# Tagged Questions

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### how is the inverse Hankel transform defined

The finite Hankel transform in my notes is defined as $$f^*(k_i) = \int_{0}^{\infty} {r f(r)J_0(rk_i)dr},$$ where $k_i$ is one of the positive roots of $J_0$. However, my notes don't say anything ...
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### Lower estimates on Mellin transform

Let $f(t)$ be a smooth decreasing function on $[0,+\infty)$. Its Mellin transform is the function $f^\ast(z)$ given by $$f^\ast(z) = \int\limits_0^\infty x^{z-1} f(x) \, \mathrm dx.$$ What are ...
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### Double Integral involving modifed bessel function

I'm try to derive a closed form of the following double integral: $\int\limits_0^x {\int\limits_0^x {{e^{ - {K_1}uv}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)du} dv}$; where $K_1$ is a constant. Do you ...
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### Is this formula for $\sum_{n} (n^{2}+z^{2})^{-s}$ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
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### A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation ...
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### Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, \int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma ...
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### Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: ...
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### Hankel Transform - Eigenfunctions and Inverse

I was reading Akhiezer's Lectures on Integral Transforms and in chapter nine, The Hankel Transform, he says that because the kernel of the Hankel transform is symmetric, its eigenfunctions ...
I'm curious as to how the Fourier transform of the various types of Bessel functions would be calculated. The Wikipedia page on the Fourier transform gives the transform of $J_o(x)$ as being ...
### How do I find the inverse Hankel transform of $k^2e^{-k^2}$?
I am trying to solve: $f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk$, where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0. Thanks in advance for any answers!