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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Replace subexpression with variable in result from Laplace transform

I'd like to clean up the result I obtained from an inverse Laplace transform: First of all, I'd like to replace the square root expressions in the hyperbolic function arguments (part encircled in ...
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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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It is not possible to build a 3D graph of the inverse Fourier transform [duplicate]

That's what I need to do. There is a function Φ[α,β,z] in which the replacement of k1 and k2 is used. Thus, an analytical representation of the integral characteristic of the solution of the problem ...
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3 votes
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115 views

FourierCosTransform bug?

FourierCosTransform[Cos[(k + p) z], z, q] gives correct result ...
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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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The inverse Laplace transform alters parameter constraints

I have this Laplace transform: $$\left( w \frac{L}{L+s}+(1-w) \frac{Q}{Q+s}\right)^n \ for \ L>0, Q>0,0<w<1.\ (1)$$ ...
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Wrong InverseLaplaceTransform? [closed]

Is this a bug? https://www.wolframalpha.com/input?i=InverseLaplaceTransform%5B1%2FSqrt%5Bt%5D%2C+t%2C+x%5D ...
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Inverse Laplace transform of two coupled equations in Mathematica [closed]

I want to solve these coupled equations in Mathematica for F1(s) and F2(s) and the inverse Laplace of each of them to find c1(t) and c2(t). the code I tried is: ...
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Inverse Fourier Transform and the sign of the Shifted Delta

I am using the sign convention of FourierParameters->{0, -2 Pi} for calculating the inverse FT of $Aexp(-2\iota\pi f K)$, where $A$, $K$ are real numbers >0 ...
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How do I make Mathematica to automatically evaluate divergent integrals using the Laplace transform?

For instance, I want Integrate[f[x],{x,0,Infinity}] to be automatically evaluated as ...
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Implementing the Shift Property of the Fourier Transform in Mathematica

I am trying to determine the Fourier tranform of a time shifted (t-b) Lorentzian function in Mathematica. With the zero centered Lorentzian (b=0), the Integrate function with the conditions that all ...
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How to calculate an `InverseMellinTransform` up to its definition in Mathematica?

I have a question about how to calculate Inverse Mellin Transformation's in Mathematica, one way or the other. Look at these findings. The following integral results in Gamma[s]. ...
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How do I make divergent integrals to be automatically evaluated?

As a first step I want the divergent integrals to be evaluated to their Laplace transform, like this: $\int_0^\infty \sin x dx\to 1$ because ...
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How to do that Fourier transform?

I want to perform Fourier Transform of $$\frac{\exp(jkr)}{r},$$ where $k=\frac{2 \pi}{\lambda}$ and $r=\sqrt{x^2+y^2+z^2}$. The result should be $\exp\left(jkz \sqrt{1-(\lambda u)^2 - (\lambda v)^2} \...
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How to let Mathematica return impulse or Dirac delta functions when computing integrals?

For example, let's say I want to compute the (continuous-time) Fourier transform of the signal/function $\cos{(3t)}$, which is given by the following improper integral: $\displaystyle\int_{-\infty}^{\...
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Are these 3 divergent integrals regularization methods equivalent?

I implemented in Mathematica 3 methods for regularizing divergent integrals, and wonder if they are equivalent. Code: ...
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What integrals can `FourierTransform` evaluate that `Integrate` cannot?

FourierTransform can make sense of integrals that diverge according to Integrate. ...
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1 answer
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Inverse Laplace Transform of function containing ArcTanh

I'm after the numerical inverse Laplace transform of a function, so I type f[t_?NumericQ] := InverseLaplaceTransform[1/(1 + s + ArcTanh[1/(s - 1)]), s, t]; This ...
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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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1 answer
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Convert complex exponential to real exponentials, sines and cosines

While taking the inverse Laplace transform of certain expressions, Mathematica yields complex exponentials. For example, using the following code: ...
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7 votes
1 answer
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LaplaceTransform works well with x[t], but doesn't recognize x[1][t], how to make it works for x[1][t]?

Bug introduced in 12.2(?), persisting through 13.0. When I am solving odes using LaplaceTransform, I found that LaplaceTransform works well with u[t] or y[t] or x[...
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5 votes
2 answers
142 views

How to guide Eliminate or GroebnerBasis to reduce a set of simple odes to a single ode (which can be done by Laplace Transform)

Recently, I am trying to use Eliminate or GroebnerBasis to simplify a system of ODEs. I don't want the solution of ODEs. What I ...
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4 votes
1 answer
61 views

How to accelerate numerial inverse laplace transform for pdes of Euler-Bernoulli beam problem

Happy new year! :) Recently, I am practicing laplace transform technique for solving pdes. In this extemely helpful post, @xzczd mentioned that "The last step is to transform the solution back, ...
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2 votes
1 answer
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Multidimensional numerical integral giving wrong answer

I am reducing a series of atomic transition amplitudes using Gaussian transforms (that allow one to combine all coordinate dependence into a single quadratic form so that one can complete the square ...
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Same integral yielding to different results

I am currently working with the following integrals \begin{equation} \int_{0}^{\infty} dk\thinspace \frac{k^{3}e^{-2kd}}{\omega^{2}+k^{4}} = \frac{1}{\omega^{2}d^{4}}\int_{0}^{\infty}d\epsilon\...
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Is there a general unifying formula for the Laplace transform of a general weibull density?

It seems that the Laplace transform of weibull density has different formulas for rational shape parameters, depending on the degree of the numerator/denominator https://idp.springer.com/authorize/...
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1 answer
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Want to realize this operation (multiplication of divergent integrals of polynomials) in Mathematica [closed]

I am currently researching divergent integrals. Definition. An extended number is an expression of the form $\int_a^b f(x)dx$, where function $f(x)$ is defined almost everywhere at $(a,b)$. Generally ...
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6 votes
2 answers
246 views

Inverting integral transform $f(s)=\int_0^\infty g(x) \exp(-s g(x)) \mathbb{d}x$

Suppose our function $f$ is defined in terms of $g$ as follows. $$f(s)=\int_0^\infty g(x) \exp(-s g(x)) \mathbb{d}x.$$ Are there tools in Mathematica that could let me obtain $g$ given knowledge of $...
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1 vote
1 answer
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I ask for help with commands TransferFunctionModel + StateSpaceModel

Given system of ODE: $\begin{cases} \dot{x}=G+u_1 \\ \dot{z}=-z+\frac{df}{dt} \\ \dot{G}=-G+z \cdot u_2 \end{cases}$ where $f=-x^2$, $u_1=\frac{d}{dt}(\alpha \sin(\omega \cdot t))$ and $u_2=\alpha \...
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4 votes
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Obtain $\{h_1,h_2,\ldots\}$ from $\{f(0),f(1),f(2),\ldots\}$ with $f(s)=\sum_{i=1}^n \exp(-h_i s)h_i$

I need to obtain $\{h_1,h_2,\ldots\}$ when given $\{f(0),f(1),f(2),\ldots\}$ with $f$ defined as follows $$f(s)=\sum_{i=1}^n \exp(-h_i s)h_i$$ We know that $h_i>0$. Is there a way to do this ...
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2 votes
2 answers
66 views

Removing transient part in equation and solving for a general form

I have the following code that I run: ...
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1 vote
0 answers
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Closed form for integral with SinIntegral

MMA can't solve this integral: $$\int_0^{\pi } \text{sinc}(x) \text{Si}(2 x) \, dx$$ The I use FourierSinTransform to find closed form,but I try a verify result is ...
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1 answer
120 views

Strange result of LaplaceTransform

I mean LaplaceTransform[BesselI[3, x], x, s] // Simplify ...
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1 vote
2 answers
87 views

Function decomposition to Fourier series using first impulse function

I have periodic function with certain first impulse function and period value. My task is to decompose it to approximate Fourier series (k_max = 10) using Laplace transform of first impulse function: ...
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2 answers
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Solve was unable to prove that the solution set found is complete

I have such a code: ...
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1 answer
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Differences in Mathematica's behaviour with identical functions

I have three functions whose identity was verified, but WM doesn't behave itself equally with them. The code is: ...
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1 vote
1 answer
83 views

Numerical Laplace Transform using Fourier transform and plotting results [closed]

I would like to take a numerical transform of interpolating functions which is a solution of at least two coupled differential equations. But for this question, I am simplifying the solution to make ...
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3 votes
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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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1 answer
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Derivative of -I (Log[-x] - Log[x]) - why is it zero? [closed]

Mathematica gives the derivative of the function -I (Log[-x] - Log[x]) as $0$, but on the real domain the expected result is $\pi\delta(x)$ and on complex domain it ...
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1 vote
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Performing operator expression on a function

What I want. I write an operator expression, and Mathematica performs it on a function. Examples: E^(b D): f[x] -> f[x+b] ...
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3 votes
0 answers
135 views

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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0 answers
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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 ...
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0 answers
69 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] ...
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0 answers
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Possible bug in InverseZTransform

Try x1n=InverseZTransform[1/(1-(1/z))//Factor,z,n] x2n=InverseZTransform[1/(1-(1/z)),z,n] DiscretePlot[x1n,{n,-10,10}] DiscretePlot[x2n, {n,-10,10}] The result ...
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Need help to understand the code

Here is the code by P. Valko and J. Abate for finding numerical inverse Laplace transform by trapezoidal's rule. Can someone willing to explain this code? I don't understand this part: ...
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How to solve this double integral analytically or numerically?

Can someone help me to solve this integral or explain me how to set it on mathematica? \begin{equation} \int_{-\infty}^{+\infty} dk_\alpha dk_\phi \frac{1}{2\pi} e^{-\frac{k_\alpha^2}{2}} e^{-\...
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3 votes
4 answers
309 views

Numerical Inverse Z Transform?

Is there a way to compute the inverse z-transform in Mathematica numerically? I'm trying to compute the following... ...
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1 vote
0 answers
104 views

Fourier transform in polar coordinates using built-in hankel transform of the function constant 1 [closed]

Like in the table of transforms https://en.wikipedia.org/wiki/Fourier_transform#Distributions,_one-dimensional the FT (Fourier transform) of $\delta$ is 1 and the FT of 1 is $\delta$, but in polar ...
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8 votes
5 answers
2k views

3D Fourier transform of 1/r^2

How can I compute the Fourier Transform of $f(r)=1/r^2$, where $r=\sqrt{x^2+y^2+z^2}$?
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1 vote
3 answers
113 views

How to calculate an iterated derivative in Mathematica?

I try to calculate an inverse Mellin transform for $s^n \Gamma(s)$: $$x\frac{\mathrm{d}}{\mathrm{d}x}\left(x\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}\right)\right)$$ for $n=2$ for $n$ as $$\left(x\...
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