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.

Filter by
Sorted by
Tagged with
6 votes
1 answer
561 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)...
8 votes
1 answer
272 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] ...
  • 19.5k
0 votes
0 answers
54 views

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 ...
2 votes
0 answers
54 views

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] ...
  • 19.5k
4 votes
1 answer
127 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 ...
2 votes
1 answer
39 views

Validation of the Laplace inversion and storage in a table or an array

I tried to validate this function from 0 to 50 but it takes very long time, is there a faster way to validate this function for t from 0 to 50 and add them in a list? ...
1 vote
0 answers
112 views

Inverse Laplace in mathematica

I can't even get an inverse Laplace for this expression numerically in mathematica, is there a way to inverse this equation below? I have also tried to use fixt talbot package for a numerical ...
0 votes
2 answers
164 views

Inverse Triple Laplace Transform of $\frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}}$

I want to find the inverse triple Laplace transform of $L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right]$. I did \begin{align*} L^{-1}_{x_{3}} L^{-1}...
0 votes
1 answer
96 views

Fourier Transform of Integral Expression

I am trying to Fourier transform an expression containing an integral like this: FourierTransform[Integrate[f[v]*Cos[w[v]*t],{v,-v_0,v_0}],t,k] where ...
  • 1
1 vote
0 answers
27 views

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 ...
  • 11
0 votes
0 answers
84 views

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=...
0 votes
0 answers
52 views

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 ...
  • 1
3 votes
1 answer
120 views

FourierCosTransform bug?

FourierCosTransform[Cos[(k + p) z], z, q] gives correct result ...
2 votes
0 answers
75 views

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] ...
  • 3,267
0 votes
1 answer
76 views

The inverse Laplace transform alters parameter constraints

I have this Laplace transform: $$\left( w \frac{L}{L+s}+(1-w) \frac{Q}{Q+s}\right)^n \ for \ L>0, Q>0,0<w<1.\ (1)$$ ...
1 vote
0 answers
78 views

Wrong InverseLaplaceTransform? [closed]

Is this a bug? https://www.wolframalpha.com/input?i=InverseLaplaceTransform%5B1%2FSqrt%5Bt%5D%2C+t%2C+x%5D ...
  • 3,267
1 vote
1 answer
77 views

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: ...
1 vote
1 answer
69 views

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 ...
  • 121
0 votes
0 answers
37 views

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 ...
  • 3,267
1 vote
0 answers
60 views

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 ...
  • 121
0 votes
0 answers
51 views

Search for fitting parameters for an inverse Laplace function

I am trying to find a suitable parameter for this function, which I got using the Talbot Inverse Laplace method. But since I'm really not a pro in Mathematica, I don't know why I'm getting the ...
2 votes
1 answer
88 views

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]. ...
0 votes
0 answers
43 views

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 ...
  • 3,267
1 vote
1 answer
308 views

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} \...
  • 19
1 vote
1 answer
232 views

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}^{\...
  • 453
0 votes
0 answers
45 views

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: ...
  • 3,267
5 votes
1 answer
280 views

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

FourierTransform can make sense of integrals that diverge according to Integrate. ...
  • 13.6k
3 votes
1 answer
105 views

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 ...
1 vote
0 answers
41 views

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. ...
1 vote
1 answer
82 views

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: ...
  • 453
8 votes
1 answer
136 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 13.1. When I am solving odes using LaplaceTransform, I found that LaplaceTransform works well with u[t] or y[t] or x[...
5 votes
2 answers
144 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 ...
4 votes
1 answer
88 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, ...
2 votes
1 answer
58 views

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 ...
  • 61
0 votes
2 answers
76 views

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\...
  • 551
1 vote
1 answer
30 views

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 ...
  • 3,267
6 votes
2 answers
255 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 $...
1 vote
1 answer
68 views

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 \...
  • 2,112
4 votes
2 answers
126 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 ...
2 votes
2 answers
69 views

Removing transient part in equation and solving for a general form

I have the following code that I run: ...
  • 1,797
1 vote
0 answers
59 views

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 ...
0 votes
1 answer
123 views

Strange result of LaplaceTransform

I mean LaplaceTransform[BesselI[3, x], x, s] // Simplify ...
  • 19.5k
1 vote
2 answers
97 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: ...
0 votes
2 answers
77 views

Solve was unable to prove that the solution set found is complete

I have such a code: ...
0 votes
1 answer
90 views

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: ...
1 vote
1 answer
96 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 ...
  • 873
3 votes
0 answers
73 views

How does Mathematica obtain this result?

FullSimplify[ Sqrt[2 π] InverseFourierTransform[1/(x^2 - a^2), x, p], Element[a, Reals]] Gives the output ...
0 votes
1 answer
125 views

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 ...
  • 3,267
1 vote
0 answers
43 views

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] ...
  • 3,267
3 votes
0 answers
138 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 ...
  • 3,267

1
2 3 4 5