# 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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### The paired domains of integral transforms. Is there a moniker?

Use the Fourier transform example. For several years I thought of Fourier transform pairs as functions in two domains which were paired by a term "conjugate domains" and significantly the dimensions ...
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### Hankel Transform Property/Inverse For a Function

I don't know much about Hankel transform properties, but I do know it is its own inverse. However, I would like to know that if the arguments were swapped, i.e.on the left hand side, $r\to k$, would ...
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### Unitary change of variables in real integral

I am interested in the solution of the folliwng multivariate Gaussian integral: \begin{equation} I=\int_{\mathbb R^N} \mathrm d x\; e^{-\frac 12 x^T\Omega x} \end{equation} where $\Omega$ is a complex ...
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### Using orthogonal transform in set of SDE's

If I have a set of SDE's in Ito form \begin{align} d\phi(z) &= \sin(\psi(z))dW_1(z)+\cos(\psi(z))d\tilde{W}_1(z) \\ d\psi(z) &= -\bigg(\sin(\psi(z))dW_1(z)+\cos(\psi(z))d\tilde{W}_1(z)\bigg) ...
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### Laplace Transform Integral

$$\mathscr{L}\left ( \int_0^t e^{t-\tau}\cos(t-\tau)e^{-\tau}d\tau \right )$$ Hi all I am trying to solve this Laplace transform but I dont know if it's correct, please tell me. Here is my attempt:...
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### Solution of Hilbert type Fredholm integral equation

Is there a formula similar to Hilbert inverse relation that would solve following integral equation in general form for $\phi$ as an integral? $$f(x) = \int_1^\infty \frac{\phi(y)dy}{x-y}$$ Provided ...
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### Difficult numerical Integral with Besselfunctions: transformation of variables?

For a physics problem that I'm trying to study I would like to exand an eigenproblem in the eigenfunctions of the laplacian over a unit disk with neuman boundry conditions. To do this I need to ...
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### Inverse Laplace transform of $s^c/log(s)$

I'm trying to solve for the inverse Laplace transform of $\frac{s^c}{\log(s)}$ where $c$ is some constant. Mathematica is apparently unable to solve it, and while I know there's a running joke that ...
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### Is there a broader definition of the convolution operation?

The convolution operator is defined as $$(f*g)(t)\triangleq \ \int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .}$$ where it shares a relationship with the Laplace transform such ...
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### Jacobi function

Im reading a material for discrete and contionus Jacobi transform by E.Y. Deeba and E.L Koh. I can't understand a proof of lemma 2.2. To be more precise i can't understand how did they bound absolute ...
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### Change of argument in Laplace Transform

The Laplace transform of a function $f=f(x)$ has the following definition: $$\mathcal{L}({f(x)})=\int^{\infty}_0e^{-sx}f(x)dx \tag{1}$$ However, when $f=f(ax-c)$, where $a$ and $c$ are arbitrary ...
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### about the solution method of a non-homogeneous heat equation

Solve the following problem $$\frac{\partial^2 u}{\partial x^2}=\frac{\partial u}{\partial t}-2x$$ subject to $u(0,t)=0$, $u(1,t)=0$ and $u(x,0)=x(1-x)$ where $0<x<1$ and $t>0$. I have two ...
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### How do summations/integrals like Fourier, Laplace, z-transforms preserve all the information about the original signal?

In normal summations, like 2+3=5, the information about the original numbers is lost. But in infinite summations like integral transforms, no information is lost and the function can still be ...
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### Looking for a rigorous treatment about Laplace transform.

I am looking for a book that deal with integral transforms in a very rigorous way, Fourier analysis I discover that functional analysis and Lebesgue Integral books cover the Fourier analysis( With a ...
So what I'd like to achieve is the following: suppose we have a time dependent signal $f(t)$ - e.g. the wave of a song. This song contains a drumbeat that (for the sake of simplicity) has a single ...