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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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Extending function from hyperplane segment to ball, estimating integral by integral on manifold

Let $B:=B_R (0) \subset \mathbb{R}^d$ be the ball with radius $R$ at $0$ and $H \subset \mathbb{R}^d$ a hyperplane that satisfies $H \cap B \neq \emptyset$ and $0 \not\in H$. Furthermore, let $f \in C^...
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Is there a name for the square of a function plus the square of its Hilbert transform?

Given a real-valued analytic function $f$ defined on the whole real line, and its Hilbert transform ${\cal H}f$, it seems that the quantity $f(x)^2+{\cal H}f(x)^2$ should have some kind of importance ...
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How can I solve this integral equation with the inverse Laplace Transform?

This question is related to Solving an integral equation with inverse Laplace transform. Let $\alpha,\beta,\mu>0$ with $\alpha/\beta>\mu$ and $X\sim\operatorname{Gamma}(\alpha,\beta)$. I am ...
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Boundary to boundary transformation of an integral

In my textbook "Mathematical analysis I" we saw something called "Boundary to boundary transformation of an integral" (Note that my textbook is a Dutch textbook, I've tried to translate the name the ...
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27 views

Binomial sum for an arbitrary function

I'm looking for some known results for sum of this type but I can't find anything. The sum is defined as: $$S(x,a,b,n)=\sum_{k=0}^n \binom{n}{k} (-1)^{k} f((a(n-k)+bk)x)$$ where $f$ is an arbitrary ...
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Bessel integral invovling algebraic and hyperbolic functions

I am desperate in evaluating the following Hankel transform $$ \int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k, $$ where $J_0(kr)$ is the Bessel function of ...
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How can the following be transformed in to a sum of complete elliptic integrals of the first and second kind

I have the following, that I known from a numerical implementation of the problem by a third party should be able to be transformed in to elliptic integrals of the first and second kind however I can'...
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1answer
61 views

Tricky integral relationship

I am trying to prove that $$\begin{equation}\int_x^{x+1}\left(\int_0^{v} (u-0)f(u)\textrm{d}u+\int_v^{1} (u-1)f(u)\textrm{d}u\right)\textrm{d}v=\\\int_0^x\int_v^{v+1}f(u)\textrm{d}u\textrm{d}v\end{...
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Laplace Transform of Complementary Error Function

I need to apply one Laplace transform formula while I have no idea how to prove it: $$\int_0^\infty e^{-st} e^{a k} e^{a^2 t} \operatorname{erfc} \left( a \sqrt{t} + \frac{k}{2 \sqrt{t}} \right) dt = ...
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Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $

After weeks of going back and forth I've been able to solve the following definite integral: $$I = \int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $$ To solve this I employ ...
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Fourier and Mellin transforms of Hilbert Transform

I am reading Hilbert transform recently and meet two questions. The book I am reading is Debnath and Bhatta "Integral Transforms and Their Applications". If we define the Hilbert transform on the ...
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Reference Request: n-dimensional Laplace Transform

I am looking for a reference, where the conditions for the existence of the n-dimensional Laplace transform are proven, i.e. when the laplace transform \begin{equation} F(\lambda_1, ..., \lambda_n) = ...
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About $ F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx $

We know for $ u > 1 $ $$ \int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) $$ Usually this is shown by using differentiation under the ...
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235 views

Double integral with Hankel transform

Let's say we have a double integral in the following form: $$I=\int_0^\infty \int_0^\infty f(x) g(y) J_0(xy) x y dx dy $$ Using the definition of the Hankel transform, we can write: $$I=\int_0^\...
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Integral Transformation from circle to unit sphere

I want to show that $\displaystyle \frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)\,\mathrm{d}s(y)=\frac{1}{|S^1|}\int_{S^1}u(x+r\theta)\,\mathrm{d}s(\theta)$ This is essentially a shift and dilation ...
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Decomposition of time series into weighted (weights known) stretched exponentials with unknown offset function

Looking for solution to decomposing function $f(t)$ into stretched exponential functions all with same meta-exponent and decay constants, but with different amplitudes $p$ (known) and offsets $\tau$ (...
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1answer
78 views

Proof of inverse Laplace transform

Why is $$f(t) = \frac{1}{2πj}\int_{\sigma-j\infty}^{\sigma+j\infty} F(s) e^{st} \, ds,$$ provided that $$F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt \ ?$$ I tried to find out myself, or searched ...
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32 views

“Bilateral Mellin convolution”

The Mellin convolution of two functions, when it exists, is of the form $$ (f \ast_M g)(t) = \int_0^\infty f\left( \frac{t}{\tau} \right) g(\tau) \frac{\mathrm{d}\tau}{\tau} $$ and ...
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What are the easiest inverse Mellin transforms?

I'm looking to familiarize myself with the inverse Mellin transform \begin{align*} \mathcal{M}^{-1}[\varphi](t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-s}\varphi(s)\, \mathrm{d}s \end{...
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1answer
62 views

On the variety of Integral Transforms

Recently I have come across this book of Integral Transforms. At this point I realized for myself that there are so many transforms with different kernel out there. Now to provide a little bit of ...
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2answers
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Laplace Transform of the incomplete Gamma Function

While looking through this ($178$,$(30)$) Table of Integral Transforms I have come across the Laplace Transform of the Incomplete Gamma Function which is given by $$\mathcal{L}\{\Gamma(\nu,at\}(p)~...
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1answer
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Kernel of Hankel Transform

I need to solve a cylindrical diffusion problem that is defined in $[1,\infty]$. I would like to use Hankel Transform that has is defined on $[0,\infty]$. So in order to apply Hankel transform in my ...
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Building an Integral Transform from an Orthonormal Basis on the L2 Circle

Background It is well known that the orthonormal basis for $L_{2}[-\pi, \pi]$ is $\Omega = \{ e^{-jmt} \}_{m \in \mathbb{Z}}$. We extend this to $L_{2}(\mathbb{R})$ via the Fourier Transform, which ...
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1answer
39 views

Which transform makes correllation a multiplication?

In Fourier analysis, a central theorem for the Fourier Transform states: $$\mathcal F\{(f*g)(t)\}(\omega)=\mathcal F \{f(t)\}(\omega)\cdot \mathcal F\{g(t)\}(\omega)$$ In other words, convolution ...
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1answer
72 views

Asymptotics of Hilbert transform from asymptotics of original function

Suppose we have a locally integrable function $f: \mathbb R^+ \to \mathbb R$ and we consider the 'Hilbert transformed' function $$ h(t) := \int_0^\infty \frac{f(\tau)}{t -\tau} \mathrm d \tau. $$ We ...
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51 views

Fix points of integral transfroms

After I have worked a little bit with different integral transform, especially with the Laplace transform, I was confronted with the fact that there are some functions which remain of the same type ...
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2answers
154 views

Mellin transform of a Gaussian Hypergeometric Function with negative $x$-argument

I am quite fascinated by the formula for the Mellin transform of the Gaussian Hypergeometric Function, which is given by: $$\mathcal M [_2F_1(\alpha,\beta;\gamma;-x)] = \frac {B(s,\alpha-s)B(s,\...
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Can an inverse Mellin transform converge conditionally?

Let $F(x)$ satisfy $$\begin{equation}\int\limits_0^\infty \vert F(x)\vert x^{c-1}dx<\infty\end{equation}\tag{1}$$ for some $c\in(a,b)$ with $a,b\in \mathbb{R}$ and $a<b$. Then, the Mellin ...
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Are there inverse Mellin transforms with two distinct strips of convergence?

Let $$F(x)=\frac{1}{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}f(s)x^{-s}ds$$ $$f(s)=\int\limits_{0}^{\infty}F(x)x^{s-1}dx$$ Real constant $c$ is from the strip $\Re(s)\in(a,b)$. $\textbf{...
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85 views

Division by t of an Inverse Laplace Transform

My problem is to find a function $g(s)$, solution of the equation: $$ \frac{1}{t} \mathcal{L}^{-1} \left\{ f(s)\right\}(t)=\mathcal{L}^{-1} \{ g(s)\}(t)$$ I know the general property: $$ \mathcal{L}^{...
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1answer
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Conditions for uniqueness of a Mellin transform

Let $f(x)$ and $F(s)$ be a Mellin pair, such that one is the Mellin inversion of the other in the fundamental strip $S_f$. Let $g(x)$ and $G(s)$ be a Mellin pair, such that one is the Mellin ...
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1answer
101 views

Laplace transform of a “heat kernel”

This question is closely releted to this question: How do we solve the laplace transform of the Heat Kernel? Let $A>0$ and $$f(t) = \frac{A^2}{2\sqrt{\pi}t^\frac{3}{2}}e^{-\frac{A^2}{4t}}$$ ...
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1answer
151 views

Solving the Heat Equation using the Fourier Transform

The Question: Solve the Heat Equation (for $u = u(x,t)$) $$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial x^2} \qquad u(x,0)=T(x)$$ by applying the Fourier Transform in the $x$ ...
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1answer
64 views

Holomorphic extension of the Laplace transform

Let $u \in L^1_{\mbox{loc}}((0,+\infty))$ be such that $e^{-\lambda t} u(t) \in L^1((0,+\infty))$ for some $\lambda >0$. Let $\mathcal{L}[u]$ be the Laplace transform of $u$, that is $$\mathcal{L}[...
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1answer
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Laplace convolution with the Bessel function

The Question: (i) Find the Laplace Transform of the Bessel Function $J_0(x)$ (ii) Hence, show that if $f(x)$ satisfies the differential equation $$f''(x)+f(x)=J_0(x) \qquad f(0)=f'(0)=0$$ then $f$ ...
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1answer
150 views

Solving Bessel Equation using Laplace Transform

The Question: Given that $J_0(x)$ satisfies $$x\frac{d^2J_0}{dx^2}+\frac{dJ_0}{dx}+xJ_0=0 \qquad J_0(0)=1 \qquad \frac{dJ_0}{dx}(0)=0$$ Show that the Laplace Transform $\bar{J_0}(p)$ of $J_0$ is ...
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Can integration contour of an inverse Mellin transform be deformed at will in fundamental strip?

Let $m(x)$ be inverse Mellin transform of $M(s)$: $m(x)=\frac{1}{2\pi i} \int\limits_{c-i\infty}^{c+i\infty}x^{-s}M(s)ds$ Mellin transform $M(s)$ is analytic on fundamental strip $a<\Re(s)<b$ ...
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1answer
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Fourier Transform of $\dfrac{1}{x^2+2x+2}$

The Question: (i) Determine the Fourier Transform of $$f(x) = \frac{1}{a^2+x^2} \qquad a>0$$ (ii) Hence determine the Fourier Transform of $$g(x) = \frac{1}{x^2+2x+2}$$ My Attempt: (i) I got $...
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Error function and its associated operation

Consider a manifold which may or may not be bounded, being the possibility space of functions for which all possible polynomial integrals exist. One could consider the integral transform as a change ...
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Fredholm integral equation of the 2nd kind with a weakly singular convolution kernel

I've reviewed the literature but could not find a solution of the integral equation $$\int_{0}^{1} dx\frac{\phi(x)}{|x-y|^{\alpha}}+a\phi(y)=b.$$ for the function $\phi$ where $\alpha\in(0,1)$ and ...
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What overarching term would include use of a Fourier transform, other similar integral transforms, and their discrete counterparts?

Question: What overarching terms could be used to refer to the use of a Fourier transform, Hankel transform, etc. as well as discrete versions of those, such as a Fast Fourier Transform (FFT), etc.? ...
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1answer
71 views

The tauberian final value Theorem

I have found many formulations of the FVT (final value theorem), but none of this with the precise assumptions that one needs. As far I understood, one needs to assume that the $\lim_{t\to+\infty}f(t)...
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80 views

When an Hilbert–Schmidt operator is invertible? How to construct the kernel of the inverse?

This question is motivated by the following example. I have the following integral operator defined on $L^2(\mathbb{T}^n)\to L^2(\mathbb{T}^n) $ $$ T f(x) = \sum_{k \in \mathbb{Z}^n} \sigma(k,x) \ \...
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63 views

Maximum/Minimum of a function defined as Laplace-Transform

Consider a function $f(t)$ that is only defined via its Laplace transform $F(s)$ $f(t) := \int\limits_{c - i\infty}^{c + i \infty} d s \; e^{st} F(s)$, where $c$ is chosen appropriately (i.e. to ...
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I want to disprove an equality involving a double integral

I want to show that the following equality does not hold: \begin{equation}\label{at3} \frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
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1answer
66 views

Solve ODE by integral transform

$$g’’(x) + f^2*g’(x) + 4 g(x) = 0$$$$ g(0) = 0$$$$ g’(\pi/2) = 0$$ where $x \in\mathbb{R}$, $f$ is frequency. If we use Laplace transform, the initial value of g'(pi/2) makes me cannot go further, ...
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39 views

Transform $\cosh(x)$ into $\exp(x^2/2)$

I'm looking for an integral transform that maps $\cosh(x)$ into $\exp(x^2/2)$. More specifically I'm looking for a kernel $K(y,x)$ and a domain $\Omega \subset \mathbb{R}$ such that $$ \int_\Omega dx ...
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84 views

two dimensional fourier transform of a bessel function multiplied by a second bessel function

$\frac{J_1 \bigl(\sqrt {x^2+y^2}\bigr)}{\sqrt {x^2+y^2}}$ and $\frac{J_1 \bigl(\sqrt {(x-a)^2+(y-b)^2}\bigr)}{\sqrt {(x-a)^2+(y-b)^2}}$ are two Bessel functions defined in the x-y coordinate plane, ...
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25 views

Master equation for a Multiplicative random process

I would like some assistance to derive a master equation for a Multiplicative random process. I suppose it should take the form of a Mellin convolution. It´s well known that for an additive random ...
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2answers
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Can the Laplace transform be inverted?

Consider the Laplace transform $\tilde{f}(s)$ of a function $f(t)$ defined as $$\tilde{f}(s)=\int_0^\infty f(t)e^{-st} \, dt.$$ Can this relation be inverted to obtain $f(t)$ in terms of $\tilde{f}(s)$...