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232 views

### On the generalisation of the Laplace transform

I consider a measure transform $A$ given by
$$
A\mu(x) = \int\limits_{\mathbb{R}^n_{+}} e^{-g(x,y)} \mu(dy)
$$
where $g(x,y)$ is some positive smooth function, $\mu$ is a Borel measure. Is it a ...

**3**

votes

**2**answers

707 views

### Inverse Hankel Transform

I was reading through Akhiezer's book Lectures on Integral Transforms and in chapter nine, he states that the Hankel transform is unitary for $\nu > -1$, so that for a suitable function, $f$,
...

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**0**answers

199 views

### Regularity class of certain diffeomorphisms of the real line.

I care about the following class of homeomorphisms of $\mathbb R$, which I'll call $\mathcal C^?$.
For simplicity, let us restrict attention to compactly supported homeomorphisms
(a homeomorphism ...

**6**

votes

**1**answer

2k views

### Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does.
The gist of my work is that I have an $N\times N$ true covariance ...

**8**

votes

**2**answers

609 views

### An identity involving an infinite integral with a sinh in the denominator

I recently encountered the rather appealing looking integral, which appears in the theory of random matrices :
$$\int_{-\infty}^{\infty}\prod_{j=1}^{p-1}(j^{2}+z^{2})\frac{zdz}{\mathrm{sinh}(2\pi z)} ...

**9**

votes

**1**answer

421 views

### Properties of a matrix-valued generalization of the $\Gamma$ function

I am interested in the following function (Mellin transform of matrix exponential):
$$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$
Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ ...

**1**

vote

**3**answers

757 views

### A definite integral

Hello,
I am trying to find an explicit form of the following definite integral. I have tried Mathematica and it failed to give an answer. I am wondering whether anyone knows this integral. It might ...