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Plotting the Bessel functions of the first kind $J_n(z)$ versus $n$ for some fixed $z\gg1$, it appears that there is a sharp cutoff just before $n=z$.

Three questions:

  1. What is a reference describing this sharp cutoff?
  2. What is an expression for the location of the maximum of $J_n(z)$ with respect to $n$, for fixed (large) $z$?
  3. What is a nice expression for the envelope of the function $J_n(z)$ with respect to $z$? I.e, what is a function that (approximately) goes through all the maxima of the following plot, and then dies off appropriately for $n>z$?

Plot of the Bessel function J_n(z) for z=100, n from 0 to 200.

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1 Answer 1

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1.

The so-called "transition region" for $J_n(z)$ is well known; among its many applications, the usual three-term recursion relation for Bessel functions is numerically sound in the direction of increasing $n$ for as long as $z>n$. As a guide of sorts,

$$J_n(z)\approx\frac{x^n}{2^n\Gamma(n+1)}\quad\mathrm{if}\quad 0 < z \ll n$$

$$J_n(z)\approx\sqrt{\frac{2}{\pi z}}\cos\left(z-\frac{\pi}{4}-\frac{n\pi}{2}\right)\quad\mathrm{if}\quad n \ll z$$

2.

The expressions for $\frac{\mathrm d}{\mathrm d\nu}J_\nu(z)$ are rather complicated; there's no reason to expect a simple expression for the solution of $\frac{\mathrm d}{\mathrm d\nu}J_\nu(z)=0$, as is usual for any transcendental equation.

3.

There are a lot of asymptotic results for Bessel functions with varying order in the DLMF; you might want to look into them. Offhand, I recall results for envelopes of $J_n(z)$ for fixed $n$ and varying $z$, but not for your situation; I'll edit this answer when I come across results of relevance to you.

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