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.
219
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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, ...
- 433
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1
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Strange result simplifying higher order BesselJ
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....
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0
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72
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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:
...
5
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1
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94
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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
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1
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117
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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 ...
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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.
...
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1
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86
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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{\...
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3
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456
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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
...
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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 ...
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1
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2
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63
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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!...
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2
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117
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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$ (...
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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(...
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1
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68
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Handling singularities like (x-y) in the denominator while evaluating double integrals
I have to solve an Integral of the following type
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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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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?
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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 $k\approx 1000$?
<...
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Incorrect result of InverseFourierTransform
When executing in 13.2 on Windows 10, the command
FourierTransform[DiracDelta@Cos[x], x, s]
results in
...
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Compute symbolic identities for FourierTransform
Mathematica computes some of the identities for Fourier transforms for an arbitrary function f[t] directly, for example:
FourierTransform[f'[t], t, w]
...
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45
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0
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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 :
...
- 561
5
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1
answer
207
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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]])
...
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6
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1
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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
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1
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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]
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0
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80
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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 ...
2
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0
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59
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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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4
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1
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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 ...
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2
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1
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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?
...
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153
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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 ...
- 141
0
votes
2
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170
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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}...
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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
...
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0
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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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0
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0
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114
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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 $$Rf(s,w)= \int_{x\cdot w=...
- 109
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144
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FourierCosTransform bug?
Bug introduced in 13.0 or earlier and persisting through 13.2.0 or later.
FourierCosTransform[Cos[(k + p) z], z, q]
gives correct result
...
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InverseLaplaceTransform gives inconsistent results
These should give the same results, but they do not:
InverseLaplaceTransform[1/Sqrt[x] HeavisideTheta[x + 1], x, y]
InverseLaplaceTransform[1/Sqrt[x], x, y]
...
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1
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The inverse Laplace transform alters parameter constraints
I have this Laplace transform:
$$\left( w \frac{L}{L+s}+(1-w) \frac{Q}{Q+s}\right)^n \ for \ L>0, Q>0,0<w<1.\ (1)$$
...
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Wrong InverseLaplaceTransform? [closed]
Is this a bug?
https://www.wolframalpha.com/input?i=InverseLaplaceTransform%5B1%2FSqrt%5Bt%5D%2C+t%2C+x%5D
...
- 3,553
1
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1
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Inverse Laplace transform of two coupled equations in Mathematica [closed]
I want to solve these coupled equations in Mathematica for F1(s) and F2(s) and the inverse Laplace of each of them to find c1(t) and c2(t).
the code I tried is:
...
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1
answer
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Inverse Fourier Transform and the sign of the Shifted Delta
I am using the sign convention of FourierParameters->{0, -2 Pi} for calculating the inverse FT of $Aexp(-2\iota\pi f K)$, where $A$, $K$ are real numbers >0 ...
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Implementing the Shift Property of the Fourier Transform in Mathematica
I am trying to determine the Fourier tranform of a time shifted (t-b) Lorentzian function in Mathematica. With the zero centered Lorentzian (b=0), the Integrate function with the conditions that all ...
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1
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How to calculate an `InverseMellinTransform` up to its definition in Mathematica?
I have a question about how to calculate Inverse Mellin Transformation's in Mathematica, one way or the other.
Look at these findings.
The following integral results in Gamma[s].
...
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380
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How to do that Fourier transform?
I want to perform Fourier Transform of $$\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}^{\...
- 453
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1
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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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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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0
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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
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159
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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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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 13.2 or later.
When I am solving odes using LaplaceTransform, I found that LaplaceTransform works well with u[t] or y[...
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2
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149
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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 ...
- 999
4
votes
1
answer
113
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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, ...
- 999
2
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1
answer
68
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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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