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.
57
questions with no upvoted or accepted answers
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answers
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Inverse Laplace transform not obtained
I can't seem to be able to compute the inverse Laplace transform of a Laplace transform:
...
- 403
6
votes
0
answers
247
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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-...
- 61
5
votes
1
answer
14k
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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 ...
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4
votes
0
answers
85
views
4
votes
0
answers
179
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Integral Representation of DiracDelta
I would like to get dirac delta from integral definition, of course if I ask
Integrate[Exp[I x t],{x,-Infinity,Infinity}]
I get that the integral does not ...
- 909
3
votes
0
answers
76
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
...
- 541
3
votes
0
answers
156
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 ...
- 3,585
3
votes
0
answers
159
views
Convolution using the Laplace integral transform of certain functions
I am trying to convolve two functions:
$f(t) = e^{- t}$
$g(t) = e^{-(e^{-t})^2}$
$(f*g)(t) = \int_{0}^{t} f(t-\tau)g(\tau) d\tau = \int_{0}^{t} e^{-(t-\tau)} e^{-(e^{-\tau})^2} d\tau$
Using the ...
- 2,394
3
votes
0
answers
872
views
Region of convergence in Inverse laplace transform
Lets say we have a function in the s domain. Let it be H(s)=1/(s+1). It has one pole.
If we consider the region to the right of the pole as the ROC, we would one function in the time domain when we ...
- 31
3
votes
0
answers
683
views
Computation of a Fresnel Diffraction pattern with Discrete Hankel Transform
In the next link: Computation of Hankel Transform using FFT (Fourier)
Rainer implemented a great solution given in the next reference: Manuel Guizar-Sicairos and Julio C. Gutiérrez-Vega, "Computation ...
- 31
2
votes
0
answers
60
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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 ...
- 7,793
2
votes
0
answers
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]
...
- 26.3k
2
votes
0
answers
81
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]
...
- 3,585
2
votes
0
answers
96
views
Solving a finite Hilbert transform from 2D thin wing theory
I have a problem where I know $w(x) =1$ and want to find $u(x)$. It's a well-known problem I am recreating, from a 1972 paper. Note the paper doesn't solve the problem, the integral equation, it just ...
- 7,153
2
votes
0
answers
39
views
Verifying a cosine FourierTransform
On page 31 of this standard reference we have the following relation:
I wanted to use mathematica to verify this transform numerically for some example values. So I type in:
...
- 12k
2
votes
0
answers
427
views
Tricky inverse Laplace transform
I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral
$$
F(t) = \frac{1}{2 \pi i} \int_{- ...
2
votes
0
answers
160
views
Incorrect output from InverseFourierTransform
Bug introduced in 8.0.4 or earlier and fixed in 9.0.1 or earlier
I am trying to compute the inverse Fourier transform of this expression
...
- 121
1
vote
0
answers
80
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:
...
1
vote
1
answer
96
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.
...
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1
vote
0
answers
58
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$?
<...
- 7,793
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
...
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1
vote
0
answers
167
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 ...
- 141
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 ...
- 11
1
vote
0
answers
88
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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 ...
- 123
1
vote
0
answers
86
views
Closed form for integral with SinIntegral
MMA can't solve this integral:
$$\int_0^{\pi } \text{sinc}(x) \text{Si}(2 x) \, dx$$
The I use FourierSinTransform to find closed form,but I try a verify result is ...
- 13.9k
1
vote
0
answers
50
views
Performing operator expression on a function
What I want. I write an operator expression, and Mathematica performs it on a function.
Examples:
E^(b D): f[x] -> f[x+b]
...
- 3,585
1
vote
0
answers
70
views
Numerically obtaining inverse 2D Laplace transform
I wonder if there is a function in Mathematica (or code) that can help obtaining inverse 2D Laplace transform of a function, f(s1,s2) which is the 2D Laplace transform of a function F(t1,t2) in (s1,s2)...
- 43
1
vote
0
answers
112
views
How to plot integral of a gradient function
For $f : \mathbb{R} \longrightarrow \mathbb{R}$ continously differentiable and $\phi \in [0,\pi/2]$ let
$$F(x) = \int \frac{f'(x)-\cot \phi}{1+ f'(x) \cot \phi} d x$$
be a clockwise 'rotation' of $f$.
...
- 1,005
1
vote
0
answers
154
views
Simplify trigonometric functions to take the Laplace transform
I have one trigonometric function which is simplified as
f[t_, RV_, H_] := 0.000308148 H Cos[0.172439 RV Sin[2 \[Pi] t]]
It is an input to one ODE which I want to ...
- 489
1
vote
0
answers
51
views
HankelTransform problem
Somewhere in the following code there is a bug. The result should not be zero!
FunctionExpand@HankelTransform[Sqrt[r] Cos[r], r, k, 1/2]
I think the answer should ...
- 234
1
vote
0
answers
63
views
Adding a module to teach Mathematica the Laplace transform of particular Mittag-Leffler functions
Mathematica 11.3 is not aware of a useful Laplace transform
LaplaceTransform[t^(-a) MittagLefflerE[a, a, t^a], t, s]
which for ...
- 1,798
1
vote
0
answers
142
views
Calculate a nested integral
I wish to plot the following probability on Mathematica:
$$
\pi=n\cdot\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\left[\int_{-\infty}^{z_{1}}G\left(z-z_{2}\right)\cdot f\left(z_{2}\right)dz_{2}\...
- 11
1
vote
0
answers
165
views
Can this boundary value problem leading to a partio-integral DE be solved using finite Fourier or any integral transform?
I asked the three-dimensional version of this problem here which lead to a trivial solution. I have now tried it in 2-D
$$\frac{\partial \theta_h}{\partial x} + b_h (\theta_h - \theta_w) = 0, \tag 1\\\...
- 297
1
vote
0
answers
114
views
Is it possible to take the Laplace transform of a Hankel function with Mathematica?
The following two functions are the fundamental solutions of the wave equation and its Laplace transform (the modified Helmholtz equation) in two dimensions, respectively, when the speed of wave ...
- 339
1
vote
0
answers
110
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Numerical Definite Integral of Numerical Laplace Transform
I'm having problem taking the numerical definite integral of a numerical Laplace transform that depends on two variables due to NIntegrate:inumr errors:
...
- 249
1
vote
0
answers
416
views
Mathematica 12 returns a "greater::nord" error when Mathematica 11 does not
In the following code Talbot's method is made use of to invert Laplace's transforms.
...
1
vote
0
answers
319
views
Taking numerical Laplace transform of a tabulated data with Mathematica?
I am relatively new to the use of Mathematica in the context of numerical evaluations, therefore I would greatly appreciate a detailed answer and would like to express my gratitude towards any help in ...
- 11
1
vote
0
answers
267
views
Can Mathematica verify the Cahen - Mellin integral?
Can Mathematica verify the Cahen-Mellin integral?
When I try
Integrate[Gamma[1 + I s]/Zeta[3]^(1 + I s), {s, -Infinity, Infinity}]
I just get the input back. ...
- 12k
1
vote
0
answers
187
views
Inverse Laplace Transform Failing: $\frac{1}{\exp (\beta \tau )-a}$
I have tried various approaches to get the solution to this inverse transform but all so far are failing.
...
1
vote
0
answers
118
views
Different Results under Inverse Laplace Transform
I am getting different results from the following commands:
...
- 160
1
vote
0
answers
129
views
InverseLaplaceTransform provides a wrong answer?
Question is revised as below:
I'm trying to define a function involving inverse of a Laplace Transform for a rational function, but Mathematica provides a wrong result. Can anyone help on this?
...
- 11
1
vote
0
answers
201
views
Discrepancy between laplace and fourier transform of gaussian
The function in question is Exp[x^2/2] for x>0
Laplace transform should be the same as the fourier transform since the function is absolutely integrable.
Fourier:
They are indeed the same
...
- 1,253
1
vote
0
answers
284
views
The inverse Fourier transform of a rather complex function
I am using Mathematica 7.0 and trying to find the Fourier transform of a rather complex function g. The function g is defined by
$$g=\left(\textrm{e}^{\left(\frac{-0.008372\textrm{i}+0.008372\textrm{...
- 11
0
votes
0
answers
90
views
LaplaceTransform doesn't work
I have code:
A = 1;
ω0 = 5*10^6;
τ0 = 3*10^-3*Sqrt[2/Log[2]]/(5.85*10^3);
s0[t_] = A*E^(-t^2/τ0^2)*Cos[ω0*t];
Phi[ω_] = LaplaceTransform[s0[t], t, ω]
The last ...
0
votes
0
answers
126
views
Implementing the 3D Radon transform
I am wondering how to implement the Radon transform, the 3D Radon transform, that is, given a 'density' function $f: \mathbb{R}^3\to \mathbb{R}$ The Radon transform of $f$ is $$Rf(s,w)= \int_{x\cdot w=...
- 109
0
votes
0
answers
46
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.
...
- 21
0
votes
0
answers
149
views
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
...
0
votes
0
answers
177
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] ...
- 69
0
votes
0
answers
46
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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 ...
0
votes
0
answers
52
views
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: ...
- 373