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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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31 votes
4 answers
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Implementing discrete and continuous Hilbert transforms

What is an efficient and accurate Mathematica implementation of the Hilbert transform, for both continuous and especially discretely sampled functions? This transform relates phase and amplitude in ...
Mr.Wizard's user avatar
  • 272k
23 votes
1 answer
1k views

Numerical inverse Laplace-Hankel transform

When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the following double integral (to be more precise, the 0-order inverse Hankel-Laplace ...
xzczd's user avatar
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18 votes
1 answer
1k views

Package for fast spherical harmonic transform in Mathematica?

Mathematica's implementation of the Fast Fourier Transform is, naturally, much faster than computing the discrete transform yourself using Sum. The analog of the ...
Jess Riedel's user avatar
  • 1,525
17 votes
5 answers
2k views

Laplace transform of $\frac{1-\cos (t)}{t}$

In the documentation, it states that The Laplace transform of a function $f(t)$ is defined to be $\int_0^{\infty } f(t) e^{-s t} \, \mathrm{d}t$. But why can Mathematica not get the Laplace ...
xslittlegrass's user avatar
17 votes
1 answer
7k views

Computation of Hankel Transform using FFT (Fourier)

To address circularly symmetric cases of 2-D Fourier Transformations, the so-called Hankel Transform can be applied (for a detailed derivation of the relation between the 2-D Fourier transform and the ...
Rainer's user avatar
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15 votes
1 answer
742 views

Workarounds for a possible bug in the linearity of FourierTransform

This is somewhat similar to this question, except the problem I am encountering is to do with Fourier transforms of scalar multiples of functions and their derivatives. I wish to input ...
anon1802's user avatar
  • 251
15 votes
1 answer
406 views

Possible bug with FourierTransform linearity

Each component is easily transformed but the sum is not: FourierTransform[f''[x], x, k] FourierTransform[f'[x], x, k] ...
Kuba's user avatar
  • 137k
14 votes
2 answers
8k views

Numerically obtaining the inverse Laplace transform of data

I have been using several Mathematica packages to do numerical inverse Laplace transforms on known (expressible in closed form) expressions, $\tilde{f}(s)$. I am now being confronted with the more ...
BeauGeste's user avatar
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13 votes
1 answer
4k views

How can I invert a Laplace transform numerically?

I have a very complicated expression, which I want to transform using the inverse Laplace transform. The built-in function InverseLaplaceTransform doesn't work. So,...
Mary's user avatar
  • 369
12 votes
2 answers
745 views

InverseRadon behaves differently from iradon of MATLAB

I have to calculate a 2-dimensional radially symmetric distribution from a single projection. I know that InverseRadon should actually do the job, but I get the ...
Quit007's user avatar
  • 1,255
12 votes
1 answer
1k views

Implement finite Fourier transforms

Recently I came across finite Fourier transforms, which can be used for solving certain type of boundary value problem (BVP) of linear partial differential equation (PDE) with constant coefficient. ...
xzczd's user avatar
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11 votes
2 answers
2k views

Compute inverse Laplace transform with Integrate

Since inverse Laplace transform is just an integral, while we already have InverseLaplaceTransform built in, can we compute it with ...
xzczd's user avatar
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9 votes
1 answer
182 views

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 14.0 or later. When I am solving odes using LaplaceTransform, I found that LaplaceTransform works well with u[t] or y[...
xinxin guo's user avatar
  • 1,401
9 votes
1 answer
917 views

Integrating over Bessel Function erroneous? (Hankel Transform)

Bug introduced in 8.0.4 or earlier and persists through 11.0.1 or later The Hankel Transform is given by Integrate[f[x] x BesselJ[0, x t], {x, 0, Infinity}] It ...
kram1032's user avatar
  • 335
8 votes
5 answers
4k views

3D Fourier transform of 1/r^2

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

Inverse Laplace Transform of Hypergeometric function

Any tips how to massage the following to get computed by Mathematica for $p>1$? I suspect the result should be expressible in terms of exponential integral ...
Yaroslav Bulatov's user avatar
8 votes
2 answers
2k views

Inverse Laplace transform

Let $r=\mu = 0.15; \sigma = 0.05; T = 1; S_0 = 100; K = 95;$ Let $\nu:=\frac{2\mu}{\sigma^2}-1$ and $\eta \equiv\eta(\alpha):=-\frac{\nu}{2}+\frac{1}{2}\sqrt{\nu^2+\frac{8\alpha}{\sigma^2}}$. ...
KNN's user avatar
  • 205
8 votes
1 answer
856 views

Correct way of simplifying the result of an integral

Many times Mathematica gives enormous results to simple problems. One uses the program more for trouble than for not knowing how to solve the problem. As an ejemplo I present this integral that ...
zeros's user avatar
  • 2,263
8 votes
2 answers
374 views

Why does InverseLaplaceTranform give an incorrect solution?

Bug introduced in 9.0.1, fixed in 11.1. I was trying to answer this question, but I ended up asking this. Method 1 A workaround is to differentiate the ODE once, ...
zhk's user avatar
  • 12k
8 votes
1 answer
325 views

Still bug in Integrate. 3

Let us consider in version 13.1 on Windows 10 r = Integrate[1/(x - a)/Sqrt[1 - x^2], {x, -1, 1}, Assumptions -> a \[Element] Reals] ...
user64494's user avatar
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8 votes
0 answers
1k views

Inverse Laplace transform not obtained

I can't seem to be able to compute the inverse Laplace transform of a Laplace transform: ...
highBandWidth's user avatar
7 votes
3 answers
4k views

Integral of function involving Dirac delta

I am performing lots of simple calculations with dirac delta functions. It would be awesome if Mathematica could do this routine exercise for me, eliminating any possible human errors. For example, ...
Prof. Legolasov's user avatar
6 votes
3 answers
917 views

Mathematica and MATLAB giving different results from inverse Laplace transform [closed]

Given the transfer function gs = (5*s + 4)/(s^4 + 4*s^3 + 2*s^2 + 3*s); and the input signal ut = Sin[t + π/6; I want to get ...
dcydhb's user avatar
  • 615
6 votes
2 answers
2k views

Bilateral Laplace Transform

I am trying to solve a linear ODE for a Kolmogorov forward equation to get a stationary distribution of a random variable. The easiest approach may be to transform the ODE with a two-sided Laplace ...
jlperla's user avatar
  • 977
6 votes
2 answers
1k views

Calculate the Bromwich Integral (Inverse Laplace Transform)

How can I calculate the Bromwich Integral in Mathematica? If I enter this as code it gives me just the same: $$\frac{1}{2\pi i}\int_{\alpha-\infty i}^{\alpha+\infty i} \left(e^{st}\cdot F_{(s)}\right)...
Jan Eerland's user avatar
  • 2,001
6 votes
1 answer
620 views

Wrong result of Laplace Transformation [closed]

I am trying to calculate the Laplace Transformation of the following function: $$f(x) = \theta(t+1)-\theta(t-1)$$ where $\theta(t)$ is the Heaviside step function defined as: $${\displaystyle \theta(x)...
Konstantinos Zafeiris's user avatar
6 votes
2 answers
315 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 $...
Yaroslav Bulatov's user avatar
6 votes
1 answer
204 views

InverseMellinTransform producing two different results for the same input?

Consider the expression expr = Gamma[1 + s]/Gamma[1 - s] Gamma[-s]^2; and a slightly simplified version of the same ...
Kagaratsch's user avatar
6 votes
1 answer
419 views

What integrals can `FourierTransform` evaluate that `Integrate` cannot?

FourierTransform can make sense of integrals that diverge according to Integrate. ...
John Doty's user avatar
  • 13.8k
6 votes
0 answers
247 views

Implementation of the Riesz transform

Implementing the Hilbert transformation \begin{equation}H(f)(x) = \frac{1}{\pi}\lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} \frac{1}{x-y}f(y) \, dy,\end{equation} appropriate for 1-...
confused's user avatar
5 votes
1 answer
886 views

How to set up a spherically symmetric Fourier transform?

I am having trouble figuring out how to set up the Fourier transform to: ...
aluuzz's user avatar
  • 67
5 votes
1 answer
617 views

Solving partial differential equation involving Hilbert transform

While solving one research paper published in Physical Review Letters, I came across the following equation and I am unable to solve it. $$\frac{\partial f}{\partial t}−(\mathcal{H}(f)\left(\frac{\...
Mohan Aditya Sabbineni's user avatar
5 votes
2 answers
672 views

Analytic expression for a Complex Hilbert Transform in Mathematica

I need to solve the following integral equations for a problem I'm working on - $\displaystyle \frac{-i}{2 \pi}$ $\int_{-a}^{a} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$ and $\displaystyle \...
Aegon's user avatar
  • 769
5 votes
1 answer
756 views

Solving PDE involving Hilbert transform numerically

I'm trying to solve the following equation numerically: $$u_{tt}-\mathcal{H}(u_x)=A^2_{xx},$$ where $\mathcal{H}$ is the Hilbert transform and $A$ is a prescribed forcing function which we assume ...
Nick P's user avatar
  • 349
5 votes
1 answer
100 views

InverseFourierSinTransform on Mathematica did not give a result

I am trying to solve the linear Schrödinger equation with the Fourier transform. I have difficulty making the corresponding graph when I solve the problem numerically. Can you explain to me what is my ...
Athanasios Paraskevopoulos's user avatar
5 votes
2 answers
152 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 ...
xinxin guo's user avatar
  • 1,401
5 votes
1 answer
107 views

NIntegrate with highly oscillatory Bessel and hypergeometric integrands

I am trying to compute the following double integral int2[n_] := NIntegrate[ n^2 * u * BesselJ[0, u]^n * r^2 * BesselJ[0, n*r*u], {r, 0, 1}, {u, 0, Infinity}] for ...
epsilone's user avatar
  • 153
5 votes
1 answer
216 views

Incorrect result of FourierTransform

Let us consider in 13.2 on Windows 10 FourierTransform[1/Sinh[x]^2, x, k] -((2 + k \[Pi] Coth[(k \[Pi])/2])/Sqrt[2 \[Pi]]) ...
user64494's user avatar
  • 26.7k
5 votes
1 answer
165 views

FourierParameters causing function not to simplify

When evaluating this expression, Mathematica simplifies the result: FourierTransform[FourierTransform[u[t], t, s], s, t] This simplifies to ...
AccidentalTaylorExpansion's user avatar
5 votes
1 answer
445 views

Fourier transform of Exp[x]/x

Could you please explain why Mathematica gives the following expression when taking Fourier transform of $\exp(\lambda z)/\lambda$? $$\frac{-\log(-z)+\log(z)}{\sqrt{2\pi}}$$ Why the answer does not ...
Ivan Gankevich's user avatar
5 votes
1 answer
14k views

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 ...
JEP's user avatar
  • 1,336
4 votes
3 answers
317 views

Numerical Laplace transform error

I'm running Mathematica 11.3.0.0. The inverse Laplace transform of $1/(s^2-1)$, evaluated at $t=1$ gives ...
Carlo Beenakker's user avatar
4 votes
2 answers
1k views

Is it possible to find the transfer function of these three differential equations using Mathematica

Imagine you are attempting to obtain a model of a device with two inputs [u1, u2] and three outputs [O1, O2, x] from the Lagrangian. After some of Lagrangian work, you find the following three ...
Mike Meyers's user avatar
4 votes
2 answers
136 views

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 ...
Yaroslav Bulatov's user avatar
4 votes
1 answer
315 views

Fourier transform question

Consider the fourier transform FourierTransform[1/((Cosh[x] + 1) (Cosh[x]^2 - 1)^(1/2)), x, w] If I execute the above line, Mathematica thinks for several ...
Kagaratsch's user avatar
4 votes
1 answer
218 views

Inverse Laplace Algorithm used in Mathematica [closed]

I have a general question. I want to know what algorithm (The name of this numerical method) is already used to calculate a numerical Inverse Laplace in Mathematica. I used the function ...
Ali AlCapone's user avatar
4 votes
1 answer
126 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, ...
xinxin guo's user avatar
  • 1,401
4 votes
2 answers
121 views

LaplaceTransfrom with time shifting

Hi I'm wondering if there's some workaround to get Mathematica to use time-shifting identities for Laplace and Inverse Laplace transforms. My examples are below. ...
user2757771's user avatar
4 votes
1 answer
445 views

How to speed up the integral in NDSolve?

On the MMA.SE, there have been several question on how to solve equations including integral using NDSolve, see example 1, example 2, and example 3. Many people, ...
user55777's user avatar
  • 671
4 votes
1 answer
113 views

Transform expression involving Erfc back and forth with Laplace transform and its inversion

It's a problem I encountered when answering this question and I think it's worth starting a new question for it. Consider the following expressions: ...
xzczd's user avatar
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