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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Mathematica 9 cannot solve this Integral. Mathematica 8 could. Is this a bug?

I was trying to (re)calculate a problem of an older Wolfram blog post (Problem 11457, by M. L. Glasser) with Mathematica 9.0.0.0 (on OS X 10.8.2). ...
10k views

Implementing discrete and continuous Hilbert transforms

What is an efficient and accurate Mathematica implementation of the Hilbert transform, for both continuous and especially discretely sampled functions? This transform relates phase and amplitude in ...
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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 ...
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Laplace transform of $\frac{1-\cos (t)}{t}$

In the documentation, it states that The Laplace transform of a function $f(t)$ is defined to be $\int_0^{\infty } f(t) e^{-s t} \, \mathrm{d}t$. But why can Mathematica not get the Laplace ...
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Computation of Hankel Transform using FFT (Fourier)

To address circularly symmetric cases of 2-D Fourier Transformations, the so-called Hankel Transform can be applied (for a detailed derivation of the relation between the 2-D Fourier transform and the ...
355 views

Possible bug with FourierTransform linearity

Each component is easily transformed but the sum is not: FourierTransform[f''[x], x, k] FourierTransform[f'[x], x, k] ...
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Numerically obtaining the inverse Laplace transform of data

I have been using several Mathematica packages to do numerical inverse Laplace transforms on known (expressible in closed form) expressions, $\tilde{f}(s)$. I am now being confronted with the more ...
446 views

Workarounds for a possible bug in the linearity of FourierTransform

This is somewhat similar to this question, except the problem I am encountering is to do with Fourier transforms of scalar multiples of functions and their derivatives. I wish to input ...
289 views

Package for fast spherical harmonic transform in Mathematica?

Mathematica's implementation of the Fast Fourier Transform is, naturally, much faster than computing the discrete transform yourself using Sum. The analog of the ...
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 ...
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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,...
341 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, ...
613 views

Correct way of simplifying the result of an integral

Many times Mathematica gives enormous results to simple problems. One uses the program more for trouble than for not knowing how to solve the problem. As an ejemplo I present this integral that ...
772 views

Integrating over Bessel Function erroreous? (Hankel Transform)

Bug introduced in 8.0.4 or earlier and persists through 11.0.1 or later The Hankel Transform is given by Integrate[f[x] x BesselJ[0, x t], {x, 0, Infinity}] It ...
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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: ...
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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 $\eta \equiv\eta(\alpha):=-\frac{\nu}{2}+\frac{1}{2}\sqrt{\nu^2+\frac{8\alpha}{\sigma^2}}$. ...
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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, ...
546 views

Implement finite Fourier transforms

Recently I came across finite Fourier transforms, which can be used for solving certain type of boundary value problem (BVP) of linear partial differential equation (PDE) with constant coefficient. ...
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How to set up a Fourier transform?

I am having trouble figuring out how to set up the Fourier transform to: ...
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How to speed up the integral in NDSolve?

On the MMA.SE, there have been several question on how to solve equations including integral using NDSolve, see example 1, example 2, and example 3. Many people, ...
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Can I use LaplaceTransform to find an $f$ such that $f(t)-f(t-a)= g(t)$ for some given $g$?

Suppose I know the difference between an unknown function $f$ and its delayed version, can I use LaplaceTransform to find $f$? I didn't succeed in this toy example:...
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Transform expression involving Erfc back and forth with Laplace transform and its inversion

It's a problem I encountered when answering this question and I think it's worth starting a new question for it. Consider the following expressions: ...
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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 ...
138 views

How do I plot a Laplace transform?

I have been looking everywhere for help on this issue and cannot find a solution that works. Here is the assignment. I have figured out how to find the Laplace transform, but I do not know how to ...
I want to do a Fourier transform to the below function by Mathematica. How can I do it? Here $c$, $d$, $a$, $L$ are constants. $$w(r)= \left\{ \begin{array}{ll} -\frac{c}{\epsilon r} & ... 2answers 145 views Numerical Laplace transform error I'm running Mathematica 11.3.0.0. The inverse Laplace transform of 1/(s^2-1), evaluated at t=1 gives ... 1answer 32 views Possible incompatibility between ReplaceAll and LaplaceTransform I have a large symbolic expression containing many terms of the form, LaplaceTransform[u2[z], z, s], with various functions for the first argument. I wish to ... 1answer 129 views PrincipalValue option of Convolve not working in 9.0.1? The PrincipalValue option of Convolve is used in this question to define the Hilbert transform. However, in Mathematica 9.0.1, ... 1answer 186 views Bilateral ZTransform Is there support for bilateral Z-transform in Mathematica, or a third-party package? 2answers 236 views Is it possible to find the transfer function of these three differential equations using Mathematica Imagine you are attempting to obtain a model of a device with two inputs [u1, u2] and three outputs [O1, O2, x] from the Lagrangian. After some of Lagrangian work, you find the following three ... 1answer 223 views Problem using Hankel transform Good day community, I try to solve the 2D heat equation in cylindrical coordinates. I wanna follow a paper (Selim et al.: Temperature rise in a semi-infinite medium heated by a disc source) and ... 1answer 190 views Mellin transform of x^p seems to miss a factor of 2\pi Bug introduced in 11.1 or earlier and fixed in 11.3 On Mathematica 11.1.1.0 the Mellin transform of x^p is evaluated as \delta(p+s), while I think it should be 2\pi\,\delta(p+s): ... 1answer 116 views How do I implement custom integral transform with kernel \cos^2 (zx) or |\cos (zx)| in Mathematica? I want a transform similar to Fourier transform but with a different kernel. With infinite limits:$$\int_{-\infty}^{\infty} f(x) \cos^2(zx)dx$$I want it to work with distributions and other things,... 1answer 292 views Why isn't Mathematica outputting the inverse fourier transform of this function? I'm brand new to Mathmematica, so please be nice. I don't understand where I'm going wrong. I just need to use the method of inverse fourier transform on this wavefunction to get the momentum-space ... 1answer 133 views Table with power law indexing I am using the Drubin numerical inversion out of Laplace space, and need to see how the function performs over a wide range of values, covering multiple logrithmic cycles. However, the inversion is ... 1answer 375 views Mittag Leffler function Laplace transforms with Mathematica Mathematica seems not to to know the basic Laplace and inverse Laplace relation$$\mathcal L(E_\alpha[−λt^α],t)(s)=\frac{s^{α-1}}{λ+s^α}$$surrounding the Mittag Leffler function (... 0answers 53 views Is InverseMellinTransform unaware of second Barnes lemma? Consider evaluating ... 0answers 132 views Integral Representation of DiracDelta I would like to get dirac delta from integral definition, of course if I ask Integrate[Exp[I x t],{x,-Infinity,Infinity}] I get that the integral does not ... 0answers 484 views Region of convergence in Inverse laplace transform Lets say we have a function in the s domain. Let it be H(s)=1/(s+1). It has one pole. If we consider the region to the right of the pole as the ROC, we would one function in the time domain when we ... 0answers 388 views Computation of a Fresnel Diffraction pattern with Discrete Hankel Transform In the next link: Computation of Hankel Transform using FFT (Fourier) Rainer implemented a great solution given in the next reference: Manuel Guizar-Sicairos and Julio C. Gutiérrez-Vega, "Computation ... 1answer 776 views Inverse Laplace Transform difficulty I am asked to find the inverse Laplace transform of:$$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} ...
I am trying to evaluate Integrate[x^2*Exp[I k (x - 1)], {k, -∞, ∞}, {x, -∞, ∞}] Since $\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ik(x-1)} d{k}$ is $\delta(x-1)$, ...