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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85 views

Inverse Laplace transform does not restore the original function

I take f[t_]:=Sqrt[t^2 - 1] HeavisideTheta[t - 1], apply LaplaceTransform[ t f[t], t, x], get ...
0
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1answer
92 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 ...
1
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0answers
39 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] ...
1
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0answers
27 views

How to represent InverseLaplaceTransform via Fourier transform? [migrated]

I currently use the following code to represent Laplace transform via Fourier transform. It turned to be more powerful. But can I represent inverse Laplace transform in a similar way? ...
3
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0answers
109 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 ...
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0answers
27 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 ...
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0answers
40 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] ...
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0answers
45 views

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 ...
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0answers
45 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: ...
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0answers
48 views

How to solve this double integral analytically or numerically?

Can someone help me to solve this integral or explain me how to set it on mathematica? \begin{equation} \int_{-\infty}^{+\infty} dk_\alpha dk_\phi \frac{1}{2\pi} e^{-\frac{k_\alpha^2}{2}} e^{-\...
3
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4answers
234 views

Numerical Inverse Z Transform?

Is there a way to compute the inverse z-transform in Mathematica numerically? I'm trying to compute the following... ...
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0answers
37 views

Fourier transform in polar coordinates using built-in hankel transform of the function constant 1 [closed]

Like in the table of transforms https://en.wikipedia.org/wiki/Fourier_transform#Distributions,_one-dimensional the FT (Fourier transform) of $\delta$ is 1 and the FT of 1 is $\delta$, but in polar ...
4
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3answers
328 views

3D Fourier transform of 1/r^2

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

How to calculate an iterated derivative in Mathematica?

I try to calculate an inverse Mellin transform for $s^n \Gamma(s)$: $$x\frac{\mathrm{d}}{\mathrm{d}x}\left(x\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}\right)\right)$$ for $n=2$ for $n$ as $$\left(x\...
1
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0answers
37 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)...
0
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2answers
127 views

Plotting Dirac Delta Function as colored arrows

I want to produce graphs of Fourier transforms for lectures. Using the answer from Calling Correct Function for Plotting DiracDelta I get a problem with the code mentioned below. Definition of Mr. ...
1
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0answers
80 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 $f$. ...
0
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1answer
64 views

Laplace Transform - Differentiation in Time property demonstration

I have a signal x[t_]=t/3 Exp[-3t] UnitStep[t] and the laplace transform as ...
1
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0answers
131 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 ...
1
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0answers
37 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 ...
0
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2answers
66 views

Fourier with 2D data [closed]

I have a set of points (x1,y1)......(xn,yn) that represent for example readings from a laser range finder of a certain obstacle. Using these points I want to draw the approximated shape. As I ...
1
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0answers
37 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 ...
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0answers
29 views

How to automate the transformation into spherical coordinates of a region

I've this region: S = { (x,y,z) ∈ R^3 | 2 * Sqrt[x^2+y^2] <= z <= 1 + x^2 + y^2} Now I would like to transform the coordinates into spherical ones, so as to ...
1
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0answers
82 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}\...
12
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2answers
404 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 ...
0
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1answer
61 views

Assume that the value of an integral is real

I would like to give a condition that the integral I am handling are not complexes. Consider ...
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1answer
149 views

Are there some programs about inverse Fourier and Laplace transfroms?

To be more exact, I have a function F[w_,s_], where $w$ is the Fourier transform of $x$ and $s$ is the Laplace transform of $t$. Now I want to perform the double inverse transforms $s\to t$ and $w\to ...
1
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0answers
153 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\\\...
2
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1answer
99 views

How to find the expectation $\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]$?

I'm trying to find the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$ where $a$ and $b$ are constant values, $\mathcal{Q}$ is the Gaussian Q-function,...
1
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0answers
58 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 ...
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0answers
53 views

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: ...
3
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1answer
74 views

Numerical Laplace Transform of InterpolatingFunction

What are some ways to find the numerical Laplace transform of an InterpolatingFunction? (I know that numerical Laplace transforms are rarely used but my application requires a numerical evaluation). ...
3
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0answers
69 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 ...
1
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1answer
80 views

Laplace transform of Heaviside step function of free variables

In A First Course in Differential Equations with Modeling Applications, Tenth Edition (Zill, 2013) it's stated that if $$\mathcal{U(x)}= \begin{cases} 0,\;x < 0 \\ 1,\;x \geq 0 \...
0
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1answer
93 views

Inverse Fourier Transform and peculiar results

In order to get Fourier and Inverse Transform of a function $f(x)$ I write the following commands: As far I can see that Nominator in the quotient of INF[x] is ...
6
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1answer
196 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 ...
4
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2answers
79 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. ...
2
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2answers
130 views

Issues with Fourier transform in M.12.1.0

I am using M. Version: 12.1.0. In version 10, I got the result right away.So, Why mathematica (12.1.0) is not able to solve this problem? ...
4
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2answers
159 views

InverseFourierTransform takes forever

I am trying to find InverseFourierTransform of the following function, but the code below does not give me a result. I do not get any error messages but the calculation just does not end. Any Help? ...
2
votes
1answer
173 views

Problem with Inverse Laplace Transform in Mathematica

I am trying to calculate the following inverse Laplace transform $$ f(t) = \mathcal{L}^{-1}\left[ \left(\frac{1}{s+a}\right)^{n+1}\left( \frac{1}{s+b}\right)^{n} \right](t) $$ with $s$ being the ...
1
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1answer
115 views

InverseLaplaceTransform not working

I am trying to find the Inverse Laplace transform of a function I previously obtained from a Laplace transform, but the result obtained does not agree with the initial function. Why is this happening? ...
0
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2answers
121 views

Impulse response function h(t) of Gauss functions as a stimulus and response

From the basics of the signal theory the impulse response function (IRF) can be calculate like (L=Laplace transformation): L[x(t)]= X(s) , L[y(t)]= Y(s) and transfer function H(s)= Y(s)/X(s), then IRF ...
0
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0answers
74 views

Teach Mathematica analytical continuation of the gamma function

If I ask Mathematica to compute the gamma function for me Integrate[Exp[-s] s^(a - 1), {s, 0, Infinity}] It dutifully returns to me ...
3
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1answer
181 views

Fourier transform of Log[RealAbs[x]]^2

At least one of the outputs (highly likely both) ...
1
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0answers
345 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
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1answer
94 views

Integrate 2D Gaussian over a segment

I have the following 2D Gaussian: ...
0
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2answers
90 views

Solving proplem by Laplace transform [closed]

I want to solve these ODE's by Laplace transformation and graph it dx1/dt = 38/1180 x3(t) + (1 + sin⁡(t)) - 38/2900 x1(t), x1(0) = 0 dx2/dt = 18/2900 x1(t) - 18/850 x2(t), x2(0) = 0 dx3/dt = 20/...
5
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3answers
812 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 ...
2
votes
2answers
183 views

Finding the global maximum of a function

I've the following code: ...
3
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1answer
300 views

Inverse Laplace transform of this complicated function

I have been solving a coupled PDE system analytically and I need to find the inverse Laplace transform of $(1)$ and get $T(x,y)$. $s$ is the Laplace domain variable and $\alpha, \beta, \gamma, T_{fi}, ...