Zeros of Bessel functions

Remarks

For $\nu > -1$, every zero of $z^{-\nu} J_\nu(z)$ in $\C$ is real and simple, so the $j_{\nu,k}$ and their negatives are all of them. Example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$, so $j_{1/2,k} = k\pi$; the check is $\sin(k\pi) = 0$ and $\sin x \ne 0$ on $(0, \pi)$. Numerical values: $j_{0,1} \approx 2.4048$ and $j_{1,1} \approx 3.8317$. The zeros of $J_\nu$ and $J_{\nu+1}$ interlace: $j_{\nu,1} < j_{\nu+1,1} < j_{\nu,2} < j_{\nu+1,2} < \cdots$. For fixed $\nu$, $j_{\nu,k+1} - j_{\nu,k} \to \pi$ as $k \to \infty$, and $j_{\nu,k} = \left( k + \frac{\nu}{2} - \frac{1}{4} \right)\pi + O\left( \frac{1}{k} \right)$ (McMahon's expansion). A non-example: $0$ is a zero of $J_\nu$ for $\nu > 0$, but it is not one of the $j_{\nu,k}$, which are positive by definition.