Analytically Solving Bessel Functions for x Giving J_m(x)=0

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Discussion Overview

The discussion centers on the challenge of analytically solving for values of x that satisfy the equation J_m(x)=0, where m is a constant. Participants explore the feasibility of finding these roots analytically, as well as alternative methods for identifying them.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the possibility of finding an analytical solution for J_m(x)=0.
  • Another participant mentions using Mathematica for solving the equation but indicates it was not helpful.
  • A different participant notes that there are infinitely many real roots for J_m(x)=0 and suggests using asymptotic formulas for large x, stating that the difference between successive roots approaches \pi.
  • It is proposed that for small or large x, there may be analytic solutions, but for the first few roots, consulting tables of Bessel function zeros might be necessary.

Areas of Agreement / Disagreement

Participants express differing views on the possibility of finding an analytical solution, with some suggesting it may not be feasible while others propose conditions under which solutions might exist. The discussion remains unresolved regarding the best approach to finding the roots.

Contextual Notes

Participants mention limitations in analytical methods and the reliance on tables for specific values of x, indicating that the discussion is contingent on the context of x being large or small.

man@SUT
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If we want to find x giving J_m(x)=0 where m=any constants, how can we analytically get x?

Thank you
 
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I don't think you can do that analytically. (from memory)
 
I also use mathematica to solve but it doesn't help.
 
You'll have an infinite number of real roots.

For large x, you can use the asymptotic formula for [tex]J_n(x)[/tex]. If I remember right, the difference between successive roots will tend to [itex]\pi[/itex] for large x.

Alternatively, you could look up tables which give the zeros for various Bessel functions in a mathematical handbook
 
There will be the analytic solution when we assume x -> infinity or x<<1. In the case of the first few values of x giving J_m(x)=0, we might have to use the table to be the last choice. Anyway, thanks mjsd and siddharth.
 

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