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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 $x$ ?

1 Answers1

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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.

Ruslan
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  • I am asking about the envelope of the family of $J_v(x)$ where the parameter is $\nu$. This is just by curiosity, and I expect the problem to be untractable. –  Mar 20 '17 at 19:38
  • @YvesDaoust ah, but then your asymptotic seems wrong: see this plot of Bessel functions with $\nu$ from $0.1$ to $14.8$. If you continue increasing $\nu$, the first (and any next) maximum will not approach the envelope you cited. – Ruslan Mar 20 '17 at 20:11
  • that's right. Actually, it seems that the envelope has multiple branches. –  Mar 20 '17 at 20:13