Envelope of the Bessel functions of the first kind

Question

By varying the order $\nu$ of the Bessel functions of the first kind $J_\nu(x)$, you can build the envelope of these curves. From the asymptotic expansion and from the half-integer case, we know that for large $x$ the envelope is

$$y=\sqrt{\frac2{\pi x}}.$$

But is the exact envelope known for any $x$ ?

Answer 1

Not sure if it's exact enough$^\dagger$, but you can get a natural envelope for Bessel function of order $\nu$ by combining Bessel and Neumann functions of this order, effectively getting envelope of a running cylindrical wave:

$$\operatorname{env}_\nu(x)=\sqrt{J_\nu(x)^2+Y_\nu(x)^2},$$

or, in terms of Hankel functions,

$$\operatorname{env}_\nu(x)=\left|H^{(i)}_\nu(x)\right|,$$

where $i$ can be either $1$ or $2$, the result doesn't depend on $i$.

_{$^\dagger$ I.e. you might be looking for an expression not in terms of Bessel-related functions, which this is not.}