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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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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 ...
0 votes
0 answers
172 views

How can I solve $\sum_{i=1}^{M-1} (M+i)^{M+i+1/2}/i^{i+1/2}$? [migrated]

I am trying to solve an equation in Mathematica: $$ \sum_{i=1}^{M-1} \frac{(M+i)^{M+i+\frac{1}{2}}}{i^{i+\frac{1}{2}}} $$ Does a general solution exist for this expression? And if $M \to \infty$, can ...
3 votes
2 answers
113 views

InverseLaplaceTransform returns the input

Initially, I was trying to invert the following expression: $$ \frac{e^{-a\sqrt s}}{s-c} $$ and got the following result: ...
1 vote
1 answer
115 views

Partial integro-differential equation

I want to solve the following partial integro-differential equation for $\delta(x,t)$: $$ 1-B \cdot \frac{\partial \delta(x, t)}{\partial t}=\frac{1}{\pi} \int_{-1}^1 \frac{\partial \delta(s, t)}{\...
1 vote
1 answer
99 views

Question about numerical integration of a double integral

I am trying to numerically integrate the following double integral in Mathematica for different values of t. ...
0 votes
1 answer
72 views

On the Laplace transform of Beta function

While trying to evaluate the Laplace transform below $$I = \int_{0}^{\infty}e^{-st}B(\frac{1}{2}-it,\frac{3}{2}+it)\mathrm{d}t,$$ invoking ...
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[...
3 votes
3 answers
188 views

Wolfram Alpha and Wolfram Engine produce different Laplace transforms

Consider the following Laplace transform: $$\mathcal{L}\{\dfrac{\sin{2x}}{x}\}$$ To calculate it, I'd write LaplaceTransform[Sin[2 x] / x, x, p] into both Wolfram ...
3 votes
1 answer
78 views

Solving Laplace Transform IVP [closed]

I want to solve the IVP Laplace transform for the following: $$y''-2y'+y=3e^t$$ with $$y(0)=1, \; y'(0)=1.$$ How would I input this in Wolfram Language? I've tried a bunch of different things and ...
1 vote
1 answer
73 views

Is there an option for InverseLaplaceTransform to make Mathematica use the convolution theorem when feasible?

By default, it appears that Mathematica won't use the convolution theorem to write an inverse Laplace transform in the form of a convolution of two functions. For example, ...
4 votes
1 answer
139 views

Strange result simplifying higher order BesselJ [duplicate]

Consider the following integral: $$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$. The solution of this should contain a single Bessel function: $$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
1 vote
0 answers
81 views

How can I write the code for an inverse integral transform?

How can I code any type of integral transform with its inverse integral transform in Mathematica which gives the correct results and properties? Below is the example for Elzaki transform: ...
0 votes
2 answers
145 views

How can I eliminate InverseLaplaceTransform returning complex

I tried this: F = InverseLaplaceTransform[ 1/((1 + s^2) (1 + (s + 1/2)^2)) + Exp[-Pi*s]*1/((1 + s^2) (1 + (s + 1/2)^2)), s, t] But it returns an answer ...
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 ...
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 ...
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 ...
1 vote
1 answer
129 views

How to deduce analytical solution from numerical solution

I came across one such integral in my calculations, for which there is no analytical solution. But it exists numerical solution, so how can I derive analytical solution of this integral from numerical ...
0 votes
1 answer
111 views

Taking the inverse Laplace Transform as a vector operation

I am solving a system of first-order equations using matrix operations and the Laplace Transform. I begin with the matrix equation that represents the solution to my system, like this: $$ \underline{\...
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 ...
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 ...
1 vote
2 answers
63 views

InverseFourierSinTransform in mathematika failed to give a result

I have to use InverseFourierSinTransForm in Mathematika for the function u[ω,t] but infortunately it does not work.It gives back the same! I tried it without the assumptions but it does not work again!...
2 votes
1 answer
122 views

Derive Parseval's theorem in one dimension

Parseval's theorem (in one dimension) is a fundamental result in the theory of Fourier transforms. If $f(t) \Leftrightarrow F(\omega )$ are Fourier transform pairs and $t$ (time) and $\omega$ (...
1 vote
1 answer
111 views

Evaluating Fourier transform in mathematica [closed]

I am trying to evaluate the expression FourierTransform[[m (a^2 - t^2 - I g t)]^-1, t, ω] in Mathematica. It gives me the error message that "Syntax: "[m(...
0 votes
1 answer
73 views

Handling singularities like (x-y) in the denominator while evaluating double integrals

I have to solve an Integral of the following type ...
2 votes
0 answers
62 views

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 ...
0 votes
1 answer
53 views

Improving quality of plot with bad numeric performance

I have a plot which suffers from poor numeric accuracy, any tips on how I can improve the quality of this plot? ...
1 vote
0 answers
61 views

Inverting Laplace transform numerically for a set of points

Computing inverse of Laplace transform numerically in Mathematica seems very slow and requires $k$ calls for $k$ points. Is there a faster way to do it for a set of $k$ points, ie $k\approx 1000$? <...
1 vote
0 answers
74 views

Incorrect result of InverseFourierTransform

When executing in 13.2 on Windows 10, the command FourierTransform[DiracDelta@Cos[x], x, s] results in ...
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 ...
3 votes
1 answer
148 views

FourierCosTransform bug?

Bug introduced in 13.0 or earlier and persisting through 13.2.0 or later. FourierCosTransform[Cos[(k + p) z], z, q] gives correct result ...
1 vote
0 answers
45 views

What is the basis of the Fourier transform in Mathematica? [duplicate]

I did this: ...
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]]) ...
0 votes
0 answers
54 views

Hilbert Transform in mathematica [duplicate]

I would like to do a Hilbert transformation in Mathematica on a function. However, it does not seem to give a right result. The Hilbert transform is given by So I did : ...
2 votes
1 answer
717 views

How to pass arguments in NIntegrate and NDSolve

I am solving PDEs with NDSolve and NIntegrate, but I do not know how to pass arguments correctly. My orignial code is very ...
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)...
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] ...
0 votes
0 answers
94 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 ...
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. ...
2 votes
0 answers
60 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] ...
4 votes
1 answer
219 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 ...
0 votes
1 answer
142 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 ...
2 votes
1 answer
45 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
184 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
172 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}...
1 vote
0 answers
34 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 ...
0 votes
0 answers
136 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=...
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 $r=\sqrt{x^2+y^2+z^2}$?
5 votes
1 answer
887 views

How to set up a spherically symmetric Fourier transform?

I am having trouble figuring out how to set up the Fourier transform to: ...
1 vote
1 answer
174 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: ...
1 vote
1 answer
421 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}^{\...

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