# 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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### How to solve this integral equation $\int_{-\infty}^\infty f(z)x^z dz = F(x)$ for f(x)?

My question is: solving $f(x)$ with known $F(x)$ and equation $$\int_{-\infty}^\infty f(z)x^z dz = F(x).$$ I met this problem when I tried to extend the idea of generating functions for discrete ...
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### How can I solve $\sum_{i=1}^{M-1} (M+i)^{M+i+1/2}/i^{i+1/2}$?

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 ...
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I have been studying the continuous wavelet transform and came across the following result on the Wikipedia: A wavelet $\psi(t) \in L^1(\mathbb{R})\cap L^2(\mathbb{R})$ is admissible if it has a ...
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### Integral over a product of polynomial, exponential and Bessel function

In a physics textbook I'm working through I found an interesting integral identity which I want to prove: \int_0^\infty t^{\nu +1} J_\nu(\beta t) e^{-\alpha t} \, dt = \frac{2\alpha (...
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### An example of Mellin transform asymptotics

Let $$F(x) = \sum_{k\ge 1}\frac{1}{1+k^2x^2}$$ and its Mellin transform $$F(s) = \frac{\pi}{2}\frac{\zeta(s)}{\sin{\frac{\pi}{2}s}}, \space\space where \space\space (1<Re(s)<2)$$ And then, its ...
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### Laplace Transform of $\frac{\psi ^{(2)}\left(\frac{1}{\sqrt{x}}\right)}{x^{3/2}}$?

Is there a closed form expression for the Laplace Transform of the following expression? $$f(x)=\frac{\psi ^{(2)}\left(\frac{1}{\sqrt{x}}\right)}{x^{3/2}}$$ where $\psi^{(k)}$ is the polygamma ...
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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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