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I'm currently in a crash course on X-ray Diffraction and Scattering Theory, and I've reached a point where I have to learn about Bessel Functions, and how they can be used as solutions to integrals of certain functions which have no solution. Or at least, that's as much as I understand. So far I've been able to learn about what Bessel Functions are, how they are derived/defined, etc., what simple functions (like the sinc function) are equal to some of them. But what I haven't found is something like a Fourier Transform Pair Table, only for Bessel Functions and their related unsolvable integrals.

The example I was given is

j_{0}(√(x^{2}+y^{2}+z^{2})) = 1/(4∏)∫∫sinθe^(i(xsinθcos[itex]\varphi[/itex]+ysinθsin[itex]\varphi[/itex]+zcosθ))dθd[itex]\varphi[/itex]

where the integration is from [itex]\varphi[/itex]=0→2∏ and θ=0→∏

I'm still not entirely sure on the geometric representation of this expression, though it seems ellipsoid (at least the expression within the exponential does).

The idea which was expressed to me was that, through memorization, one should be able to recognize which integrals of this nature (∫e^ikr) are unsolvable, and which Bessel Functions provide the 'solutions'. To me, this says there must be a Fourier-transform-pair-esque table somewhere. Am I wrong in my thinking?

Thanks for any help you can give!

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# Bessel Functions as Solutions to Scattering Integrals?

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