Fourier Bessel series

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  • #1
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Homework Statement


By appropriate limiting procedures prove the following expansion
[tex]\frac{1}{\left(\rho^2+z^2\right)^{1/2}}=\int^{\infty}_{0} e^{-k\left|z\right|}J_{0}(k\rho)dk[/tex]


Homework Equations





The Attempt at a Solution



I tried to implicate the fourier-bessel series but it turned out that there is no k dependence on coefficients of the series.

What can I do to proceed this question?
 

Answers and Replies

  • #2
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I think you should write out the integral representation of this 0-order bessel function, and then switch the integrals. With this you get a 2D fourier transform of the given function.
After all, the Hankel-transform with zero order bessel functions, is the 2D fourier transform of some cylindricaly symmetric two variable function.
 

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