Zero and first order Hankel Transforms

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SUMMARY

The discussion centers on the analytical computation of the inverse first order Hankel transform of a zero order Hankel transform, specifically the expression H_1^(-1)[H_0(P(r))/q]. The user seeks guidance on this topic, noting its relevance in cylindrical symmetry and the common occurrence of such expressions in literature related to cylindrical divergence. The response suggests referring to the definitions provided on Wikipedia for Hankel transforms as a starting point for further exploration.

PREREQUISITES
  • Understanding of Hankel transforms, specifically H_0 and H_1^(-1).
  • Familiarity with Bessel functions J_0(q*r) and J_1(q*r).
  • Knowledge of cylindrical symmetry in mathematical physics.
  • Experience with kinetic equations in applied mathematics.
NEXT STEPS
  • Study the properties and applications of Hankel transforms in mathematical physics.
  • Explore the derivation and applications of Bessel functions J_0 and J_1.
  • Investigate analytical techniques for solving kinetic equations.
  • Review literature on cylindrical divergence and its mathematical implications.
USEFUL FOR

Mathematicians, physicists, and engineers working with cylindrical coordinate systems, particularly those involved in solving kinetic equations and applying Hankel transforms in their research.

dnp
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Hello everyone,

I am dealing with the following problem. Solving and kinetic equation I came up with the expression

H_1^(-1)[H_0(P(r))/q]

where H_0 is the zero order Hankel transform, H_1^(-1) is the first order inverse Hankel transform P(r) is a function that depends on the radial coordinate in cylindrical symmetry. q is the variable of the transformation that comes from the definitions of the respective Bessel functions J_0(q*r) and J_1(q*r).

My question is is there a way to analytically take an inverse first order Hankel transform of a zero order Hankle transform of the function P(r). I have been through a lot of literature but could not find any discussion. On the other hand such expression seems to occeru pretty commonly when dealing with the radial part of cylindrical divergence.

Thanks in advance for any help

dnp
 
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