Let$(X,\mu)$ be a measure space. $f:X \to \mathbb{R}$ be a measure function. For every $t\in \mathbb{R}$ the distribution function $F$ of $f$ is defined as $ F(t)=\mu\{x \in X:f(x)<t\}.$
I have difficulties of finding distribution function of the bessel function of the first order, i.e. $F(t)=\{x \in \mathbb{R}: f(x)=\frac{|J_1(x)|}{x}<t\}$. Any ideas or references will be very helpful. Thank you.
$J_1(x)/x$ is an even function whose global maximum value is $1/2$ at $x=0$ (actually that's a removable singularity). The global minimum value is approximately $-0.06613974369805002$, attained at approximately $x = \pm 5.135622301840683$. It's unlikely that there are "closed-form" expressions for these numbers, or for any $x$ for which $J_1(x)/x$ is rational. So I don't know what you expect an answer to look like.