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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Evaluating FourierTransform like integral manually

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 $\delta(x-1)$, ...
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1answer
94 views

Solving partial differential equation involving Hilbert transform

While solving one research paper published in Physical Review Letters, I came across the following equation and I am unable to solve it. $$\frac{\partial f}{\partial t}−(\mathcal{H}(f)\left(\frac{\...
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1answer
55 views

Generate this contour plot

I have the following complex function: ...
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3answers
103 views

How to obtain that result of the integral in Mathematica?

I would like to integate this equation: Integrate[Exp[I*(t1-t2)*ω, {ω, -∞, ∞}] According to a textbook, I know the answer is $2\pi\delta(t_1 - t_2)$, where $\...
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0answers
23 views

Solving a system of two differential equation using Laplace Transform [duplicate]

I was trying to solve a system of two differential equation using Laplace Transform with Mathematica, but couldn't get any implementation for it. How do you reckon this can be solved? Consider this ...
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1answer
210 views

How to solve Laplace transform question for a system in Mathematica

I know how to use the LaplaceTransform function but am struggling to do this with a system with two ODEs. This is my question: Use Mathematica and the Laplace ...
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0answers
54 views

Solving a finite Hilbert transform from 2D thin wing theory

I have a problem where I know $w(x) =1$ and want to find $u(x)$. It's a well-known problem I am recreating, from a 1972 paper. Note the paper doesn't solve the problem, the integral equation, it just ...
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1answer
118 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 ...
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1answer
45 views

How to find discrete Z-transform of a list of complex numbers

Can we find the discrete Z-transform of a list of complex numbers in Mathematica? Similar to the function Fourier for the discrete Fourier transform, do we have a ...
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34 views

MellinTransform yields DiractDelta function

I want to do a Mellin transform for the following function psi[x_] := Sum[1 - E^(-x/2^h), {h, 0, Infinity}] If I try ...
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1answer
46 views

Peculiar result of InverseHankelTransform

In using the HankelTransform and its inverse I find that the inverse does not lead to the initial input. I begin with r (the independent variable) in the denominator but end up with r0 (a constant) in ...
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1answer
50 views

Laplace transform giving incorrect result

In Mathematica: LaplaceTransform[Exp[-Exp[-t]], t, s] Out:= Gamma[s]-Gamma[s,1] But performing InverseLaplaceTransform on the ...
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1answer
54 views

Difference between FourierTransform and LaplaceTransform [duplicate]

There is an equation for example: eqn=D[c[x, t], t] == d D[c[x, t], x, x]; When I make a LaplaceTransform of it: ...
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1answer
64 views

Collect terms in Fourier Transform

I want to collect terms in x from a product of polynomials and Fourier transform. So I try following code: ...
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0answers
43 views

Inverse Laplace transform of powers with an arbitrary index

Tried for Inverse Laplace transform (ILT) for the following: L[s] = (L /(L + s) w + Q /(Q + s) (1 - w))^n $L[s]$ can also be written as $$L[s]=\sum_{k=0}^n\...
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0answers
50 views

Is `InverseMellinTransform` unaware of second Barnes lemma?

Consider evaluating ...
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2answers
129 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 ...
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1answer
248 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 ...
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1answer
58 views

Evaluating a Fourier integral

Is it possible to evaluate the following integral? $$\int \frac{(x-x_0) \: dx \: dy \: e^{i k_x x + i k_y y}}{((x-x_0)^2+(y-y_0)^2+4 h^2)^{\frac{5}{2}}}$$ As a first try, I evaluated ...
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How can I eliminate InverseLaplaceTransform returning complex

I tried this: F = InverseLaplaceTransform[ 1/((1 + s^2) (1 + (s + 1/2)^2)) + Exp[-Pi*s]*1/((1 + s^2) (1 + (s + 1/2)^2)), s, t] But it returns an answer ...
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1answer
196 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 ...
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1answer
80 views

Plotting a sum for a series of sine and cosine all within the same height on the x-y axis? [closed]

The limits of the graph should be (0,1) on the x and y axis but mine is [1,11] on x-axis and [0,7] on the y axis. Any idea what to do? ...
4
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1answer
57 views

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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2answers
199 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 ...
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2answers
79 views

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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1answer
140 views

Can the product of an exponential and a hypergeometric function be expressed as a Meijer-G function?

Let $0<\alpha< 1$. Can the function $\qquad e^{-\alpha x}U(a,b,x),$ where $U(a,b,z)$ is the hypergeometric U function be expressed as a Meijer-G function? The closest I found was here which ...
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1answer
77 views

Inverse Laplace transform - Getting a compact solution

I had the following inverse laplace transform: ...
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1answer
75 views

Inverse Laplace transform of a symbolic expression

I'd like to get the inverse Laplace transform of a symbolic expression. The output of my code is equal to the input? How can I modify the code to obtain the correct result? All k and V1 are positives....
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1answer
273 views

2D Fourier transform of annulus

I have an annulus and I'd like to the take the 2D Fourier transform of it, my code: ...
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1answer
138 views

FourierParameters causing function not to simplify

When evaluating this expression, Mathematica simplifies the result: FourierTransform[FourierTransform[u[t], t, s], s, t] This simplifies to ...
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1answer
617 views

Solving the heat equation on the semi-infinite rod

Cross posted in scicomp.SE. I want to test the solution which is given below is right by Mathematica. Please look the post in mathstackexhange or Please look below. Question: Solve the ...
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1answer
154 views

The inverse Laplace Transform of a real proper rational function must be real

How does one get Mathematica to return a real answer when using InverseLaplaceTransform? Tried using Re and ...
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1answer
501 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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1answer
622 views

How to deal with highly oscillatory integrand when using “NIntegrate” and have a precise result

FTw][n_][w_]:=NIntegrate[ax[n][1][t]/.sol2]Exp[i w1 t],{t,0,600}] Plot[Abs[FTw[1][w1]],{w,1,3}] Here I'd like to calculate the Fourier transform of a complicated ...
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1answer
184 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 $2\pi\,\delta(p+s)$: ...
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83 views

Multidimensional DFT of function

Assume that we have a well-defined real-valued functionf[x,y,z] whose (inverse) Fourier transform is impossible to solve for symbolically and numerically using ...
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0answers
38 views

LaplaceTransform vs Integrate with inconsistent result

I am trying to evaluate the following integral, which happens to be a laplace transform $$ G(x,x';z)=\frac{1}{i\hslash}\int_0^{\infty}dt \exp\left(\frac{izt}{\hslash}\right)\sqrt{\frac{m}{2\pi i \...
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1answer
133 views

Numerical Fourier transform of an infinite-order polynomial

I was looking to numerically compute the Fourier transform of a function which can formally be represented as the infinite-order polynomial $$\varphi \, (\zeta ) = \sum_{t=0,2,4,\ldots}^\infty (-1)^t\,...
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1answer
72 views

Why doesn't this integral evaluate the laplace transform [closed]

I want to evaluate the laplace transform using "Integrate" rather than "LaplaceTransform". However, for some reason the two don't give the same output. I want to do ...
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1answer
354 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 (...
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0answers
31 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
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1answer
172 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 ...
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0answers
122 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 ...
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1answer
102 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 ...
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0answers
117 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. ...
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0answers
454 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 ...
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1answer
326 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 ...