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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)

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}. Any ideas or references will be very helpful. Thank you.

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    Sorry, I've lost denominator. Now we can consider $F$ on $(0,1)$2012-01-25

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$J_1(x)/x$ 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.

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    If $\mu$ is a finite measure with a continuous density, so $d\mu =g(x)\ dx$ where $g$ is continuous, and F(t) = \mu(\{x: f(x) < t\}) where $f$ is continuously differentiable, then at any $t$ for which $f(x)=t$ has finitely many solutions $x_j$, at all of which $f' \ne 0$, we should have $F'(t) = \sum_j g(x_j)/|f'(x_j)|$.2012-01-25