# Questions tagged [integral-transforms]

This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine transforms.

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### Inverse Laplace transform of $\frac{\sinh(\sqrt z a)}{\sqrt z}$

I was trying solve this inverse Laplace transform, given by $\frac{\sinh(\sqrt z a)}{\sqrt z}$, with $a \in \mathbb{R}$. But i dont have any good idea to solve this. Please someone have a idea?
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### Guessing the kernel of an integral

Suppose I have a function $h(x_1,x_2)$ which is a polynomial function of degree $2n$ with all even powers; $$h(x_1,x_2) = \sum_{i,j=0}^{n} c_{ij} x_1^i x_2^j$$ where $c_{ij}, x \in \mathbb{R}$. Is ...
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### Transform integral with complex bounds

Question I have a line integral given by \begin{align} I &= \int_{z_0=a+ib}^{z_1=c+id}dz~ z \\ \\ &= -\frac{a^2}{2}+\frac{b^2}{2}-\frac{d^2}{2}+\frac{c^2}{2}+i(-ab+cd) \end{align} For ...
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### How to show algebraically that the determinant of the Jacobian the scaling factor for change in variables?

I am currently trying to learn about Jacobians (self study). In particular struggling to understand the geometry of change in variables and why the determinant of the Jacobian is the scaling factor ...
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### $L^1\big((1+x)\mathrm dx\big)$ form

I am reading the article. On page 15, they wrote But I don't know why we can denote $\Phi(s) = \int_{0}^{+\infty}x^{s-1}\Phi(x)dx$. For your convenience, I also attach the picture in which there are ...
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### Looking for an inverse of an integral transform

I have an integral transform (motivated by a physics problem) $F (x) = \int_0^\infty dx' \frac {a x'} {(a x)^2 + (x - x')^2} f (x')$, where $x, x', a > 0$ real, $f : \mathbb{R}_+^0 \to \mathbb{R}$. ...
1 vote
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### Evaluate $\lim_{s \rightarrow 1^{-}} \frac{\Gamma(b-as)}{\Gamma(s)\Gamma(1-s)}$, for $a,b>0$

I encountered the following as a result of a Mellin-transform: $$\frac{\Gamma(b-as)}{\Gamma(s)\Gamma(1-s)},\quad b>0,\ 0 < a,\text{Re}(s) < 1$$ where $\Gamma(\cdot)$ is the gamma function. ...
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### Computing $\operatorname{Tr}[(\text{D}+uu')^k]$ using generating functions?

I have a large diagonal + rank1 matrix with positive entries. Can someone help me understand how to compute the following using Laplace transform? $$\operatorname{Tr}[(\text{D}+uu')^k]$$ In my ...
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### Name of discrete transform $f(x)=\sum a(n) \exp(-x a(n))$

Is there a way to express the following transform (and perhaps its inverse) in terms of known discrete transforms? For $a_i>0$ and $\sum_i a_i=1$ we have $$f(x)=\sum_i^n a(i) \exp(-x a(i))$$ ...
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### Existence of inverse fourier transform

Is it possible to evaluate an inverse fourier transform of these functions? $f(\omega)=\exp(-(k^2-\omega^2)^{1/2})$, $g(\omega)=\frac{\exp(-(k^2-\omega^2)^{1/2})}{(k^2-\omega^2)^{1/2}}$, where k is a ...
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### Transporting orbit space of vector field via integral transform

Consider a vector field which can be written as system of separable differential equations, which I will solve below: $$X=\big\langle x\log x, -y \log y \big\rangle$$ for $x,y \in (0,1).$ I would ...
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### Probability, Fourier Series and Distribution functions

A sequence of distributions $\left(F_n\right)$ converges to a distribution $F$ if $\left\langle F_n, \phi\right\rangle \rightarrow\langle F, \phi\rangle$ for all test functions $\phi$. (i) Show that ...
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### How to solve the fourier transform of this function?

Is this solution correct in order to find the Fourier transform of $\mathrm{g}(\sigma, \mathrm{t})=\mathrm{e}^{-\lambda|t|} \mathrm{e}^{\mathrm{i} \sigma \mathrm{t}} ?$ If it seems wrong, please let ...
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### Causality/Non-Causality, One/Two-sided integral transform (Laplace, Fourier), Discrete case

When introducing unilateral and bilateral integral transforms (in the context of Laplace transform), I have problems understanding how causality and non-causality are expressed when considering the (...
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### Laplace Transform to solve ODE

I'm trying to solve the BVP $f''(x)=\delta(x-a)$ where $0<a<1$ and $f(0)=f(1)=1$ but I'm really not sure where to start. I tried taking the Laplace Transform of the equation, to get (after ...
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### Inverting integral transform with kernel $x^n$

I'm looking to find the inverse of $$g(n)= \int_{-\infty}^\infty f(x)x^n dx$$ I'm not sure if an inverse exists but i suspect the kernel is well behaved enough to have an inverse.If an inverse exists ...
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### Kramers-Kroning peaks relationship

My question arise from an statement on Qualitative interpretation of Hilbert transform Kramers-Kronig relations state that for analytic (in the upper half-plane) functions with fast decay, its ...
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### Fourier transform of $\dfrac{\cos(\beta x)}{a^{4} + x^{4}}$

What I've done so far: \dfrac{1}{2\sqrt{2\pi}}\left[\int_{-\infty}^{\infty}\dfrac{e^{i(\beta + k)x}}{a^{4} + x^{4}}dx +\int_{-\infty}^{\infty}\dfrac{e^{i(k - \beta)x}}{a^{4} + x^{4}}dx\right] = \\\...
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### What is a kernel integral operator?

I was reading a paper that define the kernel integral operator as follows: We define the kernel integral operator $\mathcal{K}$. Let $\kappa^{(l)} \in C(D \times D; \mathbb{R}^{d_{l+1} \times d_l})$ ...
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