# Tagged Questions

172 views

### 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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### 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 ...
373 views

### Fourier Transform of Bessel function with square root argument

Fourier Transform of the following function: ...
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!