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.
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Inverse Laplace transform of $\frac{\sinh(\sqrt z a)}{\sqrt z}$
I was trying solve this inverse Laplace transform, given by $\frac{\sinh(\sqrt z a)}{\sqrt z}$, with $a \in \mathbb{R}$. But i dont have any good idea to solve this. Please someone have a idea?
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Guessing the kernel of an integral
Suppose I have a function $h(x_1,x_2)$ which is a polynomial function of degree $2n$ with all even powers;
$$h(x_1,x_2) = \sum_{i,j=0}^{n} c_{ij} x_1^i x_2^j$$
where $c_{ij}, x \in \mathbb{R}$. Is ...
- 117
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Transform integral with complex bounds
Question
I have a line integral given by
$$
\begin{align}
I &= \int_{z_0=a+ib}^{z_1=c+id}dz~ z \\
\\
&= -\frac{a^2}{2}+\frac{b^2}{2}-\frac{d^2}{2}+\frac{c^2}{2}+i(-ab+cd)
\end{align}
$$
For ...
- 111
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2
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How to show algebraically that the determinant of the Jacobian the scaling factor for change in variables?
I am currently trying to learn about Jacobians (self study). In particular struggling to understand the geometry of change in variables and why the determinant of the Jacobian is the scaling factor ...
- 111
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How to identify a Fox $H$ function from its Mellin transform?
I have obtained a Mellin transform
$
\mathcal{M}[f](s)= \frac{ \Gamma\left( 1-\frac{2-s}{a}\right) \Gamma\left( \frac{2-s}{a}\right) \Gamma\left( \frac{s}{2}\right) {{2}^{s-1}}}{a \Gamma\left( 1-\...
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Integral over a product of polynomial, exponential and Bessel function
In a physics textbook I'm working through I found an interesting integral identity which I want to prove:
\begin{equation}
\int_0^\infty t^{\nu +1} J_\nu(\beta t) e^{-\alpha t} \, dt = \frac{2\alpha (...
- 343
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An example of Mellin transform asymptotics
Let $$F(x) = \sum_{k\ge 1}\frac{1}{1+k^2x^2}$$ and its Mellin transform
$$F(s) = \frac{\pi}{2}\frac{\zeta(s)}{\sin{\frac{\pi}{2}s}}, \space\space where \space\space (1<Re(s)<2)$$
And then, its ...
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Laplace Transform of $\frac{\psi ^{(2)}\left(\frac{1}{\sqrt{x}}\right)}{x^{3/2}}$?
Is there a closed form expression for the Laplace Transform of the following expression?
$$f(x)=\frac{\psi ^{(2)}\left(\frac{1}{\sqrt{x}}\right)}{x^{3/2}}$$
where $\psi^{(k)}$ is the polygamma ...
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Representing matrix multiplication as integral (Integral Kernel )
I've aware that a discrete approximation of an integral can be made using matrix multiplication. I seek to find the continuous analog of the matrix multiplication $(GD^{-1})(GD^{-1})^T\vec{x}$. Where $...
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$L^1\big((1+x)\mathrm dx\big)$ form
I am reading the article.
On page 15, they wrote
But I don't know why we can denote $\Phi(s) = \int_{0}^{+\infty}x^{s-1}\Phi(x)dx$.
For your convenience, I also attach the picture in which there are ...
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inverse Laplace transform of an integral
find $$u(x,t)$$ Given
$$ U(x,s)=\frac{s+2}{(s+1)}{\int_{-\infty}^{\infty}f(x)cosh((s+1)(x-y))dx}$$
where U is the Laplace transform of the function u.
I tried substituting $$cosh((s+1)(x-y))=\frac{e^{(...
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Solving a PDE with non-zero IC
Given the function $f(x,t)$, solve the following PDE
$$\partial_{tt}f+ 2\partial_{t}f- \partial_{xx}f+f=0$$
BC: $$f(x=\pm \infty, t)=0$$
IC: $$f(x,t=0)=g(x), \quad f_t(x,t=0)=0$$
I tried solving it ...
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Proof of the "Maz identity" for solving integrals
The "Maz identity" states:
$$ \int_0^\infty f(x)g(x)\mathrm{d}x = \int_0^\infty \mathcal{L}\{f\}(u)\mathcal{L}^{-1}\{g\}(u)\mathrm{d}u, $$ where $\mathcal{L}$ is the Laplace transform.
I ...
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Proof product propriety of bessel function
The bessel function is given by:
$${J}_{n}(x) ={ \mathop{∑
}}_{k=0}^{∞} {{(−1)}^{k}\over
k!(n + k)!}{\left ({x\over
2}\right )}^{n+2k}.$$
and Translation operator can be wrighten as:
$$T_x f(y) = \...
- 21
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0
answers
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Determining if function tends to zero given its Laplace Transform
Given $h:\mathbb{R}\to \mathbb{R}^+$, I want to know whether $f(t)\to 0$ as $t\to \infty$ where $f(t)$ is defined in terms of its Laplace transform $F(s)$:
$$F(s)=\frac{\langle h, z_s\rangle}{1-\...
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I need some reference about the Fourier-bessel transform
Hi everyone I'm looking for reference about the Fourier-bessel transform and translation operator associed with the bessel operator.
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Abel transform of Gauss Function and other bell shaped functions
Could you help me compute Abel transform of Gauss function. I need
$$A_g[\sigma](x) = \int_x^\infty \frac{r}{\sqrt{r^2-x^2}} e^{-(\frac{r}{\sigma})^2} \, d\mathrm{r}, \,\,\,\,\,\, x\geq0, $$
where $\...
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Composing integral transforms means multiplying the kernels.
The exercise I'm working with is 2.1.6(b) in Conway's A Course in Functional Analysis (2/e):
Let $(X,\,\Omega,\,\mu)$ be a $\sigma$-finite measure space and let $k_1$, $k_2$ be two kernels satisfying
...
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Integral transform and Fourier Transform
I was trying to study deeply the Fourier Transform and I was wondering why the use of this kind of kernel for this integral transform. Could you also suggest me an “engineer with good math knowledge ...
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What conditions must hold to have a valid integral transform and associated inverse transform?
According to Wikipedia, an integral transform is any transform $T$ of the following form:
$$(Tf)(u) = \int_{t_1}^{t_2}f(t)K(t, u)dt.$$
The inverse transform is of the following form:
$$f(t) = \int_{...
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When is $\int_0^\infty dk \int_0^\infty dq \int_0^R dt \,f(k,q,t,r)\stackrel{?}{=}g(r)$ where both $f$ and $g$ are known functions
I would like to know under which circumstances the following triple integral can be evaluated analytically as
$$
\int_{k=0}^{k=\infty} \int_{q=0}^{q=\infty} \int_{t=0}^{t=R} f(k,q,t,r) \,\mathrm{d}t \,...
- 431
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votes
1
answer
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Inverse Laplace transform, rank1 correction of matrix exponential
Given real-valued vectors $\mathbf{a},\mathbf{u}$ in $\mathbb{R}^d$, is there a nice expression the inverse Laplace transform of $f(y)$ below?
$$f(y)=\frac{\left(\sum_i \frac{u_i}{y-a_i}\right)^2}{1-\...
- 4,739
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Preforming an explicit inverse Mellin transform
Playing around with some elementary integrals and Mellin transforms, I arrive at the following integral expression
$$(1):~~ I(A,B) = \int_{c-i\infty}^{c+i\infty} dz~(2z+1) e^{-Az} K_z(B)~,$$
where $...
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A functional that returns the function's value outside integration bounds
We have a functional over $\mathbb {R} \to \mathbb {R}$ functions $\{f\}$, that could be written as
$F (f) := \int_{-\infty}^\infty \mathbb{d}t K (t) f (t)$,
where $K(t)$ is a distribution that could ...
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1
answer
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Logarithmic-Fourier type integral transform
I'm working with transforms of the kind $$\int_{-\infty}^\infty f(x)e^{it\log(x)} dx \qquad (t\in \mathbb{R}),$$
where $f \in L^1(\mathbb{R})$ and you have fixed a branch of the complex logarithm in $\...
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Looking for an inverse of an integral transform
I have an integral transform (motivated by a physics problem)
$F (x) = \int_0^\infty dx' \frac {a x'} {(a x)^2 + (x - x')^2} f (x')$,
where $x, x', a > 0$ real, $f : \mathbb{R}_+^0 \to \mathbb{R}$.
...
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2
answers
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Evaluate $\lim_{s \rightarrow 1^{-}} \frac{\Gamma(b-as)}{\Gamma(s)\Gamma(1-s)}$, for $a,b>0$
I encountered the following as a result of a Mellin-transform:
$$
\frac{\Gamma(b-as)}{\Gamma(s)\Gamma(1-s)},\quad b>0,\ 0 < a,\text{Re}(s) < 1
$$
where $\Gamma(\cdot)$ is the gamma function.
...
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Computing $\operatorname{Tr}[(\text{D}+uu')^k]$ using generating functions?
I have a large diagonal + rank1 matrix with positive entries. Can someone help me understand how to compute the following using Laplace transform?
$$\operatorname{Tr}[(\text{D}+uu')^k]$$
In my ...
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Name of discrete transform $f(x)=\sum a(n) \exp(-x a(n))$
Is there a way to express the following transform (and perhaps its inverse) in terms of known discrete transforms? For $a_i>0$ and $\sum_i a_i=1$ we have
$$f(x)=\sum_i^n a(i) \exp(-x a(i))$$
...
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Existence of inverse fourier transform
Is it possible to evaluate an inverse fourier transform of these functions?
$f(\omega)=\exp(-(k^2-\omega^2)^{1/2})$,
$g(\omega)=\frac{\exp(-(k^2-\omega^2)^{1/2})}{(k^2-\omega^2)^{1/2}}$,
where k is a ...
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Determinant of a continuous Kernel
Before jumping to the determinant, lets consider the discrete eigenvalue equation:
$$A\text{x}=\lambda\text{x}$$
$$\sum_jA_{ij}\text{x}_j=\lambda\text{x}_i$$
And the continuous eigenvalue equation:
$$\...
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Invert Laplace Transform with Heaviside function
I'm solving the following boundary value problem
$$
y \frac{\partial u}{\partial y}+\frac{\partial u}{\partial x}=1, \quad u(x, 1)=1=u(0, y) .
$$
I've derived that
$\bar{u}(p, y)=p^{-2}+p^{-1}-p^{-2} ...
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Necessary condition for invertible integral tansform.
i have a rather complicated question (at least for me) to figure out. Let’s say that:
$$I[f(t)](u)=\int_{t_0}^{t_1}f(t)K(t,u)
\mathrm{d}t$$
Is a general intengral transform $I$ with kernel $K$. Are ...
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3D Poisson analytical solution (paper adaptations for mixed boundary conditions)
I am attempting to find an analytical solution, in terms of general Green functions, of the 3D poisson equation over a finite cylinder, with mixed boundary conditions. I have Neumann boundary ...
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1
answer
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Order of Pole (if it is indeed one) in a Laplace Transform
I'm trying to invert the following Laplace Transform using the Cauchy Residue Theorem
$$
F(s)=\frac{p(s)}{sq(s)}
$$
which is subject to the following conditions:
$$p(0)=q(0)=0$$
$$\lim_{s\rightarrow 0}...
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Transporting orbit space of vector field via integral transform
Consider a vector field which can be written as system of separable differential equations, which I will solve below:
$$ X=\big\langle x\log x, -y \log y \big\rangle$$
for $x,y \in (0,1).$
I would ...
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Probability, Fourier Series and Distribution functions
A sequence of distributions $\left(F_n\right)$ converges to a distribution $F$ if $\left\langle F_n, \phi\right\rangle \rightarrow\langle F, \phi\rangle$ for all test functions $\phi$.
(i) Show that ...
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How to solve the fourier transform of this function?
Is this solution correct in order to find the Fourier transform of $\mathrm{g}(\sigma, \mathrm{t})=\mathrm{e}^{-\lambda|t|} \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} ?$
If it seems wrong, please let ...
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Causality/Non-Causality, One/Two-sided integral transform (Laplace, Fourier), Discrete case
When introducing unilateral and bilateral integral transforms (in the context of Laplace transform), I have problems understanding how causality and non-causality are expressed when considering the (...
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1
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Laplace Transform to solve ODE
I'm trying to solve the BVP $f''(x)=\delta(x-a)$ where $0<a<1$ and $f(0)=f(1)=1$ but I'm really not sure where to start. I tried taking the Laplace Transform of the equation, to get (after ...
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Inverting integral transform with kernel $x^n$
I'm looking to find the inverse of
$$g(n)= \int_{-\infty}^\infty f(x)x^n dx$$
I'm not sure if an inverse exists but i suspect the kernel is well behaved enough to have an inverse.If an inverse exists ...
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Conditions on $\sum\limits_{n=0}^\infty f(n)=-\frac12\int_{c-i\infty}^{c+i\infty}(\cot(\pi z)+i)f(z)dz$
The Ramanujan interpolation formula states:
$$\int_0^\infty x^{s-1}\sum_{n=0}^\infty f(n) (-x)^n dx=\pi\csc(\pi s)f(-s)$$
therefore we use the Mellin inversion:
$$\text M_s\left(\sum_{n=0}^\infty f(n) ...
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Solving hypergeometric like differential equations using Integral transforms.
It is known that for 2nd order differential equations with 3 singularities that are regular, can be converted into the hypergeometric differential equation.
As an example of this, we can see the first ...
- 98
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0
answers
28
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Second Order Hankel Transform of the Polar Laplacian $\mathcal H_2 (\partial_r^2 + r^{-1} \partial_r + r^{-2} \partial_\theta^2)$
The Hankel transform is typically reserved for axisymmetric forms of Laplace's equation in cylindrical coordinates. Although the assumption of the transform relies on angular invariance of the two ...
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votes
1
answer
107
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Cosine Convolution Theorem
I have found a number of sources that suggest:
$$\int_0^\infty f(t)g(t)\sin(xt) dt = \frac{\pi}{2}\int_0^\infty \left(\int_0^\infty f(s) \sin(ts) ds\right) \left(\int_0^\infty g(s)\sin(xs)\sin(ts)ds\...
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0
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Kramers-Kroning peaks relationship
My question arise from an statement on Qualitative interpretation of Hilbert transform
Kramers-Kronig relations state that for analytic (in the upper half-plane) functions with fast decay, its ...
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0
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Fourier transform of $\dfrac{\cos(\beta x)}{a^{4} + x^{4}}$
What I've done so far:
$$\dfrac{1}{2\sqrt{2\pi}}\left[\int_{-\infty}^{\infty}\dfrac{e^{i(\beta + k)x}}{a^{4} + x^{4}}dx +\int_{-\infty}^{\infty}\dfrac{e^{i(k - \beta)x}}{a^{4} + x^{4}}dx\right] = \\\...
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1
answer
200
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What is a kernel integral operator?
I was reading a paper that define the kernel integral operator as follows:
We define the kernel integral operator $\mathcal{K}$. Let $\kappa^{(l)} \in C(D \times D; \mathbb{R}^{d_{l+1} \times d_l})$ ...
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0
answers
55
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Using Laplace transform to solve a radial differential equation.
I have been trying to solve a differential equation using the Laplace transform. I don't provide that expression here since it is quite too large IMO, and rather to asking for the solution, I am ...
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votes
1
answer
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Analytic signal of the Dirac delta function
Does anyone know of any derivations of the analytic signal of the Dirac delta function? I suppose that it can be found by first working from the definition of the Hilbert transform, since the analytic ...
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