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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2answers
69 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\...
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1answer
28 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 ...
6
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2answers
229 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 $...
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1answer
56 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 \...
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2answers
115 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 ...
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0answers
49 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 ...
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0answers
59 views

Strange result of LaplaceTransform

I mean LaplaceTransform[BesselI[3, x], x, s] // Simplify ...
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2answers
75 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: ...
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0answers
66 views

Problems with the output of the integral function

I write here because I've some problems when I want to integrate this quantity with respect to z: rH, a and z are real values, being TT4 a real value too and I especify that in the assumptions of the ...
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2answers
75 views
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1answer
82 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: ...
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0answers
68 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 ...
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1answer
116 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 ...
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0answers
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] ...
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0answers
124 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
62 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
47 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
46 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
47 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
53 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^{-\...
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4answers
264 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
70 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 ...
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3answers
921 views

3D Fourier transform of 1/r^2

How can I compute the Fourier Transform of $f(r)=1/r^2$, where $r=\sqrt{x^2+y^2+z^2}$?
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3answers
104 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\...
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0answers
45 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)...
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2answers
242 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. ...
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0answers
91 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 $f$. ...
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1answer
78 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 ...
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0answers
142 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 ...
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0answers
44 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 ...
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2answers
76 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 ...
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0answers
46 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
87 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
459 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 ...
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1answer
63 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
155 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 ...
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0answers
156 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
111 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,...
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0answers
69 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
74 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
120 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
101 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 ...
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1answer
148 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 \...
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1answer
104 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
197 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
85 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
132 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
162 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
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1answer
263 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
131 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? ...