# Questions tagged [integral-transforms]

Questions about transforms of functions by means of integration, such as Fourier transforms and Laplace transforms. Many integral transforms can be inverted by means of another integral transform.

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### Inverse Laplace transform not obtained

I can't seem to be able to compute the inverse Laplace transform of a Laplace transform: ...
247 views

### Implementation of the Riesz transform

Implementing the Hilbert transformation $$H(f)(x) = \frac{1}{\pi}\lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} \frac{1}{x-y}f(y) \, dy,$$ appropriate for 1-...
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### Fourier Transform of Absolute Value

If you ask Mathematica to provide the Fourier Transform of a singular functions it is likely to provide an answer that while nearly correct, is technically incorrect and it will do so without a word ...
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### Is InverseMellinTransform unaware of second Barnes lemma?

Consider evaluating ...
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### Integral Representation of DiracDelta

I would like to get dirac delta from integral definition, of course if I ask Integrate[Exp[I x t],{x,-Infinity,Infinity}] I get that the integral does not ...
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### How does Mathematica obtain this result?

FullSimplify[ Sqrt[2 π] InverseFourierTransform[1/(x^2 - a^2), x, p], Element[a, Reals]] Gives the output ...
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### Strange result with Laplace transform

The following code: f[x_] := HeavisideTheta[1 - x]/x InverseLaplaceTransform[ f[t], t, x]/x Returns 1/x. But this should not be ...
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### Convolution using the Laplace integral transform of certain functions

I am trying to convolve two functions: $f(t) = e^{- t}$ $g(t) = e^{-(e^{-t})^2}$ $(f*g)(t) = \int_{0}^{t} f(t-\tau)g(\tau) d\tau = \int_{0}^{t} e^{-(t-\tau)} e^{-(e^{-\tau})^2} d\tau$ Using the ...
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### Region of convergence in Inverse laplace transform

Lets say we have a function in the s domain. Let it be H(s)=1/(s+1). It has one pole. If we consider the region to the right of the pole as the ROC, we would one function in the time domain when we ...
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### Computation of a Fresnel Diffraction pattern with Discrete Hankel Transform

In the next link: Computation of Hankel Transform using FFT (Fourier) Rainer implemented a great solution given in the next reference: Manuel Guizar-Sicairos and Julio C. Gutiérrez-Vega, "Computation ...
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### Efficient powers of DPR1 matrices

I'm looking to compute the following quantity for $A$ where $A$ is a convergent positive definite $d\times d$ diagonal + rank1 matrix (DPR1). $$f(s)=\operatorname{Tr}(A^s)$$ Earlier answer answer by ...
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### Is this a bug in InverseLaplaceTransform or LaplaceTransform?

Let us consider in version 13.1 on Windows 10 ClearAll["Global`*"]; a = InverseLaplaceTransform[s*Log[(s - 1)/(s + 1)], s, x] ...
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### InverseLaplaceTransform gives inconsistent results

These should give the same results, but they do not: InverseLaplaceTransform[1/Sqrt[x] HeavisideTheta[x + 1], x, y] InverseLaplaceTransform[1/Sqrt[x], x, y] ...
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### Solving a finite Hilbert transform from 2D thin wing theory

I have a problem where I know $w(x) =1$ and want to find $u(x)$. It's a well-known problem I am recreating, from a 1972 paper. Note the paper doesn't solve the problem, the integral equation, it just ...
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### Verifying a cosine FourierTransform

On page 31 of this standard reference we have the following relation: I wanted to use mathematica to verify this transform numerically for some example values. So I type in: ...
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### LaplaceTransform doesn't work

I have code: A = 1; ω0 = 5*10^6; τ0 = 3*10^-3*Sqrt[2/Log[2]]/(5.85*10^3); s0[t_] = A*E^(-t^2/τ0^2)*Cos[ω0*t]; Phi[ω_] = LaplaceTransform[s0[t], t, ω] The last ...
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### Implementing the 3D Radon transform

I am wondering how to implement the Radon transform, the 3D Radon transform, that is, given a 'density' function $f: \mathbb{R}^3\to \mathbb{R}$ The Radon transform of $f$ is Rf(s,w)= \int_{x\cdot w=...
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### How to do variable change for a general Fourier series? I failed at first step :(

Well, firtly, I think I should define a general rule for Fourier series, but I fails. Below is my code including several attempts. Anys helps are greatly appreciated, thanks. ...
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### Solving coupled integro-differential equations using Laplace transform

I have two coupled integrodifferential equations, as shown in the image attached. I am trying to solve them using a Laplace transform, as follows ...
181 views

### How to plot this FourierTransform?

I need to find the Fourier transform and plot the function: Delta(x-xo) I've already tried to write it as: FourierTransform [DiracDelta[x - Subscript[x, 0]], x, w] ...