# 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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### 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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### 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 ...
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### 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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### 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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### Strange result of LaplaceTransform

I mean LaplaceTransform[BesselI[3, x], x, s] // Simplify ...
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### 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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### 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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### Solve was unable to prove that the solution set found is complete

I have such a code: ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$\left(x\... 0answers 45 views ### Numerically obtaining inverse 2D Laplace transform I wonder if there is a function in Mathematica (or code) that can help obtaining inverse 2D Laplace transform of a function, f(s1,s2) which is the 2D Laplace transform of a function F(t1,t2) in (s1,s2)... 2answers 242 views ### Plotting Dirac Delta Function as colored arrows I want to produce graphs of Fourier transforms for lectures. Using the answer from Calling Correct Function for Plotting DiracDelta I get a problem with the code mentioned below. Definition of Mr. ... 0answers 91 views ### How to plot integral of a gradient function For f : \mathbb{R} \longrightarrow \mathbb{R} continously differentiable and \phi \in [0,\pi/2] let$$F(x) = \int \frac{f'(x)-\cot \phi}{1+ f'(x) \cot \phi} d x$$be a clockwise 'rotation' of f. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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}\...
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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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### 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 ...