# 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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### Inverse Laplace Transform of Hypergeometric function

Any tips how to massage the following to get computed by Mathematica for $p>1$? I suspect the result should be expressible in terms of exponential integral ...
172 views

### How can I solve $\sum_{i=1}^{M-1} (M+i)^{M+i+1/2}/i^{i+1/2}$? [migrated]

I am trying to solve an equation in Mathematica: $$\sum_{i=1}^{M-1} \frac{(M+i)^{M+i+\frac{1}{2}}}{i^{i+\frac{1}{2}}}$$ Does a general solution exist for this expression? And if $M \to \infty$, can ...
113 views

### InverseLaplaceTransform returns the input

Initially, I was trying to invert the following expression: $$\frac{e^{-a\sqrt s}}{s-c}$$ and got the following result: ...
1 vote
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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 ...
107 views

### NIntegrate with highly oscillatory Bessel and hypergeometric integrands

I am trying to compute the following double integral int2[n_] := NIntegrate[ n^2 * u * BesselJ[0, u]^n * r^2 * BesselJ[0, n*r*u], {r, 0, 1}, {u, 0, Infinity}] for ...
1 vote
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### InverseFourierSinTransform in mathematika failed to give a result

I have to use InverseFourierSinTransForm in Mathematika for the function u[ω,t] but infortunately it does not work.It gives back the same! I tried it without the assumptions but it does not work again!...
122 views

### Derive Parseval's theorem in one dimension

Parseval's theorem (in one dimension) is a fundamental result in the theory of Fourier transforms. If $f(t) \Leftrightarrow F(\omega )$ are Fourier transform pairs and $t$ (time) and $\omega$ (...
1 vote
111 views

### Evaluating Fourier transform in mathematica [closed]

I am trying to evaluate the expression FourierTransform[[m (a^2 - t^2 - I g t)]^-1, t, ω] in Mathematica. It gives me the error message that "Syntax: "[m(...
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### Handling singularities like (x-y) in the denominator while evaluating double integrals

I have to solve an Integral of the following type ...
62 views

### Efficient powers of DPR1 matrices

I'm looking to compute the following quantity for $A$ where $A$ is a convergent positive definite $d\times d$ diagonal + rank1 matrix (DPR1). $$f(s)=\operatorname{Tr}(A^s)$$ Earlier answer answer by ...
53 views

### Improving quality of plot with bad numeric performance

I have a plot which suffers from poor numeric accuracy, any tips on how I can improve the quality of this plot? ...
1 vote
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### Inverting Laplace transform numerically for a set of points

Computing inverse of Laplace transform numerically in Mathematica seems very slow and requires $k$ calls for $k$ points. Is there a faster way to do it for a set of $k$ points, ie $k\approx 1000$? <...
1 vote
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### Incorrect result of InverseFourierTransform

When executing in 13.2 on Windows 10, the command FourierTransform[DiracDelta@Cos[x], x, s] results in ...
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### 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 ...
148 views

### FourierCosTransform bug?

Bug introduced in 13.0 or earlier and persisting through 13.2.0 or later. FourierCosTransform[Cos[(k + p) z], z, q] gives correct result ...
1 vote
45 views

I did this: ...
216 views

### Incorrect result of FourierTransform

Let us consider in 13.2 on Windows 10 FourierTransform[1/Sinh[x]^2, x, k] -((2 + k \[Pi] Coth[(k \[Pi])/2])/Sqrt[2 \[Pi]]) ...
54 views

### Hilbert Transform in mathematica [duplicate]

I would like to do a Hilbert transformation in Mathematica on a function. However, it does not seem to give a right result. The Hilbert transform is given by So I did : ...
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### How to pass arguments in NIntegrate and NDSolve

I am solving PDEs with NDSolve and NIntegrate, but I do not know how to pass arguments correctly. My orignial code is very ...
620 views

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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 $r=\sqrt{x^2+y^2+z^2}$?
887 views

### How to set up a spherically symmetric Fourier transform?

I am having trouble figuring out how to set up the Fourier transform to: ...
1 vote
For example, let's say I want to compute the (continuous-time) Fourier transform of the signal/function $\cos{(3t)}$, which is given by the following improper integral: \$\displaystyle\int_{-\infty}^{\...