Questions tagged [integral-transforms]
This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine transforms.
759 questions
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Mellin transform of the power of the reciprocal gamma function
Let $$f(z)=\frac{1}{\Gamma^n(z)}\text{, $n>0$: integer, and $z>0$}.$$ How to obtain the Mellin transform of $f(z)$? I couldn't find it in any table of Mellin transforms. thanks in advance.
My ...
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Hankel transformation formula derivation. (Verification! )
Suppose that $f \in L^1(\mathbb{R}^2)$ is a radially symmetric. Show that:
$$
\widetilde{f}(y) = \int_0^\infty J_0(r|y|) f(r) r \, dr,
$$
where $J_0(z)= \frac{1}{\pi}\int_0^\pi e^{iz \cos \theta} d\...
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Why are different operations in mathematics referred to as "convolution"?
Convolution appears in many mathematical contexts, such as signal processing, probability, and harmonic analysis. Each context seems to involve slightly different formulas and operations:
In standard ...
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Existence of a Restricted Set of Orthogonal Functions
Let $f: [0,1] \rightarrow \mathbb{R}$ such that $\int_0^1 f(\theta)d\theta=0$ and $f \neq 0$ ($f$ is not almost everywhere zero). Does there exist a set of function $\mathcal{G}$ such that
$\forall g ...
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Help with zero order Hankel Transform of a Complex Function
I'm trying to calculate the zero order Hankel transform of the following function:
$𝑓(𝑟)=\log(𝐵^2⋅𝑟^2)⋅\exp(−𝑖⋅𝑘⋅𝑟^2)$.
Where $B$ is a real positive constant and $k$ is a real number.
We may ...
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On the solvability of a nonlinear integral equation
Consider the integral equation
\begin{equation}
\phi(t)=\int_{t}^{\infty} f(s)\exp\left(-\int_{t}^{t+s} \ln\left(1+\frac{K\phi(u)+\eta(u)}{\phi(u)}\right)du\right)ds\tag{1}
\end{equation}
for a ...
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Alternating series of Laplace transforms is zero implies that Laplace transform is zero?
$\newcommand{\on}[1]{\operatorname{#1}}$
Suppose $\on{F}\left(s\right)$ is the Laplace transform of $\on{f}\left(x\right)$, and
$$
\sum_{k = 1}^{\infty}\left(-1\right)^{k - 1} \on{F}\left(rk\right) = ...
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Asymptotic expansion of Hankel transform of even function
I have found in this paper that the integral
$$
I(k) = \int^{\infty}_0 f(x) J_0(kx)x \,dx \tag{9}
$$
has a closed formula for the asymptotic expansion in $k$. The problem is that this doesn't work if $...
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How to calculate this integral
Is there a formula of this integral
$$\int_{S_{2n-1}} \frac{e^{i a \langle z, \zeta \rangle}}{|z - \zeta|^{2\lambda}} \, d\sigma(\zeta)$$
and how to calculate it.
Thank you in advance
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Fourier transform in finite range
I know that the Fourier transform applies to infinite intervals $(-\infty,+\infty)$:
$$\hat{f}(k)=\int_{-\infty}^{\infty}f(x)\exp\left(-2\pi kxi \right) \mathrm d x$$
I would like to see if there is ...
- 241
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Change of variables along vertical strip
I'm working on an integral involving multiple Hankel transforms. After the integrating over the Bessel functions, I'm left with a double integral over the region $\mathcal{R}= \{x,y:|x-y|<1<x+y\}...
- 59
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Distribution of random variable obtained by non-invertible integral transformation of another random Variable
My question is concerned with the following scenario: Let $X$ be a (univariate if necessary) random variable with density $f_X$. I define the new random variable $Y$ as $Y = g(X)$. To derive the ...
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Change of variables mapping (xy)/(x+y) to a sum or product
I've been working on a double integral of the form
$$
\int_\alpha^\infty dx\ f(x)\int_\beta^\infty dy\ g(y)\ e^{-a\frac{xy}{x+y}}\left(\frac{xy}{x+y}\right)^b,
$$
where $f$ and $g$ are not ...
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Mistakes taking Fourier Transforms on finite time intervals
I am trying to find the all the following Fourier Transforms on a finite time interval $[t_0,\ t_f]$ with $t_0<t_f$:
$$A(iw) = \int\limits_{t_0}^{t_f} f(t)\ \delta(t-t_0)\ e^{-iwt} dt$$
$$B(iw) = \...
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Infinite sum of $(\sqrt{n^2+1}-n)^{2r}$
I came across the following sum in a statistical physics context,
$$
\sum_{n=\frac{1}{2},\frac{3}{2}\cdots}^\infty \left(\sqrt{\left(\frac{n}{\beta}\right)^2+1}-\frac{n}{\beta}\right)^{2r}=\sum_{n=\...
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When can the Weierstrass transform be represented as $e^{D^2}$?
The Weierstrass transform $W$ of $f:\mathbb{R}\to\mathbb{R}$ can be defined as the convolution of $f$ with a Gaussian:
$$W[f](x) = \frac1{\sqrt{4\pi} } \int_{\mathbb{R}} dy\, f(x-y) e^{-y^2/4}.$$
It's ...
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Are the Y and H transforms inverses of each other for any $\nu$?
In my work I have come across a function which is defined on the whole real axis and diverges at the origin. I would like to expand it in terms of spherical Bessel funcions of the second kind. I have ...
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Integral representation of Lambert Series.
Is there a general integral representation for Lambert series, similar to those that exist for various L-series and zeta functions?
For a sequence of complex numbers $\{a_n\}$, its corresponding ...
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Does anyone know what the 2D Hilbert transform is of $\exp(-(x+y)^2)$?
I'm having no luck finding the 2D Hilbert Transform (standard principle value definition) $f(x,y)=\exp(-(x+y)^2)$. Naively I might expect this to be zero since under change of variables (rotation) to $...
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Mellin Transform of a function evaluated on the imaginary axis.
I am interested in finding the Mellin transform of a complex function evaluated on the imaginary axis.
The Mellin transform is typically evaluated over the real numbers as follows:
$$ M[ f(x), s ]=\...
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Is Hilbert transform of locally integrable function also locally integrable?
For a locally integrable function $f:\mathbb R\to \mathbb C$, define the Hilbert transform $Hf$ as $$Hf(x)=\lim_{\epsilon\to +0, R\to\infty\qquad}\frac{1}{\pi}\int_{\epsilon<|x-y|<R}\qquad\frac{...
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What integral transform is being used in this case?
I'm trying to understand a lecture where they compute the integral of $\frac{1}{|\vec x - \vec A|}$ for fixed A and integrating over all $x\in R^3$.
I'm specifically confused by the final line in this ...
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Mellin Transform of $1/(1 - x)$
What is the Mellin transform of $\frac{1}{1-x}$ ?
According to the book "Tables of Mellin Transforms" by F. Oberhettinger, page 14 (entry 2.12):
For $s$ complex such that $0 < \Re(s) < ...
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What is Laplace transform of $t^af(t)$ if Laplace transform of $f(x)$ is $F$.
What is the Laplace transform of $t^af(t),a>0$ if the Laplace transform of $f(x)$ is given to be $F(s)$. By definition, it should be like this:
$$\mathcal{L}\{t^a f(t)\}(s) = \int_{0}^{\infty} t^a ...
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Mellin transform and boundary condition
Let $k>0$, $f\in C[0,+\infty)$. Consider the problem
$$ -u''(r)-\frac 1 r u'(r)+\frac 1 {k^2}u(r)=f(r),
r>0,\tag{1} $$
$$u(0)=0, u(+\infty)=0.
\tag{2}$$
By the Green function method one can ...
user1360262
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Can the gamma function be generalized to quaternions and how? [duplicate]
The gamma function is a generalization of the operator !n. The question is: Can the concept of the gamma function be generalized to quaternion analysis and the use of quaternions, and how?
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How many bits would be required to store 2D FDCT result of 8-bit sample data?
The 2D FDCT can be expressed as a multiplication between an 8x8 matrix of 8-bit integers (lets call it A) and another 8x8 matrix containing fraction values (lets call it B) that are all less than 1. ...
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Laplace transform of exponential functions with derivatives.
I have been trying to calculate the Laplace transform of these troublesome exponential functions:
Having $\alpha \in \mathbb{R^+}$
1.$\mathcal{L}\left\{e^{n \alpha t}\frac{f(t)}{t^2} \right\},n \in \...
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Write triple integral as cylindrical coordinate of given region, confused in determining lower and upper bound.
Given $E$ is a region as follows:
$$E=\left\{0\leq x\leq 1, 0\leq y\leq \sqrt{1-x^2}, \sqrt{x^2+y^2}\leq z\leq \sqrt{2-x^2-y^2}\right\}.$$
Write triple integral
$$\iiint_\limits{E}xydzdydx$$
as triple ...
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How to derive the 1/s rule for Laplace transform of an integral?
How do we derive the rule $\mathcal{L}\{ \int_0^t f(\tau) d\tau\}=F(s)/s$ where $F(s)=\mathcal{L}\{f(t)\}$ and the symbol $\mathcal{L}$ represents the Laplace transform operator ?
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Inverse Fourier in two dimensions
I want to compute the Fourier inverse in 2D of the following integral
$\displaystyle\int_{\mathbb{R}^2}\Big(\frac{\zeta_1}{\zeta_2}\Big)^k F(\zeta_1,\zeta_2)e^{i(\zeta_1 x_1 + \zeta_2 x_2)}d\zeta_1d\...
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How to solve this integral equation $ \int_{-\infty}^\infty f(z)x^z dz = F(x)$ for f(x)?
My question is: solving $f(x)$ with known $F(x)$ and equation
$$ \int_{-\infty}^\infty f(z)x^z dz = F(x).$$
I met this problem when I tried to extend the idea of generating functions for discrete ...
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How can I solve $\sum_{i=1}^{M-1} (M+i)^{M+i+1/2}/i^{i+1/2}$?
I am trying to solve an equation in Mathematica:
$$
\sum_{i=1}^{M-1} \frac{(M+i)^{M+i+\frac{1}{2}}}{i^{i+\frac{1}{2}}}
$$
Does a general solution exist for this expression?
And if $M \to \infty$, can ...
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Wavelet admissibility and orthogonality
I have been studying the continuous wavelet transform and came across the following result on the Wikipedia:
A wavelet $\psi(t) \in L^1(\mathbb{R})\cap L^2(\mathbb{R})$ is admissible if it has a ...
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Relation between the inverse Laplace and inverse Mellin transforms
If we have the answer to the inverse Melin transformation of an expression, can we arrive at the inverse Laplace transform of that expression?
$$M^{-1}\left (\frac{1}{\Gamma (c+s)\Gamma (d-s)} \right ...
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Radon Transform of Gaussian function
I am trying to find the radon transform of the gaussian function
$$f(x,y) = e^{-(x^2 + y^2)}$$
Now, I am using the formula for radon transform as
$$
[\mathcal{R}f]{(\rho, \theta)} = \int_{-\infty}^{\...
- 1
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What is the inverse of an integral transformation that turns second order ODEs into first order ones?
Let $\mathcal{S}_f:\mathcal{C^\infty\rightarrow C^\infty}$ be an integral transform such that, for any $\{f,\psi\}\subseteq\mathcal{C}^\infty$, $\int_0^\infty f(x)dx=\infty$, we have that:
$$\mathcal{...
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How can i find the kernel of this integral transform?
I'm trying to define a class of integral transforms $\mathfrak{S}:\mathcal{C}^\infty\rightarrow\mathcal{C}^\infty$ with the following property:
$$\mathfrak{S}_{\psi}\{f(x)\psi(x)\}(t)=\alpha_f(t)\...
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Definition of Convolution of functions of two variables
We know that, If $f$ and $g$ are functions then their convolution is defined as,
$(f*g)(x) =\int_{-∞}^{∞} f(t)g(x-t) dt$
(This is the convolution structure for Fourier transform)
But, what if $f$ and $...
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Laplace Transform of the Product of a function and the Step Function
I need to find the Laplace transform of the product of a function $f(t)$ with the Unit Heaviside Step Function $H(t-c)$, i.e., $\text{L}[H(t-c)f(t)]$.
Given that
$$\text{L}[H(t)f(t-c)] = e^{-sc}F(s)\...
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Inverse transform of a sine kernel
I’m not a mathematician and I’m working with some transforms in physical chemistry.
I use a transform to pass from the time domain of phase domain in a process that use a square wave to perturb and ...
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0
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Inverse kernel of a sine kernel
I’m not a mathematician and I’m working with some transforms in physical chemistry.
I use a transform to pass from the time domain of phase domain in a process that use a square wave to perturb and ...
- 21
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0
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35
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Generalizing a Laplace transform?
Consider the function given by
$$
F(x)=\int_0^\infty e^{-\int_0^t\int_{x-vs}^{x+vs}f(y)\,dyds} \,dt
$$
Is it possible to invert this and write $f$ in terms of $F$?
Some thoughts: If $F$ was in the ...
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How fast does a holomorphic function have to decay at $\infty$ in order to satisfy Titchmarsh's theorem?
Titchmarch's theorem says that a complex function $f(z)$ is analytic on the (closed) upper half of the complex plane and decays rapidly as $|z| \to \infty$ iff its real and imaginary parts are Hilbert ...
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Is there an integral transform that simplifies functional composition?
Laplace, Fourier and Mellin integral transforms simplify operations like multiplication by a variable, taking derivatives, shifts, dilatations. Are there integral transforms that, in some way, ...
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Hilbert transform of the integral of a function
Given that f and g are Hilbert transform pair
$$Hf(x) = g(x)$$
Although the derivative will maintain the transform pair relation
$$Hf'(x) = g'(x)$$
Does the Hilbert transform pair relation apply to ...
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Sine transform of $f(x) = \dfrac{1}{e^{\sqrt{2 \pi} x}-1} - \dfrac{1}{\sqrt{2 \pi} x}$
Quoted from Titchmarsh's book The Theory of the Riemann Zeta-Function:
--
Now it is known that the function $$f(x) = \dfrac{1}{e^{\sqrt{2 \pi} x}-1} - \dfrac{1}{\sqrt{2 \pi} x}$$ is self-reciprocal ...
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2
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Intuition for how the Fourier transform "preserves information" and "changes basis"?
I am an undergrad that has taken a few courses in real and complex analysis. I am trying to understand the Fourier transform better at a level of abstraction somewhere between "it moves from ...
- 701
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Hankel transform of arbitrary order of $f(r)=1$
The Hankel transform of order $\nu$ of a function $f(r)$ is defined by
\begin{equation*}
F_\nu(k) = \int_0 ^\infty r dr f(r) J_\nu(kr ),
\end{equation*}
where $J_v$ is a Bessel function of the first ...
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How to define a linear operator in Maple that commutes with derivatives?
I would like to simplify an expression involving the Hilbert transform in Maple. The Hilbert transform is defined by $$ Hf(x) = \frac{1}{\pi} \ \mathrm{p.v.} \int_{-\infty}^{+\infty} \frac{f(z)}{z-x} \...
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