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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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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 \...
Jmtz's user avatar
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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 ...
Ongky Denny Wijaya's user avatar
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1 answer
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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 ?
drC1Ron's user avatar
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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\...
Mary's user avatar
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2 votes
1 answer
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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 ...
Jie Zhu's user avatar
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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 ...
No Yeah's user avatar
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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 ...
Isaac Mortiboy's user avatar
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38 views

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 ...
3pi.sahagh's user avatar
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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}^{\...
Subham's user avatar
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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{...
Simón Flavio Ibañez's user avatar
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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)\...
Simón Flavio Ibañez's user avatar
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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 $...
General Mathematics's user avatar
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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)\...
Sharat V Chandrasekhar's user avatar
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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 ...
PierT's user avatar
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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 ...
PierT's user avatar
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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 ...
tparker's user avatar
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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, ...
Daigaku no Baku's user avatar
1 vote
1 answer
65 views

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 ...
A AlOmar's user avatar
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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 ...
Ali's user avatar
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3 votes
2 answers
362 views

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 ...
Tanishq Kumar's user avatar
1 vote
1 answer
79 views

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 ...
pot plant's user avatar
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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} \...
Liu's user avatar
  • 11
14 votes
2 answers
562 views

An unzipping problem

Imagine a continuous one-dimensional line, which is duplicated exactly once. Duplication starts at random spacetime points. Once a point is duplicated, it starts a double duplication wave moving in ...
sam wolfe's user avatar
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Does this term vanish after doing the Fourier transform of a function under a derivative even in a bounded domain?

The Fourier transform of a function under a derivative is: $$ \mathcal{F}\Bigg({df (t) \over dt}\Bigg) = {1\over \sqrt{2 \pi}}\int _{-\infty}^{\infty}{df (t) \over dt} e^{-iwt}dt \tag 1$$ Using ...
FriendlyNeighborhoodEngineer's user avatar
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Solutions and graphs of partial integro differential equations

Consider $$\partial_tu+ \partial_xu-D\partial_{yy}u=0,$$ subject to the initial condition $u(x,y,0) = v(x,y)$ and the boundary conditions at the infinities, $ u(0,0,t) = u(\pm\infty,\pm\infty,t) = 0,$ ...
Ola's user avatar
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1 vote
1 answer
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Inverse Mellin transform of $a^{-s} \zeta (s)$

The Mellin inverse is given by $$ \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $$ Is it possible to compute the inverse transform of: $$F(s)=a^{-s} \zeta (s) $$ where: $a>0$ Mathematica ...
Mariusz Iwaniuk's user avatar
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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?
Impetus's user avatar
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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 ...
Michael Williams's user avatar
1 vote
1 answer
80 views

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 ...
t.o.'s user avatar
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2 answers
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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 ...
gowerc's user avatar
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2 votes
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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-\...
user48672's user avatar
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4 votes
2 answers
305 views

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 (...
Pascal S.'s user avatar
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0 answers
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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 ...
David Lee's user avatar
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1 answer
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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 ...
Yaroslav Bulatov's user avatar
3 votes
0 answers
67 views

$L^1\big((1+x)\mathrm dx\big)$ 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 ...
Pipnap's user avatar
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1 vote
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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^{(...
ochem1's user avatar
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2 votes
1 answer
198 views

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 ...
ochem1's user avatar
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1 vote
2 answers
835 views

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 ...
Jonathan Huang's user avatar
2 votes
0 answers
191 views

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-\...
Yaroslav Bulatov's user avatar
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0 answers
100 views

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 $\...
VojtaK's user avatar
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0 answers
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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_{...
user572780's user avatar
9 votes
1 answer
186 views

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 \,...
Siegfriedenberghofen's user avatar
4 votes
1 answer
308 views

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-\...
Yaroslav Bulatov's user avatar
1 vote
0 answers
45 views

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 $...
z.v.'s user avatar
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1 vote
1 answer
39 views

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 ...
Radek Vavřička's user avatar
0 votes
1 answer
131 views

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 $\...
javi1996's user avatar
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1 vote
0 answers
43 views

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, $f : \mathbb{R}_+^0 \to \mathbb{R}$. ...
Radek Vavřička's user avatar
1 vote
2 answers
46 views

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. ...
rxs2p0's user avatar
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3 votes
1 answer
168 views

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 ...
Yaroslav Bulatov's user avatar

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