How to Integrate Bessel Functions Over z?

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Discussion Overview

The discussion revolves around the integration of Bessel functions, specifically the integral of H0(z)²/z and related forms involving H0(z)²/(z-a) and J0(z)Y0(z)/(z-a). The scope includes mathematical reasoning and exploration of special functions.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • One participant seeks assistance with the integral ∫H0(z)²/z dz.
  • Another participant provides a link to a resource that may contain relevant information.
  • A participant expresses difficulty in finding the integral and specifies interest in more complex forms involving H0(z)²/(z-a) or J0(z)Y0(z)/(z-a).
  • It is noted that the closed form for the integral may involve complicated special functions, with skepticism about the existence of a closed form for the case with (z-a) in the denominator.
  • A later reply mentions that encountering a Meijer-G function typically suggests a preference for numerical solutions or abandoning the problem.

Areas of Agreement / Disagreement

Participants express varying levels of difficulty in finding closed forms for the integrals, with some skepticism about their existence. No consensus is reached regarding the feasibility of closed forms or preferred methods of approach.

Contextual Notes

Limitations include the complexity of special functions involved and the potential for numerical solutions being more practical than analytical ones.

besselevil
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Hi, I am trying to find the following integral of bessel functions, any help would be great:
∫H0(z)2/z dz

Thanks
 
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http://www.fh-jena.de/~rsh/Forschung/Stoer/besint.pdf

Perhaps this has the one you are looking for.
 
Thanks but I have tried it and could not find it. Actually I am interested in integrals involving H0(z)2/(z-a) or J0(z)Y0(z)/(z-a)
 
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The website is very useful, thanks a lot!
 
JJacquelin said:
The closed form involves very complicated special functions :
http://www.wolframalpha.com/input/?i=integrate+BesselH(0,x)^2*dx/x
If the denominator is (z-a) instead of z, I doubt that a closed form might exist with the standard special functions.

Yeah, when I run into a Meijer-G function I usually take that as a sign that I want to solve the problem numerically (or, better yet, move on to another problem entirely. :-p ).
 

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