# Questions tagged [integral-transforms]

For questions about integral transforms, inlcuding the Fourier transform, Laplace transform, Radon transform, Mellin transform, Hankel transform etc.

265
questions

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### Inverse Laplace if products of hyperbolic function to other function

I want to calculate $\sinh(as) F(s)$. We have
$$L^{-1}\left [ \left ( \frac{e^{as}-e^{-as}}{2} \right )F(s) \right]=\frac{1}{2}L^{-1}\left ( e^{as} F(s)\right )-\frac{1}{2}L^{-1}\left ( e^{-as}F(s) \...

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105
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### Continuity of Radon transform w.r.t the angle

Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by
$$
R[f](w,b) := ...

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77
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### Is there a continuum-dimensional analog of complex numbers?

Consider the set of functions $f:\mathbb{R}\to\mathbb{R}$. This set is similar to the set $\mathbb{R}^n$ with Hadamard (element-wise) multiplication: ${a}{b}={a_n b_n}$, similar to $fg=f(x)g(x)$.
Now, ...

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49
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### The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...

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87
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### What's wrong with this Laplace transform?

The following operators keep the area under the convergent integrals unchanged:
$$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,...

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66
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### Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...

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### Most general space for the Wigner-Weyl transform

The Wigner-Weyl transform $\mathfrak{W}$ is a bijective mapping between functions on a phase space and Hilbert space operators in order to map quantum mechanics into a phase-space formulation. Then ...

5
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122
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### Inverse Mellin transform of 3 gamma functions product

I want to calculate the inverse Mellin transform of products of 3 gamma functions.
$$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$
Above contour integral has ...

1
vote

1
answer

88
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### Convergence of oscillatory integrals

I'm considering integrals of the (Hilbert transform) type
$$p.v.\int_{-\infty}^\infty\frac{f(r)}{r}\,dr$$
where $f(r)$ is periodic, say, with period $2\pi$. I'm assuming very little regularity on $f$. ...

2
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101
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### Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation

Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...

1
vote

1
answer

36
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### Inverse transform of $\int K(s,t) \rho(s) dS $

Assume that $V$ is a n-dimensional surface, or the disjoint union such surfaces.
Let $K(s,t)$ be a function $V^2\to \mathbb R$. For the sake of simplicity, it can be assumed that $K$ is continuous, ...

2
votes

1
answer

77
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### Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$

Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by
$$
R[f](w,b) := ...

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49
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### Hyperplanes which equalize the Radon transforms of two distributions

Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...

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1
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### Consistent empirical estimation of Radon transform of a multivariate density function

Let $P$ be a "nice" distribution on $\mathbb R^m$ (e.g., multivariate Gaussian, etc.), with density $p$. Let $H := \{x \in \mathbb R^m \mid x^\top w = b\}$ be a hyperplane in $\mathbb R^m$ ...

3
votes

1
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111
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### On an asymptotic integral

Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that
$$
\int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}.
$$
Does it follow that $\phi$ is a ...

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0
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### Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...

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381
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### How to solve the following $0= \int_{-\infty}^\infty e^{-\frac{(bt+\omega)^2}{2}} f(t+\omega) \frac{1}{i t} dt, \forall \omega \in \mathbb{R}$

Suppose that for a given $b\in \mathbb{R}$
\begin{align}
0= \int_{-\infty}^\infty e^{-\frac{(bt+\omega)^2}{2}} f(t+\omega) \frac{1}{i t} dt, \forall \omega \in \mathbb{R}
\end{align}
where $i =\sqrt{...

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1
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549
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### Equivalent action of convolution of directional derivative

I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented.
Let $k*f(x)...

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1
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113
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### Fourier transform of the indicator function of the semi-ball

I wonder if it is possible to derive an analytical form for the Fourier transform of the indicator function of a semi-ball:
$$g_1(\underline{\xi}) = \int_{\mathcal{B}_+(R)} e^{i \underline{\xi} \cdot \...

2
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1
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179
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### Cauchy's integral formula and essential singularities

Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral
$$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$
where $\gamma_{z_0}$ is a small circle around $z_0$.
By ...

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95
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### Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$

I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral:
$$\frac{1}{\pi}...

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1
answer

76
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### How to prove that $\int (1-z)^{u} z^{v} dz$ is equal to $\frac{z^{v+1}}{v+1}_2F_1(-u, v+1; v+2; z)$?

How to prove that
$$\int (1-z)^{u} z^{v} dz = \frac{z^{v+1}}{v+1} \, _2F_1(-u, v+1; v+2; z)?$$

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1
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135
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### What are the steps involved in the solution to $\int{x^{-a} (b -cx^{-d})^e }dx$? [closed]

Mathematica gives me the following solution to $\int{x^{-a} (b -cx^{-d})^e }$:
$$\int{x^{-a} (b -cx^{-d})^e dx} = -\frac{b^{e}x^{1-a} \, _2F_1\left(\frac{a-1}{d},-e;\frac{a+d-1}{d};\frac{c x^{-d}}{b}\...

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1
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79
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### Solution or approximation to $\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx$?

I'm looking for a closed solution or an approximation to $$\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx,$$
where $a, b, c, d > 0$.

1
vote

1
answer

69
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### Solution to $\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx$

I'm looking for a solution to $$\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx.$$
Mathematica gives me the following solution, but I'd like to know/understand the steps involved in finding it.
$$\int ...

3
votes

0
answers

65
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### Injectivity of the Mellin Transform and discontinuities

I am reading Stopple's A Primer of Analytic Number Theory. On page 234, the Mellin Transform $\mathcal{M} f$ of a function $f$ is defined as
$$\mathcal{M} f (s) = \int_1^{\infty} f(x) x^{-s - 1} dx$$
...

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votes

2
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130
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### Integrals can sometimes be computed through their saddle points. Are there examples of converse, when saddle points are found via integrals?

Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g.
$$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$...

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1
answer

170
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### Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?

I'm looking for a solution to the following integral. However, it seems it doesn't have a solution.
$$\int\limits_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{\displaystyle1}{\...

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67
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### Differential properties of the Radon transform

I am looking for a reference for the following fact:
Let $f \in L_1(\mathbb{R}^{d})$ that has compact support and that is also in $C^{s}(\mathbb{R}^{d})$. We define the Radon transform of $f$ as:
\...

4
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0
answers

219
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### A tricky integral equation

In the context of reconstruction of climate data from ice cores (see related question at MSE) I came about the following problem. (More background and motivation can be given on demand.)
Let $f:\...

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0
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100
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### Are Poisson integrals uniquely determined by their radial limits?

Let $\mu$ be a complex Borel measure on the unit circle, and suppose its Poisson integral $u$ satisfies $\lim_{r\to 1-}u(re^{i\theta})=0$ for every $\theta$. Does it follow that $\mu=0$?
This is of ...

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0
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100
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### Hilbert's fourth problem and a non-linear integral transform

The following nonlinear integral transform takes continuous functions defined on the cylinder
$\mathbb{R} \times S^1$ to $C^2$ functions defined on the plane $\mathbb{R}^2$:
$$
\mathcal{A}f (x,y) := \...

1
vote

3
answers

205
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### Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$

Is there a solution to this integral?
$$\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx,$$
where $a > 0$ and $d > 0$.

0
votes

1
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70
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### Unique zero solution to a difference equation via Laplace transform

We want to prove that
the unique solution to the following difference equation is the null one:
$$
au(x)+b\mathbf{1}_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}_{(\frac{1%
}{2},1)}(x)u(x-\frac{1}{...

0
votes

1
answer

121
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### Laplace transform injectivity for different values of $p$

Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty
).$ Assume that there exist $p_{0},p_{1}\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
,$ $p_{0}\neq ...

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34
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### Solving a difference functional equation by using Laplace transform

Consider the operator $T:L^{2}(0,r+1)\longrightarrow L^{2}(0,r+1)$:
\begin{equation*}
Tu(x)=:u(x)+a\mathbf{1}_{(0,1)}(x)u(x+r)+b\mathbf{1}_{(1,1+r)}(x)u(x-1),%
\text{ }x\in (0,r+1),
\end{equation*}
...

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0
answers

32
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### Solving an equation containing Laplace transform

Consider the equation
\begin{equation}
\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}%
(y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),
\end{equation}
where $\mathcal{L}$ is the ...

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vote

1
answer

275
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### What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...

2
votes

1
answer

99
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### Solving an integral involving a Bessel function, Laguerre function and Gaussian

We want to calculate the expectation value $\langle q^2\rangle$ in polar coordinates which gives us the following integral, for integer values of $p$:
\begin{equation}\int_0^\infty dq~q^3 \left(\int_0^...

2
votes

1
answer

196
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### Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?

Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation
$$g(v)=\mathbb{E}[f(v+W)]$$
is defined for all $v \in \...

1
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0
answers

13
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### Graph of the Hilbert transform of a function extended on Boehmians

If a function is defined on set of Reals, we can draw the graph of function on plane, similarly we can draw the graph of its Hilbert transform. Now when we extend Hilbert transform to the space of ...

3
votes

1
answer

158
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### Generalization(s) of variation diminishing property to multivariate case

Let us first define the variation diminishing property for the Gaussian kernel. Consider a function $f: \mathbb{R} \to \mathbb{R}$ that is sufficiently smooth and define
\begin{align}
F(x)= \int_{-\...

2
votes

0
answers

26
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### finite fractional integral transform

Several integral transforms are generalized by introducing their fractional counterparts,e.g., fractional Fourier transform is a very popular one. Similarly, their discrete versions are also ...

7
votes

1
answer

298
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### Explicit isomorphism between $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$?

As Hilbert spaces, $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$ are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I ...

3
votes

1
answer

155
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### Fast computation of convolution integral of a gaussian function

Given a convolution integral
$$
g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx
$$
where
$\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...

2
votes

0
answers

62
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### On a question relating integral equation:

I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...

0
votes

0
answers

41
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### Inversion of integral transforms

We have the following theorem statement of this Paper "An inversion formula for a Class of integral transforms, 1975, C.Nasim" page 8.
Let
$\phi(x)= \frac{1}{x} \int_{0}^{x} \psi(t) dt$, ...

2
votes

0
answers

63
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### Reference request for type of specific integral equation in two variable:

Consider the following integral equation:
$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$
Here, $K(t,y)$ is a trigonometric kernel and
$\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$).
I want to find the ...

2
votes

0
answers

35
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### Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous

Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...

4
votes

1
answer

251
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### Calculation of an inverse Mellin transform

Let $z \in C$ and consider the following integral equation:
$$-\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,b-a;z)}= \int_{0}^{+ \infty}x^{z-1}K(x) \mathrm{d}x$$...