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 $n=z$.
- What is a reference describing this sharp cutoff?
- What is an expression for the location of the maximum of $J_n(z)$ with respect to $n$, for fixed (large) $z$?
- 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 $n>z$?