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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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1answer
381 views

Mittag Leffler function Laplace transforms with Mathematica

Mathematica seems not to to know the basic Laplace and inverse Laplace relation $$\mathcal L(E_\alpha[−λt^α],t)(s)=\frac{s^{α-1}}{λ+s^α}$$ surrounding the Mittag Leffler function (...
2
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0answers
32 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: ...
2
votes
1answer
190 views

Fourier transform question

Consider the fourier transform FourierTransform[1/((Cosh[x] + 1) (Cosh[x]^2 - 1)^(1/2)), x, w] If I execute the above line, Mathematica thinks for several ...
3
votes
0answers
135 views

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 ...
1
vote
1answer
112 views

Mathematica 11: Mellin transform conditions?

Consider the expression fun = 1/(y^2-1)^a; If I do the Mellin Transform explicitly, I get a result that involves a few conditions on the parameters ...
1
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0answers
125 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. ...
3
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0answers
507 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 ...
-3
votes
1answer
351 views

How to find the Inverse Laplace transform of $Exp[(1-s)(1+1/s)]$? [closed]

How do i find the inverse laplace transform of following function: f[s]=Exp[(1-s)(1+1/s)] Mathematica returns exactly the simplified function without computing ...
0
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0answers
76 views

Understanding Mathematica's InverseZTransform Function

I'm trying to find the Inverse Z Transform of the following: InverseZTransform[(1 - ρ)/(1 - ρ z), z, n] and I get from Mathematica the result ...
3
votes
1answer
117 views

How do I implement custom integral transform with kernel $\cos^2 (zx)$ or $|\cos (zx)|$ in Mathematica?

I want a transform similar to Fourier transform but with a different kernel. With infinite limits: $$\int_{-\infty}^{\infty} f(x) \cos^2(zx)dx$$ I want it to work with distributions and other things,...
8
votes
2answers
342 views

Why does InverseLaplaceTranform give an incorrect solution?

Bug introduced in 9.0.1, fixed in 11.1. I was trying to answer this question, but I ended up asking this. Method 1 A workaround is to differentiate the ODE once, ...
5
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0answers
166 views

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-...
3
votes
1answer
293 views

Why isn't Mathematica outputting the inverse fourier transform of this function?

I'm brand new to Mathmematica, so please be nice. I don't understand where I'm going wrong. I just need to use the method of inverse fourier transform on this wavefunction to get the momentum-space ...
1
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0answers
158 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. ...
5
votes
1answer
259 views

Fourier transform of Exp[x]/x

Could you please explain why Mathematica gives the following expression when taking Fourier transform of $\exp(\lambda z)/\lambda$? $$\frac{-\log(-z)+\log(z)}{\sqrt{2\pi}}$$ Why the answer does not ...
1
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0answers
254 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_{- ...
3
votes
0answers
397 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 ...
22
votes
1answer
1k views

Numerical inverse Laplace-Hankel transform

When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the following double integral (to be more precise, the 0-order inverse Hankel-Laplace ...
1
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0answers
87 views

Different Results under Inverse Laplace Transform

I am getting different results from the following commands: ...
3
votes
1answer
32 views

Possible incompatibility between ReplaceAll and LaplaceTransform

I have a large symbolic expression containing many terms of the form, LaplaceTransform[u2[z], z, s], with various functions for the first argument. I wish to ...
0
votes
0answers
35 views

How can I get left-sided-sequence with InverseZTransform

InverseZTransform[z^2/((4 - z)*(z - 1/4)), z, n, Assumptions -> {1/4 < Abs[z] < 4}] (* -(1/15) 4^-n (-1 + 16^(1 + n)) *) the result is not true.It only ...
3
votes
1answer
133 views

Table with power law indexing

I am using the Drubin numerical inversion out of Laplace space, and need to see how the function performs over a wide range of values, covering multiple logrithmic cycles. However, the inversion is ...
0
votes
1answer
139 views

InverseFourierTransform very slow

I determined a distribution dist = SmoothKernelDistribution[ListOfValues];. Then I determined the FT of the PDF, which is the ...
5
votes
1answer
539 views

Solving PDE involving Hilbert transform numerically

I'm trying to solve the following equation numerically: $$u_{tt}-\mathcal{H}(u_x)=A^2_{xx},$$ where $\mathcal{H}$ is the Hilbert transform and $A$ is a prescribed forcing function which we assume ...
2
votes
0answers
125 views

How to do the Fourier transform of a picture? [duplicate]

I want to generate Fourier transforms of a picture. the transform can represent the diffraction pattern of the picture. Basically, if I have a picture. Then, the result should look like this: For a ...
1
vote
1answer
866 views

Laplace Transform of a Multidimensional Partial Differential Equation

I'm trying to use Mathematica to solve a multidimensional partial differential equation using Laplace tranforms. The PDE is as follows: $$ \frac{{\partial T}}{{\partial t}} = \frac{{{\partial ^2}T}}{{...
7
votes
2answers
1k views

Inverse Laplace transform

Let $r=\mu = 0.15; \sigma = 0.05; T = 1; S_0 = 100; K = 95;$ Let $\nu:=\frac{2\mu}{\sigma^2}-1$ and $\eta \equiv\eta(\alpha):=-\frac{\nu}{2}+\frac{1}{2}\sqrt{\nu^2+\frac{8\alpha}{\sigma^2}}$. ...
2
votes
0answers
298 views

How to Mellin transform a complicated Log integrand?

I got a question concerning an integral. I need to know the analytical expression. I have to Mellin transforn a function and the integral is then sth. like this: $$ \int x^{N-1} \frac{Ln(a -x)}{1-x} ...
1
vote
0answers
103 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? ...
9
votes
1answer
2k views

How can I invert a Laplace transform numerically?

I have a very complicated expression, which I want to transform using the inverse Laplace transform. The built-in function InverseLaplaceTransform doesn't work. So,...
4
votes
2answers
667 views

Calculate the Bromwich Integral (Inverse Laplace Transform)

How can I calculate the Bromwich Integral in Mathematica? If I enter this as code it gives me just the same: $$\frac{1}{2\pi i}\int_{\alpha-\infty i}^{\alpha+\infty i} \left(e^{st}\cdot F_{(s)}\right)...
1
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0answers
263 views

Convergence conditions of the Laplace Transform [closed]

I have to calculate some Laplace Integrals but if I use LaplaceTransform then it does not give the conditions when the integral converges. Is there an option to get ...
1
vote
0answers
83 views

Transfer function in recursive form [closed]

To get transfer function in recursive form, I tried this: ...
1
vote
1answer
86 views
1
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0answers
118 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 ...
3
votes
1answer
197 views

Bilateral ZTransform

Is there support for bilateral Z-transform in Mathematica, or a third-party package?
1
vote
0answers
265 views

Inverse Laplace computation Mathematica [closed]

I have a transfer function such as following, which I would like to find its inverse: F=(a*s^2)/(b*s^3+c*s^2+d*s+e) I have a numeric value for a b c d and e but ...
5
votes
2answers
547 views

Analytic expression for a Complex Hilbert Transform in Mathematica

I need to solve the following integral equations for a problem I'm working on - $\displaystyle \frac{-i}{2 \pi}$ $\int_{-a}^{a} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$ and $\displaystyle \...
1
vote
2answers
310 views

Why does InverseFourierTransform gives different result to doing it by hand? [duplicate]

When I can calculate an inverse fourier transform by the built-in function InverserFourierTransform, the result is different from what I calculate by definition ...
2
votes
1answer
789 views

Inverse Laplace Transform difficulty

I am asked to find the inverse Laplace transform of: $$F(s)=\frac{2s^2+s+13}{(s-1)((s+1)^2+4)}$$ I did the partial fraction decomposition by hand and got: $$F(s)=\frac{2}{s-1}-\frac{3}{(s+1)^2+4}$$ ...
10
votes
1answer
1k views

Compute inverse Laplace transform with Integrate

Since inverse Laplace transform is just an integral, while we already have InverseLaplaceTransform built in, can we compute it with ...
1
vote
1answer
1k views

Discrete Fourier Transform of a vector defined on a 3D lattice

In my research I need to compute the Discrete Fourier transform of a vector defined on a 3D lattice (a cube) to the "reciprocal" lattice. This is quite new to me so before proceeding i wanted to be ...
6
votes
3answers
2k views

Integral of function involving Dirac delta

I am performing lots of simple calculations with dirac delta functions. It would be awesome if Mathematica could do this routine exercise for me, eliminating any possible human errors. For example, ...
1
vote
0answers
237 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{...
4
votes
0answers
7k views

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 ...
1
vote
1answer
173 views

Inverse Laplace transform only returning exponentials

The inverse Laplace transform is only returning exponentials. I know it is from the definition $$ \frac{1}{2\pi i}\int_{\gamma - i\infty}^{\gamma + i\infty}F(s)e^{st}ds = \sum\text{Res} $$ However, I ...
1
vote
1answer
763 views

Trouble with Fourier transform of Exp[-Sqrt[x]]

First time trying to do something "real" in Mathematica, I am having trouble getting it to calculate this Fourier transform. It runs for a long time, then just prints the input expression. ...
2
votes
0answers
135 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 ...
0
votes
2answers
192 views

Indirect transformation based on a list of data points

I have a set of data which describes the Intensity function $Int(q)$ as a function of $q$. The data list provided below provides the shape of $Int(q)$ versus $q$.How can I use this information to ...