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

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 J_n(x). If I remember right, the difference between successive roots will tend to \pi 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.
 

Thread 'Volterra equation of the second kind'

There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...
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